null  null
the Essential
Standards
High School StudyText, Math BC,
Volume 2
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted
under the United States Copyright Act, no part of this publication may be reproduced or
distributed in any form or by any means, or stored in a database or retrieval system, without
the prior permission of the publisher.
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1 2 3 4 5 6 7 8 9 10 QWD 16 15 14 13 12 11 10
North Carolina StudyText, Math BC, Volume 2
Using Your North Carolina StudyText
North Carolina StudyText, Math BC, Volume 2, is a practice workbook designed to help you
master the North Carolina Essential Standards for High School Math BC. By mastering the
mathematics standards, you will be prepared to do well on your end-of-course (EOC) test.
This StudyText is divided into two sections.
Chapter Resources
• Each chapter contains four pages for each key lesson in your North Carolina Algebra 2
Student Edition. Your teacher may ask you to complete one or more of these worksheets
as an assignment.
Mastering the EOC
This section of StudyText is composed of three parts. Each part can help you study for your
EOC test.
• The Diagnostic Test can help you determine which standards you might need to review
before taking the EOC test. Each question lists the standard that it is assessing. Your
teacher may assign review pages based on the questions that you did not answer
correctly.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
• Practice by Standard gives you more practice problems to help you become a better
test-taker. The problems are organized by the North Carolina High School Math BC
Essential Standards. You can also use these pages as a general review before you take the
EOC test.
• The Practice Test can be used to simulate what an EOC test might be like so that you
will be better prepared to take it in the spring.
iii
North Carolina StudyText, Math BC, Volume 2
Contents in Brief
Chapter Resources
Chapter 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
Chapter 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5
Chapter 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Chapter 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Chapter 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Chapter 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Chapter 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Chapter 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Chapter 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Chapter 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Concepts and Skills Bank . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Mastering the EOC, Algebra 2 / Math BC, Volume 2
Diagnostic Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .A1
Practice by Standard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A23
Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A64
v
North Carolina StudyText, Math BC, Volume 2
NAME
1-1
DATE
PERIOD
Study Guide
SCS
MBC.A.9.2
Expressions and Formulas
Order of Operations
Order of
Operations
Step 2
Step 3
Step 4
Example 1
Evaluate expressions inside grouping symbols.
Evaluate all powers.
Multiply and/or divide from left to right.
Add and/or subtract from left to right.
Evaluate [18 - (6 + 4)] ÷ 2.
[18 - (6 + 4)] ÷ 2 = [18 - 10] ÷ 2
=8÷2
=4
Example 2
Evaluate 3x2 + x(y - 5)
if x = 3 and y = 0.5.
Replace each variable with the given value.
3x2 + x(y - 5) = 3 (3)2 + 3(0.5 - 5)
= 3 (9) + 3(-4.5)
= 27 - 13.5
= 13.5
Exercises
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Evaluate each expression.
1. 14 + (6 ÷ 2)
2. 11 - (3 + 2)2
3. 2 + (4 - 2)3 - 6
4. 9(32 + 6)
5. (5 + 23)2 - 52
1
6. 52 + −
+ 18 ÷ 2
8. (7 - 32)2 + 62
9. 20 ÷ 22 + 6
16 + 2 3 ÷ 4
1-2
7. −
2
10. 12 + 6 ÷ 3 - 2(4)
11. 14 ÷ (8 - 20 ÷ 2)
13. 8(42 ÷ 8 - 32)
14. −
6+4÷2
4÷6-1
4
12. 6(7) + 4 ÷ 4 - 5
6 + 9 ÷ 3 + 15
8-2
15. −
1
Evaluate each expression if a = 8.2, b = -3, c = 4, and d = - −
.
2
ab
16. −
17. 5(6c - 8b + 10d)
c2 - 1
18. −
19. ac - bd
20. (b - c)2 + 4a
a
21. −
+ 6b - 5c
c
22. 3 −
-b
b
23. cd + −
24. d(a + c)
25. a + b ÷ c
26. b - c + 4 ÷ d
a
27. −
-d
d
(d)
Chapter 1
d
1
b-d
d
b+c
North Carolina StudyText, Math BC, Volume 2
Lesson 1-1
Step 1
NAME
1-1
DATE
Study Guide
PERIOD
SCS
(continued)
MBC.A.9.2
Expressions and Formulas
Formulas
A formula is a mathematical sentence that expresses the
relationship between certain quantities. If you know the value of every
variable in the formula except one, you can use substitution and the
order of operations to find the value of the remaining variable.
Example
The formula for the number of reams of paper
needed to print n copies of a booklet that is p pages long is
np
r = −, where r is the number of reams needed. How many
500
reams of paper must you buy to print 172 copies of a 25-page
booklet?
np
500
(172)(25)
=−
500
4300
=−
500
r=−
= 8.6
Formula for paper needed
n = 172 and p = 25
Evaluate (172)(25).
Divide.
You cannot buy 8.6 reams of paper. You will need to buy 9 reams to
print 172 copies.
Exercises
a. Her beach ball has a radius of 9 inches. First she converts the
radius to centimeters using the formula C = 2.54I, where C is a
length in centimeters and I is the same length in inches. How many
centimeters are there in 9 inches?
4 3
b. The volume of a sphere is given by the formula V = −
πr , where
3
V is the volume of the sphere and r is its radius. What is the volume of
the beach ball in cubic centimeters? (Use 3.14 for π.)
c. Sarah takes 40 breaths to blow up the beach ball. What is the
average volume of air per breath?
2. A person’s basal metabolic rate (or BMR) is the number of Calories needed
to support his or her bodily functions for one day. The BMR of an 80-yearold man is given by the formula BMR = 12w - (0.02)(6)12w, where w is
the man’s weight in pounds. What is the BMR of an 80-year-old man who
weighs 170 pounds?
Chapter 1
2
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. For a science experiment, Sarah counts the number of breaths needed
for her to blow up a beach ball. She will then find the volume of the
beach ball in cubic centimeters and divide by the number of breaths
to find the average volume of air per breath.
NAME
DATE
1-1
PERIOD
Practice
SCS
MBC.A.9.2
Expressions and Formulas
1. 3(4 - 7) - 11
2. 4(12 - 42)
3. 1 + 2 - 3(4) ÷ 2
4. 12 - [20 - 2(62 ÷ 3 × 22)]
5. 20 ÷ (5 - 3) + 52(3)
6. (-2)3 - (3)(8) + (5)(10)
7. 18 - {5 - [34 - (17 - 11)]}
8. [4(5 - 3) - 2(4 - 8)] ÷ 16
1
[6 - 42]
9. −
1
10. −
[-5 + 5(-3)]
2
4
(-8)2
5-9
-8(13 - 37)
6
12. − - (-1)2 + 4(-9)
11. −
3
1
, b = -8, c = -2, d = 3, and g = −
.
Evaluate each expression if a = −
3
4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
13. ab - d
14. (c + d)b
ab
2
15. −
c +d
16. −
ac
17. (b - dg)g2
18. ac3 - b2dg
19. -b[a + (c - d) 2]
ac4
c
20. −
-−
2
1
21. 9bc - −
g
22. 2ab2 - (d 3 - c)
d(b - c)
d
g
9
23. TEMPERATURE The formula F = −
C + 32 gives the temperature in
5
degrees Fahrenheit for a given temperature in degrees Celsius. What
is the temperature in degrees Fahrenheit when the temperature is
-40 degrees Celsius?
24. PHYSICS The formula h = 120t - 16t2 gives the height h in feet of an
object t seconds after it is shot upward from Earth’s surface with an
initial velocity of 120 feet per second. What will the height of the object
be after 6 seconds?
25. AGRICULTURE Faith owns an organic apple orchard. From her
experience the last few seasons, she has developed the formula
P = 20x - 0.01x2 - 240 to predict her profit P in dollars this season
if her trees produce x bushels of apples. What is Faith’s predicted
profit this season if her orchard produces 300 bushels of apples?
Chapter 1
3
North Carolina StudyText, Math BC, Volume 2
Lesson 1-1
Evaluate each expression.
NAME
1-1
DATE
Word Problem Practice
PERIOD
SCS
MBC.A.9.2
Expressions and Formulas
1. ARRANGEMENTS The chairs in an
auditorium are arranged into two
rectangles. Both rectangles are 10 rows
deep. One rectangle has 6 chairs per
row and the other has 12 chairs per row.
Write an expression for the total number
of chairs in the auditorium.
4. GAS MILEAGE Rick has d dollars. The
formula for the number of gallons of
gasoline that Rick can buy with d dollars
d
is given by g = −
. The formula for the
3
number of miles that Rick can drive on g
gallons of gasoline is given by m = 21g.
How many miles can Rick drive on $8
worth of gasoline?
2. GEOMETRY The formula for the area
of a ring-shaped object is given by A =
π(R2 - r2), where R is the radius of the
outer circle and r is the radius of the
inner circle. If R = 10 inches and r = 5
inches, what is the area rounded to the
nearest square inch?
r
5. COOKING A steak has thickness w
inches. Let T be the time it takes to
broil the steak. It takes 12 minutes to
broil a one-inch-thick steak. For every
additional inch of thickness, the steak
should be broiled for 5 more minutes.
a. Write a formula for T in terms of w.
R
3. GUESS AND CHECK Amanda
received a worksheet from her teacher.
Unfortunately, one of the operations in
an equation was covered by a blot.
What operation is hidden by the blot?
10 + 3(4 + 6) = 4
Chapter 1
4
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
b. Use your formula to compute the
number of minutes it would take to
broil a 2-inch-thick steak.
NAME
2-1
DATE
Study Guide
SCS
PERIOD
MBC.A.1.2, MBC.A.8.1, MBC.A.8.3
Relations and Functions
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0OUP'VODUJPO
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Example
State the domain and range of the relation.
Does the relation represent a function?
The domain and range are both all real numbers. Each element
of the domain corresponds with exactly one element of the range,
so it is a function.
X
Y
1
2
3
D
B
C
A
X
Y
1
2
3
4
D
B
C
X
Y
1
2
3
C
B
A
x
y
-1
-5
0
-3
1
-1
2
1
3
3
Exercises
State the domain and range of each relation. Then determine whether each
relation is a function. If it is a function, determine if it is one-to-one, onto, both,
or neither.
1. {(0.5, 3), (0.4, 2), (3.1, 1), (0.4, 0)}
2. {(–5, 2), (4, –2), (3, –11), (–7, 2)}
3. {(0.5, –3), (0.1, 12), (6, 8)}
4. {(–15, 12), (–14, 11), (–13, 10), (–12, 12)}
Chapter 2
5
North Carolina StudyText, Math BC, Volume 2
Lesson 2-1
Relations and Functions A relation can be represented as a set of ordered pairs or
as an equation; the relation is then the set of all ordered pairs (x, y) that make the equation
true. A function is a relation in which each element of the domain is paired with exactly
one element of the range.
NAME
DATE
2-1
Study Guide
PERIOD
SCS
(continued)
MBC.A.1.2, MBC.A.8.1, MBC.A.8.3
Relations and Functions
Equations of Relations and Functions Equations that represent functions are
often written in functional notation. For example, y = 10 - 8x can be written as
f (x) = 10 - 8x. This notation emphasizes the fact that the values of y, the dependent
variable, depend on the values of x, the independent variable.
To evaluate a function, or find a functional value, means to substitute a given value in the
domain into the equation to find the corresponding element in the range.
Example
Given f (x) = x2 + 2x, find each value.
a. f(3)
f (x) = x2 + 2x
f (3) = 32 + 2(3)
= 15
Original function
Substitute.
Simplify.
b. f(5a)
f(x) = x2 + 2x
f(5a) = (5a)2 + 2(5a)
= 25a2 + 10a
Original function
Substitute.
Simplify.
Exercises
2. y = x2 - 1
1. y = 3
3. y = 3x + 2
y
y
y
O
O
x
x
x
O
Find each value if f(x) = -2x + 4.
4. f(12)
6. f (2b)
5. f (6)
Find each value if g(x) = x3 - x.
7. g(5)
Chapter 2
9. g (7c)
8. g(-2)
6
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Graph each relation or equation and determine the domain and range. Determine
whether the relation is a function, is one-to-one, onto, both, or neither. Then state
whether it is discrete or continuous.
NAME
DATE
2-1
Practice
PERIOD
SCS
MBC.A.1.2, MBC.A.8.1, MBC.A.8.3
Relations and Functions
State the domain and range of each relation. Then determine whether each
relation is a function. If it is a function, determine if it is one-to-one, onto, both
or neither.
Domain
Range
2
21
25
30
8
3.
x
y
-3
2.
4.
Domain
Range
5
10
15
105
110
x
y
0
-2
-1
-1
-1
-2
1
0
0
-1
0
2
-2
1
0
3
4
2
1
Lesson 2-1
1.
Graph each equation and determine the domain and range. Determine whether
the relation is a function, is one-to-one, onto, both, or neither. Then state whether
it is discrete or continuous.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5. x = -1
6. y = 2x - 1
y
y
O
O
x
x
5
Find each value if f(x) = −
and g(x) = -2x + 3.
x+2
7. f(3)
8. f(-4)
10. f(-2)
11. g(-6)
(2)
1
9. g −
12. f(m - 2)
13. MUSIC The ordered pairs (1, 16), (2, 16), (3, 32), (4, 32), and (5, 48) represent the cost of
buying various numbers of CDs through a music club. Identify the domain and range of
the relation. Is the relation discrete or continuous? Is the relation a function?
14. COMPUTING If a computer can do one calculation in 0.0000000015 second, then the
function T(n) = 0.0000000015n gives the time required for the computer to do n
calculations. How long would it take the computer to do 5 billion calculations?
Chapter 2
7
North Carolina StudyText, Math BC, Volume 2
NAME
DATE
2-1
Word Problem Practice
PERIOD
SCS
MBC.A.1.2, MBC.A.8.1, MBC.A.8.3
Relations and Functions
1. PLANETS The table below gives the
mean distance from the Sun and orbital
period of the eight major planets in our
Solar System. Think of the mean
distance as the domain and the orbital
period as the range of a relation. Is this
relation a function? Explain.
Planet
3. SCHOOL The number of students N in
Vassia’s school is given by N = 120 +
30G, where G is the grade level. Is 285
in the range of this function?
Mean Distance from Orbital Period
Sun (AU)
(years)
Mercury
0.387
0.241
Venus
0.723
0.615
Earth
1.0
1.0
Mars
1.524
1.881
Jupiter
5.204
11.75
Saturn
9.582
29.5
Uranus
19.201
84
Neptune
30.047
165
4. FLOWERS Anthony decides to decorate
a ballroom with r = 3n + 20 roses,
where n is the number of dancers. It
occurs to Anthony that the dancers
always come in pairs. That is, n = 2p,
where p is the number of pairs. What is
r as a function of p?
5. SALES Cool Athletics introduced the
new Power Sneaker in one of their
stores. The table shows the sales for the
first 6 weeks.
1
2
3
4
5
6
Pairs Sold
8
10
15
22
31
44
a. Graph the data.
2. PROBABILITY Martha rolls a number
cube several times and makes the
frequency graph shown. Write a relation
to represent this data.
7
Frequency
6
5
4
3
b. Identify the domain and range.
2
1
0
1
2 3 4 5 6
Number
c. Is the relation a function? Explain.
Chapter 2
8
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Week
NAME
DATE
2-2
Study Guide
PERIOD
SCS
MBC.A.1.1, MBC.A.1.2
Linear Relations and Functions
Linear Relations and Functions A linear equation has no operations other than
addition, subtraction, and multiplication of a variable by a constant. The variables may not
be multiplied together or appear in a denominator. A linear equation does not contain
variables with exponents other than 1. The graph of a linear equation is always a line.
A linear function is a function with ordered pairs that satisfy a linear equation. Any
linear function can be written in the form f (x) = mx + b, where m and b are real numbers.
If an equation is linear, you need only two points that satisfy the equation in order to graph
the equation. One way is to find the x-intercept and the y-intercept and connect these two
points with a line.
Example 1
x
Is f(x) = 0.2 - −
a linear function? Explain.
5
Yes; it is a linear function because it can be written in the form
1
f(x) = -−
x + 0.2.
5
Is 2x + xy - 3y = 0 a linear function? Explain.
No; it is not a linear function because the variables x and y are multiplied together in the
middle term.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
State whether each function is a linear function. Write yes or
no. Explain.
1. 6y - x = 7
18
2. 9x = −
y
x
3. f (x) = 2 - −
x
4. 2y- −
-4=0
5. 1.6x - 2.4y = 4
0.4
6. 0.2x = 100 - −
y
7. f(x) = 4 - x3
4
8. f(x) = −
x
9. 2yx - 3y + 2x = 0
6
Chapter 2
11
9
North Carolina StudyText, Math BC, Volume 2
Lesson 2-2
Example 2
NAME
DATE
2-2
Study Guide
PERIOD
SCS
(continued)
MBC.A.1.1, MBC.A.1.2
Linear Relations and Functions
Standard Form The standard form of a linear equation is Ax + By = C, where
A, B, and C are integers whose greatest common factor is 1.
Example 1
Write each equation in standard form. Identify A, B, and C.
a. y = 8x - 5
y = 8x - 5
-8x + y = -5
8x - y = 5
b. 14x = -7y + 21
14x = -7y + 21
14x + 7y = 21
2x + y = 3
Original equation
Subtract 8x from each side.
Multiply each side by -1.
So A = 8, B = -1, and C = 5.
Original equation
Add 7y to each side.
Divide each side by 7.
So A = 2, B = 1, and C = 3.
Example 2
Find the x-intercept and the y-intercept of the graph of 4x - 5y = 20.
Then graph the equation.
The x-intercept is the value of x when y = 0.
4x - 5y = 20
Original equation
4x - 5(0) = 20
Substitute 0 for y.
x=5
y
2
O
4
x
6
−2
Simplify.
−4
So the x-intercept is 5. Similarly, the y-intercept is -4.
Exercises
Write each equation in standard form. Identify A, B, and C.
2. 5y = 2x + 3
3. 3x = -5y + 2
4. 18y = 24x - 9
3
2
y=−
x+5
5. −
6. 6y - 8x + 10 = 0
7. 0.4x + 3y = 10
8. x = 4y - 7
9. 2y = 3x + 6
4
3
Find the x-intercept and the y-intercept of the graph of each equation. Then graph
the equation using the intercepts.
10. 2x + 7y = 14
11. 5y - x = 10
y
O
12. 2.5x - 5y + 7.5 = 0
y
x
y
x
O
O
Chapter 2
10
x
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. 2x = 4y -1
NAME
DATE
2-2
PERIOD
Practice
SCS
MBC.A.1.1, MBC.A.1.2
Linear Relations and Functions
State whether each function is a linear function. Write yes or no. Explain.
1. h(x) = 23
2
2. y = −
x
5
3. y = −
x
4. 9 - 5xy = 2
3
Write each equation in standard form. Identify A, B, and C.
5. y = 7x - 5
3
6. y = −
x+5
7. 3y - 5 = 0
3
2
y+−
8. x = -−
8
4
7
9. y = 2x + 4
10. 2x + 7y = 14
y
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 2-2
Find the x-intercept and the y-intercept of the graph of each equation. Then graph
the equation using the intercepts.
y
O
x
x
O
11. y = -2x - 4
12. 6x + 2y = 6
y
O
y
x
O
x
13. MEASURE The equation y = 2.54x gives the length y in centimeters corresponding to a
length x in inches. What is the length in centimeters of a 1-foot ruler?
14. LONG DISTANCE For Meg’s long-distance calling plan, the monthly cost C in dollars is
given by the linear function C(t) = 6 + 0.05t, where t is the number of minutes talked.
a. What is the total cost of talking 8 hours? of talking 20 hours?
b. What is the effective cost per minute (the total cost divided by the number of minutes
talked) of talking 8 hours? of talking 20 hours?
Chapter 2
11
North Carolina StudyText, Math BC, Volume 2
NAME
DATE
2-2
PERIOD
Word Problem Practice
SCS
MBC.A.1.1, MBC.A.1.2
Linear Relations and Functions
1. WORK RATE The linear equation
n = 10t describes n, the number of
origami boxes that Holly can fold in
t hours. How many boxes can Holly
fold in 3 hours?
4. RAMP A ramp is described by the
equation 5x + 7y = 35. What is the area
of the shaded region?
8
6
2. BASKETBALL Tony tossed a basketball.
Below is a graph showing the height of
the basketball as a function of time. Is
this the graph of a linear function?
Explain.
Height (ft)
y
4
5x + 7y = 35
2
2
O
4
6
8x
10
8
6
5. SWIMMING POOL A swimming pool is
shaped as shown below. The total
perimeter is 500 feet.
4
2
0.2
x ft
0.6
1.0
Time (s)
y ft
5 ft
a. Write an equation that relates x
and y.
3. PROFIT Paul charges people $25 to
test the air quality in their homes.
The device he uses to test air quality
cost him $500. Write an equation that
describes Paul’s net profit as a function
of the number of clients he gets. How
many clients does he need to break
even?
b. Write the linear equation from
part a in standard form.
c. Graph the equation.
d. Olympic swimming pools are 164 feet
long. If this pool is an olympic pool,
what is the value of y?
Chapter 2
12
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
10 ft
NAME
DATE
2-5
PERIOD
Study Guide
SCS
MBC.S.2.2, MBC.S.2.4
Scatter Plots and Lines of Regression
Scatter Plots and Prediction Equations
A set of data points graphed as ordered
pairs in a coordinate plane is called a scatter plot. A scatter plot can be used to determine
if there is a relationship among the data. A line of fit is a line that closely approximates a
set of data graphed in a scatter plot. The equation of a line of fit is called a prediction
equation because it can be used to predict values not given in the data set.
Example
STORAGE COSTS According to a certain prediction equation, the
cost of 200 square feet of storage space is $60. The cost of 325 square feet of
storage space is $160.
a. Find the slope of the prediction equation. What does it represent?
Since the cost depends upon the square footage, let x represent the amount of storage
space in square feet and y represent the cost in dollars. The slope can be found using the
y -y
160 - 60
100
2
1
formula m = −
= −
= 0.8
x2 - x1 . So, m = −
325 - 200
125
The slope of the prediction equation is 0.8. This means that the price of storage increases
80¢ for each one-square-foot increase in storage space.
y - y1 = m(x - x1)
y - 60 = 0.8(x - 200)
y - 60 = 0.8x - 160
y = 0.8x - 100
Point-slope form
(x1, y1) = (200, 60), m = 0.8
Distributive Property
Add 60 to both sides.
A prediction equation is y = 0.8x - 100.
Exercises
1. SALARIES The table below shows the years of experience for eight technicians at Lewis
Techomatic and the hourly rate of pay each technician earns.
9
4
3
1
10
6
12
8
Hourly Rate of Pay
$17
$10
$10
$7
$19
$12
$20
$15
a. Draw a scatter plot to show how years of experience are
related to hourly rate of pay. Draw a line of fit and
describe the correlation.
b. Write a prediction equation to show how years of
experience (x) are related to hourly rate of pay (y).
Technician Salaries
24
20
16
12
8
4
0
c. Use the function to predict the hourly rate of pay for
15 years of experience.
Chapter 2
13
2 4 6 8 10 12 14
Experience (years)
North Carolina StudyText, Math BC, Volume 2
Lesson 2-5
Experience (years)
Hourly Pay ($)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
b. Find a prediction equation.
Using the slope and one of the points on the line, you can use the point-slope form to find
a prediction equation.
NAME
DATE
2-5
Study Guide
PERIOD
SCS
(continued)
MBC.S.2.2, MBC.S.2.4
Scatter Plots and Lines of Regression
Lines of Regression Another method for writing a line of fit is to use a line of
regression. A regression line is determined through complex calculations to ensure that
the distance of all the data points to the line of fit are at the minimum.
Example
WORLD POPULATION The following table gives the United Nations
estimates of the world population (in billions) every five years from 1980-2005.
Find the equation and graph the line of regression. Then predict the population
in 2010.
Year
Population (billions)
1980
4.451
1985
4.855
1990
5.295
1995
5.719
2000
6.124
2005
6.515
2010
?
[1980, 2010] scl: 5 by [3, 7] scl: 0.5
Source: UN 2006 Revisions Population database
Exercise
1. The table below shows the number of women who served in
the United States Congress during the years 1995–2006.
Find an equation for and graph a line of regression. Then
use the function to predict the number of women in
Congress in the 112th Congressional Session.
Congressional Session
Number of Women
104
59
105
65
106
67
107
75
108
77
109
83
[103, 113] scl: 1 by [55, 100] scl: 5
Source: U. S. Senate
Chapter 2
14
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Step 1 Use your calculator to make a scatter plot.
Step 2 Find the equation of the line of regression.
The equation is about y = 0.083x - 160.180.
Step 3 Graph the regression equation.
Step 4 Predict using the function.
In 2010 the population will be approximately 6.948 billion.
NAME
DATE
2-5
PERIOD
Practice
SCS
MBC.S.2.2, MBC.S.2.4
Scatter Plots and Lines of Regression
For Exercises 1 and 2, complete parts a–c.
a. Make a scatter plot and a line of fit, and describe the correlation.
b. Use two ordered pairs to write a prediction equation.
c. Use your prediction equation to predict the missing value.
1. FUEL ECONOMY The table gives the
weights in tons and estimates the fuel
economy in miles per gallon for
several cars.
Weight (tons)
1.3 1.4 1.5 1.8 2.0 2.1 2.4
Miles per Gallon
29
24
23
21
?
17
15
Fuel Economy (mi/gal)
Fuel Economy Versus Weight
30
25
20
15
10
5
0
0.5
1.0
1.5
Weight (tons)
2.0
2.5
Temperature (°F)
7500
8200
8600
9200
9700
10,400
12,000
61
58
56
53
50
46
?
Temperature (°F)
Altitude (ft)
65
Temperature
Versus Altitude
60
55
50
45
0 7000
8000 9000 10,000
Altitude (ft)
3. HEALTH Alton has a treadmill that uses the time on the treadmill to estimate the
number of Calories he burns during a workout. The table gives workout times and
Calories burned for several workouts. Find an equation for and graph a line of
regression. Then use the function to predict the number of Calories burned in
a 60-minute workout.
Time (min)
18
24
30
40
42
48
52
60
Calories Burned
260
280
320
380
400
440
475
?
Burning Calories
500
Lesson 2-5
Calories Burned
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. ALTITUDE As Anchara drives into the mountains, her car thermometer registers the
temperatures (°F) shown in the table at the given altitudes (feet).
400
300
200
100
0
Chapter 2
5 10 15 20 25 30 35 40 45 50 55
Time (min)
15
North Carolina StudyText, Math BC, Volume 2
NAME
DATE
2-5
PERIOD
Word Problem Practice
SCS
MBC.S.2.2, MBC.S.2.4
Scatter Plots and Lines of Regression
1. AIRCRAFT The table shows the
maximum speed and altitude of different
aircraft. Draw a scatter plot of this data.
Max. Speed
(knots)
121
123
137
173
153
Max. Altitude
(1000 feet)
14.2
17.0
15.3
20.7
16.0
4. ALGAE One type of algae grows
fastest at 31°C. The scatter plot shows
data recording the amount of algae and
the temperature of the water in various
aquarium tanks. Draw a line of fit for
this data and write a prediction
equation. Will this prediction equation
be accurate for temperatures above
31°C?
22
20
Temperature (°C)
Maximum Altitude
(thousands of ft)
Source: RisingUp Aviation
18
16
14
12
0
120 140 160 180
Maximum Speed (knots)
30
25
20
15
10
5
0
2. TESTING The scatter plot shows the
height and test scores of students in a
math class. Describe the correlation
between heights and test scores.
Test Score
5. SPORTS The scatter plot shows the
height and score of different contestants
shooting darts.
12
0
62 64 66 68 70 72
Height (in.)
Score
10
2
3
4
5
8.30
8.60
8.55
8.90
9.30
Stock Price ($)
Price
1
2
0
1 2 3 4 5 6
Height (ft)
a. What is the equation of the line of fit?
9.35
9.20
9.05
8.90
8.75
8.60
8.45
8.30
8.15
0
6
4
3. STOCKS The prices of a technology
stock over 5 days are shown in the table.
Draw a scatter plot of the data and a
line of fit.
Day
8
b. What do you predict someone 5 feet
tall would score?
1
2
3
4
5
Day
Chapter 2
16
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
100
90
80
70
60
50
40
1 2 3 4 5 6
Algae (pounds)
NAME
DATE
2-6
PERIOD
Study Guide
SCS
MBC.A.1.1, MBC.A.1.2
Piecewise-Defined Functions A piecewise-defined function is written using two
or more expressions. Its graph is often disjointed.
Example
Graph f(x) =
x -2x1 ifif xx <≥ 22.
4
First, graph the linear function f (x) = 2x for x < 2. Since 2 does not
satisfy this inequality, stop with a circle at (2, 4). Next, graph the
linear function f(x) = x - 1 for x ≥ 2. Since 2 does satisfy this
inequality, begin with a dot at (2, 1).
f(x)
2
-4
O
-2
2
4x
-2
-4
Exercises
Graph each function. Identify the domain and range.
1. f (x) =
y
x + 2 if x < 0
2x + 5 if 0 ≤ x ≤ 2
-x + 1 if x > 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
x
y
-x - 4 if x < -7
2. f (x) =
5x - 1 if -7 ≤ x ≤ 0
2x + 1 if x > 0
x
0
3. h(x) =
Chapter 2
h(x)
x
−
if x ≤ 0
3
O
2x - 6 if 0 < x < 2
1 if x ≥ 2
17
x
North Carolina StudyText, Math BC, Volume 2
Lesson 2-6
Special Functions
NAME
DATE
2-6
Study Guide
PERIOD
SCS
(continued)
MBC.A.1.1, MBC.A.1.2
Special Functions
Step Functions and Absolute Value Functions
Name
Written as
Greatest Integer Function
f(x) = x
Graphed as
y
4
2
-4
-2
0
2
4
x
-2
-4
Absolute Value Function
Example
f (x ) = ⎪x ⎥
two rays that are mirror images of each other and meet at a point,
the vertex
Graph f(x) = 3⎪x⎥ - 4.
Find several ordered pairs. Graph the points and
connect them. You would expect the graph to look
similar to its parent function, f(x) = ⎪x⎥.
x
3⎪x⎥ - 4
4
0
-4
2
1
-1
2
2
-1
-2
2
-2
O
4x
2
-2
-4
Exercises
Graph each function. Identify the domain and range.
2. h(x) = ⎪2x + 1⎥
1. f(x) = 2x
f(x)
0
h(x)
f(x)
x
O
Chapter 2
3. f(x) = x + 4
18
x
0
x
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
-1
-4
f (x)
NAME
DATE
2-6
PERIOD
Practice
SCS
MBC.A.1.1, MBC.A.1.2
Graph each function. Identify the domain and range.
1. f(x) =
x +3x2 ifif xx ≤> -2
-2
2. h(x) =
f (x)
-2x4 -- 2x ifif xx >< 00
h (x)
O
x
x
O
3. f(x) = 0.5x
4. f (x) = x - 2
f (x)
f (x)
x
O
x
5. g(x) = -2⎪x⎥
6. f(x) = ⎪x + 1⎥
f(x)
g(x)
x
O
O
7. BUSINESS A Stitch in Time charges
$40 per hour or any fraction thereof
for labor. Draw a graph of the step
function that represents this situation.
Labor Costs
280
x
8. BUSINESS A wholesaler charges a store
$3.00 per pound for less than 20 pounds
of candy and $2.50 per pound for 20 or
more pounds. Draw a graph of the function
Candy Costs
that represents
105
this situation.
240
90
200
75
Cost ($)
Total Cost ($)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
O
160
120
60
45
80
30
40
15
0
Chapter 2
1
2
3 4 5
Hours
6
0
7
19
5 10 15 20 25 30 35
Pounds
North Carolina StudyText, Math BC, Volume 2
Lesson 2-6
Special Functions
NAME
2-6
DATE
PERIOD
Word Problem Practice
SCS
MBC.A.1.1, MBC.A.1.2
Special Functions
1. SAVINGS Nathan puts $200 into a
checking account as soon as he gets his
paycheck. The value of his checking
account is modeled by the formula
200m, where m is the number of
months that Nathan has been working.
After 105 days, how much money is in
the account?
4. ARCHITECTURE The cross-section of a
roof is shown in the figure. Write an
absolute value function that models the
shape of the roof.
y
O
2. FINANCE A financial advisor handles
the transactions for a customer. The
median annual earnings for financial
advisors is around $60,000. For every
transaction, a certain financial advisor
gets a 5% commission, regardless of
whether the transaction is a deposit or
withdrawal. Write a formula using the
absolute value function for the advisor’s
commission. Let D represent the value
of one transaction.
a. Write a formula that gives the
horizontal distance from the center of
the dartboard.
b. Write a formula using the greatest
integer function that can be used to
find the person’s score.
20
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5. GAMES Some young people are playing
a game where a wooden plank is used as
a target. It is marked off into 6 equal
parts. A value is written in each section
to represent the score earned if the dart
lands in that section. Let x denote the
horizontal position of a dart on the
board, where the center of the board is
the origin. Negative values correspond
to the left half of the dart board, and
positive values correspond to the right
half. A player’s score depends on the
distance of the dart from the origin.
3. ROUNDING A science teacher instructs
students to round their measurements
as follows: If a number is less than 0.5
of a millimeter, students are instructed
to round down. If a number is exactly
0.5 or greater, students are told to round
up to the next millimeter. Write a
formula that takes a measurement
x millimeters and yields the rounded off
number.
Chapter 2
x
NAME
DATE
2-7
PERIOD
Study Guide
SCS
Parent Functions and Transformations
MBC.A.1.1, MBC.A.3.1,
MBC.A.3.2, MBC.A.10.1,
MBC.A.10.2
Parent Graphs The parent graph, which is the graph of the parent function,
is the simplest of the graphs in a family. Each graph in a family of graphs has
similar characteristics.
Characteristics
Parent Function
Constant Function
Straight horizontal line
y = a, where a is a real number
Linear Function
Straight diagonal line
Identity function y = x
Absolute Value Function
Diagonal lines shaped
like a V
y = ⎪x⎥
Quadratic Function
Curved like a parabola
y = x2
Example
a.
Identify the type of function represented by each graph.
b.
y
4
y
4
2
−4
−2
0
2
4x
2
−4
−2
0
−2
−2
−4
−4
2
4x
The graph is a parabolic curve. The
graph represents a quadratic function.
The graph is a diagonal line. The graph
represents a linear function.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 2-7
Name
Exercises
Identify the type of function represented by each graph.
y
8
1.
y
4
2.
4
−8
−4
0
2
4
8x
−4
−2
Chapter 2
−2
2
4x
−4
−2
0
−2
−2
−8
−4
−4
y
4
5.
0
2
4x
−4
−2
0
4x
2
4x
2
2
4x
−4
−2
0
−2
−2
−2
−4
−4
−4
21
2
y
4
6.
2
2
−4
0
2
−4
y
4
4.
y
4
3.
North Carolina StudyText, Math BC, Volume 2
NAME
DATE
2-7
Study Guide
PERIOD
SCS
(continued)
MBC.A.1.1, MBC.A.3.1
MBC.A.3.2, MBC.A.10.1
MBC.A.10.2
Parent Functions and Transformations
Transformations
Transformations of a parent graph may appear in a different location,
may flip over an axis, or may appear to have been stretched or compressed.
Example
Describe the reflection in y = - ⎪x⎥. Then graph the function.
y
4
3
2
1
The graph of y = - ⎪x⎥ is a reflection of
the graph of y = ⎪x⎥ in the x-axis.
−4−3−2−1 0
−1
−2
−3
−4
y = |x|
1 2 3 4x
y = −|x|
Exercises
Describe the translation in each function. Then graph the function.
2. y = ⎪x + 5⎥
1. y = x - 4
y
3. y = x2 - 3
y
y
x
0
0
x
x
0
1⎪ ⎥
x
5. y = −
4. y = 5x
y
0
Chapter 2
6. y = 2x2
2
y
x
y
0
22
x
0
x
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Describe the dilation in each function. Then graph the function.
NAME
DATE
2-7
PERIOD
Practice
SCS
Parent Functions and Transformations
MBC.A.1.1, MBC.A.3.1,
MBC.A.3.2, MBC.A.10.1,
MBC.A.10.2
Describe the translation in each function. Then graph the function.
2. y = x2 - 3
y
y
x
0
0
x
Describe the reflection in each function. Then graph the function.
3. y = (-x)2
4. y = -(3)
y
y
x
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
0
0
x
Describe the dilation in each function. Then graph the function.
5. y = ⎪2x⎥
6. 4y = x2
y
y
x
0
0
x
7. CHEMISTRY A scientist tested how fast a chemical reaction occurred at different
temperatures. The data made this graph. What type of function shows the relation of
temperature and speed of the chemical reaction?
y
12
4
-12
-4
-4
4
12
x
-12
Chapter 2
23
North Carolina StudyText, Math BC, Volume 2
Lesson 2-7
1. y = x + 3
NAME
DATE
2-7
PERIOD
Word Problem Practice
SCS
MBC.A.1.1, MBC.A.3.1
MBC.A.3.2, MBC.A.10.1
MBC.A.10.2
Parent Functions and Transformations
1. GAMES Pedro decided to measure how
close to a target he and his friends could
throw a football. They counted 1 point
for each foot away from the target that
the football landed. The graph of points
versus distance thrown is shown here.
What type of function had Pedro and his
friends followed?
150
Net Salary ($)
75
4. BUSINESS Maria earns an hourly wage
of $10. She drew the following graph to
show the relation of her income as a
function of the hours she works. How
did she modify the identity function to
create her graph?
y
50
y
100
50
x
25
0
x
0
25
50
75
100
100
Height (meters)
y
25,000
x
75
y
50
25
x
5 10 15 20 25 30 35 40
Time (seconds)
0
25
50
75
100
Time (seconds)
3. GEOMETRY Chen made this graph to
show how the perimeter of a square
changes as the length of one side is
increased. The original graph showed
an identity function. How has it
been dilated?
Perimeter
30
a. What type of function does the
graph show?
b. In which axis has the function
been reflected?
y
c. Which directions has the graph been
translated? How many units?
20
10
x
0
10
20
30
40
d. What is the equation for the curve
shown on the graph?
Side Length
Chapter 2
24
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Velocity (miles per hour)
75
5. HOBBIES Laura launched a model
rocket into the air. The height of her
rocket over time is shown by the graph.
50,000
0
50
Hours Worked
2. ASTRONOMY The graph shows the
velocity of the space probe Cassini as it
passed Saturn. What type of function
best models Cassini’s velocity?
75,000
25
NAME
3-2
DATE
PERIOD
Study Guide
SCS
MBC.A.9.2
Solving Systems of Equations Algebraically
Substitution
To solve a system of linear equations by substitution, first solve for one
variable in terms of the other in one of the equations. Then substitute this expression into
the other equation and simplify.
Use substitution to solve the system of equations.
2x - y = 9
x + 3y = -6
Solve the first equation for y in terms of x.
2x - y = 9
First equation
-y = -2x + 9
Subtract 2x from both sides.
y = 2x - 9
Multiply both sides by -1.
Substitute the expression 2x - 9 for y into the second equation and solve for x.
Second equation
x + 3y = -6
x + 3(2x - 9) = -6
Substitute 2x - 9 for y.
x + 6x - 27 = -6
Distributive Property
7x - 27 = -6
Simplify.
7x = 21
Add 27 to each side.
x=3
Divide each side by 7.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Now, substitute the value 3 for x in either original equation and solve for y.
2x - y = 9
First equation
2(3) - y = 9
Replace x with 3.
6-y=9
Simplify.
-y = 3
Subtract 6 from each side.
y = -3
Multiply each side by -1.
The solution of the system is (3, -3).
Exercises
Solve each system of equations by using substitution.
1. 3x + y = 7
4x + 2y = 16
2. 2x + y = 5
3x - 3y = 3
3. 2x + 3y = -3
x + 2y = 2
4. 2x - y = 7
6x - 3y = 14
5. 4x - 3y = 4
2x + y = -8
6. 5x + y = 6
3-x=0
7. x + 8y = -2
x - 3y = 20
8. 2x - y = -4
4x + y = 1
9. x - y = -2
2x - 3y = 2
10. x - 4y = 4
2x + 12y = 13
11. x + 3y = 2
4x + 12 y = 8
12. 2x + 2y = 4
x - 2y = 0
Chapter 3
25
North Carolina StudyText, Math BC, Volume 2
Lesson 3-2
Example
NAME
3-2
DATE
Study Guide
PERIOD
SCS
(continued)
MBC.A.9.2
Solving Systems of Equations Algebraically
Elimination
To solve a system of linear equations by elimination, add or subtract the
equations to eliminate one of the variables. You may first need to multiply one or both of
the equations by a constant so that one of the variables has the opposite coefficient in one
equation as it has in the other.
Example 1
Use the elimination method to solve the system of equations.
2x - 4y = -26
3x - y = -24
Multiply the second equation by -4. Then add
the equations to eliminate the y variable.
2x - 4y = -26
2x - 4y = -26
3x - y = -24 Multiply by -4. -12x + 4y = 96
-10x
= 70
x
= -7
Example 2
Replace x with -7 and solve for y.
2x - 4y = -26
2(-7) -4y = -26
-14 - 4y = -26
-4y = -12
y=3
The solution is (-7, 3).
Use the elimination method to solve the system of equations.
3x - 2y = 4
5x + 3y = -25
Replace x with -2 and solve for y.
3x - 2y = 4
3(-2) - 2y = 4
-6 - 2y = 4
-2y = 10
y = -5
The solution is (-2, -5).
Exercises
Solve each system of equations by using elimination.
1. 2x - y = 7
3x + y = 8
2. x - 2y = 4
-x + 6y = 12
3. 3x + 4y = -10
x - 4y = 2
4. 3x - y = 12
5x + 2y = 20
5. 4x - y = 6
6. 5x + 2y = 12
7. 2x + y = 8
8. 7x + 2y = -1
y
2x - − = 4
2
9. 3x + 8y = -6
x-y=9
Chapter 3
3
3x + −
y = 12
-6x - 2y = -14
10. 5x + 4y = 12
7x - 6y = 40
2
11. -4x + y = -12
4x + 2y = 6
26
4x - 3y = -13
12. 5x + 2y = -8
4x + 3y = 2
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Multiply the first equation by 3 and the second
equation by 2. Then add the equations to
eliminate the y variable.
3x - 2y = 4
Multiply by 3.
9x - 6y = 12
5x + 3y = -25 Multiply by 2. 10x + 6y = -50
19x
= -38
x
= -2
NAME
3-2
DATE
PERIOD
Practice
SCS
MBC.A.9.2
Solving Systems of Equations Algebraically
Solve each system of equations by using substitution.
1. 2x + y = 4
3x + 2y = 1
2. x - 3y = 9
x + 2y = -1
3. g + 3h = 8
1
−
g+h=9
3
4. 2a - 4b = 6
-a + 2b = -3
5. 2m + n = 6
5m + 6n = 1
6. 4x - 3y = -6
-x - 2y = 7
1
7. u - 2v = −
8. x - 3y = 16
9. w + 3z = 1
2
-u + 2v = 5
4x - y = 9
3w - 5z = -4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
10. 2r + t = 5
3r - t = 20
11. 2m - n = -1
3m + 2n = 30
12. 6x + 3y = 6
8x + 5y = 12
13. 3j - k = 10
4j - k = 16
14. 2x - y = -4
-4x + 2y = 6
15. 2g + h = 6
3g - 2h = 16
16. 2t + 4v = 6
- t - 2v = -3
17. 3x - 2y = 12
1
18. −
x + 3y = 11
2
2x + −
y = 14
8x - 5y = 17
3
2
Solve each system of equations.
19. 8x + 3y = -5
10x + 6y = -13
20. 8q - 15r = -40
4q + 2r = 56
21. 3x - 4y = 12
1
4
4
−
x-−
y=−
3
9
3
22. 4b - 2d = 5
-2b + d = 1
23. x + 3y = 4
x=1
24. 4m - 2p = 0
-3m + 9p = 5
25. 5g + 4k = 10
-3g - 5k = 7
26. 0.5x + 2y = 5
x - 2y = -8
27. h - z = 3
-3h + 3z = 6
28. SPORTS Last year the volleyball team paid $5 per pair for socks and $17 per pair for
shorts on a total purchase of $315. This year they spent $342 to buy the same number of
pairs of socks and shorts because the socks now cost $6 a pair and the shorts cost $18.
a. Write a system of two equations that represents the number of pairs of socks and
shorts bought each year.
b. How many pairs of socks and shorts did the team buy each year?
Chapter 3
27
North Carolina StudyText, Math BC, Volume 2
Lesson 3-2
Solve each system of equations by using elimination.
NAME
3-2
DATE
PERIOD
Word Problem Practice
SCS
MBC.A.9.2
Solving Systems of Equations Algebraically
1. SUPPLIES Kirsta and Arthur both need
pens and blank CDs. The equation that
represents Kirsta’s purchases is
y = 27 - 3x. The equation that
represents Arthur’s purchases is
y = 17 - x. If x represents the price
of the pens, and y represents the price of
the CDs, what are the prices of the pens
and the CDs?
5. GAMES Mark and Stephanie are
playing darts according to World Darts
Federation rules. Each toss earns points
depending on where the dart lands.
Each ring and sector is worth a different
number of points. Mark and Stephanie
both toss three darts in the first round.
It happens that their darts only land in
two areas of the board.
12
5 20 1
18
4
9
13
14
2. WALKING Amy is walking a straight
path that can be represented by the
equation y = 2x + 3. At the same time
Kendra is walking the straight path that
has the equation 3y = 6x + 6.
What is the solution to the system of
equations that represents the paths
the two girls walked? Explain.
10
8
15
16
7
2
19 3 17
b. Solve the equations. How many
points is the first area worth?
How many points is the second
area worth?
c. If Mark and Stephanie are playing to
150 points and their darts continue to
land in only the two areas, which
different combinations of darts
landing in the two areas will add up
to 150? How many darts landing in
the first area would total 150 points?
How many darts landing in the
second area?
4. PRICES At a store, toothbrushes cost
x dollars and bars of soap cost y dollars.
One customer bought 2 toothbrushes
and 1 bar of soap for $11. Another
customer bought 6 toothbrushes and
5 bars of soap for $38. Both amounts
do not include tax. Write and solve a
system of equations for x and y.
28
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
a. Stephanie earned a total of 60 points
with two darts landing in the first
area and one dart landing in the
second area. One of Mark’s darts
landed in the first area and two in
the second. He scored 75 points.
Write a system of equations for their
scores. Use x for the first area and y
for the second area.
3. CAFETERIA To furnish a cafeteria, a
school can spend $5200 on tables and
chairs. Tables cost $200 and chairs cost
$40. Each table will have 8 chairs
around it. How many tables and chairs
will the school purchase?
Chapter 3
6
11
NAME
DATE
3-4
PERIOD
Study Guide
SCS
MBC.D.2.1
Optimization with Linear Programming
Maximum and Minimum Values When a system of linear inequalities produces a
bounded polygonal region, the maximum or minimum value of a related function will occur
at a vertex of the region.
Example
Graph the system of inequalities. Name the coordinates of the
vertices of the feasible region. Find the maximum and minimum values of the
function f(x, y) = 3x + 2y for this polygonal region.
y≤4
y ≤ -x + 6
1
3
y≥−
x-−
2
2
y ≤ 6x + 4
y
(x, y )
3x + 2y
f (x, y)
(0, 4)
3(0) + 2(4)
8
(2, 4)
3(2) + 2(4)
14
(5, 1)
3(5) + 2(1)
17
(-1, -2)
3(-1) + 2(-2)
-7
6
4
2
x
-4
-2
2
O
4
6
-2
The maximum value is 17 at (5, 1). The minimum value is -7 at (-1, -2).
Exercises
Graph each system of inequalities. Name the coordinates of the vertices of the
feasible region. Find the maximum and minimum values of the given function for
this region.
1. y ≥ 2
1≤x≤5
y≤x+3
f(x, y) = 3x - 2y
2. y ≥ -2
y ≥ 2x - 4
x - 2y ≥ -1
f(x, y) = 4x - y
y
3. x + y ≥ 2
4y ≤ x + 8
y ≥ 2x - 5
f(x, y) = 4x + 3y
y
y
O
O
O
Chapter 3
Lesson 3-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
First find the vertices of the bounded region. Graph
the inequalities.
The polygon formed is a quadrilateral with vertices at
(0, 4), (2, 4), (5, 1), and (-1, -2). Use the table to find the
maximum and minimum values of f(x, y) = 3x + 2y.
x
x
x
29
North Carolina StudyText, Math BC, Volume 2
NAME
3-4
DATE
Study Guide
PERIOD
SCS
(continued)
MBC.D.2.1
Optimization with Linear Programming
Optimization
When solving linear programming problems, use the following
procedure.
1.
2.
3.
4.
5.
6.
7.
Define variables.
Write a system of inequalities.
Graph the system of inequalities.
Find the coordinates of the vertices of the feasible region.
Write an expression to be maximized or minimized.
Substitute the coordinates of the vertices in the expression.
Select the greatest or least result to answer the problem.
Example
A painter has exactly 32 units of yellow dye and 54 units of green
dye. He plans to mix as many gallons as possible of color A and color B. Each
gallon of color A requires 4 units of yellow dye and 1 unit of green dye. Each
gallon of color B requires 1 unit of yellow dye and 6 units of green dye. Find the
maximum number of gallons he can mix.
40
Step 1 Define the variables.
x = the number of gallons of color A made
y = the number of gallons of color B made
Color B (gallons)
35
30
25
Exercises
1. FOOD A delicatessen has 12 pounds of plain sausage and 10 pounds of spicy sausage.
3
1
A pound of Bratwurst A contains −
pound of plain sausage and −
pound of spicy
4
4
1
pound of each sausage.
sausage. A pound of Bratwurst B contains −
2
Find the maximum number of pounds of bratwurst that can be made.
2. MANUFACTURING Machine A can produce 30 steering wheels per hour at a cost of $8
per hour. Machine B can produce 40 steering wheels per hour at a cost of $12 per hour.
The company can use either machine by itself or both machines at the same time. What
is the minimum number of hours needed to produce 380 steering wheels if the cost must
be no more than $108?
Chapter 3
30
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Step 2 Write a system of inequalities.
20
Since the number of gallons made cannot be
15
negative, x ≥ 0 and y ≥ 0.
(6, 8)
10
There are 32 units of yellow dye; each gallon of
(0, 9) 5
color A requires 4 units, and each gallon of
(8, 0)
color B requires 1 unit.
0
5 10 15 20 25 30 35 40 45 50 55
Color A (gallons)
So 4x + y ≤ 32.
Similarly for the green dye, x + 6y ≤ 54.
Steps 3 and 4 Graph the system of inequalities and find the coordinates of the vertices of
the feasible region. The vertices of the feasible region are (0, 0), (0, 9), (6, 8), and (8, 0).
Steps 5–7 Find the maximum number of gallons, x + y, that he can make. The maximum
number of gallons the painter can make is 14, 6 gallons of color A and 8 gallons of color B.
NAME
3-4
DATE
PERIOD
Practice
SCS
MBC.D.2.1
Optimization with Linear Programming
Graph each system of inequalities. Name the coordinates of the vertices of the
feasible region. Find the maximum and minimum values of the given function for
this region.
1. 2x - 4 ≤ y
-2x - 4 ≤ y
y≤2
f (x, y) = -2x + y
2. 3x - y ≤ 7
2x - y ≥ 3
y≥x-3
f (x, y) = x - 4y
y
3. x ≥ 0
y≥0
y≤6
y ≤ -3x + 15
f(x, y) = 3x + y
y
y
x
x
O
x
O
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4. x ≤ 0
y≤0
4x + y ≥ -7
f (x, y) = -x - 4y
5. y ≤ 3x + 6
4y + 3x ≤ 3
x ≥ -2
f (x, y) = -x + 3y
y
O
6. 2x + 3y ≥ 6
2x - y ≤ 2
x≥0
y≥0
f (x, y) = x + 4y + 3
y
y
x
O
x
O
x
7. PRODUCTION A glass blower can form 8 simple vases or 2 elaborate vases in an hour.
In a work shift of no more than 8 hours, the worker must form at least 40 vases.
a. Let s represent the hours forming simple vases and e the hours forming elaborate
vases. Write a system of inequalities involving the time spent on each type of vase.
b. If the glass blower makes a profit of $30 per hour worked on the simple vases and
$35 per hour worked on the elaborate vases, write a function for the total profit on
the vases.
c. Find the number of hours the worker should spend on each type of vase to maximize
profit. What is that profit?
Chapter 3
31
North Carolina StudyText, Math BC, Volume 2
Lesson 3-4
O
NAME
3-4
DATE
PERIOD
Word Problem Practice
SCS
MBC.D.2.1
Optimization with Linear Programming
1. REGIONS A region in the plane is
formed by the equations x - y < 3,
x - y > -3, and x + y > -3. Is this
region bounded or unbounded? Explain.
4. ELEVATION A trapezoidal park is built
on a slight incline. The function for the
ground elevation above sea level is
f(x, y) = x - 3y + 20 feet. What are
the coordinates of the highest point in
the park?
y
5
2. MANUFACTURING Eighty workers are
available to assemble tables and chairs.
It takes 5 people to assemble a table and
3 people to assemble a chair. The
workers always make at least as many
tables as chairs because the tables are
easier to make. If x is the number of
tables and y is the number of chairs, the
system of inequalities that represent
what can be assembled is x > 0, y > 0,
y ≤ x, and 5x + 3y ≤ 80. What is the
maximum total number of chairs and
tables the workers can make?
O
x
a. Write linear inequalities to represent
the number of pots p and plates a
Josh may bring to the fair.
b. List the coordinates of the vertices of
the feasible region.
c. How many pots and how many plates
should Josh make to maximize his
potential profit?
32
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5. CERAMICS Josh has 8 days to make
pots and plates to sell at a local fair.
Each pot weighs 2 pounds and each
plate weighs 1 pound. Josh cannot carry
more than 50 pounds to the fair. Each
day, he can make at most 5 plates and
at most 3 pots. He will make $12 profit
for every plate and $25 profit for every
pot that he sells.
3. FISH An aquarium is 7000 cubic inches.
Nathan wants to populate the aquarium
with neon tetras and catfish. It is
recommended that each neon tetra be
allowed 170 cubic inches and each
catfish be allowed 700 cubic inches of
space. Nathan would like at least one
catfish for every 4 neon tetras. Let n be
the number of neon tetra and c be the
number of catfish. The following
inequalities form the feasible region for
this situation: n > 0, c > 0, 4c ≥ n, and
170n + 700c ≤ 7000. What is the
maximum number of fish Nathan can
put in his aquarium?
Chapter 3
5
NAME
DATE
4-1
PERIOD
Study Guide
SCS
MBC.N.2.1
Introduction to Matrices
Organize and Analyze Data
a rectangular array of variables or constants in horizontal rows and vertical columns, usually
enclosed in brackets.
Matrix
Example 1
Owls’ eggs incubate for 30 days and their fledgling period is also
30 days. Swifts’ eggs incubate for 20 days and their fledgling period is 44 days.
Pigeon eggs incubate for 15 days, and their fledgling period is 17 days. Eggs of the
king penguin incubate for 53 days, and the fledgling time for a king penguin is
360 days. Write a 2 × 4 matrix to organize this information. Source: The Cambridge Factfinder
Owl
Incubation
Fledgling
⎡ 30
⎢
⎣ 30
Swift Pigeon King Penguin
20
44
15
17
53
360
⎤
⎦
Example 2
⎡ 13 10 -3 45⎤
What are the dimensions of matrix A if A = ⎢
?
⎣ 2 8 15 80⎦
Since matrix A has 2 rows and 4 columns, the dimensions of A are 2 × 4.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
State the dimensions of each matrix.
⎡ 15 5 27 -4⎤
23 6 0 5
1.
2. [16 12 0]
14 70 24 -3
⎣63 3 42 90 ⎦
⎢
⎡ 71
⎢
3. 39
⎢
⎢ 45
⎢ 92
⎣ 78
44 ⎤
27
16 53 65 ⎦
4. A travel agent provides for potential travelers the normal high temperatures for the
months of January, April, July, and October for various cities. In Boston these figures are
36°, 56°, 82°, and 63°. In Dallas they are 54°, 76°, 97°, and 79°. In Los Angeles they are
68°, 72°, 84°, and 79°. In Seattle they are 46°, 58°, 74°, and 60°. In St. Louis they are
38°, 67°, 89°, and 69°. Organize this information in a 4 × 5 matrix. Source: The New York Times Almanac
Chapter 4
33
North Carolina StudyText, Math BC, Volume 2
Lesson 4-1
A matrix can be described by its dimensions. A matrix with m rows and n columns is an
m × n matrix.
NAME
4-1
DATE
Study Guide
PERIOD
SCS
(continued)
MBC.N.2.1
Introduction to Matrices
Elements of a Matrix
A matrix is a rectangular array of variables or constants in
horizontal rows and vertical columns. The values are called elements and are identified by
their location in the matrix. The location of an element is written as a subscript with the
number of its row followed by the number of its column. For example, a12 is the element in
the first row and second column of matrix A.
In the matrices below, 11 is the value of a12 in the first matrix. The value of b32 in the
second matrix is 7.
⎡ 3
⎢ 5
B=⎢ 8
⎢
⎢ 11
⎣ 4
⎡ 7 11 2 8 ⎤
A = 5 4 10 1
⎣ 9 3 6 12 ⎦
⎢
Example 1
9 12 ⎤
10 15 7 6 13 1
2 14 ⎦
Example 2
Find the value of c23.
Find the value of d54.
⎡ 25
⎢ 7
matrix D = ⎢ 17
⎢
⎢ 22
⎣ 5
⎡2 5 3⎤
C=⎢
⎣3 4 1⎦
Since c23 is the element in row 2,
column 3, the value of c23 is 1.
Exercises
Identify each element for the following matrices.
⎡ 12 7 5 ⎤
F=
⎢
9 2 11
,
6 14 8
⎣ 1 4 3⎦
⎡
⎢
G=⎢
⎢
⎢
⎣
1
2
3
4
5
14
15
16
17
6
13 12 ⎤
20 11 19 10 ,
18 9
7 8⎦
⎡ 5 9 11 4 ⎤
H = 3 7 2 10 .
⎣8 2 6 1 ⎦
⎢
1. f32
2. g51
3. h22
4. g43
5. h34
6. f23
7. h14
8. f42
9. g14
Chapter 4
34
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
11 4 1 20 ⎤
8 9 12 13 6 15 18 2 16 21 24 19
23 3 14 10 ⎦
Since d54 is the element in row 5, column 4,
the value of d54 is 14.
NAME
4-1
DATE
PERIOD
Practice
SCS
MBC.N.2.1
Introduction to Matrices
State the dimensions of each matrix.
⎡-2 2 -2 3 ⎤
3. 5 16 0 0
⎣ 4 7 -1 4⎦
⎡ 5 8 -1⎤
2. ⎢
⎣-2 1 8⎦
1. [-3 -3 7]
⎢
⎡ 4
9
A=
3
⎣-1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
⎢
7 0⎤
8 -4
,
0 5
2 6⎦
Lesson 4-1
Identify each element for the following matrices.
⎡ 2 6 -1 0 ⎤
B=⎢
.
⎣9 5 7 2⎦
4. b23
5. a42
6. b11
7. a32
8. b14
9. a23
10. TICKET PRICES The table at the right gives
ticket prices for a concert. Write a 2 × 3 matrix
that represents the cost of a ticket.
Child Student Adult
Cost Purchased in
Advance
$6
$12
$18
Cost Purchased at
the Door
$8
$15
$22
11. CONSTRUCTION During each of the last
Week 1
Week 2
Week 3
three weeks, a road-building crew has used
Load 1 40 tons Load 1 40 tons Load 1 32 tons
three truckloads of gravel. The table at the
right shows the amount of gravel in each load. Load 2 32 tons Load 2 40 tons Load 2 24 tons
Load 3 24 tons Load 3 32 tons
Load 3 24 tons
a. Write a matrix for the amount of gravel
in each load.
b. What are the dimensions of the matrix?
Chapter 4
35
North Carolina StudyText, Math BC, Volume 2
NAME
DATE
4-1
PERIOD
Word Problem Practice
SCS
MBC.N.2.1
Introduction to Matrices
1. HAWAII The table shows the
population and area of some of the
islands in Hawaii. What would be the
dimensions of a matrix that represented
this information?
Island
Population
Area(mi2)
Hawaii
120,317
4038
Maui
91,361
729
Oahu
836,231
594
Kauai
50,947
549
Lanai
2426
140
4. INVENTORY A store manager records
the number of light bulbs in stock for 3
different brands over a five-day period.
The manager decides to make a matrix
of this information. Each row represents
a different brand, and each column
represents a different day. The entry in
column N represents the inventories at
the beginning of day N.
⎡ 25 24 22 20 19 ⎤
30 27 25 22 21
⎣ 28 25 21 19 19 ⎦
⎢
Source: Virtual Tour of Hawaii
Assuming that the inventories were
never replenished, which brand holds the
record for most light bulbs sold on a
given day?
2. LAUNDRY Carl is looking for a
Laundromat. SuperWash has 20 small
washers, 10 large washers, and 20
dryers. QuickClean has 40 small
washers, 5 large washers, and 50 dryers.
ToughSuds has 15 small washers, 40
large washers, and 100 dryers. Write a
matrix to organize this information.
a. Organize this data in a 3 by 5 matrix.
3. CITY DISTANCES The incomplete
matrix shown gives the approximate
distances between Chicago, Los Angeles,
and New York City. Complete
the matrix.
Los
NYC Chicago Angeles
NYC
Chicago
0
810
b. Which salesperson made the most
money that week?
2790
2050
Los Angeles
Chapter 4
36
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5. SHOE SALES A shoe store manager
keeps track of the amount of money
made by each of three salespeople for
each day of a workweek. Monday
through Friday, Carla made $40, $70,
$35, $50, and $20. John made $30, $60,
$20, $45, and $30. Mary made $35, $90,
$30, $40, and $30.
NAME
DATE
4-2
PERIOD
Study Guide
SCS
MBC.N.2.2
Operations with Matrices
Add and Subtract Matrices Matrices with the same dimensions can be added
together or one can be subtracted from the other.
⎡a
Addition of Matrices
Subtraction of Matrices
b c⎤ ⎡ j k l⎤ ⎡ a+j
d e f + m n o = d+m
⎣g h i ⎦ ⎣ p q r ⎦ ⎣ g + p
b+k c+l⎤
e+n f+o
h+q i+r⎦
⎡a
b-k c-l⎤
e-n f-o
h-q i-r⎦
⎢
⎢
⎢
b c⎤ ⎡ j k l⎤ ⎡ a-j
d e f - m n o = d-m
⎣g h i ⎦ ⎣ p q r ⎦ ⎣ g - p
⎢
⎢
⎢
Example 1
Lesson 4-2
⎡ 6 -7 ⎤
⎡ 4 2⎤
Find A + B if A = ⎢
and B = ⎢
.
⎣ 2 -12 ⎦
⎣ -5 -6 ⎦
⎡ 6 -7 ⎤ ⎡ 4 2 ⎤
A+B=⎢
+⎢
⎣ 2 -12 ⎦ ⎣-5 -6 ⎦
⎡6 + 4
⎤
-7 + 2
=⎢
⎣ 2 + (-5) -12 + (-6) ⎦
⎡ 10 -5 ⎤
=⎢
⎣ -3 -18 ⎦
⎡-2 8 ⎤
⎡ 4 -3 ⎤
Find A - B if A =
3 -4 and B = -2 1 .
⎣ 10 7 ⎦
⎣-6 8 ⎦
⎡-2 8 ⎤ ⎡ 4 -3 ⎤
3 -4 - -2 1
A-B=
⎣ 10 7 ⎦ ⎣-6 8 ⎦
8 - (-3) ⎤ ⎡-6 11⎤
⎡-2 - 4
3 - (-2) -4 - 1
=
=
5 -5
⎣ 10 - (-6)
⎦
⎣
7-8
16 -1⎦
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Example 2
⎢
⎢
⎢
⎢
⎢
⎢
Exercises
Perform the indicated operations. If the matrix does not exist, write impossible.
⎡ 8
7⎤
1. ⎢
⎣-10 -6 ⎦
⎡-4
3⎤
⎢
⎣ 2 -12 ⎦
⎡ 6 -5 9⎤
⎡-4 3 2 ⎤
2. ⎢
+ ⎢
⎣-3 4 5⎦
⎣ 6 9 -4 ⎦
⎡ 6⎤
3. -3 + [-6 3 -2]
⎣ 2⎦
⎡ 5 -2⎤ ⎡-11 6 ⎤
4. -4 6 +
2 -5
⎣ 7 9⎦ ⎣ -4 -7 ⎦
⎢ ⎢
⎡ 8 0 -6⎤ ⎡-2 1 7⎤
5. 4 5 -11 3 -4 3
⎣-7 3
4⎦ ⎣-8 5 6⎦
⎢
Chapter 4
⎢
⎢
3 2⎤
⎡ −
−
4
6.
1
⎣- −
2
⎢
37
2⎤
1
⎡−
−
5
3
2
4
1
2
−⎦ ⎣− -−
3
2⎦
3
⎢
North Carolina StudyText, Math BC, Volume 2
NAME
DATE
4-2
Study Guide
PERIOD
SCS
(continued)
MBC.N.2.2
Operations with Matrices
Scalar Multiplication
Scalar Multiplication
k
You can multiply an m × n matrix by a scalar k.
⎡a
⎢
⎣d
b c ⎤ ⎡ ka kb kc ⎤
=⎢
e f ⎦ ⎣ kd ke kf ⎦
⎡ 4 0⎤
⎡ -1 5 ⎤
If A = ⎢
and B = ⎢
, find 3B - 2A.
⎣ -6 3 ⎦
⎣ 7 8⎦
Example
⎡-1 5 ⎤
⎡ 4 0⎤
3B - 2A = 3 ⎢
-2⎢
⎣ 7 8⎦
⎣-6 3 ⎦
Substitution
⎡ 3(-1) 3(5)⎤ ⎡ 2(4) 2(0)⎤
=⎢
-⎢
⎣ 3(7) 3(8)⎦ ⎣ 2(-6) 2(3)⎦
Multiply.
⎡ -3 15⎤ ⎡ 8 0 ⎤
=⎢
-⎢
⎣ 21 24⎦ ⎣ -12 6 ⎦
Simplify.
⎡ -3 - 8
=⎢
⎣ 21 - (-12)
Subtract.
15 - 0 ⎤
24 - 6 ⎦
⎡-11 15 ⎤
=⎢
⎣ 33 18 ⎦
Simplify.
Exercises
⎡ 2 -5 3 ⎤
1. 6 0 7 -1
⎣-4 6 9 ⎦
⎢
1
2. - −
3
⎡
6 15 9 ⎤
51 -33 24
⎣-18
3 45 ⎦
⎢
⎡ 25 -10 -45 ⎤
3. 0.2 5 55 -30
⎣ 60 35 -95 ⎦
⎢
⎡ -4 5⎤
⎡ -1 2⎤
4. 3 ⎢
-2⎢
⎣ 2 3⎦
⎣ -3 5⎦
⎡ 3 -1 ⎤
⎡-2 0 ⎤
5. -2 ⎢
+ 4⎢
⎣ 0 7⎦
⎣ 2 5⎦
⎡ 6 -10 ⎤
⎡2 1⎤
6. 2 ⎢
+ 5⎢
⎣-5
⎣4 3⎦
8⎦
⎡ 1 -2 5⎤
⎡ 4 3 -4 ⎤
7. 4 ⎢
- 2⎢
⎣-3 4 1⎦
⎣ 2 -5 -1 ⎦
⎡ 2 1⎤
⎡ 4 0⎤
8. 8 3 -1 + 3 -2 3
⎣-2 4 ⎦
⎣ 3 -4 ⎦
⎢
Chapter 4
⎢
(
)
1 ⎡ 9 1⎤ ⎡ 3 -5 ⎤
9. −
⎢
+⎢
4 ⎣ -7 0 ⎦ ⎣ 1
7⎦
38
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Perform the indicated operations. If the matrix does not exist, write impossible.
NAME
4-2
DATE
PERIOD
Practice
SCS
MBC.N.2.2
Operations with Matrices
Perform the indicated operations. If the matrix does not exist, write impossible.
⎡ 2 -1 ⎤ ⎡-6
9⎤
1. 3 7 + 7 -11
⎣14 -9 ⎦ ⎣-8 17 ⎦
⎡ 4⎤ ⎡-67 ⎤
2. -71 - 45
⎣ 18⎦ ⎣-24 ⎦
⎡-1
⎡ -3 16 ⎤
0⎤
3. -3 ⎢
+ 4⎢
⎣ 17 -11 ⎦
⎣-21 12 ⎦
⎡2 -1 8 ⎤
⎡-1 4 -3 ⎤
4. 7 ⎢
- 2⎢
⎣4 7 9 ⎦
⎣ 7 2 -6 ⎦
⎡1 ⎤
⎡0 ⎤ ⎡10⎤
5. -2 ⎢ + 4 ⎢ - ⎢ ⎣ 2⎦
⎣ 5⎦ ⎣18⎦
8 12⎤ 2 ⎡ 27 -9 ⎤
3⎡
6. −
⎢
+ −⎢
4 ⎣-16 20⎦
3 ⎣ 54 -18 ⎦
⎢
⎢
⎢
⎢
7. A - B
8. A - C
10. 4B - A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
9. -3B
11. -2B - 3C
12. A + 0.5C
13. ECONOMICS Use the table that
shows loans by an economic
development board to women
and men starting new businesses.
a. Write two matrices that
represent the number of new
businesses and loan amounts,
one for women and one for men.
Lesson 4-2
⎡ 4 -1 0 ⎤
⎡-2 4 5 ⎤
⎡ 10 -8 6 ⎤
Use matrices A = ⎢
, B = ⎢
, and C = ⎢
to find
⎣ -3 6 2 ⎦
⎣ 1 0 9⎦
⎣-6 -4 20 ⎦
the following.
Women
Men
Businesses
Loan
Amount ($)
Businesses
Loan
Amount ($)
2008
27
$567,000
36
$864,000
2009
41
$902,000
32
$672,000
2010
35
$777,000
28
$562,000
b. Find the sum of the numbers of new businesses and loan amounts
for both men and women over the three-year period expressed as
a matrix.
14. PET NUTRITION Use the table that gives nutritional
information for two types of dog food. Find the
difference in the percent of protein, fat, and fiber
between Mix B and Mix A expressed as a matrix.
Chapter 4
39
% Protein
% Fat
% Fiber
Mix A
22
12
5
Mix B
24
8
8
North Carolina StudyText, Math BC, Volume 2
NAME
DATE
4-2
PERIOD
Word Problem Practice
SCS
MBC.N.2.2
Operations with Matrices
1. FARES The matrix below gives general
admission and film fares at New York’s
Museum of Modern Art.
4. SUNFLOWERS Matrix H is a 3 by 1
matrix that contains the initial heights
of three sunflowers. Matrix G is a 3 by 1
matrix that contains the numbers of
inches the corresponding sunflowers
grow in a week. What does matrix
H + 4G represent?
Child Adult
⎡ 20 12 ⎤
⎢
Film ⎣ 12
8.5⎦
What can you do to this matrix in order
to create another matrix that represents
fares for 5 people?
General Admission
SOURCE: Museum of Modern Art
5. DINNER The menu shows prices for
some dishes at a restaurant.
2. NEGATION Two engineers need to
negate all the entries of a matrix. One
engineer tries to do this by multiplying
the matrix by -1. The other engineer
tries to do this by subtracting twice the
matrix from itself. Which engineer, if
either, will get the correct result?
Il Ristorante Menu
0
660
Los Angeles
850
700
0
$9.00
Chicken
$14.00
$7.00
Steak
$22.00
$11.00
b. Let M be the matrix you wrote for
part a. Write an expression involving
M that would give prices that include
an additional 20% to cover tax and
tip.
Los
NYC Chicago Angeles
0
40
70
NYC
Chicago
46
0
60
Los Angeles
85
70
0
c. Compute the matrix you described in
part b.
Write a matrix that represents the full
cost for travel between these cities.
Chapter 4
40
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Los
NYC Chicago Angeles
0
440
700
NYC
460
Half-portion
$17.00
a. Make a 3 by 2 matrix to organize
these data.
3. PLANE FARES The airfares for travel
between New York, Chicago, and Los
Angeles are organized in the first
matrix. The second matrix gives the tax
surcharges for corresponding flights.
Chicago
Regular
Lamb
NAME
DATE
4-3
PERIOD
Study Guide
SCS
MBC.N.2.3
Multiplying Matrices
Multiply Matrices You can multiply two matrices if and only if the number of columns
in the first matrix is equal to the number of rows in the second matrix.
·
A
Multiplication of Matrices
⎡a
⎢
⎣c
b⎤
d⎦
B
⎡
· ⎢e
⎣g
=
AB
f ⎤ ⎡ae + bg af + bh ⎤
=⎢
h⎦ ⎣ ce + dg cf + dh⎦
Example
⎡ -4 3 ⎤
⎡ 5 -2 ⎤
Find AB if A =
.
2 -2 and B = ⎢
⎣ -1 3 ⎦
⎣ 1 7 ⎦
⎡ -4 3 ⎤
⎡ 5 -2 ⎤
AB =
Substitution
2 -2 · ⎢
⎣ -1 3 ⎦
⎣ 1 7⎦
⎢
⎢
-4(-2) + 3(3)
⎡ - 4(5) + 3(-1)
⎤
=
2(5) + (-2)(-1)
2(-2) + (-2)(3)
⎣ 1(5) + 7(-1)
⎦
1(-2) + 7(3)
Multiply columns by rows.
⎡ -23 17 ⎤
=
12 -10
⎣ -2 19 ⎦
Simplify.
⎢
⎢
Find each product, if possible.
⎡ 4 1⎤ ⎡ 3 0 ⎤
⎡-1 0 ⎤ ⎡ 3 2 ⎤
1. ⎢
2. ⎢
·⎢
·⎢
⎣-2 3 ⎦ ⎣ 0 3 ⎦
⎣ 3 7 ⎦ ⎣-1 4 ⎦
Lesson 4-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
⎡3 -1 ⎤ ⎡3 -1⎤
3. ⎢
·⎢
⎣2
4 ⎦ ⎣2 4⎦
⎡-3 1⎤ ⎡ 4 0 -2⎤
4. ⎢
·⎢
⎣ 5 -2⎦ ⎣-3 1 1⎦
⎡ 3 -2⎤
⎡1 2 ⎤
5. 0 4 · ⎢
⎣2 1 ⎦
⎣-5 1⎦
⎡5 -2⎤ ⎡ 4 -1⎤
6. ⎢
·⎢
⎣2 -3⎦ ⎣-2 5⎦
⎡ 6 10⎤
7. -4 3 · [0 4 -3]
⎣-2 7⎦
⎡7 -2⎤ ⎡ 1 -3⎤
8. ⎢
·⎢
⎣5 -4⎦ ⎣-2 0⎦
⎡ 2 0 -3⎤ ⎡ 2 -2⎤
9. 1 4 -2 · 3 1
⎣-1 3 1⎦ ⎣-2 4⎦
⎢
Chapter 4
⎢
41
⎢
⎢
North Carolina StudyText, Math BC, Volume 2
NAME
4-3
DATE
Study Guide
PERIOD
SCS
(continued)
MBC.N.2.3
Multiplying Matrices
Multiplicative Properties
The Commutative Property of Multiplication does not hold
for matrices.
Properties of Matrix Multiplication
For any matrices A, B, and C for which the matrix product is
defined, and any scalar c, the following properties are true.
Associative Property of Matrix Multiplication
(AB)C = A(BC)
Associative Property of Scalar Multiplication
c(AB) = (cA)B = A(cB)
Left Distributive Property
C(A + B) = CA + CB
Right Distributive Property
(A + B)C = AC + BC
Example
⎡2 0 ⎤
⎡1 -2⎤
⎡ 4 -3⎤
Use A = ⎢
, B = ⎢
, and C = ⎢
to find each product.
⎣5 -3⎦
⎣6 3⎦
⎣ 2 1⎦
a. (A + B)C
(
)
⎡4 -3⎤ ⎡2 0⎤ ⎡1 -2⎤
(A + B) C = ⎢
+⎢
·⎢
⎣2 1⎦ ⎣5 -3⎦ ⎣6 3⎦
⎡6 -3⎤ ⎡1 -2⎤
=⎢
·⎢
⎣7 -2⎦ ⎣6 3⎦
⎡6(1) + (-3)(6)
=⎢
⎣ 7(1) + (-2)(6)
6(-2) + (-3)(3) ⎤
7(-2) + (-2)(3)⎦
⎡-12 -21⎤
=⎢
⎣ -5 -20⎦
⎡4 -3⎤ ⎡1 -2⎤ ⎡ 2 0⎤ ⎡ 1 -2⎤
AC + BC = ⎢
·⎢
+⎢
·⎢
⎣2 1⎦ ⎣6 3⎦ ⎣ 5 -3⎦ ⎣ 6 3⎦
⎡4(1) + (-3)(6) 4(-2) + (-3)(3) ⎤ ⎡2(1) + 0(6)
⎤
2(-2) + 0(3)
=⎢
+⎢
⎣ 2(1) + 1(6)
2(-2) + 1(3) ⎦ ⎣ 5(1) + (-3)(6) 5(-2) + (-3)(3)⎦
⎡-14 -17⎤ ⎡ 2 -4⎤ ⎡-12 -21⎤
=⎢
+⎢
=⎢
⎣ 8 -1⎦ ⎣-13 -19⎦ ⎣ -5 -20⎦
Note that although the results in the example illustrate the Right Distributive Property,
they do not prove it.
Exercises
1
⎡- −
-2⎤
⎡3 2⎤
⎡6 4⎤
2
Use A = ⎢
, and scalar c = -4 to determine whether
, B = ⎢
, C =
⎣5 -2⎦
⎣2 1⎦
1 -3
⎣
⎦
the following equations are true for the given matrices.
⎢
1. c(AB) = (cA)B
2. AB = BA
3. BC = CB
4. (AB)C = A(BC)
5. C(A + B) = AC + BC
6. c(A + B) = cA + cB
Chapter 4
42
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
b. AC + BC
NAME
4-3
DATE
PERIOD
Practice
SCS
MBC.N.2.3
Multiplying Matrices
Determine whether each matrix product is defined. If so, state the dimensions of
the product.
1. A7 × 4 · B4 × 3
2. A3 × 5 · M5 × 8
3. M2 × 1 · A1 × 6
4. M3 × 2 · A3 × 2
5. P1 × 9 · Q9 × 1
6. P9 × 1 · Q1 × 9
Find each product, if possible.
⎡ 2 4 ⎤ ⎡ 3 -2 7⎤
7. ⎢
·⎢
⎣ 3 -1 ⎦ ⎣ 6 0 -5⎦
⎡-3 0 ⎤ ⎡ 2 4⎤
9. ⎢
·⎢
⎣ 2 5 ⎦ ⎣ 7 -1⎦
11. [4 0 2] ·
⎡ 1⎤
3
⎣-1 ⎦
⎢ ⎡ 3 -2 7 ⎤ ⎡ 3 -2 7⎤
10. ⎢
·⎢
⎣ 6 0 -5 ⎦ ⎣ 6 0 -5⎦
12.
⎡ 1⎤
3 · [4 0 2]
⎣-1 ⎦
⎢ ⎡ 6 11⎤
14. [-15 -9] · ⎢
⎣23 -10⎦
⎡1 3⎤
⎡ 4 0⎤
⎡-1 0⎤
Use A = ⎢
, B = ⎢
, C = ⎢
, and c = 3 to determine whether the
⎣3 1⎦
⎣-2 -1⎦
⎣ 0 -1⎦
following equations are true for the given matrices.
15. AC = CA
16. A(B + C) = BA + CA
17. (AB)c = c(AB)
18. (A + C)B = B(A + C)
19. RENTALS For their one-week vacation,
the Montoyas can rent a 2-bedroom
condominium for $1796, a 3-bedroom
condominium for $2165, or a 4-bedroom
condominium for $2538. The table
shows the number of units in each of
three complexes.
Lesson 4-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
⎡-6 2⎤ ⎡5 0⎤
13. ⎢
·⎢
⎣ 3 -1⎦ ⎣0 5⎦
⎡ 2 4⎤ ⎡-3 0⎤
8. ⎢
· ⎢
⎣ 7 -1⎦ ⎣ 2 5⎦
2-Bedroom
3-Bedroom
4-Bedroom
Sun Haven
36
24
22
Surfside
29
32
42
Seabreeze
18
22
18
a. Write a matrix that represents the number of each type
of unit available at each complex and a matrix that
represents the weekly charge for each type of unit.
b. If all of the units in the three complexes are rented for
the week at the rates given the Montoyas, express the
income of each of the three complexes as a matrix.
c. What is the total income of all three complexes for the week?
Chapter 4
43
North Carolina StudyText, Math BC, Volume 2
NAME
4-3
DATE
PERIOD
Word Problem Practice
SCS
MBC.N.2.3
Multiplying Matrices
1. FIND THE ERROR Both A and B are
2 by 2 matrices. Maggie made the
following derivation. Is this derivation
valid? If not, what error did she make?
a. (A + B)2 = (A + B)(A + B)
b.
= (A + B)A + (A + B)B
c.
= AA + BA + AB + BB
d.
= A2 + BA + AB + B2
e.
= A2 + AB + AB + B2
f.
= A2 + 2AB + B2
4. POWERS Thad just learned about
matrix multiplication. He began to
wonder what happens when you take
powers of a matrix. He computed the
first few powers of the matrix
⎡1 1⎤
M=⎢
and noticed a pattern. What
⎣0 1⎦
is M n?
5. COST COMPARISONS The average
family spends more than $500 on school
supplies at the beginning of each school
year. Barbara and Lance need to buy
pens, pencils, and erasers. They make a
2 by 3 matrix that represents the
numbers of each item they would like to
purchase.
2. EXAM SCORES Mr. Farey recorded the
exam scores of his students in a 20 by 3
matrix. Each row listed the scores of a
different student. The first exam scores
were listed in the first column, and the
second exam scores were listed in the
second column. The final exam scores
were listed in the third column. Mr.
Farey needed to create a 20 by 1 matrix
that contained the weighted scores of
each student. The first two exams
account for 25% of the weighted score,
and the final exam counted 50%. To
make the matrix of weighted scores,
what matrix can Mr. Farey multiply his
20 by 3 matrix by on the right?
Pens Pencils Erasers
⎡ 10
15
3 ⎤
⎢
Lance ⎣ 5
20
5 ⎦
They call this matrix M. Barbara and
Lance find two stores that sell the items
at different prices and record this
information in a second matrix that they
call P.
Barbara
Pens
Pencils
Erasers
⎡2.20
0.85
⎣0.60
⎢
1.90⎤
0.95
0.65⎦
a. Compute MP.
3. SPECIAL MATRICES Mandy has a 3 by
3 matrix M. She notices that for any 3
by 3 matrix X, MX = X. What must M
be?
Chapter 4
b. What do the entries in MP mean?
44
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Store 1 Store 2
NAME
DATE
4-4
PERIOD
Study Guide
SCS
MBC.N.2.1
Transformations with Matrices
Translations and Dilations
Matrices that represent coordinates of points on a plane
are useful in describing transformations.
Translation
a transformation that moves a figure from one location to another on the coordinate plane
You can use matrix addition and a translation matrix to find the coordinates of the
translated figure.
Dilation
a transformation in which a figure is enlarged or reduced
You can use scalar multiplication to perform dilations.
Example
Find the coordinates of the vertices of the
image of ABC with vertices A(-5, 4), B(-1, 5), and
C(-3, -1) if it is moved 6 units to the right and 4 units
down. Then graph ABC and its image A' B' C'.
⎡-5
Write the vertex matrix for ABC. ⎢
⎣ 4
⎡ 6
Add the translation matrix ⎢
⎣-4
matrix of ABC.
-1
5
-3⎤ ⎡ 6
+⎢
-1⎦ ⎣-4
6
-4
B
A
-3⎤
-1⎦
x
O
C
6⎤
to the vertex
-4⎦
C
6⎤ ⎡1 5
3⎤
=⎢
-4⎦ ⎣0 1 -5⎦
The coordinates of the vertices of A'B'C' are A'(1, 0), B'(5, 1), and C'(3, -5).
Exercises
1. Quadrilateral QUAD with vertices Q(-1, -3), U(0, 0), A(5, -1), and D(2, -5) is
translated 3 units to the left and 2 units up.
a. Write the translation matrix.
Lesson 4-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
⎡-5
⎢
⎣ 4
6
-4
-1
5
y
B
A
b. Find the coordinates of the vertices of Q' U' A' D'.
2. The vertices of ABC are A(4, -2), B(2, 8), and C(8, 2). The triangle is dilated so that
its perimeter is one-fourth the original perimeter.
y
a. Write the coordinates of the vertices of ABC in a vertex
matrix.
b. Find the coordinates of the vertices of image A'B'C'.
O
x
c. Graph the preimage and the image.
Chapter 4
45
North Carolina StudyText, Math BC, Volume 2
NAME
4-4
DATE
Study Guide
PERIOD
SCS
(continued)
MBC.N.2.1
Transformations with Matrices
Reflections and Rotations
Reflection
Matrices
Rotation
Matrices
For a reflection in the:
x-axis
y-axis
⎡1 0⎤
⎢
⎣0 -1⎦
multiply the vertex matrix on the left by:
For a counterclockwise rotation about the
origin of:
90°
0⎤
1⎦
⎡0 1⎤
⎢
⎣1 0⎦
180°
⎡0 -1⎤
⎢
⎣1
0⎦
multiply the vertex matrix on the left by:
⎡-1
⎢
⎣ 0
line y = x
⎡-1
⎢
⎣ 0
0⎤
-1⎦
270°
⎡ 0 1⎤
⎢
⎣-1 0⎦
Example
Find the coordinates of the vertices of the image of ABC with
A(3, 5), B(-2, 4), and C(1, -1) after a reflection in the line y = x.
Write the ordered pairs as a vertex matrix. Then multiply the vertex matrix by the
reflection matrix for y = x.
⎡0 1⎤ ⎡3 -2 1⎤ ⎡5
4
⎢
·⎢
=⎢
⎣1 0⎦ ⎣5
4 -1⎦ ⎣3 -2
-1⎤
-1⎦
The coordinates of the vertices of A'B'C' are A'(5, 3), B'(4, -2), and C'(-1, 1).
Exercises
2. DEF with vertices D(-2, 5), E(1, 4), and F(0, -1) is rotated 90° counterclockwise about
the origin.
a. Write the coordinates of the triangle in a vertex matrix.
b. Write the rotation matrix for this situation.
y
c. Find the coordinates of the vertices of D'E'F'.
O
x
d. Graph the preimage and the image.
Chapter 4
46
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. The coordinates of the vertices of quadrilateral ABCD are A(-2, 1), B(-1, 3), C(2, 2), and
D(2, -1). Find the coordinates of the vertices of the image A'B'C'D' after a
reflection in the y-axis.
NAME
4-4
DATE
Practice
PERIOD
SCS
MBC.N.2.1
Transformations with Matrices
1. Quadrilateral WXYZ with vertices W(-3, 2), X(-2, 4), Y(4, 1), and
Z(3, 0) is translated 1 unit left and 3 units down.
y
a. Write the translation matrix.
x
O
b. Find the coordinates of quadrilateral W' X' Y' Z'.
c. Graph the preimage and the image.
2. The vertices of RST are R(6, 2), S(3, -3), and T(-2, 5).
The triangle is dilated so that its perimeter is one half
the original perimeter.
y
a. Write the coordinates of RST in a vertex matrix.
b. Find the coordinates of the image R'S'T '.
x
O
3. The vertices of quadrilateral ABCD are A(-3, 2), B(0, 3), C(4, -4),
and D(-2, -2). The quadrilateral is reflected in the y-axis.
a. Write the coordinates of ABCD in a
vertex matrix.
y
b. Write the reflection matrix for this situation.
O
c. Find the coordinates of A'B'C'D'.
x
d. Graph ABCD and A'B'C'D'.
4. ARCHITECTURE Using architectural design software, the Bradleys plot their kitchen
plans on a grid with each unit representing 1 foot. They place the corners of an island at
(2, 8), (8, 11), (3, 5), and (9, 8). If the Bradleys wish to move the island 1.5 feet to the
right and 2 feet down, what will the new coordinates of its corners be?
5. BUSINESS The design of a business logo calls for locating the vertices of a triangle at
(1.5, 5), (4, 1), and (1, 0) on a grid. If design changes require rotating the triangle 90º
counterclockwise, what will the new coordinates of the vertices be?
Chapter 4
47
North Carolina StudyText, Math BC, Volume 2
Lesson 4-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
c. Graph RST and R'S'T'.
NAME
4-4
DATE
PERIOD
Word Problem Practice
SCS
MBC.N.2.1
Transformations with Matrices
4. PHOTOGRAPHY Alejandra used a
digital camera to take a picture. Because
she held the camera sideways, the image
on her computer screen appeared
sideways. In order to transform the
picture, she needed to perform a 90°
clockwise rotation. What matrix
represents this transformation?
1. ICONS Louis needs to perform many
matrix transformations to the basic
house icon shown in the graph.
y
5
O
5
x
What is the vertex matrix for
this image?
5. ARROWS A compass arrow is pointing
Northeast.
y
2. LANDSCAPING Using the center
as the origin, a landscaper placed
features at the given coordinates in
the northern half of the Great Lawn of
Grand Central Park in New York City:
a fountain (75, 200), a rock sculpture
(150, 175), a bench (–150, 130), and a
plaque (0, 260). Use a reflection matrix
to find the coordinates for features the
landscaper can place so that the south
half will be a reflection of the north.
5
x
a. What is the vertex matrix for
the arrow?
b. What would the vertex matrix be for
the arrow if it were pointing
Northwest? (Hint: Rotate 90° around
the origin.)
3. MIRROR SYMMETRY A detective
found only half of an image with mirror
symmetry about the line y = x. The
vertex matrix of the visible part is
⎡ 4 5 -2 ⎤
⎢
. What are the coordinates
⎣ 2 -5 -4 ⎦
of the hidden vertices?
Chapter 4
5
48
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
O
NAME
DATE
4-5
PERIOD
Study Guide
SCS
MBC.N.2.1, MBC.A.2.1
Determinants and Cramer’s Rule
Determinants
A 2×2 matrix has a second-order determinant; a 3×3 matrix has a
third-order determinant.
Second-Order
Determinant
⎡a b ⎤
For the matrix ⎢
, the determinant is
⎣ c d⎦
Third-Order
Determinant
⎡a b c⎤
For the matrix d e f , the determinant is found using the diagonal rule.
⎣ g h i⎦
⎡a b c⎤ a b
⎡a b c⎤ a b
d e f d e
d e f d e
⎣ g h i⎦ g h
⎣ g h i⎦ g h
⎢
⎢
⎪ ac db ⎥ = ad – bc.
⎢
The area of a triangle having vertices (a, b), (c, d ), and (e, f ) is ⎪A⎥,
Area of a Triangle
a.
⎪
6 3
-8 5
⎪ ⎥
Evaluate each determinant.
4 5 -2
b. 1 3 0
2 -3 6
⎥
⎪-86 35⎥ = 6 (5) - 3 (-8)
= 54
⎪
⎪
⎥
⎥
4 5 -2
1 3 0
2 -3 6
4 5
1 3
2 -3
4 5 -2
1 3 0
2 -3 6
⎪
⎥
4 5
1 3
2 -3
= [4(3)6 + 5(0)2 + (-2)1(-3)] - [(-2)3(2) +
4(0)(-3) + 5(1)6]
= [72 + 0 + 6] - [-12 + 0 + 30]
= 78 - 16 or 60
Exercises
Evaluate each determinant.
6 -2
1.
5 7
⎪
⎥
3 -2 -2
3. 0 4 1
-1 4 -3
⎪
3 2
2.
9 6
⎪ ⎥
⎥
4. Find the area of a triangle with vertices (2, –3), (7, 4), and (–5, 5).
Chapter 4
49
North Carolina StudyText, Math BC, Volume 2
Lesson 4-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Example
a b 1
1
where A = −
c d 1 .
2
e f 1
NAME
DATE
4-5
Study Guide
PERIOD
SCS
(continued)
MBC.N.2.1, MBC.A.2.1
Determinants and Cramer’s Rule
Cramer’s Rule
Determinants provide a way for solving systems of equations.
Let C be the coefficient matrix of the system
fx + gy = n
Cramer’s Rule for
Two-Variable Systems
⎪
m b
n g
a m
⎪
f n ⎥
The solution of this system is x = − , y = − , if C ≠ 0.
a m
y=
Cramer’s Rule
⎪C⎥
8 -10
⎪−
-2 25 ⎥
⎪ 105 -1025 ⎥
8(25) - (-2)(-10)
5(25) - (-10)(10)
5(-2) - 8(10)
5(25) - (-10)(10)
(5
225
5
)
4
2
The solution is −
, -−
.
5
Exercises
Use Cramer’s Rule to solve each system of equations.
1. 3x - 2y = 7
2x + 7y = 38
2. x - 4y = 17
3x - y = 29
3. 2x - y = -2
4x - y = 4
4. 2x - y = 1
5x + 2y = -29
5. 4x + 2y = 1
5x - 4y = 24
6. 6x - 3y = -3
2x + y = 21
7. 2x + 7y = 16
x - 2y = 30
8. 2x - 3y = -2
3x - 4y = 9
x
+−=2
9. −
10. 6x - 9y = -1
11. 3x - 12y = -14
3
12. 8x + 2y = −
3x + 18y = 12
Chapter 4
9x + 6y = -7
50
y
3
5
y
x
−
- − = -8
4
6
7
27
5x - 4y = - −
7
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
90
2
= -−
or - −
Simplify.
5
⎪C⎥
= −−
Evaluate each determinant.
180
4
or −
= −
⎪f n ⎥
−
5 8
⎪
10 -2 ⎥
= −
⎪105 -1025⎥
a = 5, b = -10, f = 10, g = 25, m = 8, n = -2
= −−
225
⎪C ⎥
Use Cramer’s Rule to solve the system of equations. 5x - 10y = 8
10x + 25y = -2
m b
⎪
n g⎥
x=−
=
⎪ af gb ⎥
→
⎥
⎪C ⎥
Example
ax + by = m
NAME
4-5
DATE
PERIOD
Practice
SCS
MBC.N.2.1, MBC.A.2.1
Determinants and Cramer’s Rule
Evaluate each determinant.
1.
⎪12 67⎥
2.
⎪93 62⎥
3.
⎪-24 -51⎥
4.
⎪-142 -3
-2⎥
5.
⎪-124 -34⎥
6.
-5
⎪25 -11
⎥
7.
⎪33.75 -45⎥
8.
⎪23 -1
-9.5⎥
9.
⎪0.5
0.4
-0.7
-0.3
⎥
Evaluate each determinant using expansion by diagonals.
-2 3 1
10. 0 4 -3
2 5 -1
2 -4 1
11. 3 0 9
-1 5 7
0 -4 0
13. 2 -1 1
3 -2 5
2 7 -6
14. 8 4 0
1 -1 3
⎪
⎥
⎪
⎪
⎥
⎥
-12 0 3
15.
7 5 -1
4 2 -6
⎪
⎥
⎪
⎥
Use Cramer’s Rule to solve each system of equation.
16. 4x - 2y = -6
3x + y = 18
19. 6x + 6y = 9
4x - 4y = -42
17. 5x + 4y = 10
-3x - 2y = -8
20. 5x - 6 = 3y
5y = 54 + 3x
18. -2x - 3y = -14
4x - y = 0
y
2
4
y
x
− - − = -6
4
6
x
21. −
+−=2
25. GEOMETRY Find the area of a triangle whose vertices have coordinates (3, 5), (6, -5),
and (-4, 10).
26. LAND MANAGEMENT A fish and wildlife management organization uses a GIS
(geographic information system) to store and analyze data for the parcels of land it
manages. All of the parcels are mapped on a grid in which 1 unit represents 1 acre. If
the coordinates of the corners of a parcel are (-8, 10), (6, 17), and (2, -4), how many
acres is the parcel?
Chapter 4
51
North Carolina StudyText, Math BC, Volume 2
Lesson 4-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
⎪
⎥
2 1 1
12. 1 -1 -2
1 1 -1
NAME
DATE
4-5
PERIOD
Word Problem Practice
SCS
MBC.N.2.1, MBC.A.2.1
Determinants and Cramer’s Rule
1. FIND THE ERROR Mark’s determinant
computation has sign errors. Circle the
signs that must be reversed.
4. ITALY The figure shows a map of Italy
overlaid on a graph. The coordinates of
Milan, Venice, and Pisa are about
(-4, 5), (3.25, 4.8), and (-1.4, -0.8),
respectively. Each square unit on the
map represents about 400 square miles.
1 2 3
4 5 6 = 1(5) (9) - 2(6)(7) + 3(4)(8)
- 3(5)(7) + 1(6)(8) - 2(4)(9)
7 8 9
⎪ ⎥
y
Milan
2. POOL An architect has a pool in
the floor plans for a home. Set up a
determinant that gives the unit area
of the pool.
5
5
Venice
Genoa
-5
Florence
Pisa
y
O
5
x
-5
5
-5
O
What is the area of the triangular
region? Round your answer to the
nearest square mile.
x
-5
3. HALF-UNIT TRIANGLES For a school
art project, students had to decorate a
pegboard by looping strings around the
pegs. Ronald wanted to make triangles
with areas of one half square unit.
Because Ronald had studied
determinants, he knew that this was
essentially the same as finding the
coordinates of the vertices of a triangle
(a, b), (c, d) and (e, f ), so that the
a b 1
determinant c d 1 is 1 or -1.
a. Write the determinant that gives the
area of this triangle.
b. Evaluate the determinant you wrote
for part a and determine the value of
x that results in a $60 triangle.
⎪ ⎥
e f 1
Give an example of such a triangle.
Chapter 4
52
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5. ARROWS Kyle is making a triangle
with vertices at (-6, 0), (0, -x), and
(0, x), and x > 0. He plans to make the
triangle using a material that costs $2
for every square unit.
NAME
4-6
DATE
PERIOD
Study Guide
SCS
MBC.N.2.1, MBC.A.2.1
Inverse Matrices and Systems of Equations
Identity and Inverse Matrices
Identity Matrix for
Multiplication
Lesson 4-6
The identity matrix for matrix multiplication is a
square matrix with 1s for every element of the main diagonal and zeros elsewhere.
If A is an n × n matrix and I is the identity matrix,
then A I = A and I A = A.
If an n × n matrix A has an inverse A-1, then A A-1 = A-1 A = I.
⎡ 3 -2 ⎤
⎡ 7 4⎤
Determine whether X = ⎢
and Y =
7 are
-5 −
⎣ 10 6 ⎦
inverse matrices.
2⎦
⎣
Example
⎢
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find X · Y.
⎤
⎡
⎡ 7 4 ⎤ ⎢ 3 -2 X·Y=⎢
·
7
⎣ 10 6 ⎦ ⎢ -5
−
⎣
2⎦
⎡ 21 - 20 -14 + 14 ⎤
⎡ 1 0⎤
=⎢
or ⎢
⎣ 30 - 30 -20 + 21 ⎦
⎣ 0 1⎦
Find Y · X.
⎤
⎡
⎢ 3 -2 ·
Y·X=
⎢ -5
7
−
⎣
2⎦
⎡ 21 - 20
=⎢
⎣ -35 + 35
⎡ 7 4⎤
⎢
⎣ 10 6 ⎦
⎡ 1 0⎤
12 - 12 ⎤
or ⎢
⎣ 0 1⎦
-20 + 21 ⎦
Since X · Y = Y · X = I, X and Y are inverse matrices.
Exercises
Determine whether the matrices in each pair are inverses of each other.
⎡4 5⎤
⎡ 4 -5 ⎤
1. ⎢
and ⎢
⎣3 4⎦
⎣ -3 4 ⎦
⎡
⎤
⎡3 2⎤
2 -1
2. ⎢
and
5
3
⎣5 4⎦
⎣- −2 −2 ⎦
3. ⎢
⎡ 8 11 ⎤
⎡ -4 11 ⎤
4. ⎢
and ⎢
⎣ 3 14 ⎦
⎣ 3 -8 ⎦
⎡ 4 -1 ⎤
⎡1 2⎤
5. ⎢
and ⎢
⎣5 3⎦
⎣3 8⎦
⎡-2
1⎤
⎡ 5 2⎤
6. ⎢
and 11
5
− -−
⎣ 11 4 ⎦
2 ⎦
⎣ 2
⎡5 8⎤
-3
4⎤
8. ⎢
and ⎡
⎢
5
⎣4 6⎦
2 -−
2⎦
⎣
⎡ −7 - −3 ⎤
⎡3 7⎤
2
9. ⎢
and 2
⎣2 4⎦
⎣ 1 -2 ⎦
⎡
⎡4 2⎤
7. ⎢
and
⎣ 6 -2 ⎦
⎢
⎣
⎢
1
1
-−
-−
5
3
−
10
Chapter 4
⎡ 7 2⎤
⎡ 5 -2 ⎤
and ⎢
⎣ 17 5 ⎦
⎣-17 7 ⎦
11. ⎢
53
⎡2 3⎤
⎡ 2 3⎤
and ⎢
⎣ 5 -1 ⎦
⎣ -1 -2 ⎦
⎢
⎤
10
1
−
10 ⎦
⎡3 2⎤
⎡ 3 2⎤
and ⎢
⎣ 4 -6 ⎦
⎣ -4 -3 ⎦
10. ⎢
⎢
⎡4 3⎤
⎡ -5 3 ⎤
and ⎢
⎣7 5⎦
⎣ 7 -4 ⎦
12. ⎢
North Carolina StudyText, Math BC, Volume 2
NAME
4-6
DATE
Study Guide
PERIOD
SCS
(continued)
MBC.N.2.1, MBC.A.2.1
Inverse Matrices and Systems of Equations
Matrix Equations A matrix equation for a system of equations consists of the product
of the coefficient and variable matrices on the left and the constant matrix on the right of
the equals sign.
Example
Use a matrix equation to solve a system of equations.
3x - 7y = 12
x + 5y = -8
Determine the coefficient, variable, and constant matrices.
⎡ 3 -7 ⎤ ⎡ x ⎤ ⎡ 12 ⎤
·⎢ =⎢
⎢
⎣1
5 ⎦ ⎣ y ⎦ ⎣ -8 ⎦
⎢
⎢
⎢ Exercises
Use a matrix equation to solve each system of equations.
1. 2x + y = 8
5x - 3y = -12
2. 4x - 3y = 18
x + 2y = 12
3. 7x - 2y = 15
3x + y = -10
4. 4x - 6y = 20
3x + y + 8= 0
5. 5x + 2y = 18
x = -4y + 25
6. 3x - y = 24
3y = 80 - 2x
Chapter 4
54
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find the inverse of the coefficient matrix.
⎡ 5
7 ⎤
− −
⎡ 5 7⎤
22
22
1
−
⎢
=
3(5) - 1(-7) ⎣ -1 3 ⎦
1
3
-−
−
22
22
⎦
⎣
Rewrite the equation in the form of X = A-1B
⎡ 5
7 ⎤
− −
⎡x⎤
22
22 ⎡ 12 ⎤
⎢ =
⎢
1
3
⎣y⎦
⎣ -8 ⎦
-−
−
⎣ 22 22 ⎦
Solve.
⎡ 2 ⎤
−
⎡x⎤
11
⎢ =
18
⎣y⎦
-−
11
⎦
⎣
NAME
DATE
4-6
PERIOD
Practice
SCS
MBC.N.2.1, MBC.A.2.1
Inverse Matrices and Systems of Equations
⎡2
⎣3
⎡ -2
1⎤
, N = ⎢
⎣ 3
2⎦
1. M = ⎢
1⎤
-2 ⎦
⎡ -3
⎣ 5
2. X = ⎢
⎡1
1 ⎤
− -−
⎡ 3 1⎤
5
10
3. A = ⎢
, B =
2
3
⎣ -4 2 ⎦
−
−
5
10
⎣
⎦
⎢
Lesson 4-6
Determine whether each pair of matrices are inverses.
⎡3 2⎤
2⎤
, Y = ⎢
⎣5 3⎦
-3 ⎦
⎡ 3 1⎤
− −
⎡ 6 -2 ⎤
14 7
4. P = ⎢
, Q =
1 3
⎣ -2 3 ⎦
−
−
⎣ 7 7⎦
⎢ Determine whether each statement is true or false.
5. All square matrices have multiplicative inverses.
6. All square matrices have multiplicative identities.
Find the inverse of each matrix, if it exists.
⎡ 4 5⎤
⎡2
7. ⎢
8. ⎢
⎣ -4 -3 ⎦
⎣3
⎡ -1 3 ⎤
⎣ 4 -7 ⎦
10. ⎢
⎡ 2
⎣ -1
⎡ 2 -5 ⎤
⎣3 1⎦
12. ⎢
9. ⎢
5⎤
3⎦
⎡4 6⎤
⎣6 9⎦
11. ⎢
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
0⎤
5⎦
13. GEOMETRY Use the figure at the right.
y
a. Write the vertex matrix A for the rectangle.
(4, 4)
⎡ 1.5 0 ⎤
b. Use matrix multiplication to find BA if B = ⎢
.
⎣ 0 1.5 ⎦
(1, 2)
(5, 1)
x
O
(2, –1)
c. Graph the vertices of the transformed quadrilateral on the previous graph.
Describe the transformation.
d. Make a conjecture about what transformation B-1 describes on a coordinate plane.
⎡1 2⎤
14. CODES Use the alphabet table below and the inverse of coding matrix C = ⎢
⎣2 1⎦
to decode this message:
19 | 14 | 11 | 13 | 11 | 22 | 55 | 65 | 57 | 60 | 2 | 1 | 52 | 47 | 33 | 51 | 56 | 55.
CODE
A
1
B
2
C
3
D
4
E
5
H
8
I
9
J
10
K 11
L
12 M 13
N 14
O 15
P
16
Q 17
R 18
S
19
T
20
U 21
X
Y
Z
26
–
0
V
22 W 23
Chapter 4
24
25
F
6
G
55
7
North Carolina StudyText, Math BC, Volume 2
NAME
DATE
4-6
Word Problem Practice
PERIOD
SCS
MBC.N.2.1, MBC.A.2.1
Inverse Matrices and Systems of Equations
1. TEACHING Paula is explaining
matrices to her father. She writes down
the following system of equations.
3. AGES Hank, Laura, and Ned are ages
h, l, and n, respectively. The sum of
their ages is 15 years. Laura is one year
younger than the sum of Hank and
Ned’s ages. Ned is three times as old as
Hank. Use matrices to determine the
age of each sibling.
2x + y = 4
3x + y = 5.
Next, Paula shows her father the
matrices that correspond to this system
of equations. What are the matrices?
4. SELF-INVERSES Phillip notices that any
matrix with ones and negative ones on
the diagonal and zeroes everywhere else
has the property that it is its own
inverse. Give an example of a 2 by 2
matrix that is its own inverse but has at
least 1 nonzero number off the diagonal.
2. TRANSPORTATION Paula wrote the
following matrix equation to show the
costs of two trips by water taxi to Logan
Airport in Boston. She used x for the
cost of round trips and y for the cost one
one-way trips.
5. MATRIX OPERATIONS Garth is
studying determinants and inverses of
matrices in math class. His teacher
suggests that there are some matrices
with unique properties, and challenges
the class to find such matrices and
describe the properties found. Garth is
⎡0 1 ⎤
curious about the matrix G = ⎢
.
⎣ 0 0⎦
Next, she found the inverse.
⎡ 3 1 ⎤-1 1 ⎡ 2 -1 ⎤
= −4 ⎢
⎢
⎣ -2 3 ⎦
⎣2 2⎦
Then she computed her answer.
⎡ 1 1⎤
− -−
⎡x⎤
⎡ 34 ⎤
2
4 ⎡ 61 ⎤
⎢ =⎢ ⎢y =
1
3 ⎣ 54 ⎦
⎣ 10 ⎦
⎣ ⎦
-−
−
2
4
⎣
⎦
⎢
a. What is the determinant of G?
b. Does the inverse of G exist? Explain.
When she checked her answer, the total
cost of the trips came out as $112 and
$88. Where did she make a mistake?
c. Determine a matrix operation that
could be used to transform G into its
Additive Identity matrix.
Chapter 4
56
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
⎡ 3 1 ⎤ ⎡ x ⎤ ⎡ 61 ⎤
⎢ =⎢ ⎢
⎣ 2 2 ⎦ ⎣ y ⎦ ⎣ 54 ⎦
NAME
DATE
5-1
PERIOD
Study Guide
SCS
MBC.A.1.1, MBC.A.1.2,
MBC.A.8.1, MBC.A.8.5
Graphing Quadratic Functions
Graph Quadratic Functions
Quadratic Function
a function defined by an equation of the form f(x) = ax2 + bx + c, where a ≠ 0
Graph of a
Quadratic Function
-b
;
a parabola with these characteristics: y-intercept: c; axis of symmetry: x = −
2a
-b
x- coordinate of vertex: −
2a
Find the y-intercept, the equation of the axis of symmetry, and the
x-coordinate of the vertex for the graph of f(x) = x2 - 3x + 5. Use this information
to graph the function.
a = 1, b = -3, and c = 5, so the y-intercept is 5. The equation of the axis of symmetry is
-(-3)
3
3
x = − or −
. The x-coordinate of the vertex is −
.
2
2(1)
2
3
.
Next make a table of values for x near −
2
x2 - 3x + 5
f(x)
(x, f(x))
0
02 - 3(0) + 5
5
(0, 5)
6
1
12 - 3(1) + 5
3
(1, 3)
4
2
2
3
−
2
(−32 ) - 3(−32 ) + 5
11
−
11
(−32 , −
4)
2
22 - 3(2) + 5
3
(2, 3)
5
(3, 5)
3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
f (x)
x
2
3 - 3(3) + 5
4
-2
2
O
4
x
Exercises
Complete parts a–c for each quadratic function.
a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate
of the vertex.
b. Make a table of values that includes the vertex.
c. Use this information to graph the function.
1. f(x) = x2 + 6x + 8
2. f(x) = -x2 - 2x + 2
f(x)
f (x)
O
O
3. f(x) = 2x2 - 4x + 3
f (x)
x
x
O
Chapter 5
57
x
North Carolina StudyText, Math BC, Volume 2
Lesson 5-1
Example
NAME
DATE
5-1
Study Guide
PERIOD
SCS
(continued)
MBC.A.1.1, MBC.A.1.2,
MBC.A.8.1, MBC.A.8.5
Graphing Quadratic Functions
Maximum and Minimum Values
The y-coordinate of the vertex of a quadratic
function is the maximum value or minimum value of the function.
Maximum or Minimum Value
of a Quadratic Function
The graph of f(x ) = ax 2 + bx + c, where a ≠ 0, opens up and has a minimum
when a > 0. The graph opens down and has a maximum when a < 0.
Example
Determine whether each function has a maximum or minimum
value, and find that value. Then state the domain and range of the function.
a. f(x) = 3x 2 - 6x + 7
b. f(x) = 100 - 2x - x 2
For this function, a = -1 and b = -2.
Since a < 0, the graph opens down, and
the function has a maximum value.
The maximum value is the y-coordinate of
the vertex. The x-coordinate of the vertex
-b
-2
is −
=-−
= -1.
2a
2(-1)
Evaluate the function at x = -1 to find
the maximum value.
f (-1) = 100 - 2(-1) - (-1)2 = 101, so the
maximum value of the function is 101. The
domain is all real numbers. The range is
all reals less than or equal to the
maximum value, that is {f(x) ⎪ f(x) ≤ 101}.
For this function, a = 3 and b = -6.
Since a > 0, the graph opens up, and the
function has a minimum value.
The minimum value is the y-coordinate of
the vertex. The x-coordinate of the
-(-6)
-b
vertex is −
= − = 1.
2a
2(3)
Exercises
Determine whether each function has a maximum or minimum value, and find
that value. Then state the domain and range of the function.
1. f(x) = 2x2 - x + 10
2. f(x) = x2 + 4x - 7
3. f(x) = 3x2 - 3x + 1
4. f(x) = x2 + 5x + 2
5. f(x) = 20 + 6x - x2
6. f(x) = 4x2 + x + 3
7. f(x) = -x2 - 4x + 10
8. f(x) = x2 - 10x + 5
9. f(x) = -6x2 + 12x + 21
Chapter 5
58
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Evaluate the function at x = 1 to find the
minimum value.
f(1) = 3(1)2 - 6(1) + 7 = 4, so the
minimum value of the function is 4. The
domain is all real numbers. The range is
all reals greater than or equal to the
minimum value, that is {f(x) | f(x) ≥ 4}.
NAME
DATE
5-1
Practice
PERIOD
SCS
MBC.A.1.1, MBC.A.1.2,
MBC.A.8.1, MBC.A.8.5
Graphing Quadratic Functions
Complete parts a–c for each quadratic function.
a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate
of the vertex.
b. Make a table of values that includes the vertex.
c. Use this information to graph the function.
2. f(x) = -x2 - 4x + 12
f(x)
f (x)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
O
x
O
3. f(x) = 2x2 - 2x + 1
Lesson 5-1
1. f(x) = x2 - 8x + 15
f(x)
x
O
x
Determine whether each function has a maximum or minimum value, and find
that value. Then state the domain and range of the function.
4. f(x) = x2 + 2x - 8
5. f (x) = x2 - 6x + 14
6. v(x) = -x2 + 14x - 57
7. f(x) = 2x2 + 4x - 6
8. f (x) = -x2 + 4x - 1
2 2
9. f (x) = - −
x + 8x - 24
3
10. GRAVITATION From 4 feet above a swimming pool, Susan throws a ball upward with a
velocity of 32 feet per second. The height h(t) of the ball t seconds after Susan throws it
is given by h(t) = -16t2 + 32t + 4. For t ≥ 0, find the maximum height reached by the
ball and the time that this height is reached.
11. HEALTH CLUBS Last year, the SportsTime Athletic Club charged $20 to participate in
an aerobics class. Seventy people attended the classes. The club wants to increase the
class price this year. They expect to lose one customer for each $1 increase in the price.
a. What price should the club charge to maximize the income from the aerobics classes?
b. What is the maximum income the SportsTime Athletic Club can expect to make?
Chapter 5
59
North Carolina StudyText, Math BC, Volume 2
NAME
5-1
DATE
PERIOD
Word Problem Practice
SCS
MBC.A.1.1, MBC.A.1.2,
MBC.A.8.1, MBC.A.8.5
Graphing Quadratic Functions
1. TRAJECTORIES A cannonball is
launched from a cannon on the wall of
Fort Chambly, Quebec. If the path of the
cannonball is
traced on a
piece of graph
paper aligned so
that the cannon
is situated on
the y-axis, the
equation that
describes the
path is
4. FRAMING A frame company offers a
line of square frames. If the side length
of the frame is s, then the area of the
opening in the frame is given by the
function a(s) = s2 - 10s + 24.
Graph a(s).
O
1
1
y = -−
x2 + −
x + 20,
1600
2
where x is the horizontal distance from
the cliff and y is the vertical distance
above the ground in feet. How high
above the ground is the cannon?
2. TICKETING The manager of a
symphony computes that the symphony
will earn -40P2 + 1100P dollars per
concert if they charge P dollars for
tickets. What ticket price should the
symphony charge in order to maximize
its profits?
a. When Jack is x miles east of the
intersection, where is Evita?
3. ARCHES An architect decides to use a
parabolic arch for the main entrance of a
science museum. In one of his plans, the
top edge of the arch is described by the
5
1 2
graph of y = - −
x +−
x + 15. What are
2
4
the coordinates of the vertex of this
parabola?
b. The distance between Jack and Evita is
given by the formula √
x2 + (5 - x) 2 .
For what value of x are Jack and
Evita at their closest?
(Hint: Minimize the square of the
distance.)
c. What is the distance of closest
approach?
Chapter 5
60
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5. WALKING Canal Street and Walker
Street are perpendicular to each other.
Evita is driving south on Canal Street
and is currently 5 miles north of the
intersection with Walker Street. Jack is
at the intersection of Canal and Walker
Streets and heading east on Walker.
Jack and Evita are both driving 30 miles
per hour.
NAME
DATE
5-2
Study Guide
PERIOD
SCS
MBC.A.7.1, MBC.A.8.2
Solving Quadratic Equations by Graphing
Solve Quadratic Equations
Quadratic Equation
A quadratic equation has the form ax 2 + bx + c = 0, where a ≠ 0.
Roots of a Quadratic Equation
solution(s) of the equation, or the zero(s) of the related quadratic function
The zeros of a quadratic function are the x-intercepts of its graph. Therefore, finding the
x-intercepts is one way of solving the related quadratic equation.
Example
Solve x2 + x - 6 = 0 by graphing.
f (x)
Graph the related function f(x) = x2 + x - 6.
-4
-2
-b
1 , and the equation of the
= -−
The x-coordinate of the vertex is −
-2
1
.
axis of symmetry is x = - −
-4
2a
2
O
x
2
2
1
Make a table of values using x-values around - −
.
-6
x
f (x)
1
-−
-1
2
1
-6 -6 −
4
0
1
2
0
-6 -4
From the table and the graph, we can see that the zeros of the function are 2 and -3.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
Use the related graph of each equation to determine its solution.
1. x2 + 2x - 8 = 0
2. x2 - 4x - 5 = 0
f(x)
f (x)
O
3. x2 - 5x + 4 = 0
x
f (x)
x
O
x
O
4. x2 - 10x + 21 = 0
5. x2 + 4x + 6 = 0
f (x)
f (x)
f (x)
O
6. 4x2 + 4x + 1 = 0
x
O
x
O
Chapter 5
61
x
North Carolina StudyText, Math BC, Volume 2
Lesson 5-2
2
NAME
DATE
5-2
Study Guide
PERIOD
SCS
(continued)
MBC.A.7.1, MBC.A.8.2
Solving Quadratic Equations by Graphing
Estimate Solutions
Often, you may not be able to find exact solutions to quadratic
equations by graphing. But you can use the graph to estimate solutions.
Example
Solve x2 - 2x - 2 = 0 by graphing. If exact roots cannot be found,
state the consecutive integers between which the roots are located.
The equation of the axis of symmetry of the related function is
4
-2
x = -−
= 1, so the vertex has x-coordinate 1. Make a table of values.
2
2(1)
x
-1
0
1
2
3
f (x)
1
-2
-3
-2
1
f(x)
O
-2
4x
2
-2
The x-intercepts of the graph are between 2 and 3 and between 0 and
-1. So one solution is between 2 and 3, and the other solution is
between 0 and -1.
-4
Exercises
Solve the equations. If exact roots cannot be found, state the consecutive integers
between which the roots are located.
1. x2 - 4x + 2 = 0
2. x2 + 6x + 6 = 0
f(x)
f(x)
f (x)
x
x
4. -x2 + 2x + 4 = 0
5. 2x2 - 12x + 17 = 0
5
1 2
6. - −
x +x+−
=0
2
2
f(x)
f(x)
f(x)
O
O
Chapter 5
x
x
O
x
x
O
62
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
O
O
3. x2 + 4x + 2= 0
NAME
DATE
5-2
PERIOD
Practice
SCS
MBC.A.7.1, MBC.A.8.2
Solving Quadratic Equations By Graphing
Use the related graph of each equation to determine its solutions.
1. -3x2 + 3 = 0
4
2. 3x2 + x + 3 = 0
f(x)
2
-4
-2
O
x
2
-2
-4
-4
-2
3. x2 - 3x + 2 = 0
f(x)
8
8
6
6
4
4
2
2
O
4x
2
-4
-2
O
f (x)
2
4x
Solve each equation. If exact roots cannot be found, state the consecutive integers
between which the roots are located.
4. -2x2 - 6x + 5 = 0
5. x2 + 10x + 24 = 0
6. 2x2 - x - 6 = 0
O
O
x
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
O
7. -x2 + x + 6 = 0
x
8. -x2 + 5x - 8 = 0
f(x)
f (x)
O
x
O
x
x
9. GRAVITY Use the formula h(t) = v0t - 16t2, where h(t) is the height of an object in feet,
v0 is the object’s initial velocity in feet per second, and t is the time in seconds.
a. Marta throws a baseball with an initial upward velocity of 60 feet per second.
Ignoring Marta’s height, how long after she releases the ball will it hit the ground?
b. A volcanic eruption blasts a boulder upward with an initial velocity of
240 feet per second. How long will it take the boulder to hit the ground if it lands at
the same elevation from which it was ejected?
Chapter 5
63
North Carolina StudyText, Math BC, Volume 2
Lesson 5-2
f(x)
f(x)
f(x)
NAME
DATE
5-2
PERIOD
Word Problem Practice
SCS
MBC.A.7.1, MBC.A.8.2
Solving Quadratic Equations by Graphing
1. TRAJECTORIES David threw a baseball
into the air. The function of the height
of the baseball in feet is h = 80t -16t2,
where t represents the time in seconds
after the ball was thrown. Use this
graph of the function to determine how
long it took for the ball to fall back to
the ground.
4. RADIO TELESCOPES The cross-section
of a large radio telescope is a parabola.
The dish is set into the ground. The
equation that describes the cross-section
32
2 2 4
is d = −
x -−x -−
, where d gives
75
3
3
the depth of the dish below ground and
x is the distance from the control center,
both in meters. If the dish does not
extend above the ground level, what is
the diameter of the dish? Solve by
graphing.
h
80
d
40
-1
0
1
2
3
4
5
t
-40
0
x
5. BOATS The distance between two
boats is
d = √
t2 - 10t + 35,
where d is distance in meters and t is
time in seconds.
h
a. Make a graph of d2 versus t.
0
d
x
O
3. LOGIC Wilma is thinking of two
numbers. The sum is 2 and the product
is -24. Use a quadratic equation to find
the two numbers.
Chapter 5
t
b. Do the boats ever collide?
64
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. BRIDGES In 1895, a brick arch railway
bridge was built on North Avenue in
Baltimore, Maryland. The arch is
1 2
described by the equation h = 9 – −
x,
50
where h is the height in yards and x is
the distance in yards from the center of
the bridge. Graph this equation and
describe, to the nearest yard, where the
bridge touches the ground.
NAME
DATE
5-3
PERIOD
Study Guide
SCS
MBC.A.7.1
Solving Quadratic Equations by Factoring
Factored Form To write a quadratic equation with roots p and q, let (x - p)(x - q) = 0.
Then multiply using FOIL.
Example
Write a quadratic equation in standard form with the given roots.
7 1
b. - −
,−
a. 3, -5
8 3
(x - p)(x - q) = 0
(x - 3)[x - (-5)] = 0
(x - 3)(x + 5) = 0
x2 + 2x - 15 = 0
(x - p)(x - q) = 0
Write the pattern.
Replace p with 3, q with -5.
Simplify.
Use FOIL.
The equation x2 + 2x - 15 = 0 has roots
3 and -5.
x - (-−78 ) (x - −13 ) = 0
(x + −78 )(x - −13 ) = 0
(8x + 7)
8
(3x - 1)
3
−− =0
24 (8x + 7)(3x - 1)
−−
24
= 24 0
24x2 + 13x - 7 = 0
The equation 24x2 + 13x - 7 = 0 has
7
1
and −
.
roots - −
8
3
Write a quadratic equation in standard form with the given root(s).
1. 3, -4
2. -8, -2
3. 1, 9
4. -5
5. 10, 7
6. -2, 15
1
7. - −
,5
2
8. 2, −
3
9. -7, −
3
2
10. 3, −
3
4
4
11. - −
, -1
1
12. 9, −
2
2
, -−
13. −
5
1
14. −
, -−
3 1
15. −
,−
7 7
16. - −
,−
1 3
17. −
,−
1 1
18. −
,−
5
3
3
8 2
Chapter 5
6
9
4
7 5
2
2 4
8 6
65
North Carolina StudyText, Math BC, Volume 2
Lesson 5-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
NAME
5-3
DATE
Study Guide
PERIOD
SCS
(continued)
MBC.A.7.1
Solving Quadratic Equations by Factoring
Solve Equations by Factoring
When you use factoring to solve a quadratic equation,
you use the following property.
Zero Product Property
Example
For any real numbers a and b, if ab = 0, then either a = 0 or b =0, or both a and b = 0.
Solve each equation by factoring.
a. 3x2 = 15x
3x2 = 15x Original equation
3x2 - 15x = 0
Subtract 15x from both sides.
3x(x - 5) = 0
Factor the binomial.
3x = 0 or x - 5 = 0
Zero Product Property
x = 0 or
x=5
Solve each equation.
b. 4x2 - 5x = 21
4x2 - 5x = 21 Original equation
4x2 - 5x - 21 = 0 Subtract 21 from both sides.
(4x + 7)(x - 3) = 0 Factor the trinomial.
4x + 7 = 0 or x - 3 = 0 Zero Product Property
7
x = -−
or
x=3
4
The solution set is {0, 5}.
{
Solve each equation.
}
7
The solution set is - −
,3 .
4
Exercises
Solve each equation by factoring.
2. x2 = 7x
3. 20x2 = -25x
4. 6x2 = 7x
5. 6x2 - 27x = 0
6. 12x2 - 8x = 0
7. x2 + x - 30 = 0
8. 2x2 - x - 3 = 0
9. x2 + 14x + 33 = 0
10. 4x2 + 27x - 7 = 0
11. 3x2 + 29x - 10 = 0
12. 6x2 - 5x - 4 = 0
13. 12x2 - 8x + 1 = 0
14. 5x2 + 28x - 12 = 0
15. 2x2 - 250x + 5000 = 0
16. 2x2 - 11x - 40 = 0
17. 2x2 + 21x - 11 = 0
18. 3x2 + 2x - 21 = 0
19. 8x2 - 14x + 3 = 0
20. 6x2 + 11x - 2 = 0
21. 5x2 + 17x - 12 = 0
22. 12x2 + 25x + 12 = 0
23. 12x2 + 18x + 6 = 0
24. 7x2 - 36x + 5 = 0
Chapter 5
66
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. 6x2 - 2x = 0
NAME
DATE
5-3
PERIOD
Practice
SCS
MBC.A.7.1
Solving Quadratic Equations by Factoring
Write a quadratic equation in standard form with the given root(s).
1. 7, 2
2. 0, 3
3. -5, 8
4. -7, -8
5. -6, -3
6. 3, -4
1
7. 1, −
1
8. −
,2
7
9. 0, - −
2
3
2
Factor each polynomial.
10. r3 + 3r2 - 54r
11. 8a2 + 2a - 6
12. c2 - 49
13. x3 + 8
14. 16r2 - 169
15. b4 - 81
16. x2 - 4x - 12 = 0
17. x2 - 16x + 64 = 0
18. x2 - 6x + 8 = 0
19. x2 + 3x + 2 = 0
20. x2 - 4x = 0
21. 7x2 = 4x
22. 10x2 = 9x
23. x2 = 2x + 99
24. x2 + 12x = -36
25. 5x2 - 35x + 60 = 0
26. 36x2 = 25
27. 2x2 - 8x - 90 = 0
28. NUMBER THEORY Find two consecutive even positive integers whose product is 624.
29. NUMBER THEORY Find two consecutive odd positive integers whose product is 323.
30. GEOMETRY The length of a rectangle is 2 feet more than its width. Find the
dimensions of the rectangle if its area is 63 square feet.
31. PHOTOGRAPHY The length and width of a 6-inch by 8-inch photograph are reduced by
the same amount to make a new photograph whose area is half that of the original. By
how many inches will the dimensions of the photograph have to be reduced?
Chapter 5
67
North Carolina StudyText, Math BC, Volume 2
Lesson 5-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Solve each equation by factoring.
NAME
5-3
DATE
PERIOD
Word Problem Practice
SCS
MBC.A.7.1
Solving Quadratic Equations by Factoring
1. FLASHLIGHTS When Dora shines her
flashlight on the wall at a certain angle,
the edge of the lit area is in the shape of
a parabola. The equation of the parabola
is y = 2x2 + 2x - 60. Factor this
quadratic equation.
4. PROGRAMMING Ray is a computer
programmer. He needs to find the
quadratic function of this graph for an
algorithm related to a game involving
dice. Provide such a function.
y
4
2
O
2
4
6
8
10
12 x
-2
2. SIGNS David was looking through an
old algebra book and came across this
equation.
x2
-4
6x + 8 = 0
The sign in front of the 6 was blotted
out. How does the missing sign depend
on the signs of the roots?
a. What are the solutions of f(x) = 0?
3. ART The area in square inches of the
drawing Maisons prés de la mer by
Claude Monet is approximated by the
equation y = x2 – 23x + 130. Factor the
equation to find the two roots, which are
equal to the approximate length and
width of the drawing.
Chapter 5
b. Write f (x) in standard form.
c. If the animator changes the equation
to f(x) = -0.2x2 + 20, what are the
solutions of f(x) = 0?
68
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5. ANIMATION A computer graphics
animator would like to make a realistic
simulation of a tossed ball. The animator
wants the ball to follow the parabolic
trajectory represented by the quadratic
equation f(x) = -0.2(x + 5) (x - 5).
NAME
5-5
DATE
Study Guide
PERIOD
SCS
MBC.A.7.1
Completing the Square
Square Root Property
Use the Square Root Property to solve a quadratic equation
that is in the form “perfect square trinomial = constant.”
Example
Solve each equation by using the Square Root Property. Round to
the nearest hundredth if necessary.
a. x2 - 8x + 16 = 25
x2 - 8x + 16 = 25
(x - 4)2 = 25
x - 4 = √
25
or
x = 5 + 4 = 9 or
x - 4 = - √25
x = -5 + 4 = -1
b. 4x2 - 20x + 25 = 32
4x2 - 20x + 25 = 32
(2x - 5)2 = 32
2x - 5 = √
32 or 2x - 5 = - √
32
2x - 5 = 4 √
2 or 2x - 5 = -4 √
2
5 ± 4 √2
2
The solution set is {9, -1}.
x=−
The solution set is
{
5 ± 4 √2
}
− .
2
Exercises
1. x2 - 18x + 81 = 49
2. x2 + 20x + 100 = 64
3. 4x2 + 4x + 1 = 16
4. 36x2 + 12x + 1 = 18
5. 9x2 - 12x + 4 = 4
6. 25x2 + 40x + 16 = 28
7. 4x2 - 28x + 49 = 64
8. 16x2 + 24x + 9 = 81
9. 100x2 - 60x + 9 = 121
10. 25x2 + 20x + 4 = 75
11. 36x2 + 48x + 16 = 12
12. 25x2 - 30x + 9 = 96
Lesson 5-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Solve each equation by using the Square Root Property. Round to the nearest
hundredth if necessary.
Chapter 5
69
North Carolina StudyText, Math BC, Volume 2
NAME
DATE
5-5
Study Guide
PERIOD
SCS
(continued)
MBC.A.7.1
Completing the Square
Complete the Square
To complete the square for a quadratic expression of the form
x2 + bx, follow these steps.
b
1. Find −
.
2
(2)
b
2. Square −
.
b
3. Add −
2
Example 1
Find the value
of c that makes x2 + 22x + c a
perfect square trinomial.
Then write the trinomial as the
square of a binomial.
Step 1
b
b = 22; −
= 11
2
Step 2 112 = 121
Step 3
c = 121
The trinomial is x2 + 22x + 121,
which can be written as (x + 11)2.
2
to x2 + bx.
Example 2
Solve 2x2 - 8x - 24 = 0 by
completing the square.
2x2 - 8x - 24 = 0
0
2x - 8x - 24
− =−
2
2
Original equation
2
Divide each side by 2.
x2 - 4x - 12 = 0
x 2 - 4x - 12 is not a perfect square.
x2 - 4x = 12
Add 12 to each side.
2
2
x - 4x + 4 = 12 + 4 Since (−24 ) = 4, add 4 to each side.
(x - 2)2 = 16
Factor the square.
x - 2 = ±4
Square Root Property
x = 6 or x = - 2
Solve each equation.
The solution set is {6, -2}.
Exercises
4. x2 + 3.2x + c
1
5. x2 + −
x+c
2
Solve each equation by completing the square.
7. y2 - 4y - 5 = 0
8. x2 - 8x - 65 = 0
6. x2 - 2.5x + c
9. w2 - 10w + 21 = 0
10. 2x2 - 3x + 1 = 0
11. 2x2 - 13x - 7 = 0
12. 25x2 + 40x - 9 = 0
13. x2 + 4x + 1 = 0
14. y2 + 12y + 4 = 0
15. t2 + 3t - 8 = 0
Chapter 5
70
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find the value of c that makes each trinomial a perfect square. Then write the
trinomial as a perfect square.
1. x2 - 10x + c
2. x2 + 60x + c
3. x2 - 3x + c
NAME
DATE
5-5
Practice
PERIOD
SCS
MBC.A.7.1
Completing the Square
Solve each equation by using the Square Root Property. Round to the nearest
hundredth if necessary.
1. x2 + 8x + 16 = 1
2. x2 + 6x + 9 = 1
3. x2 + 10x + 25 = 16
4. x2 - 14x + 49 = 9
5. 4x2 + 12x + 9 = 4
6. x2 - 8x + 16 = 8
7. x2 - 6x + 9 = 5
8. x2 - 2x + 1 = 2
9. 9x2 - 6x + 1 = 2
Find the value of c that makes each trinomial a perfect square. Then write the
trinomial as a perfect square.
10. x2 + 12x + c
11. x2 - 20x + c
12. x2 + 11x + c
13. x2 + 0.8x + c
14. x2 - 2.2x + c
15. x2 - 0.36x + c
5
x+c
16. x2 + −
1
17. x2 - −
x+c
5
18. x2 - −
x+c
4
3
Solve each equation by completing the square.
19. x2 + 6x + 8 = 0
20. 3x2 + x - 2 = 0
21. 3x2 - 5x + 2 = 0
22. x2 + 18 = 9x
23. x2 - 14x + 19 = 0
24. x2 + 16x - 7 = 0
25. 2x2 + 8x - 3 = 0
26. x2 + x - 5 = 0
27. 2x2 - 10x + 5 = 0
28. x2 + 3x + 6 = 0
29. 2x2 + 5x + 6 = 0
30. 7x2 + 6x + 2 = 0
31. GEOMETRY When the dimensions of a cube are reduced by 4 inches on each side, the
surface area of the new cube is 864 square inches. What were the dimensions of the
original cube?
32. INVESTMENTS The amount of money A in an account in which P dollars are invested
for 2 years is given by the formula A = P(1 + r)2, where r is the interest rate
compounded annually. If an investment of $800 in the account grows to $882 in two
years, at what interest rate was it invested?
Chapter 5
71
North Carolina StudyText, Math BC, Volume 2
Lesson 5-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
6
NAME
5-5
DATE
Word Problem Practice
PERIOD
SCS
MBC.A.7.1
Completing the Square
1. COMPLETING THE SQUARE
Samantha needs to solve the equation
5. PARABOLAS A parabola is modeled by
y = x2 - 10x + 28. Jane’s homework
problem requires that she find the
vertex of the parabola. She uses the
completing square method to express
the function in the form
y = (x - h)2 + k, where (h, k) is the
vertex of the parabola. Write the
function in the form used by Jane.
x2 - 12x = 40.
What must she do to each side of the
equation to complete the square?
2. ART The area in square inches of the
drawing Foliage by Paul Cézanne is
approximated by the equation
y = x2 – 40x + 396. Complete the square
and find the two roots, which are equal
to the approximate length and width of
the drawing.
6. AUDITORIUM SEATING The seats in
an auditorium are arranged in a square
grid pattern. There are 45 rows and 45
columns of chairs. For a special concert,
organizers decide to increase seating by
adding n rows and n columns
to make a square pattern of seating
45 + n seats on a side.
3. COMPOUND INTEREST Nikki invested
$1000 in a savings account with interest
compounded annually. After two years
the balance in the account is $1210.
Use the compound interest formula
A = P(1 + r)t to find the annual
interest rate.
a. How many seats are there after
the expansion?
4. REACTION TIME Lauren was eating
lunch when she saw her friend Jason
approach. The room was crowded and
Jason had to lift his tray to avoid
obstacles. Suddenly, a glass on Jason’s
lunch tray tipped and fell off the tray.
Lauren lunged forward and managed to
catch the glass just before it hit the
ground. The height h, in feet, of the
glass t seconds after it was dropped is
given by h = -16t2 + 4.5. Lauren caught
the glass when it was six inches off the
ground. How long was the glass in the
air before Lauren caught it?
Chapter 5
c. If organizers do add 1000 seats,
what is the seating capacity of
the auditorium?
72
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
b. What is n if organizers wish to add
1000 seats?
NAME
5-6
DATE
Study Guide
PERIOD
SCS
MBC.A.7.1
Quadratic Formula The Quadratic Formula can be used to solve any quadratic
equation once it is written in the form ax2 + bx + c = 0.
Quadratic Formula
Example
2
-b ± √b
- 4ac
2a
The solutions of ax 2 + bx + c = 0, with a ≠ 0, are given by x = − .
Solve x2 - 5x = 14 by using the Quadratic Formula.
Rewrite the equation as x2 - 5x - 14 = 0.
-b ± √
b2 - 4ac
2a
-(-5) ± √
(-5)2 - 4(1)(-14)
= −−
2(1)
5 ± √
81
= −
2
5±9
= −
2
x= −
Quadratic Formula
Replace a with 1, b with -5, and c with -14.
Simplify.
= 7 or -2
The solutions are -2 and 7.
Exercises
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Solve each equation by using the Quadratic Formula.
1. x2 + 2x - 35 = 0
2. x2 + 10x + 24 = 0
3. x2 - 11x + 24 = 0
4. 4x2 + 19x - 5 = 0
5. 14x2 + 9x + 1 = 0
6. 2x2 - x - 15 = 0
7. 3x2 + 5x = 2
8. 2y2 + y - 15 = 0
9. 3x2 - 16x + 16 = 0
10. 8x2 + 6x - 9 = 0
3r
2
11. r2 - −
+−
=0
12. x2 - 10x - 50 = 0
13. x2 + 6x - 23 = 0
14. 4x2 - 12x - 63 = 0
15. x2 - 6x + 21 = 0
Chapter 5
5
25
73
North Carolina StudyText, Math BC, Volume 2
Lesson 5-6
The Quadratic Formula and the Discriminant
NAME
DATE
5-6
Study Guide
PERIOD
SCS
(continued)
MBC.A.7.1
The Quadratic Formula and the Discriminant
Roots and the Discriminant
Discriminant
The expression under the radical sign, b2 - 4ac, in the Quadratic Formula is called
the discriminant.
Discriminant
Type and Number of Roots
2
2 rational roots
2
b - 4ac > 0, but not a perfect square
2 irrational roots
b 2 - 4ac = 0
1 rational root
b - 4ac > 0 and a perfect square
2
2 complex roots
b - 4ac < 0
Example
Find the value of the discriminant for each equation. Then describe
the number and type of roots for the equation.
a. 2x2 + 5x + 3
The discriminant is
b2 - 4ac = 52 - 4(2) (3) or 1.
The discriminant is a perfect square, so
the equation has 2 rational roots.
b. 3x2 - 2x + 5
The discriminant is
b2 - 4ac = (-2)2 - 4(3) (5) or -56.
The discriminant is negative, so the
equation has 2 complex roots.
Exercises
1. p2 + 12p = -4
2. 9x2 - 6x + 1 = 0
3. 2x2 - 7x - 4 = 0
4. x2 + 4x - 4 = 0
5. 5x2 - 36x + 7 = 0
6. 4x2 - 4x + 11 = 0
7. x2 - 7x + 6 = 0
8. m2 - 8m = -14
9. 25x2 - 40x = -16
11. 6x2 + 26x + 8 = 0
12. 4x2 - 4x - 11 = 0
10. 4x2 + 20x + 29 = 0
Chapter 5
74
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Complete parts a-c for each quadratic equation.
a. Find the value of the discriminant.
b. Describe the number and type of roots.
c. Find the exact solutions by using the Quadratic Formula.
NAME
5-6
DATE
PERIOD
Practice
SCS
MBC.A.7.1
The Quadratic Formula and the Discriminant
Lesson 5-6
Solve each equation by using the Quadratic Formula.
1. 7x2 - 5x = 0
2. 4x2 - 9 = 0
3. 3x2 + 8x = 3
4. x2 - 21 = 4x
5. 3x2 - 13x + 4 = 0
6. 15x2 + 22x = -8
7. x2 - 6x + 3 = 0
8. x2 - 14x + 53 = 0
9. 3x2 = -54
10. 25x2 - 20x - 6 = 0
11. 4x2 - 4x + 17 = 0
12. 8x - 1 = 4x2
13. x2 = 4x - 15
14. 4x2 - 12x + 7 = 0
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Complete parts a-c for each quadratic equation.
a. Find the value of the discriminant.
b. Describe the number and type of roots.
c. Find the exact solutions by using the Quadratic Formula.
15. x2 - 16x + 64 = 0
16. x2 = 3x
17. 9x2 - 24x + 16 = 0
18. x2 - 3x = 40
19. 3x2 + 9x - 2 = 0
20. 2x2 + 7x = 0
21. 5x2 - 2x + 4 = 0
22. 12x2 - x - 6 = 0
23. 7x2 + 6x + 2 = 0
24. 12x2 + 2x - 4 = 0
25. 6x2 - 2x - 1 = 0
26. x2 + 3x + 6 = 0
27. 4x2 - 3x2 - 6 = 0
28. 16x2 - 8x + 1 = 0
29. 2x2 - 5x - 6 = 0
30. GRAVITATION The height h(t) in feet of an object t seconds after it is propelled straight
up from the ground with an initial velocity of 60 feet per second is modeled by the equation
h(t) = -16t2 + 60t. At what times will the object be at a height of 56 feet?
31. STOPPING DISTANCE The formula d = 0.05s2 + 1.1s estimates the minimum stopping
distance d in feet for a car traveling s miles per hour. If a car stops in 200 feet, what is the
fastest it could have been traveling when the driver applied the brakes?
Chapter 5
75
North Carolina StudyText, Math BC, Volume 2
NAME
5-6
DATE
Word Problem Practice
PERIOD
SCS
MBC.A.7.1
The Quadratic Formula and the Discriminant
1. PARABOLAS The graph of a quadratic
equation of the form y = ax2 + bx + c is
shown below.
4. EXAMPLES Give an example of a
quadratic function f(x) that has the
following properties.
I. The discriminant of f is zero.
y
II. There is no real solution of the
equation f(x) = 10.
5
Sketch the graph of x = f(x).
-5
O
x
O y
2
Is the discriminant b - 4ac positive,
negative, or zero? Explain.
5. TANGENTS The graph of y = x2 is a
parabola that passes through the point
at (1, 1). The line y = mx - m + 1,
where m is a constant, also passes
through the point at (1, 1).
a. To find the points of intersection
between the line y = mx - m + 1
and the parabola y = x2, set x2 =
mx - m + 1 and then solve for x.
Rearranging terms, this equation
becomes x2 - mx + m - 1 = 0. What
is the discriminant of this equation?
3. SPORTS In 1990, American Randy
Barnes set the world record for the shot
put. His throw can be described by the
equation y = –16x2 + 368x. Use the
Quadratic Formula to find how far his
throw was to the nearest foot.
b. For what value of m is there only one
point of intersection? Explain the
meaning of this in terms of the
corresponding line and the parabola.
76
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. TANGENT Kathleen is trying to find b
so that the x-axis is tangent to the
parabola y = x2 + bx + 4. She finds one
value that works, b = 4. Is this the only
value that works? Explain.
Chapter 5
x
NAME
DATE
5-7
Study Guide
PERIOD
SCS
MBC.A.3.1, MBC.A.3.2,
MBC.A.10.2
Transformations with Quadratic Functions
Write Quadratic Equations in Vertex Form A quadratic function is easier to
graph when it is in vertex form. You can write a quadratic function of the form
y = ax2 + bx + c in vertex from by completing the square.
Example
=
=
=
=
2x2 - 12x + 25
2(x2 - 6x) + 25
2(x2 - 6x + 9) + 25 - 18
2(x - 3)2 + 7
y
8
6
4
The vertex form of the equation is y = 2(x - 3)2 + 7.
2
2
O
4
x
6
Exercises
Write each equation in vertex form. Then graph the function.
1. y = x2 - 10x + 32
2. y = x2 + 6x
3. y = x2 - 8x + 6
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
y
y
y
O
x
x
O
x
O
4. y = -4x2 + 16x - 11
5. y = 3x2 - 12x + 5
y
y
y
x
O
O
6. y = 5x2 - 10x + 9
x
O
Chapter 5
77
x
North Carolina StudyText, Math BC, Volume 2
Lesson 5-7
y
y
y
y
Write y = 2x2 - 12x + 25 in vertex form. Then graph the function.
NAME
5-7
DATE
Study Guide
PERIOD
SCS
(continued)
MBC.A.3.1, MBC.A.3.2,
MBC.A.10.2
Transformations with Quadratic Functions
Transformations of Quadratic Functions Parabolas can be transformed by
changing the values of the constants a, h, and k in the vertex form of a quadratic equation:
y = a(x – h)2 + k.
• The sign of a determines whether the graph opens upward (a > 0) or downward (a < 0).
• The absolute value of a also causes a dilation (enlargement or reduction) of the
parabola. The parabola becomes narrower if ⎪a⎥ >1 and wider if ⎪a⎥ < 1.
• The value of h translates the parabola horizontally. Positive values of h slide the graph
to the right and negative values slide the graph to the left.
• The value of k translates the graph vertically. Positive values of k slide the graph
upward and negative values slide the graph downward.
Example
Graph y = (x + 7)2 + 3.
y
15
• Rewrite the equation as y = [x – (–7)]2 + 3.
• Because h = –7 and k = 3, the vertex is at (–7, 3). The axis
of symmetry is x = –7. Because a = 1, we know that the
graph opens up, and the graph is the same width as the
graph of y = x2.
5
-15
-5
-5
0
5
x
15
-15
• Translate the graph of y = x2 seven units to the left and
three units up.
Graph each function.
1. y = –2x2 + 2
2. y = – 3(x – 1)2
y
y
y
0
Chapter 5
3. y = 2(x + 2)2 + 3
x
0
78
0
x
x
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
NAME
5-7
DATE
Practice
PERIOD
SCS
MBC.A.3.1, MBC.A.3.2,
MBC.A.10.2
Transformations with Quadratic Functions
4. y = x2 + 10x + 20
5. y = 2x2 + 12x + 18
6. y = 3x2 - 6x + 5
7. y = -2x2 - 16x - 32
8. y = -3x2 + 18x - 21
9. y = 2x2 + 16x + 29
Graph each function.
10. y = (x + 3)2 - 1
11. y = -x2 + 6x - 5
y
y
y
x
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
O
O
12. y = 2x2 - 2x + 1
x
x
O
13. Write an equation for a parabola with vertex at (1, 3) that passes through (-2, -15).
14. Write an equation for a parabola with vertex at (-3, 0) that passes through (3, 18).
15. BASEBALL The height h of a baseball t seconds after being hit is given by
h(t) = -16t2 + 80t + 3. What is the maximum height that the baseball reaches, and
when does this occur?
16. SCULPTURE A modern sculpture in a park contains a parabolic arc that
starts at the ground and reaches a maximum height of 10 feet after a
horizontal distance of 4 feet. Write a quadratic function in vertex form
that describes the shape of the outside of the arc, where y is the height
of a point on the arc and x is its horizontal distance from the left-hand
starting point of the arc.
10 ft
4 ft
Chapter 5
79
North Carolina StudyText, Math BC, Volume 2
Lesson 5-7
Write each equation in vertex form. Then identify the vertex, axis of symmetry,
and direction of opening.
1. y = -6x2 - 24x - 25
2. y = 2x2 + 2
3. y = -4x2 + 8x
NAME
DATE
5-7
PERIOD
Word Problem Practice
SCS
MBC.A.3.1, MBC.A.3.2,
MBC.A.10.2
Transformations with Quadratic Functions
1. ARCHES A parabolic arch is used as a
bridge support. The graph of the arch is
shown below.
y
y
5
5
O
-5
4. WATER JETS The graph shows the path
of a jet of water.
5
x
O
x
If the equation that corresponds to
this graph is written in the form
y + a(x - h)2 + k, what are h and k?
The equation corresponding to this
graph is y = a(x - h)2 + k. What are a,
h, and k?
2. TRANSLATIONS For a computer
animation, Barbara uses the quadratic
function f(x) = -42(x - 20)2 + 16800 to
help her simulate an object tossed on
another planet. For one skit, she had to
use the function f(x + 5) - 8000 instead
of f (x). Where is the vertex of the graph
of y = f (x + 5) - 8000?
5. PROFIT A theater operator predicts that
the theater can make -4x2 + 160x
dollars per show if tickets are priced at x
dollars.
a. Rewrite the equation y = -4x2 + 160x
in the form y = a(x - h)2 + k.
3. BRIDGES The shape formed by the
main cables of the Golden Gate Bridge
approximately follows the equation
y = 0.0002x2 - 0.23x + 227. Graph the
parabola formed by one of the cables.
c. Graph the parabola.
y
Height
y
O
0
x
x
Width
Chapter 5
80
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
b. What is the vertex of the parabola
and what is its axis of symmetry?
NAME
6-3
DATE
Study Guide
PERIOD
SCS
MBC.A.8.4, MBC.A.8.5
Polynomial Functions
Polynomial Functions
A polynomial of degree n in one variable x is an expression of the form
Polynomial in
One Variable
anx n + an - 1x n - 1 + … + a2 x 2 + a1x + a0 ,
where the coefficients an -1 , an - 2 , an - 3 , …, a0 represent real numbers, an is not zero,
and n represents a nonnegative integer.
The degree of a polynomial in one variable is the greatest exponent of its variable. The
leading coefficient is the coefficient of the term with the highest degree.
Polynomial
Function
A polynomial function of degree n can be described by an equation of the form
P(x ) = anx n + an-1x n - 1 + … + a2x 2 + a1x + a0,
where the coefficients an - 1, an - 2, an - 3, …, a0 represent real numbers, an is not zero,
and n represents a nonnegative integer.
Example 1
What are the degree and leading coefficient of 3x2 - 2x4 - 7 + x3 ?
Rewrite the expression so the powers of x are in decreasing order.
-2x4 + x3 + 3x2 - 7
This is a polynomial in one variable. The degree is 4, and the leading coefficient is -2.
Find f(-5) if f(x) = x3 + 2x2 - 10x + 20.
f (x) = x3 + 2x2 - 10x + 20
f(-5) = (-5)3 + 2(-5)2 - 10(-5) + 20
= -125 + 50 + 50 + 20
= -5
Example 3
g(x) =
g(a - 1) =
=
=
2
Original function
Replace x with -5.
Evaluate.
Lesson 6-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Example 2
Simplify.
Find g(a2 - 1) if g(x) = x2 + 3x - 4.
x2 + 3x - 4
(a2 - 1)2 + 3(a2 - 1) - 4
a4 - 2a2 + 1 + 3a2 - 3 - 4
a4 + a2 - 6
Original function
Replace x with a 2 - 1.
Evaluate.
Simplify.
Exercises
State the degree and leading coefficient of each polynomial in one variable. If it is
not a polynomial in one variable, explain why.
1. 3x4 + 6x3 - x2 + 12
2. 100 - 5x3 + 10 x7
3. 4x6 + 6x4 + 8x8 - 10x2 + 20
4. 4x2 - 3xy + 16y2
5. 8x3 - 9x5 + 4x2 - 36
x2
x6
x3
1
6. −
-−
+−
-−
18
25
36
72
Find f(2) and f(-5) for each function.
7. f(x) = x2 - 9
Chapter 6
8. f (x) = 4x3 - 3x2 + 2x - 1
81
9. f (x) = 9x3 - 4x2 + 5x + 7
North Carolina StudyText, Math BC, Volume 2
NAME
DATE
6-3
Study Guide
PERIOD
SCS
(continued)
MBC.A.8.4, MBC.A.8.5
Polynomial Functions
Graphs of Polynomial Functions
End Behavior of
Polynomial
Functions
If the degree is even and the leading coefficient is positive, then
f(x) → +∞ as x → -∞
f(x) → +∞ as x → +∞
If the degree is even and the leading coefficient is negative, then
f(x) → -∞ as x → -∞
f(x) → -∞ as x → +∞
If the degree is odd and the leading coefficient is positive, then
f(x) → -∞ as x → -∞
f(x) → +∞ as x → +∞
If the degree is odd and the leading coefficient is negative, then
f(x) → +∞ as x → -∞
f(x) → -∞ as x → +∞
The maximum number of zeros of a polynomial function is equal to the degree of the polynomial.
A zero of a function is a point at which the graph intersects the x-axis.
On a graph, count the number of real zeros of the function by counting the number of times the
graph crosses or touches the x-axis.
Real Zeros of a
Polynomial
Function
Example
Determine whether the graph represents an odd-degree polynomial
or an even-degree polynomial. Then state the number of real zeros.
4
As x → -∞, f(x) → -∞ and as x → +∞, f(x) → +∞,
so it is an odd-degree polynomial function.
The graph intersects the x-axis at 1 point,
so the function has 1 real zero.
f (x)
2
-4
-2
4x
2
O
-2
Exercises
For each graph,
a. describe the end behavior,
b. determine whether it represents an odd-degree or an even-degree function, and
c. state the number of real zeroes.
1.
2.
4
3.
f (x)
4
2
-4
Chapter 6
-2
O
f(x)
4
2
2
4x
-4
-2
O
2
2
4x
-4
-2
O
-2
-2
-2
-4
-4
-4
82
f(x)
2
4x
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
-4
NAME
DATE
6-3
PERIOD
Practice
SCS
MBC.A.8.4, MBC.A.8.5
Polynomial Functions
State the degree and leading coefficient of each polynomial in one variable. If it is
not a polynomial in one variable, explain why.
1. (3x2 + 1)(2x2 - 9)
3 2
1 3
4
2. −
a -−
a +−
a
2
+ 3m - 12
3. −
2
4. 27 + 3xy3 - 12x2y2 - 10y
5
m
5
5
Find p(-2) and p(3) for each function.
5. p(x) = x3 - x5
6. p(x) = -7x2 + 5x + 9
8. p(x) = 3x3 - x2 + 2x - 5
1 3
1
9. p(x) = x4 + −
x -−
x
2
7. p(x) = -x5 + 4x3
1
2
10. p(x) = −
+−
+ 3x
3
2
2
3x
3x
11. p(8a)
12. r(a2)
13. -5r(2a)
14. r(x + 2)
15. p(x2 - 1)
16. 5p(x + 2)
For each graph,
a. describe the end behavior,
b. determine whether it represents an odd-degree or an even-degree function, and
c. state the number of real zeroes.
17.
4
f (x)
18.
4
-2
O
19.
4
2
2
-4
f (x)
2
4x
-4
-2
f (x)
2
2
O
4x
-4
-2
O
-2
-2
-2
-4
-4
-4
2
4x
20. WIND CHILL The function C(w) = 0.013w2 - w - 7 estimates the wind chill
temperature C(w) at 0°F for wind speeds w from 5 to 30 miles per hour. Estimate the
wind chill temperature at 0°F if the wind speed is 20 miles per hour.
Chapter 6
83
North Carolina StudyText, Math BC, Volume 2
Lesson 6-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
If p(x) = 3x2 - 4 and r(x) = 2x2 - 5x + 1, find each value.
NAME
6-3
DATE
PERIOD
Word Problem Practice
SCS
MBC.A.8.4, MBC.A.8.5
Polynomial Functions
1. MANUFACTURING A metal sheet is
curved according to the shape of the
graph of f(x) = x4 - 9x2. What is the
degree of this polynomial?
4. DRILLING The volume of a drill bit can
be estimated by the formula for a cone,
1
πhr2, where h is the height of the
V=−
3
√3
3
bit and r is its radius. Substituting − r
for h, the volume of the drill bit is
2. GRAPHS Kendra graphed the
polynomial f (x) shown below.
6
√3
estimated as − π r3. Graph the function
9
of drill bit volume. Describe the end
behavior, degree, and sign of the leading
coefficient.
y
4
V
40
2
-4
-2
O
2
4
30
6x
-2
20
-4
10
-6
0
10
20
30
40 r
From this graph, describe the end
behavior, degree, and sign of the
leading coefficient.
3. PENTAGONAL NUMBERS The nth
pentagonal number is given by the
expression
n(3n - 1)
2
−.
6
What is the degree of this polynomial?
What is the seventh pentagonal
number?
a. What is the degree of f ?
b. If Dylan drew 15 dots, how many
triangles can be made?
Chapter 6
84
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5. TRIANGLES Dylan drew n dots on a
piece of paper making sure that no line
contained 3 of the dots. The number of
triangles that can be made using the
dots as vertices is equal to
1 3
f(n) = −
(n - 3n2 + 2n).
NAME
DATE
6-4
Study Guide
PERIOD
SCS
MBC.A.8.1, MBC.A.8.3
Analyzing Graphs of Polynomial Functions
Graphs of Polynomial Functions
Location Principle
Suppose y = f (x) represents a polynomial function and a and b are two numbers such that
f (a) < 0 and f(b) > 0. Then the function has at least one real zero between a and b.
Example
Determine consecutive integer values of x between which each real
zero of f(x) = 2x4 - x3 - 5 is located. Then draw the graph.
Make a table of values. Look at the values of f(x) to locate the zeros. Then use the points to
sketch a graph of the function.
f(x)
The changes in sign indicate that there are zeros
x
f(x)
between x = -2 and x = -1 and between x = 1 and
2
35
-2
x = 2.
-1
-2
0
-5
-2
1
-4
-4
2
19
O
-2
2
4x
Exercises
1. f(x) = x3 - 2x2 + 1
2. f (x) = x4 + 2x3 - 5
f (x)
3. f (x) = -x4 + 2x2 - 1
f (x)
f(x)
x
O
x
O
4. f(x) = x3 - 3x2 + 4
5. f (x) = 3x3 + 2x - 1
f(x)
6. f (x) = x4 - 3x3 + 1
f(x)
f (x)
O
x
O
O
Chapter 6
x
O
Lesson 6-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Graph each function by making a table of values. Determine the values of x
between which each real zero is located.
x
85
x
North Carolina StudyText, Math BC, Volume 2
NAME
DATE
6-4
Study Guide
PERIOD
SCS
(continued)
MBC.A.8.1, MBC.A.8.3
Analyzing Graphs of Polynomial Functions
Maximum and Minimum Points
A quadratic function has either a maximum or a
minimum point on its graph. For higher degree polynomial functions, you can find turning
points, which represent relative maximum or relative minimum points.
Example
Graph f(x) = x3 + 6x2 - 3. Estimate the x-coordinates at which the
relative maxima and minima occur.
Make a table of values and graph the function.
f(x)
A relative maximum occurs
x
f(x)
at x = -4 and a relative
24
22
-5
minimum occurs at x = 0.
16
← indicates a relative maximum
-4
29
-3
24
-2
13
-1
2
← zero between x = -1, x = 0
0
-3
← indicates a relative minimum
1
4
2
29
8
O
-4
2
-2
x
Exercises
Graph each polynomial function. Estimate the x-coordinates at which the relative
maxima and relative minima occur.
2. f (x) = 2x3 + x2 - 3x
f (x)
3. f (x) = 2x3 - 3x + 2
f(x)
f(x)
x
O
x
O
4. f (x) = x4 - 7x - 3
5. f (x) = x5 - 2x2 + 2
Chapter 6
6. f (x) = x3 + 2x2 - 3
f(x)
f (x)
O
x
O
x
O
86
f(x)
x
O
x
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. f (x) = x3 - 3x2
NAME
DATE
6-4
PERIOD
Practice
SCS
MBC.A.8.1, MBC.A.8.3
Analyzing Graphs of Polynomial Functions
Complete each of the following.
a. Graph each function by making a table of values.
b. Determine the consecutive values of x between which each real zero is located.
c. Estimate the x-coordinates at which the relative maxima and minima occur.
1. f(x) = -x3 + 3x2 - 3
f(x)
f(x)
x
-2
-2
-1
-1
x
1
1
2
2
3
3
4
4
3. f( x) = 0.75 x4 + x3 - 3x2 + 4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4. f(x) = x4 + 4x3 + 6x2 + 4x - 3
f(x)
f(x)
O
x
O
0
x
f (x)
f(x)
x
O
5. PRICES The Consumer Price Index (CPI) gives the relative
price for a fixed set of goods and services. The CPI from
September, 2000 to July, 2001 is shown in the graph.
Source: U. S. Bureau of Labor Statistics
a. Describe the turning points of the graph.
b. If the graph were modeled by a polynomial equation,
what is the least degree the equation could have?
x
179
178
177
176
175
174
173
0
1 2 3 4 5 6 7 8 9 10 11
Months Since September, 2000
6. LABOR A town’s jobless rate can be modeled by (1, 3.3), (2, 4.9), (3, 5.3), (4, 6.4), (5, 4.5),
(6, 5.6), (7, 2.5), and (8, 2.7). How many turning points would the graph of a polynomial
function through these points have? Describe them.
Chapter 6
87
North Carolina StudyText, Math BC, Volume 2
Lesson 6-4
O
0
x
f (x)
f(x)
Consumer Price Index
x
2. f(x) = x3 - 1.5x2 - 6x + 1
NAME
6-4
DATE
Word Problem Practice
PERIOD
SCS
MBC.A.8.1, MBC.A.8.3
Analyzing Graphs of Polynomial Functions
1. LANDSCAPES Jalen uses a fourthdegree polynomial to describe the shape
of two hills in the background of a video
game that he is helping to write. The
graph of the polynomial is shown below.
3. VALUE A banker models the expected
value of a company in millions of dollars
by the formula n3 - 3n2, where n is the
number of years in business. Sketch a
graph of v = n3 - 3n2.
v
y
x
O
n
O
4. CONSECUTIVE NUMBERS Ms. Sanchez
asks her students to write expressions to
represent five consecutive integers. One
solution is x - 2, x - 1, x, x + 1, and
x + 2. The product of these five
consecutive integers is given by the fifth
degree polynomial f (x) = x5 - 5x3 + 4x.
Estimate the x-coordinates at which
the relative maxima and relative
minima occur.
a. For what values of x is f(x) = 0?
b. Sketch the graph of y = f (x).
Graph Modeling Mount Rushmore
y
y
0
x
O
x
What is the smallest degree possible for
the equation that corresponds with
this graph?
Chapter 6
88
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. NATIONAL PARKS The graph models
the cross-section of Mount Rushmore.
NAME
6-7
DATE
Study Guide
PERIOD
SCS
MBC.A.8.2
Roots and Zeros
Synthetic Types of Roots
Fundamental Theorem
of Algebra
Every polynomial equation with degree greater than zero has at least one root in the
set of complex numbers.
Corollary to the
Fundamental Theorem
of Algebras
A polynomial equation of the form P(x) = 0 of degree n with complex coefficients has
exactly n roots in the set of complex numbers, including repeated roots.
If P(x) is a polynomial with real coefficients whose terms are arranged in descending
powers of the variable,
• the number of positive real zeros of y = P (x) is the same as the number of changes
in sign of the coefficients of the terms, or is less than this by an even number, and
• the number of negative real zeros of y = P(x) is the same as the number of
changes in sign of the coefficients of the terms of P (-x), or is less than this number
by an even number.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Descartes’ Rule of
Signs
Example 1
Solve the
equation 6x3 + 3x = 0. State the
number and type of roots.
Example 2
State the number of positive
real zeros, negative real zeros, and imaginary
zeros for p(x) = 4x4 - 3x3 - x2 + 2x - 5.
6x3 + 3x = 0
3x(2x2 + 1) = 0
Use the Zero Product Property.
3x = 0 or 2x2 + 1 = 0
x=0
or
2x2 = -1
Since p(x) has degree 4, it has 4 zeros.
Since there are three sign changes, there are 3 or 1
positive real zeros.
Find p(-x) and count the number of changes in
sign for its coefficients.
p(-x) = 4(-x)4 - 3(-x)3 + (-x)2 + 2(-x) - 5
= 4x4 + 3x3 + x2 - 2x - 5
Since there is one sign change, there is exactly 1
negative real zero.
Thus, there are 3 positive and 1 negative real
zero or 1 positive and 1 negative real zeros and
2 imaginary zeros.
i √2
2
x=±−
The equation has one real root, 0,
i √2
2
and two imaginary roots, ± −.
Exercises
Solve each equation. State the number and type of roots.
1. x2 + 4x - 21= 0
2. 2x3 - 50x = 0
3. 12x3 + 100x = 0
State the possible number of positive real zeros, negative real zeros, and
imaginary zeros for each function.
4. f(x) = 3x3 + x2 - 8x - 12
Chapter 6
5. f(x) = 3x5 - x4 - x3 + 6x2 - 5
89
North Carolina StudyText, Math BC, Volume 2
Lesson 6-7
The following statements are equivalent for any polynomial
function f (x).
• c is a zero of the polynomial function f (x).
• c is a root or solution of the polynomial equation f (x) = 0.
• (x - c) is a factor of the polynomial f(x).
• If c is real, then (c, 0) is an intercept of the graph of f (x).
NAME
DATE
6-7
Study Guide
PERIOD
SCS
(continued)
MBC.A.8.2
Roots and Zeros
Find Zeros
Suppose a and b are real numbers with b ≠ 0. If a + bi is a zero of a polynomial
function with real coefficients, then a - bi is also a zero of the function.
Complex Conjugate
Theorem
Example
Find all of the zeros of f(x) = x4 - 15x2 + 38x - 60.
Since f(x) has degree 4, the function has 4 zeros.
f(x) = x4 - 15x2 + 38x - 60 f(-x) = x4 - 15x2 - 38x - 60
Since there are 3 sign changes for the coefficients of f(x), the function has 3 or 1 positive real
zeros. Since there is + sign change for the coefficients of f(-x), the function has 1 negative
real zero. Use synthetic substitution to test some possible zeros.
2
1
0 -15
38 -60
2
4 -22
32
2 -11
16 -28
1
3
1
0 -15
38 -60
3
9 -18
60
1
3 -6
20
0
So 3 is a zero of the polynomial function. Now try synthetic substitution again to find a zero
of the depressed polynomial.
-2
1
-4
1
1
-6
-2
-8
20
16
36
3
-4
-1
-6
4
-2
20
8
28
3 -6
20
-5
10 -20
1 -2
4
0
So - 5 is another zero. Use the Quadratic Formula on the depressed polynomial x2 - 2x + 4
to find the other 1 zeros, 1 ± i √
3.
The function has two real zeros at 3 and -5 and two imaginary zeros at 1 ± i √
3.
-5
1
Exercises
Find all zeros of each function.
1. f(x) = x3 + x2 + 9x + 9
2. f(x) = x3 - 3x2 + 4x - 12
3. p(a) = a3 - 10a2 + 34a - 40
4. p(x) = x3 - 5x2 + 11x - 15
5. f(x) = x3 + 6x + 20
6. f(x) = x4 - 3x3 + 21x2 - 75x - 100
Chapter 6
90
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1
3
-2
1
NAME
6-7
DATE
PERIOD
Practice
SCS
MBC.A.8.2
Roots and Zeros
1. -9x - 15 = 0
2. x4 - 5x2 + 4 = 0
3. x5 - 81x = 0
4. x3 + x2 - 3x - 3 = 0
5. x3 + 6x + 20 = 0
6. x4 - x3 - x2 - x - 2 = 0
State the possible number of positive real zeros, negative real zeros, and
imaginary zeros of each function.
7. f(x) = 4x3 - 2x2 + x + 3
9. q(x) = 3x4 + x3 - 3x2 + 7x + 5
8. p(x) = 2x4 - 2x3 + 2x2 - x - 1
10. h(x) = 7x4 + 3x3 - 2x2 - x + 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find all zeros of each function.
11. h(x) = 2x3 + 3x2 - 65x + 84
12. p(x) = x3 - 3x2 + 9x - 7
13. h(x) = x3 - 7x2 + 17x - +5
14. q(x) = x4 + 50x2 + 49
15. g(x) = x4 + 4x3 - 3x2 - 14x - 8
16. f(x) = x4 - 6x3 + 6x2 + 24x - 40
Write a polynomial function of least degree with integral coefficients that has the
given zeros.
17. -5, 3i
18. -2, 3 + i
19. -1, 4, 3i
20. 2, 5, 1 + i
21. CRAFTS Stephan has a set of plans to build a wooden box. He wants to reduce the
volume of the box to 105 cubic inches. He would like to reduce the length of each
dimension in the plan by the same amount. The plans call for the box to be 10 inches by
8 inches by 6 inches. Write and solve a polynomial equation to find out how much
Stephan should take from each dimension.
Chapter 6
91
North Carolina StudyText, Math BC, Volume 2
Lesson 6-7
Solve each equation. State the number and type of roots.
NAME
DATE
6-7
Word Problem Practice
PERIOD
SCS
MBC.A.8.2
Roots and Zeros
1. TABLES Li Pang made a table of values
for the polynomial p(x). Her table is
shown below.
x
p(x)
-4
-3
-3
-1
-2
0
-1
2
0
0
1
4
2
0
3
2
4
5
4. COMPLEX ROOTS Eric is a statistician.
During the course of his work, he had to
find something called the “eigenvalues of
a matrix,” which was basically the same
as finding the roots of a polynomial. The
polynomial was x4 + 6x2 + 25. One of the
roots of this polynomial is 1 + 2i. What
are the other 3 roots? Explain.
Name three roots of p(x).
5. QUADRILATERALS Shayna plotted the
four vertices of a quadrilateral in the
complex plane and then encoded the
points in a polynomial p(x) by making
them the roots of p(x). The polynomial
p(x) is x4 - 9x3 + 27x2 + 23x - 150.
3. REAL ROOTS There are more than a
thousand roller coasters around the
world. Roller coaster designers can use
polynomial functions to model the
shapes of possible roller coasters.
Madison is studying a roller coaster
modeled by the polynomial
f(x) = x6 - 14x4 + 49x2 - 36. She knows
that all of the roots of f(x) are real. How
many positive and how many negative
roots are there? How are the set of
positive roots and negative roots related
to each other? Explain.
a. The polynomial p(x) has one positive
real root, and it is an integer. Find
the integer.
b. Find the negative real root(s) of p(x).
c. Find the complex roots of p(x).
Chapter 6
92
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. ROOTS Ryan is an electrical engineer.
He often solves polynomial equations
to work out various properties of the
circuits he builds. For one circuit, he
must find the roots of a polynomial p(x).
He finds that p(2 - 3i) = 0. Give two
different roots of p(x).
NAME
DATE
6-8
PERIOD
Study Guide
SCS
MBC.A.8.2
Rational Zero Theorem
Identify Rational Zeros
Rational Zero
Theorem
Let f(x) = an x n + an - 1x n - 1 + … + a2 x 2 + a1x + a0 represent a polynomial function with
p
integral coefficients. If −
q is a rational number in simplest form and is a zero of y = f(x),
then p is a factor of a0 and q is a factor of an.
Corollary (Integral
Zero Theorem)
If the coefficients of a polynomial are integers such that an = 1 and a0 ≠ 0, any rational
zeros of the function must be factors of a0.
Example
List all of the possible rational zeros of each function.
a. f(x) = 3x4 - 2x2 + 6x - 10
p
If −
q is a rational root, then p is a factor of -10 and q is a factor of 3. The possible values
for p are ±1, ±2, ±5, and ±10. The possible values for q are 61 and 63. So all of the
p
5
10
1
2
possible rational zeros are −
q = ±1, ±2, ±5, ±10, ± −, ± −, ± −, and ± −.
3
3
3
2
b. q(x) = x - 10x + 14x - 36
Since the coefficient of x3 is 1, the possible rational zeros must be the factors of the
constant term -36. So the possible rational zeros are ±1, ±2, ±3, ±4, ±6, ±9, ±12, ±18,
and ±36.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
List all of the possible rational zeros of each function.
1. f(x) = x3 + 3x2 - x + 8
2. g(x) = x5 - 7x4 + 3x2 + x - 20
3. h(x) = x4 - 7x3 - 4x2 + x - 49
4. p(x) = 2x4 - 5x3 + 8x2 + 3x - 5
5. q(x) = 3x4 - 5x3 + 10x + 12
6. r(x) = 4x5 - 2x + 18
7. f(x) = x7 - 6x5 - 3x4 + x3 + 4x2 - 120
8. g(x) = 5x6 - 3x4 + 5x3 + 2x2 - 15
9. h(x) = 6x5 - 3x4 + 12x3 + 18x2 - 9x + 21
Chapter 6
10. p(x) = 2x7 - 3x6 + 11x5 - 20x2 + 11
93
North Carolina StudyText, Math BC, Volume 2
Lesson 6-8
3
3
NAME
DATE
6-8
Study Guide
PERIOD
SCS
(continued)
MBC.A.8.2
Rational Zero Theorem
Find Rational Zeros
Example 1
Find all of the rational zeros of f(x) = 5x3 + 12x2 - 29x + 12.
From the corollary to the Fundamental Theorem of Algebra, we know that there are
exactly 3 complex roots. According to Descartes’ Rule of Signs there are 2 or 0 positive
real roots and 1 negative real root. The possible rational zeros are ±1, ±2, ±3, ±4, ±6, ±12,
3
6
1
2
4
12
±−
, ±−
, ±−
, ±−
, ±−
, ±−
. Make a table and test some possible rational zeros.
5
5
5
5
5
5
p
−
q
5
12
-29
12
1
5
17
-12
0
Since f(1) = 0, you know that x = 1 is a zero.
The depressed polynomial is 5x2 + 17x - 12, which can be factored as (5x - 3)(x + 4).
3
By the Zero Product Property, this expression equals 0 when x = −
or x = -4.
5
3
The rational zeros of this function are 1, −, and -4.
5
Example 2
Find all of the zeros of f(x) = 8x4 + 2x3 + 5x2 + 2x - 3.
There are 4 complex roots, with 1 positive real root and 3 or 1 negative real roots. The
3
3
3
1
1
1
possible rational zeros are ±1, ±3, ± −
, ±−
, ±−
, ±−
, ±−
, and ± −
.
4
2
4
2
8
−
q
8
2
5
2
-3
1
8
10
15
17
14
−
q
8
6
8
6
2
8
18
41
84
165
1
-−
8
4
7
1
4−
8
0
8
0
1
−
2
8
6
8
6
p
4
3
-−
4
0
4
1
1 = 0, we know that x = −
Since f −
(2)
3
x = -−
is another rational root.
4
is a root.
The depressed polynomial is 8x2 + 8 = 0,
which has roots ±i.
3
1
The zeros of this function are −, - −, and ±i.
2
2
4
Exercises
Find all of the rational zeros of each function.
1. f(x) = x3 + 4x2 - 25x - 28
2. f(x) = x3 + 6x2 + 4x + 24
Find all of the zeros of each function.
3. f(x) = x4 + 2x3 - 11x2 + 8x - 60
Chapter 6
4. f(x) = 4x4 + 5x3 + 30x2 + 45x - 54
94
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The depressed polynomial is 8x3 + 6x2 + 8x + 6.
Try synthetic substitution again. Any remaining
rational roots must be negative.
Make a table and test some
possible values.
p
8
NAME
6-8
DATE
PERIOD
Practice
SCS
MBC.A.8.2
Rational Zero Theorem
List all of the possible rational zeros of each function.
1. h(x) = x3 - 5x2 + 2x + 12
2. s(x) = x4 - 8x3 + 7x - 14
3. f(x) = 3x5 - 5x2 + x + 6
4. p(x) = 3x2 + x + 7
5. g(x) = 5x3 + x2 - x + 8
6. q(x) = 6x5 + x3 - 3
Find all of the rational zeros of each function.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
9. c(x) = x3 - x2 - 8x + 12
8. v(x) = x3 - 9x2 + 27x - 27
10. f(x) = x4 - 49x2
Lesson 6-8
7. q(x) = x3 + 3x2 - 6x - 8
11. h(x) = x3 - 7x2 + 17x - 15
12. b(x) = x3 + 6x + 20
13. f(x) = x3 - 6x2 + 4x - 24
14. g(x) = 2x3 + 3x2 - 4x - 4
15. h(x) = 2x3 - 7x2 - 21x + 54
16. z(x) = x4 - 3x3 + 5x2 - 27x - 36
17. d(x) = x4 + x3 + 16
18. n(x) = x4 - 2x3 - 3
19. p(x) = 2x4 - 7x3 + 4x2 + 7x - 6
20. q(x) = 6x4 + -9x3 + 40x2 + 7x - 12
Find all of the zeros of each function.
21. f(x) = 2x4 + 7x3 - 2x2 - 19x - 12
22. q(x) = x4 - 4x3 + x2 + 16x - 20
23. h(x) = x6 - 8x3
24. g(x) = x6 - 1
25. TRAVEL The height of a box that Joan is shipping is 3 inches less than the width of the
box. The length is 2 inches more than twice the width. The volume of the box is 1540 in3.
What are the dimensions of the box?
26. GEOMETRY The height of a square pyramid is 3 meters shorter than the side of its base.
1
If the volume of the pyramid is 432 m3, how tall is it? Use the formula V = −
Bh.
3
Chapter 6
95
North Carolina StudyText, Math BC, Volume 2
NAME
6-8
DATE
PERIOD
Word Problem Practice
SCS
MBC.A.8.2
Rational Zero Theorem
1. ROOTS Paul was examining an old
algebra book. He came upon a page
about polynomial equations and saw the
polynomial below.
x9
4. PYRAMIDS The Great Pyramid in Giza,
Egypt has a square base with side
lengths of 5x yards and a height of
4x - 50 yards. The volume of the Great
Pyramid is 3,125,000 cubic yards. Use a
calculator to find the value of x and the
dimensions of the pyramid.
+8
As you can see, all the middle terms
were blotted out by an ink spill. What
are all the possible rational roots of this
polynomial?
5. BOXES Devon made a box with length
x + 1, width x + 3, and height x - 3.
2. IRRATIONAL CONSTANTS Cherie was
given a polynomial whose constant term
was √
2 . Is it possible for this polynomial
to have a rational root? If it is not,
explain why not. If it is possible, give an
example of such a polynomial with a
rational root.
x–3
x+3
x+1
b. What is x if the volume of the box is
equal to 1001 cubic inches?
c. What is x if the volume of the box is
5
equal to 14 −
cubic inches?
3. MARKOV CHAINS Tara is a
mathematician who specializes in
probability. In the course of her work,
she needed to find the roots of the
polynomial
p(x) = 288x4 - 288x3 + 106x2 - 17x + 1.
What are the roots of p(x)?
Chapter 6
8
96
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
a. What is the volume of Devon’s box as
a function of x?
NAME
DATE
7-1
PERIOD
Study Guide
SCS
MBC.A.5.1
Operations on Functions
Arithmetic Operations
Sum
(f + g)(x) = f(x) + g(x)
Difference
(f - g)(x) = f(x) - g(x)
Product
(f g)(x) = f(x) g(x)
Quotient
f(x)
, g(x) ≠ 0
( −gf )(x) = −
g(x)
()
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
f
2
Example
Find (f + g)(x), (f - g)(x), (f g)(x), and −
g (x) for f(x) = x + 3x - 4
and g(x) = 3x - 2.
Addition of functions
( f + g)(x) = f(x) + g(x)
2
= (x + 3x - 4) + (3x - 2)
f(x) = x 2 + 3x - 4, g(x) = 3x - 2
2
= x + 6x - 6
Simplify.
( f - g)(x) = f(x) - g(x)
Subtraction of functions
2
= (x + 3x - 4) - (3x - 2)
f(x) = x 2 + 3x - 4, g(x) = 3x - 2
= x2 - 2
Simplify.
․
( f g)(x) = f(x) g(x)
Multiplication of functions
2
= (x + 3x - 4)(3x - 2)
f(x) = x 2 + 3x - 4, g(x) = 3x - 2
2
= x (3x - 2) + 3x(3x - 2) - 4(3x - 2)
Distributive Property
3
2
2
= 3x - 2x + 9x - 6x - 12x + 8
Distributive Property
3
2
= 3x + 7x - 18x + 8
Simplify.
(−gf )(x)
f(x)
g(x)
x 2 + 3x - 4
2
= −, x ≠ −
3x - 2
3
= −
Division of functions
f(x) = x 2
Exercises
+ 3x - 4 and g(x) = 3x - 2
(f)
Find (f + g)(x), (f - g)(x), (f g)(x), and −
g (x) for each f(x) and g(x).
1. f(x) = 8x - 3; g(x) = 4x + 5
2. f(x) = x2 + x - 6; g(x) = x - 2
3. f(x) = 3x2 - x + 5; g(x) = 2x - 3
4. f(x) = 2x - 1; g(x) = 3x2 + 11x - 4
1
5. f(x) = x2 - 1; g(x) = −
x+1
Chapter 7
97
North Carolina StudyText, Math BC, Volume 2
Lesson 7-1
Operations with Functions
NAME
7-1
DATE
Study Guide
PERIOD
SCS
(continued)
MBC.A.5.1
Operations on Functions
Composition of Functions Suppose f and g are functions such that the range of g is
a subset of the domain of f. Then the composite function f ◦ g can be described by the
equation [f ◦ g](x) = f[g(x)].
Example 1
For f = {(1, 2), (3, 3), (2, 4), (4, 1)} and g = {(1, 3), (3, 4), (2, 2), (4, 1)},
find f º g and g º f if they exist.
f [ g(1)] = f(3) = 3
f[ g(2)] = f(2) = 4
So f ◦ g = {(1, 3), (2, 4), (3, 1), (4, 2)}
g[ f (1)] = g(2) = 2
g[ f (2)] = g(4) = 1
So g ◦ f = {(1, 2), (2, 1), (3, 4), (4, 3)}
Example 2
f [ g(3)] = f(4) = 1
f [ g(4)] = f (1) = 2,
g[ f(3)] = g(3) = 4
g[ f (4)] = g(1) = 3,
Find [g º h](x) and [h º g](x) for g(x) = 3x - 4 and h(x) = x2 - 1.
[g ◦ h](x) = g[ h(x)]
= g(x2 - 1)
= 3(x2 - 1) - 4
= 3x2 - 7
[h ◦ g](x) = h[ g(x)]
= h(3x - 4)
= (3x - 4)2 - 1
= 9x2 - 24x + 16 - 1
= 9x2 - 24x + 15
Exercises
1. f = {(-1, 2), (5, 6), (0, 9)},
g = {(6, 0), (2, -1), (9, 5)}
2. f = {(5, -2), (9, 8), (-4, 3), (0, 4)},
g = {(3, 7), (-2, 6), (4, -2), (8, 10)}
Find [f º g](x) and [g º f ](x), if they exist.
3. f(x) = 2x + 7; g(x) = -5x - 1
4. f(x) = x2 - 1; g(x) = -4x2
5. f(x) = x2 + 2x; g(x) = x - 9
6. f(x) = 5x + 4; g(x) = 3 - x
Chapter 7
98
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
For each pair of functions, find f º g and g º f , if they exist.
NAME
7-1
DATE
PERIOD
Practice
SCS
MBC.A.5.1
Operations on Functions
(f)
Find (f + g)(x), (f - g)(x), (f g)(x), and −
g (x) for each f(x) and g(x).
2. f(x) = 8x2
3. f(x) = x2 + 7x + 12
1
g(x) = −
2
g(x) = x - 3
g(x) = x2 - 9
x
Lesson 7-1
1. f(x) = 2x + 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
For each pair of functions, find f º g and g º f , if they exist.
4. f = {(-9, -1), (-1, 0), (3, 4)}
g = {(0, -9), (-1, 3), (4, -1)}
5. f = {(-4, 3), (0, -2), (1, -2)}
g = {(-2, 0), (3, 1)}
6. f = {(-4, -5), (0, 3), (1, 6)}
g = {(6, 1), (-5, 0), (3, -4)}
7. f = {(0, -3), (1, -3), (6, 8)}
g = {(8, 2), (-3, 0), (-3, 1)}
Find [g º h](x) and [h º g](x), if they exist.
8. g(x) = 3x
h(x) = x - 4
11. g(x) = x + 3
h(x) = 2x2
9. g(x) = -8x
h(x) = 2x + 3
10. g(x) = x + 6
h(x) = 3x2
12. g(x) = -2x
h(x) = x2 + 3x + 2
13. g(x) = x - 2
h(x) = 3x2 + 1
If f(x) = x2, g(x) = 5x, and h(x) = x + 4, find each value.
14. f[ g(1)]
15. g[h(-2)]
16. h[f(4)]
17. f[h(-9)]
18. h[ g(-3)]
19. g[ f(8)]
20. BUSINESS The function f(x) = 1000 - 0.01x2 models the manufacturing cost per item
when x items are produced, and g(x) = 150 - 0.001x2 models the service cost per item.
Write a function C(x) for the total manufacturing and service cost per item.
f
n
21. MEASUREMENT The formula f = −
converts inches n to feet f, and m = − converts
12
5280
feet to miles m. Write a composition of functions that converts inches to miles.
Chapter 7
99
North Carolina StudyText, Math BC, Volume 2
NAME
7-1
DATE
PERIOD
Word Problem Practice
SCS
MBC.A.5.1
Operations on Functions
1. AREA Bernard wants to know the
area of a figure made by joining an
equilateral triangle and square along an
4. ENGINEERING A group of engineers is
designing a staple gun. One team
determines that the speed of impact s of
the staple (in feet per second) as a
function of the handle length (in
inches) is given by s() = 40 + 3. A
second team determines that the
number of sheets N that can be stapled
as a function of the impact speed is
s - 10 .
given by N(s) = −
What function
3
gives N as a function of ?
√3
edge. The function f(s) = − s2 gives the
4
area of an equilateral
triangle with side s.
s
The function g(s) = s2
gives the area of a
square with side s.
What function h(s)
gives the area of the
figure as a function of
its side length s?
5. HOT AIR BALLOONS Hannah and
Terry went on a one-hour hot air balloon
ride. Let T(A) be the outside air
temperature as a function of altitude
and let A(t) be the altitude of the balloon
as a function of time.
Chapter 7
100
5
Altitude (km)
6
50
40
30
20
10
0
1
2 3 4 5 6
Altitude (km)
4
3
2
1
0
10 20 30 40 50 60
Time (minutes)
a. What function describes the air
temperature Hannah and Terry felt
at different times during their trip?
b. Sketch a graph of the function you
wrote for part a based on the graphs
for T(A) and A(t) that are given.
Temperature (ºF)
Temperature (ºF)
3. LAVA The temperature of lava has been
measured at up to 2000°F. A freshly
ejected lava rock immediately begins to
cool down. The temperature of the
lava rock in degrees Fahrenheit as a
function of time is given by T(t). Let
C(F) be the function that gives degrees
Celsius as a function of degrees
Fahrenheit. What function gives the
temperature of the lava rock in degrees
Celsius as a function of time?
60
60
50
40
30
20
10
0
10 20 30 40 50 60
Time (minutes)
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. PRICING A computer company decides
to continuously adjust the pricing of and
discounts to its products in an effort to
remain competitive. The function P(t)
gives the sale price of its Super2000
computer as a function of time. The
function D(t) gives the value of a special
discount it offers to valued customers.
How much would valued customers have
to pay for one Super2000 computer?
NAME
DATE
7-2
PERIOD
Study Guide
SCS
MBC.A.5.2, MBC.A.5.3
Inverse Functions and Relations
Find Inverses
Inverse Relations
Two relations are inverse relations if and only if whenever one relation contains the
element (a, b), the other relation contains the element (b, a).
Property of Inverse
Functions
Suppose f and f -1 are inverse functions.
Then f(a) = b if and only if f -1(b) = a.
Example
2
1
Find the inverse of the function f(x) = −
x-−
. Then graph the
5
5
function and its inverse.
Step 1 Replace f(x) with y in the original equation.
2
1
x-−
f(x) = −
5
5
f ( x)
4
2
1
y=−
x-−
5
5
→
f(x) = 2–5x - 1–5
Step 2 Interchange x and y.
-4
2
1
y-−
x=−
2
4x
-2
f –1(x) = 5–2x + 1–2
-4
5
Step 3 Solve for y.
2
1
x=−
y-−
5
5
5x = 2y - 1
5x + 1 = 2y
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
O
Lesson 7-2
5
-2
2
2
1
Inverse of y = −
x-−
5
5
Multiply each side by 5.
Add 1 to each side.
1
−
(5x + 1) = y
Divide each side by 2.
2
2
1
1
The inverse of f(x) = −
x-−
is f -1(x) = −
(5x + 1).
5
5
2
Exercises
Find the inverse of each function. Then graph the function and its inverse.
2
1. f(x) = −
x-1
1
3. f(x) = −
x-2
2. f(x) = 2x - 3
3
f (x)
4
f (x)
f (x)
O
O
Chapter 7
x
O
101
x
x
North Carolina StudyText, Math BC, Volume 2
NAME
DATE
7-2
Study Guide
PERIOD
SCS
(continued)
MBC.A.5.2, MBC.A.5.3
Inverse Functions and Relations
Verifying Inverses
Inverse Functions
Two functions f(x) and g(x) are inverse functions if and only if [f ◦ g](x) = x and [g ◦ f ](x) = x.
Example 1
1
Determine whether f(x) = 2x - 7 and g(x) = −
(x + 7) are
2
inverse functions.
[ f ◦ g](x) = f [ g(x)]
[ g ◦ f ](x) = g[ f(x)]
1 (x + 7)
= f −
= g(2x - 7)
2
1
= 2 − (x + 7) - 7
2
1
=−
(2x - 7 + 7)
2
=x+7-7
=x
=x
The functions are inverses since both [ f ◦ g](x) = x and [ g ◦ f ](x) = x.
Example 2
1
1
Determine whether f(x) = 4x + −
and g(x) = −
x - 3 are
3
4
inverse functions.
[ f ◦ g](x) = f [ g(x)]
(4 )
1x - 3 + −
= 4(−
) 13
4
1x - 3
=f −
3
2
= x - 11 −
3
Since [ f ◦ g](x) ≠ x, the functions are not inverses.
Exercises
Determine whether each pair of functions are inverse functions. Write yes or no.
1. f(x) = 3x - 1
1
1
x+−
g(x) = −
3
3
4. f(x) = 2x + 5
g(x) = 5x + 2
1
7. f(x) = 4x - −
2
1
1
g(x) = − x + −
4
8
x
10. f(x) = 10 - −
2
g(x) = 20 - 2x
Chapter 7
1
2. f(x) = −
x+5
4
1
3. f(x) = −
x - 10
2
g(x) = 4x - 20
1
g(x) = 2x + −
5. f(x) = 8x - 12
6. f(x) = -2x + 3
1
x + 12
g(x) = −
8
3
8. f(x) = 2x - −
5
1
g(x) = −
(5x + 3)
10
4
11. f(x) = 4x - −
5
x
1
g(x) = − + −
4
5
102
10
3
1
g(x) = -−
x+−
2
2
1
9. f(x) = 4x + −
2
3
1
g(x) = − x - −
2
2
3
12. f(x) = 9 + −
x
2
2 g(x) = − x 6
3
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1
= x - 12 + −
NAME
DATE
7-2
Practice
PERIOD
SCS
MBC.A.5.2, MBC.A.5.3
Inverse Functions and Relations
Find the inverse of each relation.
1. {(0, 3), (4, 2), (5, -6)}
2. {(-5, 1), (-5, -1), (-5, 8)}
3. {(-3, -7), (0, -1), (5, 9), (7, 13)}
4. {(8, -2), (10, 5), (12, 6), (14, 7)}
5. {(-5, -4), (1, 2), (3, 4), (7, 8)}
6. {(-3, 9), (-2, 4), (0, 0), (1, 1)}
3
x
7. f(x) = −
4
8. g(x) = 3 + x
9. y = 3x - 2
g(x)
y
f (x)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
O
x
O
x
O
x
Determine whether each pair of functions are inverse functions. Write yes or no.
10. f(x) = x + 6
11. f(x) = -4x + 1
g(x) = x - 6
1
(1 - x)
g(x) = −
13. f(x) = 2x
g(x) = -2x
4
6
14. f(x) = −
x
7
7
g(x) = −
x
6
12. g(x) = 13x - 13
1
h(x) = −
x-1
13
15. g(x) = 2x - 8
1
h(x) = −
x+4
2
16. MEASUREMENT The points (63, 121), (71, 180), (67, 140), (65, 108), and (72, 165) give
the weight in pounds as a function of height in inches for 5 students in a class. Give the
points for these students that represent height as a function of weight.
17. REMODELING The Clearys are replacing the flooring in their 15 foot by 18 foot kitchen.
The new flooring costs $17.99 per square yard. The formula f(x) = 9x converts square
yards to square feet.
a. Find the inverse f -1(x). What is the significance of f-1(x) for the Clearys?
b. What will the new flooring cost the Clearys?
Chapter 7
103
North Carolina StudyText, Math BC, Volume 2
Lesson 7-2
Find the inverse of each function. Then graph the function and its inverse.
NAME
7-2
DATE
PERIOD
Word Problem Practice
SCS
MBC.A.5.2, MBC.A.5.3
Inverse Functions and Relations
1. VOLUME Jason wants to make a
spherical water cooler that can hold half
a cubic meter of water. He knows that
4 3
V=−
πr , but he needs to know how to
3
find r given V. Find this inverse
function.
4. SELF-INVERTIBLE Karen finds the
incomplete graph of a function in the
back of her engineering handbook. The
function is graphed in the figure below.
Karen knows that this function is
its own inverse. Armed with this
knowledge, extend the graph for
values of x between -7 and 2.
y
5
-5
2. EXERCISE Alex began a new exercise
routine. To gain the maximum benefit
from his exercise, Alex calculated his
maximum target heart rate using the
function f(x) = 0.85(220 - x), where x
represents his age. Find the inverse of
this function.
O
5 x
-5
a. Solve for T in terms of d.
3. ROCKETS The altitude of a rocket in
feet as a function of time is given by
f(t) = 49t2, where t ≥ 0. Find the inverse
of this function and determine the times
when the rocket will be 10, 100, and
1000 feet high. Round your answers to
the nearest hundredth of a second.
Chapter 7
b. Pluto is about 39.44 times as far
from the Sun as Earth. About how
many years does it take Pluto to orbit
the Sun?
104
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5. PLANETS The approximate distance of
a planet from the Sun is given by
2
d = T −3 , where d is distance in
astronomical units and T is the period of
its orbit in Earth years. An astronomical
unit is the distance between Earth and
the Sun.
NAME
DATE
7-3
Study Guide
SCS
PERIOD
MBC.A.1.2, MBC.A.8.1, MBC.A.8.2, MBC.A.8.3,
MBC.A.8.4, MBC.A.8.5, MBC.A.10.2
Square Root Functions and Inequalities
Square Root Functions
A function that contains the square root of a variable
expression is a square root function. The domain of a square root function is those values
for which the radicand is greater than or equal to 0.
Example
Graph y = √
3x - 2 . State its domain and range.
2
Since the radicand cannot be negative, the domain of the function is 3x - 2 ≥ 0 or x ≥ −
.
3
2
. The range is y ≥ 0.
The x-intercept is −
3
Make a table of values and graph the function.
y
x
y
2
−
3
0
1
1
2
2
3
√
7
4
2
-2
O
y = √⎯
3x - 2
2
4
6x
-2
Exercises
2x
1. y = √
x
3. y = - −
√2
2. y = -3 √
x
y
y
y
x
O
x
0
O
x
x-3
4. y = 2 √
5. y = - √
2x - 3
y
y
y
x
O
O
Chapter 7
6. y = √
2x + 5
x
O
105
x
North Carolina StudyText, Math BC, Volume 2
Lesson 7-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Graph each function. State the domain and range.
NAME
DATE
7-3
Study Guide
(continued)
Square Root Functions and
PERIOD
SCS
MBC.A.1.2, MBC.A.8.1, MBC.A.8.2,
MBC.A.8.3, MBC.A.8.4, MBC.A.8.5,
MBC.A.10.2
Inequalities
Square Root Inequalities A square root inequality is an inequality that contains
the square root of a variable expression. Use what you know about graphing square root
functions and graphing inequalities to graph square root inequalities.
Example
Graph y ≤ √
2x - 1 + 2.
2x - 1 + 2. Since the boundary
Graph the related equation y = √
should be included, the graph should be solid.
-6
1
, so the graph is to the right
The domain includes values for x ≥ −
-4
1
.
of x = −
2
-2
y
2
O
y = √⎯
2x - 1 + 2
2
4
6
x
Exercises
Graph each inequality.
y
y
y
x
O
O
4. y < √
3x - 4
6. y > 2 √
2x - 3
5. y ≥ √
x+1-4
y
y
y
x
O
x
O
x
O
8. y ≤ √
4x - 2 + 1
3x + 1 - 2
7. y ≥ √
y
9. y < 2 √
2x - 1 - 4
y
y
O
O
x
x
x
O
Chapter 7
x
106
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
x
O
3. y < 3 √
2x - 1
2. y > √
x+3
1. y < 2 √x
NAME
DATE
7-3
Practice
SCS
PERIOD
MBC.A.1.2, MBC.A.8.1, MBC.A.8.2, MBC.A.8.3,
MBC.A.8.4, MBC.A.8.5, MBC.A.10.2
Square Root Functions and Inequalities
Graph each function. State the domain and range.
1. y = √5x
2. y = - √
x-1
3. y = 2 √
x+2
y
y
y
x
O
x
O
x
O
4. y = √3x
-4
5. y = √
x+7-4
y
y
O
y
x
x
O
Lesson 7-3
x
O
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
6. y = 1 - √
2x + 3
Graph each inequality.
7. y ≥ - √
6x
8. y ≤ √
x-5+3
y
y
O
9. y > -2 √
3x + 2
y
x
O
x
x
O
10. ROLLER COASTERS The velocity of a roller coaster as it moves down a hill is
v = √
v02 + 64h , where v0 is the initial velocity and h is the vertical drop in feet. If
v = 70 feet per second and v0 = 8 feet per second, find h.
11. WEIGHT Use the formula d =
39602 W
−E - 3960, which relates distance from Earth d
√
Ws
in miles to weight. If an astronaut’s weight on Earth WE is 148 pounds and in space Ws is
115 pounds, how far from Earth is the astronaut?
Chapter 7
107
North Carolina StudyText, Math BC, Volume 2
NAME
DATE
7-3
Word Problem Practice
Square Root Functions
1. SQUARES Cathy is building a square
roof for her garage. The roof will occupy
625 square feet. What are the
dimensions of the roof?
PERIOD
SCS
MBC.A.1.2, MBC.A.8.1,
MBC.A.8.2, MBC.A.8.3,
MBC.A.8.5,
and Inequalities MBC.A.8.4, MBC.A.10.2
4. DISTANCE Lance is standing at the
side of a road watching a cyclist go by.
The distance between Lance and the
cyclist as a function of time is given by
d = √
9 + 36t2 . Graph this function.
Find the distance between Lance and
the cyclist after 3 seconds.
d
2. PENDULUMS The period of a
pendulum, or the time it takes to
complete one swing, is given by the
formula
L
−
p = 2π ,
√g
O
where L is the length in meters of the
pendulum and g is acceleration due to
gravity, 9.8 m/s2. Find the period of a
pendulum that is 0.65 meters long.
Round to the nearest tenth.
t
5. STARS The intensity of the light from
an object varies inversely with the
square of the distance. In other words,
k
I=−
.
2
d
3. REFLEXES Rachel and Ashley are
testing one another’s reflexes. Rachel
drops a ruler from a given height so that
it falls between Ashley’s thumb and
index finger. Ashley tries to catch the
ruler before it falls through her hand.
The time required to catch the ruler is
b. The stars Antares and Spica have the
same apparent magnitudes. However,
their absolute magnitudes, or
intensities, differ. Let I1 and I2 be
their absolute magnitudes and let d1
and d2 be their respective distances
from Earth. What is the ratio of d2
to d1?
√
d
4
given by t = − where d is measured
in feet. Complete the table. Round your
answers to the nearest hundredth.
Distance (in.)
Reflex Time (seconds)
3
6
9
12
Chapter 7
108
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
a. Solve the equation to find d in terms
of I.
NAME
DATE
7-6
PERIOD
Study Guide
SCS
MBC.N.1.1, MBC.N.1.2
Rational Exponents
1
Definition of b −n
m
Definition of b −n
Example 1
Lesson 7-6
Rational Exponents and Radicals
For any real number b and any positive integer n,
1
n
b , except when b < 0 and n is even.
b −n = √
For any nonzero real number b, and any integers m and n, with n > 1,
m
n
n
m
)m, except when b < 0 and n is even.
= ( √b
b −n = √b
1
−
Write 28 2 in radical form.
Notice that 28 > 0.
Example 2
( -125 )
1
−
-8 3
Evaluate −
.
Notice that -8 < 0, -125 < 0, and 3 is odd.
1
−
28
28 = √
2
( -125 )
-8
−
2 7
= √2
1
−
3
3
√
-8
√
-125
=−
3
-2
=−
2 √
7
= √2
-5
2
=−
5
7
= 2 √
Exercises
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Write each expression in radical form, or write each radical in exponential form.
1
−
1
−
3
−
1. 11 7
2. 15 3
3. 300 2
47
4. √
3
5 2
5. √3a
b
4
5
6. √162p
Evaluate each expression.
2
−
7. -27 3
Chapter 7
1
−
1
−
8. 216 3
9. (0.0004) 2
109
North Carolina StudyText, Math BC, Volume 2
NAME
DATE
7-6
Study Guide
PERIOD
SCS
(continued)
MBC.N.1.1, MBC.N.1.2
Rational Exponents
Simplify Expressions
All the properties of powers from Lesson 6-1 apply to rational
exponents. When you simplify expressions with rational exponents, leave the exponent in
rational form, and write the expression with all positive exponents. Any exponents in the
denominator must be positive integers.
When you simplify radical expressions, you may use rational exponents to simplify, but your
answer should be in radical form. Use the smallest index possible.
2
−
Example 1
3
−
2
−
3
2
−
+−
y3 y8 = y3
8
3
−
Example 2
Simplify y 3 y 8 .
4
Simplify √
144x6 .
1
25
−
4
−
√
144x6 = (144x6) 4
= y 24
= (24 32
1
−
x6) 4
1
−
1
−
1
−
= (24) 4 (32) 4 (x6) 4
1
−
1
−
3
−
3x
= 2 3 2 x 2 = 2x (3x) 2 = 2x √
Exercises
Simplify each expression.
6
−
4
−
1. x 5 x 5
p
7. −1
−
p3
4
10. √49
3
−
7
4
−
−
3. p 5 ․ p 10
4
1 −
−
6. (s 6 ) 3
4
−
5. x 8 x 3
1
−
x2
8. −
1
6
9. √
128
−
x3
5
11. √
288
12. √
32 3 √
16
3
3
25 √
125
13. √
Chapter 7
a √
b4
√
ab3
6
14. √
16
15. −
110
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
6 −
−
4. (m5 ) 5
3
2 −
−
2. (y 3 ) 4
NAME
DATE
7-6
Practice
PERIOD
SCS
MBC.N.1.1, MBC.N.1.2
Write each expression in radical form, or write each radical in exponential form.
1
−
2
−
4
−
2
−
1. 5 3
2. 6 5
5. √79
6. √
153
4. (n3) 5
3. m 7
3
4
5
7. √
27m6n4
8. √
2a10b
Evaluate each expression.
10. 1024 5
( 216 )
11. 8 3
2
−
3
−
1
2
−
2
−
17. (25 2 )(-64
1
−
64
16. −
2
3
3
4
−
−
14. 27 3 ․ 27 3
13. (-64) 3
12. -256 4
125
15. −
5
−
1
−
1
−
9. 81 4
−
1
-−
3
)
343 3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Simplify each expression.
4
3
−
−
18. g 7 ․ g 7
22. b
3
−
5
3
1
−
3
−
10
21. y
2
−
q5
t
24. −
3
1
3
23. −2
−
−
q5
5
26. √8
4
−
20. (u 3 ) 5
13
−
−
19. s 4 ․ s 4
5t 2 ․ t
5
3
27. √
12 √12
-−
4
4
4
3√
28. √6
6
1
-−
2
1
−
2z
25. −
1
2
−
z2 - 1
a
29. −
√
3b
30. ELECTRICITY The amount of current in amps I that an appliance uses can be
( )
1
−
P 2
, where P is the power in watts and R is the
calculated using the formula I = −
R
resistance in ohms. How much current does an appliance use if P = 500 watts and
R = 10 ohms? Round your answer to the nearest tenth.
1
−
31. BUSINESS A company that produces DVDs uses the formula C = 88n 3 + 330 to
calculate the cost C in dollars of producing n DVDs per day. What is the company’s cost
to produce 150 DVDs per day? Round your answer to the nearest dollar.
Chapter 7
111
North Carolina StudyText, Math BC, Volume 2
Lesson 7-6
Rational Exponents
NAME
7-6
DATE
Word Problem Practice
PERIOD
SCS
MBC.N.1.1, MBC.N.1.2
Rational Exponents
1. SQUARING THE CUBE A cube has side
length s. What side length of the square
will cause its area to have the same
numerical value as the volume of
the cube? Write your answer using
rational exponents.
4. INTEREST Rita opened a bank account
that accumulated interest at the rate of
1% compounded annually. Her money
accumulated interest in that account for
8 years. She then took all of her money
out of that account and placed it into
another account that paid 5% interest
compounded annually. After 4 years,
she took all of her money out of that
account. What single interest rate when
compounded annually would give her
the same outcome for those 12 years?
Round your answer to the nearest
hundredth of a percent.
2. WATER TOWER Typically, drinking
water for towns is stored in water
towers. A water tower in Edmond,
Oklahoma is 218 feet high and holds
half a million
gallons. One town
is replacing its
water tower.
Residents of the
town insist that
their new tower be
a sphere. If the new
tank will hold
10 times as much
water as the old
tank, how many
times longer should the radius of the
new tank be compared to the old tank?
Write your answer using rational
exponents.
5. CELLS The number of cells in a cell
culture grows exponentially. The
number of cells in the culture as a
function of time is given by the
(5)
6 t, where t is measured
expression N −
a. After 3 hours, there were 1728 cells in
the culture. What is N?
b. How many cells were in the culture
after 20 minutes? Express your
answer in simplest form.
3. BALLOONS A spherical balloon is being
inflated faster and faster. The volume of
the balloon as a function of time is 9πt2.
What is the radius of the balloon as a
function of time? Write your answer
using rational exponents.
Chapter 7
c. How many cells were in the culture
after 2.5 hours? Express your answer
in simplest form.
112
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
in hours and N is the initial size of
the culture.
NAME
DATE
8-1
PERIOD
Study Guide
SCS
Graphing Exponential Functions
MBC.A.1.2, MBC.A.3.1,
MBC.A.3.2, MBC.A.8.1,
MBC.A.8.2, MBC.A.8.3,
MBC.A.8.4, MBC.A.8.5
Exponential Growth
An exponential growth function has the form y = bx,
where b > 1. The graphs of exponential equations can be transformed by changing the value
of the constants a, h, and k in the exponential equation: f (x) = abx – h + k.
Example
The
The
The
The
The
function is continuous, one-to-one, and increasing.
domain is the set of all real numbers.
x-axis is the asymptote of the graph.
range is the set of all non-zero real numbers.
graph contains the point (0, 1).
Graph y = 4x + 2. State the domain and range.
Make a table of values. Connect the points to form a smooth curve.
Lesson 8-1
1.
2.
3.
4.
5.
Parent Function of
Exponential Growth
Functions,
f(x) = bx, b > 1
y
6
4
x
-1
0
1
2
3
y
2.25
3
6
18
66
2
The domain of the function is all real numbers, while the range is
the set of all positive real numbers greater than 2.
0
2
x
4
Exercises
Graph each function. State the domain and range.
1. y = 3(2)
x
1
2. y = −
(3)
x
3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
y
O
x
x
5. y = 4 x - 2
y
x
y
y
0
x
4. y = 2(3)
0
3. y = 0.25(5)
O
x
6. y = 2 x + 5
y
x
0
y
x
0
Chapter 8
113
x
North Carolina StudyText, Math BC, Volume 2
NAME
DATE
8-1
Study Guide
PERIOD
SCS
(continued)
MBC.A.1.2, MBC.A.3.1,
MBC.A.3.2, MBC.A.8.1,
MBC.A.8.2, MBC.A.8.3,
MBC.A.8.4, MBC.A.8.5
Graphing Exponential Functions
Exponential Decay
The following table summarizes the characteristics of exponential
decay functions.
1.
2.
3.
4.
5.
Parent Function of
Exponential Decay
Functions,
f(x) = bx, 0 < b < 1
Example 1
The
The
The
The
The
function is continuous, one-to-one, and decreasing.
domain is the set of all real numbers.
x-axis is the asymptote of the graph.
range is the set of all positive real numbers.
graph contains the point (0, 1).
(2)
x
1
Graph y = −
. State the domain and range.
Make a table of values. Connect the points to form a smooth curve.
The domain is all real numbers and the range is the set of all
positive real numbers.
y
4
2
-4
4x
2
-2
x
-2
-1
0
1
2
-2
y
4
2
1
0.5
0.25
-4
Exercises
Graph each function. State the domain and range.
(2)
1
1. y = 6 −
x
(4)
1
2. y = -2 −
3. y = -0.4(0.2)x
y
y
O
( 5 )( 2 )
x-1
(5)
1
5. y = 4 −
+2
O
x+3
( 3 ) (−34 )
1
6. y = - −
-1
x
O
114
x-5
+6
y
y
y
Chapter 8
x
0
x
0
2 1
−
4. y = −
x
x
O
x
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
y
x
NAME
DATE
8-1
Practice
PERIOD
SCS
MBC.A.1.2, MBC.A.3.1,
MBC.A.3.2, MBC.A.8.1,
MBC.A.8.2, MBC.A.8.3,
MBC.A.8.4, MBC.A.8.5
Graphing Exponential Functions
Graph each function. State the domain and range.
2. y = 4(3)x
y
y
O
(2)
1
4. y = 5 −
3. y = 3(0.5)x
y
O
x
x
(4)
1
5. y = -2 −
-8
y
O
x
x-3
Lesson 8-1
1. y = 1.5(2)x
x
1 ( ) x+4
6. y = −
3
-5
2
y
y
x
x
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
x
7. BIOLOGY The initial number of bacteria in a culture is 12,000. The culture doubles
each day.
a. Write an exponential function to model the population y of bacteria after x days.
b. How many bacteria are there after 6 days?
8. EDUCATION A college with a graduating class of 4000 students in the year 2008 predicts
that its graduating class will grow 5% per year. Write an exponential function to model
the number of students y in the graduating class t years after 2008.
Chapter 8
115
North Carolina StudyText, Math BC, Volume 2
NAME
8-1
DATE
Word Problem Practice
SCS
Graphing Exponential Functions
1. GOLF BALLS A golf ball manufacturer
packs 3 golf balls into a single package.
Three of these packages make a gift
box. Three gift boxes make a value pack.
The display shelf is high enough to stack
3 value packs one on top of the other.
Three such columns of value packs make
up a display front. Three display fronts
can be packed in a single shipping box
and shipped to various retail stores.
How many golf balls are in a single
shipping box?
Subscriptions
40
48
2
3
5. MONEY Sam opened a savings account
that compounds interest at a rate of 3%
annually. Let P be the initial amount
Sam deposited and let t be the number
of years the account has been open.
c. What is the least number of years it
would take for such an account to
double in value?
4
Make a graph of the number of
subscribers over the first 5 years of the
club’s existence.
Chapter 8
116
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1
4. TENNIS SHOES The cost of a pair of
tennis shoes increases about 5.1% every
year. About how much would a $50 pair
of tennis shoes cost 25 years from now?
b. If Sam opened the account with $500
and made no deposits or withdrawals,
how much is in the account 10 years
later?
3. SUBSCRIPTIONS Subscriptions to an
online arts and crafts club have been
increasing by 20% every year. The club
began with 40 members.
0
MBC.A.1.2, MBC.A.3.1,
MBC.A.3.2, MBC.A.8.1,
MBC.A.8.2, MBC.A.8.3,
MBC.A.8.4, MBC.A.8.5
a. Write an equation to find A, the
amount of money in the account after
t years. Assume that Sam made no
more additional deposits and no
withdrawals.
2. FOLDING Paper thickness ranges from
0.0032 inch to 0.0175 inch. Kay folds a
piece of paper 0.01 inch thick in half
over and over until it is at least 25
layers thick. How many times does she
fold the paper in half and how many
layers are there? How thick is the
folded paper?
Year
PERIOD
NAME
8-2
DATE
PERIOD
Study Guide
SCS
MBC.A.7.4
Solving Exponential Equations and Inequalities
Solve Exponential Equations
All the properties of rational exponents that you know
also apply to real exponents. Remember that am · an = am + n, (am)n = amn, and am ÷ an = am - n.
Property of Equality for
Exponential Functions
If b is a positive number other than 1, then bx = by if and only if x = y.
Example 2
Write an exponential
function whose graph passes through the
points (0, 3) and (4, 81).
Example 1
Solve 4 x - 1 = 2 x + 5.
4x - 1 = 2x + 5
Original equation
2 x-1
(2 )
= 2x + 5
Rewrite 4 as 22.
2(x - 1) = x + 5 Prop. of Inequality for Exponential
Distributive Property
The y-intercept is (0, 3), so a = 3. Since the
4 81
−.
other point is (4, 81), b = Subtract x and add 2 to
each side.
Simplifying
Functions
2x - 2 = x + 5
x=7
√3
81
−
= √
27 ≈ 2.280, the equation
√
3
4
4
Lesson 8-2
is y = 3(2.280)x.
Exercises
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Solve each equation.
1. 3 2x - 1 = 3 x + 2
2. 23x = 4x + 2
3. 3 2x - 1 = −19
4. 4 x + 1 = 8 2x + 3
1
5. 8x - 2 = −
16
6. 25 2x = 125 x + 2
7. 9 x + 1 = 27 x + 4
8. 362x + 4 = 216
x+5
( 64 )
1
9. −
x-2
= 16
3x + 1
Write an exponential function for the graph that passes through the given points.
10. (0, 4) and (2, 36)
11. (0, 6) and (1, 81)
13. (0, 2) and (5, 486)
27
14. (0, 8) and 3, −
16. (0, 3) and (3, 24)
17. (0, 12) and (4, 144)
Chapter 8
(
117
8
)
12. (0, 5) and (6, 320)
15. (0, 1) and (4, 625)
18. (0, 9) and (2, 49)
North Carolina StudyText, Math BC, Volume 2
NAME
DATE
8-2
Study Guide
PERIOD
SCS
(continued)
MBC.A.7.4
Solve Exponential Equations and Inequalities
Solve Exponential Inequalities
An exponential inequality is an inequality
involving exponential functions.
Property of Inequality for
Exponential Functions
1
.
Solve 52x - 1 > −
Example
125
1
52x - 1 > −
Original inequality
125
2x - 1
5
If b > 1
then bx > by if and only if x > y
and bx < by if and only if x < y.
1
Rewrite −
as 5-3.
> 5-3
125
2x - 1 > -3
Prop. of Inequality for Exponential Functions
2x > -2
Add 1 to each side.
x > -1
Divide each side by 2.
The solution set is {x | x > -1}.
Exercises
Solve each inequality.
2. 42x - 2 > 2 x + 1
3. 5 2x < 125 x - 5
4. 10 4x + 1 > 100 x - 2
5. 7 3x < 49 1 - x
6. 8 2x - 5 < 4 x + 8
7. 16 ≥ 4x + 5
1
8. −
27
( 27 )
2x - 4
1
10. −
<9
2x - 1
( 25 )
Chapter 8
≤ 125
( 243 )
1
≤ −
3x + 1
x-3
( 343 )
7
14. −
118
12. 27
2x + 1
( 49 )
1
≥ −
x-3
> 82 x
2x - 5
1
< −
(2)
1
9. −
3x - 2
11. 323x - 4 > 1284x + 3
81
1
13. −
2x + 1
6x - 1
( 27 )
9
15. −
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1
1. 3 x - 4 < −
5x
(9)
-x + 6
(9)
27
≥ −
North Carolina StudyText, Math BC, Volume 2
NAME
DATE
8-2
PERIOD
Practice
SCS
MBC.A.7.4
Solving Exponential Equations and Inequalities
Solve each equation.
( 64 )
1. 4 x + 35 = 64x – 3
1
2. −
3. 3x - 4 = 9x + 28
1
4. −
(2)
1
5. −
x-3
(4)
0.5x – 3
2x + 2
= 8 9x – 2
= 64 x - 1
x+1
(9)
1
6. 36x - 2 = −
= 163x + 1
7. (0, 5) and (4, 3125)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
10. (0, –0.2) and (–3, –3.125)
8. (0, 8) and (4, 2048)
(
15
11. (0, 15) and 2, −
16
)
3
9. (0, −
) and (2, 36.75)
4
(2
1
12. (0, 0.7) and −
, 3.5
)
Solve each inequality.
7x + 8
( 20 )
1
13. 400 > −
x-6
(8)
1
16. −
4x + 5
<4
2x + 7
14. 10
x+8
( 36 )
1
17. −
≥ 1000
≤ 216
x
x-3
( 16 )
1
15. −
18. 128
3x - 4
x+3
≤ 64x - 1
2x
( 1024 )
1
< −
19. At time t, there are 216 t + 18 bacteria of type A and 36 2t + 8 bacteria of type B organisms in
a sample. When will the number of each type of bacteria be equal?
Chapter 8
119
North Carolina StudyText, Math BC, Volume 2
Lesson 8-2
Write an exponential function for the graph that passes through the given points.
NAME
DATE
8-2
Word Problem Practice
PERIOD
SCS
MBC.A.7.4
Solving Exponential Equations and Inequalities
1. BANKING The certificate of deposit that
Siobhan bought on her birthday pays
interest according to the formula
0.052 48
A = 1200 1 + −
. What is the
(
12
)
annual interest rate?
4. POPULATION In 2000, the world
population was calculated to be
6,071,675,206. In 2008, it was
6,679,493,893. Write an exponential
equation to model the growth of the
world population over these 8 years.
Round the base to the nearest
thousandth.
Source: U.S. Census Bureau
2. INTEREST Marty invested $2000 in an
account that pays at least 4% annual
interest. He wants to see how much
money he will have over the next few
years. Graph the inequality
y ≥ 2000(1 + 0.04) x to show his
potential earnings.
5. BUSINESS Ingrid and Alberto each
opened a business in 2000. Ingrid started
with 2 employees and in 2003 she had
50 employees. Alberto began with 32
employees and in 2007 he had 310
employees. Since 2000, each company
has experienced exponential growth.
y
x
b. Calculate the number of employees
each company had in 2005.
3. BUSINESS Ahmed’s consulting firm
began with 23 clients. After 7 years, he
now has 393 clients. Write an
exponential equation describing the
firm’s growth.
c. Is it reasonable to expect that a
business can experience exponential
growth? Explain your answer.
Chapter 8
120
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
0
a. Write an exponential equation
representing the growth for each
business.
NAME
DATE
8-3
Study Guide
PERIOD
SCS
MBC.A.5.4, MBC.A.6.1,
MBC.A.8.1, MBC.A.8.2,
MBC.A.8.3, MBC.A.8.4,
MBC.A.10.1, MBC.A.10.2
Logarithms and Logarithmic Functions
Logarithmic Functions and Expressions
Definition of Logarithm
with Base b
Let b and x be positive numbers, b ≠ 1. The logarithm of x with base b is denoted
logb x and is defined as the exponent y that makes the equation by = x true.
The inverse of the exponential function y = bx is the logarithmic function x = by.
This function is usually written as y = logb x.
Example 1
Write an exponential equation equivalent to log3 243 = 5.
35 = 243
Example 2
1
Write a logarithmic equation equivalent to 6-3 = −
.
216
1
= -3
log6 −
216
Example 3
Evaluate log8 16.
4
.
8 = 16, so log8 16 = −
−34
3
Exercises
1. log15 225 = 2
1
2. log3 −
= -3
27
5
3. log4 32 = −
2
Write each equation in logarithmic form.
4. 27 = 128
1
5. 3-4 = −
1 3
1
6. −
=−
1
7. 7-2 = −
8. 29 = 512
−2
9. 64 3 = 16
49
(7 )
81
343
Evaluate each expression.
10. log4 64
11. log2 64
12. log100 100,000
13. log5 625
14. log27 81
15. log25 5
1
16. log2 −
17. log10 0.00001
1
18. log4 −
128
Chapter 8
121
32
North Carolina StudyText, Math BC, Volume 2
Lesson 8-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Write each equation in exponential form.
NAME
DATE
8-3
Study Guide
PERIOD
SCS
(continued)
MBC.A.5.4, MBC.A.6.1,
MBC.A.8.1, MBC.A.8.2,
MBC.A.8.3, MBC.A.8.4,
MBC.A.10.1, MBC.A.10.2
Logarithms of Logarithmic Functions
Graphing Logarithmic Functions The function y = logb x, where b ≠ 1, is called a
logarithmic function. The graph of f(x) = logb x represents a parent graph of the
logarithmic functions. Properties of the parent function are described in the following table.
1.
2.
3.
4.
5.
Parent function of
Logarithmic Functions,
f(x) = logbx
The
The
The
The
The
function is continuous and one-to-one.
domain is the set of all positive real numbers.
y-axis is an asymptote of the graph.
range is the set of all real numbers.
graph contains the point (1, 0).
The graphs of logarithmic functions can be transformed by changing the value of the
constants a, h, and k in the equation f(x) = a logb (x – h) + k.
Example
Graph f(x) = -3 log10 (x - 2) + 1.
This is a transformation of the graph of f(x) = log10 x.
10
y
5
• |a| = 3: The graph expands vertically.
x
• a < 0: The graph is reflected across the x-axis.
-10
-5
0
5
10
-5
• h = 2: The graph is translated 2 units to the right.
• k = 1: The graph is translated 1 unit up.
-10
Graph each function.
1. f(x) = 4 log2 x
10
2. f(x) = 4 log3 (x - 1)
f(x)
10
f(x)
Chapter 8
0
5
10
f(x)
2
x
x
-5
4
5
5
-10
3. f(x) = 2 log4 (x + 3) - 2
-10
-5
0
5
10
-4
-2
0
-5
-5
-2
-10
-10
-4
122
2
4x
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
NAME
DATE
8-3
PERIOD
Practice
SCS
MBC.A.5.4, MBC.A.6.1,
MBC.A.8.1, MBC.A.8.2,
MBC.A.8.3, MBC.A.8.4,
MBC.A.10.1, MBC.A.10.2
Logarithms and Logarithmic Functions
Write each equation in exponential form.
1. log6 216 = 3
2. log2 64 = 6
1
3. log3 −
= -4
4. log10 0.00001 = -5
1
5. log25 5 = −
3
6. log32 8 = −
81
2
5
Write each equation in logarithmic form.
7. 53 = 125
8. 70 = 1
(4)
1
11. −
1
10. 3-4 = −
81
3
9. 34 = 81
1
−
1
=−
12. 7776 5 = 6
64
13. log3 81
14. log10 0.0001
1
15. log2 −
16
16. log −1 27
17. log9 1
18. log8 4
1
19. log7 −
20. log6 64
3
49
Lesson 8-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Evaluate each expression.
Graph each function.
21. f(x) = log2 (x – 2)
22. f(x) = –2 log4 x
f(x)
f (x)
0
x
0
x
23. SOUND An equation for loudness, in decibels, is L = 10 log10 R, where R is the relative
intensity of the sound. Sounds that reach levels of 120 decibels or more are painful to
humans. What is the relative intensity of 120 decibels?
24. INVESTING Maria invests $1000 in a savings account that pays 4% interest
compounded annually. The value of the account A at the end of five years can be
determined from the equation log10 A = log10[1000(1 + 0.04)5]. Write this equation in
exponential form.
Chapter 8
123
North Carolina StudyText, Math BC, Volume 2
NAME
8-3
DATE
PERIOD
Word Problem Practice
SCS
Logarithms and Logarithmic Functions
1. CHEMISTRY The pH of a solution is
found by the formula pH = -log H,
where H stands for the hydrogen ion
concentration in the formula. What is
the pH of a solution to the nearest
hundredth when H is 1356?
MBC.A.5.4, MBC.A.6.1,
MBC.A.8.1, MBC.A.8.2,
MBC.A.8.3, MBC.A.8.4,
MBC.A.10.1, MBC.A.10.2
4. EARTHQUAKES The intensity of an
earthquake can be measured on the
Richter scale using the formula
y = 10 R - 1, where y is the absolute
intensity of the earthquake and R is its
Richter scale measurement.
Absolute Intensity
1
1
2
10
3
100
4
1000
5
10,000
An earthquake in San Francisco in 1906
had an absolute intensity of 6,000,000.
What was that earthquake’s
measurement on the Richter scale?
5. GAMES Julio and Natalia decided to
play a game in which they each selected
a logarithmic function and compare
their functions to see which gave larger
values. Julio selected the function
f (x) = 10 log2 x and Natalia selected the
function 2 log10 x.
a. Which of the functions has a larger
value when x = 7?
b. Which of their functions has a larger
value when x = 1?
3. SOUND The decibel level L of a sound
is determined by the formula
I
. Find I in terms of M for
L = 10 log10 −
M
a noise with a decibel level of 120.
c. Do you think the base or the
multiplier is more important in
determining the value of a
logarithmic function?
Chapter 8
124
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. FIND THE ERROR Michio wanted to
find the value of x in the equation
2(3)x = 34. He first converted the
equation to log3 2x = 17. Next he wrote
2x = 317 and used a calculator to find
x = 64,570,081. Was his answer correct?
If not, what was his mistake and what is
the right answer?
Richter Scale Number
NAME
DATE
8-4
PERIOD
Study Guide
SCS
MBC.A.9.2
Solving Logarithmic Equations and Inequalities
Solving Logarithmic Equations
Property of Equality for
Logarithmic Functions
Example 1
log2 2x = 3
3
If b is a positive number other than 1,
then logb x = logb y if and only if x = y.
Solve log2 2x = 3.
Original equation
2x = 2
Definition of logarithm
2x = 8
Simplify.
x=4
Simplify.
Example 2
Solve the equation
log2 (x + 17) = log2 (3x + 23).
Since the bases of the logarithms are
equal, (x + 17) must equal (3x + 23).
(x + 17) = (3x + 23)
-6 = 2x
The solution is x = 4.
x = -3
Exercises
1. log2 32 = 3x
2. log3 2c = -2
3. log2x 16 = -2
x
1
4. log25 −
=−
5. log4 (5x + 1) = 2
2
6. log8 (x - 5) = −
7. log4 (3x - 1) = log4 (2x + 3)
8. log2 (x2 - 6) = log2 (2x + 2)
9. logx + 4 27 = 3
(2)
2
3
10. log2 (x +3) = 4
11. logx 1000 = 3
12. log8 (4x + 4) = 2
13. log2 x = log2 12
14. log3 (x - 5) = log3 13
15. log10 x = log10 (5x - 20)
16. log5 x = log5 (2x - 1)
17. log4 (x+12) = log4 4x
18. log6 (x - 3) = log6 2x
Chapter 8
125
North Carolina StudyText, Math BC, Volume 2
Lesson 8-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Solve each equation.
NAME
8-4
DATE
Study Guide
PERIOD
SCS
(continued)
MBC.A.9.2
Solving Logarithmic Equations and Inequalities
Solving Logarithmic Inequalities
If b > 1, x > 0, and logb x > y, then x > b y.
Property of Inequality for If b > 1, x > 0, and logb x < y, then 0 < x < by.
Logarithmic Functions
If b > 1, then logb x > logb y if and only if x > y,
and logb x < logb y if and only if x < y.
Example 1
Solve log5 (4x - 3) < 3.
log5 (4x - 3) < 3
Original equation
0 < 4x - 3 < 53
Property of Inequality
3 < 4x < 125 + 3
Simplify.
3
−
< x < 32
Simplify.
4
{|
Example 2
Solve the inequality
log3(3x - 4) < log3 ( x + 1).
Since the base of the logarithms are equal to or
greater than 1, 3x - 4 < x + 1.
2x < 5
5
x<−
2
3
The solution set is x −
< x < 32 .
4
}
Since 3x - 4 and x + 1 must both be positive
numbers, solve 3x - 4 = 0 for the lower bound of
the inequality.
5
4
The solution is x −
<x<−
.
{ |3
2
}
Exercises
Solve each inequality.
2. log5 x > 2
3. log2 (3x + 1) < 4
1
4. log4 2x > - −
5. log3 (x + 3) < 3
2
6. log27 6x > −
7. log10 5x < log10 30
8. log10 x < log10 (2x - 4)
9. log10 3x < log10 (7x - 8)
11. log10 (3x + 7) < log10 (7x - 3)
Chapter 8
2
3
10. log2 (8x + 5) > log2 (9x - 18)
12. log2 (3x - 4) < log2 (2x + 7)
126
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. log2 2x > 2
NAME
8-4
DATE
PERIOD
Practice
SCS
MBC.A.9.2
Solving Logarithmic Equations and Inequalities
Solve each equation.
1. x + 5 = log4 256
2. 3x - 5 = log2 1024
3. log3 (4x - 17) = 5
4. log5 (3 - x) = 5
5. log13 (x2 - 4) = log13 3x
6. log3 (x - 5) = log3 (3x - 25)
Solve each inequality.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
9. log11 (x + 7) < 1
8. log9 (x + 2) > log9 (6 - 3x)
10. log81 x ≤ 0.75
11. log2 (x + 6) < log2 17
12. log12 (2x - 1) > log12 (5x - 16)
13. log9 (2x - 1) < 0.5
14. log10 (x - 5) > log10 2x
15. log3 (x + 12) > log3 2x
16. log3 (0.3x + 5) > log3 (x - 2)
17. log2 (x + 3) < log2 (1 - 3x)
18. log6 (3 - x) ≤ log6 (x - 1)
19. WILDLIFE An ecologist discovered that the population of a certain endangered species
has been doubling every 12 years. When the population reaches 20 times the current
level, it may no longer be endangered. Write the logarithmic expression that gives the
number of years it will take for the population to reach that level.
Chapter 8
127
North Carolina StudyText, Math BC, Volume 2
Lesson 8-4
7. log8 (-6x) < 1
NAME
DATE
8-4
Word Problem Practice
PERIOD
SCS
MBC.A.9.2
Solving Logarithmic Equations and Inequalities
1. FISH The population of silver carp has
been growing in the Mississippi River.
About every 3 years, the population
doubles. Write logarithmic expression
that gives the number of years it will
take for the population to increase by
a factor of ten.
4. LOGARITHMS Pauline knows that
logb x = 3 and logb y = 5. She knows that
this is the same as knowing that b3 = x
and b5 = y. Multiply these two equations
together and rewrite it as an equation
involving logarithms. What is logb xy?
2. POWERS Haley tries to solve the
equation log4 2x = 5. She got the wrong
answer. What was her mistake? What
should the correct answer be?
5. MUSIC The first note on a piano
keyboard corresponds to a pitch with a
frequency of 27.5 cycles per second.
1.
log4 2x = 5
2.
2x = 45
3.
x = 25
4.
x = 32
With every successive note you go up the
white and black keys of a piano, the
12
pitch multiplies by a factor of √
2 . The
formula for the frequency of the pitch
sounded when the nth note up the
keyboard is played is given by
3. DIGITS A computer programmer wants
to write a formula that tells how many
digits there are in a number n, where
n is a positive integer. For example, if
n = 343, the formula should evaluate to
3 and if n = 10,000, the formula should
evaluate to 5. Suppose 8 ≤ log10 n < 9.
How many digits does n have?
Chapter 8
a. The pitch that orchestras tune to is
the A above middle C. It has a
frequency of 440 cycles per second.
How many notes up the piano
keyboard is this A?
b. Another pitch on the keyboard has
a frequency of 1760 cycles per second.
How many notes up the keyboard will
this be found?
128
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
f
27.5
n = 1 + 12 log2 − .
NAME
DATE
8-5
PERIOD
Study Guide
SCS
MBC.A.6.2
Properties of Logarithms
Properties of Logarithms Properties of exponents can be used to develop the
following properties of logarithms.
Product Property
of Logarithms
For all positive numbers a, b, and x, where x ≠ 1,
logx ab = logx a + logx b.
Quotient Property
of Logarithms
For all positive numbers a, b, and x, where x ≠ 1,
a
logx −
= logx a - logx b.
Power Property
of Logarithms
For any real number p and positive numbers m and b, where b ≠ 1,
logb m p = p logb m.
b
Example
Use log3 28 ≈ 3.0331 and log3 4 ≈ 1.2619 to approximate
the value of each expression.
b. log3 7
c. log3 256
a. log3 36
log3 36 = log3 (32 · 4)
= log3 32 + log3 4
= 2 + log3 4
≈ 2 + 1.2619
≈ 3.2619
(4)
28
log3 7 = log3 −
= log3 28 - log3 4
≈ 3.0331 - 1.2619
≈ 1.7712
log3 256 =
=
≈
≈
log3 (44)
4 · log3 4
4(1.2619)
5.0476
Use log12 3 ≈ 0.4421 and log12 7 ≈ 0.7831 to approximate the value of each expression.
1. log12 21
7
2. log12 −
3. log12 49
4. log12 36
5. log12 63
27
6. log12 −
81
7. log12 −
8. log12 16,807
9. log12 441
49
3
49
Use log5 3 ≈ 0.6826 and log5 4 ≈ 0.8614 to approximate the value of each expression.
10. log5 12
11. log5 100
12. log5 0.75
13. log5 144
27
14. log5 −
15. log5 375
−
16. log5 1.3
9
17. log5 −
81
18. log5 −
Chapter 8
16
16
5
129
North Carolina StudyText, Math BC, Volume 2
Lesson 8-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
NAME
DATE
8-5
Study Guide
PERIOD
SCS
(continued)
MBC.A.6.2
Properties of Logarithms
Solve Logarithmic Equations
You can use the properties of logarithms to solve
equations involving logarithms.
Solve each equation.
Example
a. 2 log3 x - log3 4 = log3 25
2 log3 x - log3 4 = log3 25
Original equation
log3 x2 - log3 4 = log3 25
log3
x2
−
4
x2
−
4
Power Property
= log3 25
Quotient Property
= 25
Property of Equality for Logarithmic Functions
x2 = 100
Multiply each side by 4.
x = ±10
Take the square root of each side.
Since logarithms are undefined for x < 0, -10 is an extraneous solution.
The only solution is 10.
b. log2 x + log2 (x + 2) = 3
log2 x + log2 (x + 2) = 3
Original equation
log2 x(x + 2) = 3
x(x + 2) = 2
Product Property
3
2
x + 2x = 8
2
x + 2x - 8 = 0
x = 2 or x = -4
Distributive Property
Subtract 8 from each side.
Factor.
Zero Product Property
Since logarithms are undefined for x < 0, -4 is an extraneous solution.
The only solution is 2.
Exercises
Solve each equation. Check your solutions.
1. log5 4 + log5 2x = log5 24
2. 3 log4 6 - log4 8 = log4 x
1
log6 25 + log6 x = log6 20
3. −
4. log2 4 - log2 (x + 3) = log2 8
5. log6 2x - log6 3 = log6 (x - 1)
6. 2 log4 (x + 1) = log4 (11 - x)
7. log2 x - 3 log2 5 = 2 log2 10
8. 3 log2 x - 2 log2 5x = 2
2
9. log3 (c + 3) - log3 (4c - 1) = log3 5
Chapter 8
10. log5 (x + 3) - log5 (2x - 1) = 2
130
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
(x + 4)(x - 2) = 0
Definition of logarithm
NAME
8-5
DATE
PERIOD
Practice
SCS
MBC.A.6.2
Properties of Logarithms
Use log10 5 ≈ 0.6990 and log10 7 ≈ 0.8451 to approximate the value of each expression.
1. log10 35
2. log10 25
7
3. log10 −
5
4. log10 −
5. log10 245
6. log10 175
7. log10 0.2
8. log10
5
7
25
−
7
Solve each equation. Check your solutions.
3
10. log10 u = −
log10 4
2
9. log7 n = −
log7 8
3
2
11. log6 x + log6 9 = log6 54
12. log8 48 - log8 w = log8 4
13. log9 (3u + 14) - log9 5 = log9 2u
14. 4 log2 x + log2 5 = log2 405
1
15. log3 y = -log3 16 + −
log3 64
16. log2 d = 5 log2 2 - log2 8
17. log10 (3m - 5) + log10 m = log10 2
18. log10 (b + 3) + log10 b = log10 4
19. log8 (t + 10) - log8 (t - 1) = log8 12
20. log3 (a + 3) + log3 (a + 2) = log3 6
21. log10 (r + 4) - log10 r = log10 (r + 1)
22. log4 (x2 - 4) - log4 (x + 2) = log4 1
23. log10 4 + log10 w = 2
24. log8 (n - 3) + log8 (n + 4) = 1
25. 3 log5 (x2 + 9) - 6 = 0
1
26. log16 (9x + 5) - log16 (x2 - 1) = −
27. log6 (2x - 5) + 1 = log6 (7x + 10)
28. log2 (5y + 2) - 1 = log2 (1 - 2y)
29. log10 (c2 - 1) - 2 = log10 (c + 1)
30. log7 x + 2 log7 x - log7 3 = log7 72
2
31. SOUND Recall that the loudness L of a sound in decibels is given by L = 10 log10 R,
where R is the sound’s relative intensity. If the intensity of a certain sound is tripled,
by how many decibels does the sound increase?
32. EARTHQUAKES An earthquake rated at 3.5 on the Richter scale is felt by many people,
and an earthquake rated at 4.5 may cause local damage. The Richter scale magnitude
reading m is given by m = log10 x, where x represents the amplitude of the seismic wave
causing ground motion. How many times greater is the amplitude of an earthquake that
measures 4.5 on the Richter scale than one that measures 3.5?
Chapter 8
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North Carolina StudyText, Math BC, Volume 2
Lesson 8-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3
NAME
DATE
8-5
PERIOD
Word Problem Practice
SCS
MBC.A.6.2
Properties of Logarithms
1. MENTAL COMPUTATION Jessica has
memorized log5 2 ≈ 0.4307 and
log5 3 ≈ 0.6826. Using this information,
to the nearest ten-thousandth, what
power of 5 is equal to 6?
5. SIZE Alicia wanted to try to quantify the
terms tiny, small, medium, large, big,
huge, and humongous. She picked a
number of objects and classified them
with these adjectives of size.
She noticed that the scale seemed
exponential. Therefore, she came up
with the following definition. Define S to
2. POWERS A chemist is testing a soft
drink. The pH of a solution is given by
1
be −
log3 V, where V is volume in cubic
3
feet. Then use the following table to find
the appropriate adjective.
-log10 C,
where C is the concentration of hydrogen
ions. The pH of a popular soft drink is
2.5. If the concentration of hydrogen ions
is increased by a factor of 100, what is
the new pH of the solution?
3. LUCKY MATH Frank is solving a
problem involving logarithms. He does
everything correctly except for one thing.
He mistakenly writes
-1 ≤ S < 0
tiny
small
0≤S<1
medium
1≤S<2
large
2≤S<3
big
3≤S<4
huge
4≤S<5
humongous
a. Derive an expression for S applied to
a cube in terms of where is the
side length of a cube.
However, after substituting the values
for a and b in his problem, he amazingly
still gets the right answer! The value
of a was 11. What must the value of b
have been?
b. How many cubes, each one foot on
a side, would have to be put together
to get an object that Alicia would
call “big”?
4. LENGTHS Charles has two poles. One
pole has length equal to log7 21 and
the other has length equal to log7 25.
Express the length of both poles joined
end to end as the logarithm of a
single number.
Chapter 8
-2 ≤ S < -1
Adjective
c. How likely is it that a large object
attached to a big object would result
in a huge object, according to
Alicia’s scale?
132
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
log2 a + log2 b = log2 (a + b).
S satisfies
NAME
DATE
8-6
PERIOD
Study Guide
SCS
MBC.A.6.2
Common Logarithms Base 10 logarithms are called common logarithms. The
expression log10 x is usually written without the subscript as log x. Use the LOG key on
your calculator to evaluate common logarithms.
The relation between exponents and logarithms gives the following identity.
Inverse Property of Logarithms and Exponents
Example 1
Use the
LOG
Evaluate log 50 to the nearest ten-thousandth.
key on your calculator. To four decimal places, log 50 = 1.6990.
Example 2
Solve 3 2x + 1 = 12.
32x + 1 = 12
Original equation
log 32x + 1 = log 12
Property of Equality for Logarithmic Functions
(2x + 1) log 3 = log 12
Power Property of Logarithms
log 12
log 3
log 12
2x = − - 1
log 3
1 log 12
−-1
x=−
2 log 3
2x + 1 = −
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
10log x = x
Divide each side by log 3.
Subtract 1 from each side.
(
)
(
)
1 1.0792
−-1
x≈−
2 0.4771
1
Multiply each side by −
.
2
Use a calculator.
x ≈ 0.6309
Exercises
Use a calculator to evaluate each expression to the nearest ten-thousandth.
1. log 18
2. log 39
3. log 120
4. log 5.8
5. log 42.3
6. log 0.003
Solve each equation or inequality. Round to the nearest ten-thousandth.
7. 43x = 12
8. 6x + 2 = 18
9. 54x - 2 = 120
10. 73x - 1 ≥ 21
11. 2.4x + 4 = 30
12. 6.52x ≥ 200
13. 3.64x - 1 = 85.4
14. 2x + 5 = 3x - 2
15. 93x = 45x + 2
16. 6x - 5 = 27x + 3
Chapter 8
133
North Carolina StudyText, Math BC, Volume 2
Lesson 8-6
Common Logarithms
NAME
DATE
8-6
Study Guide
PERIOD
SCS
(continued)
MBC.A.6.2
Common Logarithms
Change of Base Formula
The following formula is used to change expressions with
different logarithmic bases to common logarithm expressions.
Change of Base Formula
log n
log b a
b
For all positive numbers a, b, and n, where a ≠ 1 and b ≠ 1, loga n = −
.
Example
Express log8 15 in terms of common logarithms. Then round to the
nearest ten-thousandth.
log 15
log 10 8
10
log8 15 = −
Change of Base Formula
≈ 1.3023
Simplify.
The value of log8 15 is approximately 1.3023.
Exercises
Express each logarithm in terms of common logarithms. Then approximate its
value to the nearest ten-thousandth.
2. log2 40
3. log5 35
4. log4 22
5. log12 200
6. log2 50
7. log5 0.4
8. log3 2
9. log4 28.5
10. log3 (20)2
11. log6 (5)4
12. log8 (4)5
13. log5 (8)3
14. log2 (3.6)6
15. log12 (10.5)4
16. log3 √
150
3
17. log4 √
39
4
18. log5 √
1600
Chapter 8
134
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. log3 16
NAME
DATE
8-6
PERIOD
Practice
SCS
MBC.A.6.2
Use a calculator to evaluate each expression to the nearest ten-thousandth.
1. log 101
2. log 2.2
3. log 0.05
Use the formula pH = -log [H+] to find the pH of each substance given its
concentration of hydrogen ions. Round to the nearest tenth.
4. milk: [H+] = 2.51 × 10-7 mole per liter
5. acid rain: [H+] = 2.51 × 10-6 mole per liter
6. black coffee: [H+] = 1.0 × 10-5 mole per liter
7. milk of magnesia: [H+] = 3.16 × 10-11 mole per liter
Solve each equation or inequality. Round to the nearest ten-thousandth.
8. 2x < 25
9. 5a = 120
11. 9m ≥ 100
12. 3.5x = 47.9
13. 8.2 y = 64.5
14. 2b + 1 ≤ 7.31
15. 42x = 27
16. 2a - 4 = 82.1
17. 9z - 2 > 38
18. 5w + 3 = 17
19. 30x = 50
21. 42x = 9x + 1
22. 2n + 1 = 52n - 1
2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
20. 5x
-3
= 72
10. 6z = 45.6
2
Express each logarithm in terms of common logarithms. Then approximate its
value to the nearest ten-thousandth.
23. log5 12
24. log8 32
25. log11 9
26. log2 18
27. log9 6
28. log7 √
8
29. HORTICULTURE Siberian irises flourish when the concentration of hydrogen ions [H+]
in the soil is not less than 1.58 × 10-8 mole per liter. What is the pH of the soil in which
these irises will flourish?
30. ACIDITY The pH of vinegar is 2.9 and the pH of milk is 6.6. Approximately how many
times greater is the hydrogen ion concentration of vinegar than of milk?
31. BIOLOGY There are initially 1000 bacteria in a culture. The number of bacteria doubles
each hour. The number of bacteria N present after t hours is N = 1000(2) t. How long will
it take the culture to increase to 50,000 bacteria?
32. SOUND An equation for loudness L in decibels is given by L = 10 log R, where R is the
sound’s relative intensity. An air-raid siren can reach 150 decibels and jet engine noise
can reach 120 decibels. How many times greater is the relative intensity of the air-raid
siren than that of the jet engine noise?
Chapter 8
135
North Carolina StudyText, Math BC, Volume 2
Lesson 8-6
Common Logarithms
NAME
8-6
DATE
PERIOD
Word Problem Practice
SCS
MBC.A.6.2
Common Logarithms
1. OTHER BASES Jamie needs to figure
out what log2 3 is, but she only has
a table of common logarithms. In the
table, she finds that log10 2 ≈ 0.3010
and log10 3 ≈ 0.4771. Using this
information, to the nearest thousandth,
what is log2 3?
4. SCIENTIFIC NOTATION When a
number n is written in scientific
notation, it has the form n = s × 10 p,
where s is a number greater than or
equal to 1 and less than 10 and p is an
integer. Show that p ≤ log10 n < p + 1.
2. pH The pH of a solution is given by
5. LOG TABLE Marjorie is looking through
some old science books owned by her
grandfather. At the back of one of them,
there is a table of logarithms base 10.
However, the book is worn out and some
of the entries are unreadable.
-log10 C,
where C is the concentration of
hydrogen ions in moles per liter. A
solution of baking soda creates a
hydrogen ion concentration 5 × 10-9
of mole per liter. What is the pH of a
solution of baking soda? Round your
answer to the nearest tenth.
Table of Common Logarithms
(4 decimal places of accuracy)
log10 x
2
0.3010
3
0.4771
4
?
5
0.6989
6
?
a. Approximately what are the missing
entries in the table? Round off your
answers to the nearest thousandth.
3. GRAPHING The graph of y = log10 x
is shown below. Use the fact that
1
−
≈ 3.32 to sketch a graph of
log 10 2
y = log2 x on the same graph.
6
y
b. How can you use this table to
determine log10 1.5?
5
4
3
2
1
O 20 40 60 80 100 120x
Chapter 8
136
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
x
NAME
8-7
DATE
PERIOD
Study Guide
SCS
MBC.A.6.2
Base e and Natural Logarithms
Base e and Natural Logarithms
The irrational number e ≈ 2.71828… often occurs
as the base for exponential and logarithmic functions that describe real-world phenomena.
Natural Base e
(
1
As n increases, 1+ −
n
ln x = log e x
)
n
approaches e ≈ 2.71828….
Inverse Property of Base e and Natural Logarithms
e ln x = x
ln e x = x
Natural base expressions can be evaluated using the ex and ln keys on your calculator.
Example 1
Write a logarithmic equation equivalent to e 2x = 7.
e2x = 7 → loge 7 = 2x
2x = ln 7
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Example 2
Write each logarithmic equation in exponential form.
a. ln x ≈ 0.3345
ln x ≈ 0.3345 → loge x ≈ 0.3345
x ≈ e0.3345
b. ln 42 = x
ln 42 = x → loge 42 = x
42 = ex
Exercises
Write an equivalent exponential or logarithmic equation.
1. e15 = x
2. e3x = 45
3. ln 20 = x
4. ln x = 8
5. e-5x = 0.2
6. ln (4x) = 9.6
7. e8.2 = 10x
8. ln 0.0002 = x
Evaluate each logarithm to the nearest ten-thousandth.
9. ln 12,492
10. ln 50.69
13. ln 943 - ln 181 14. ln 67 + ln 103
Chapter 8
11. ln 9275
12. ln 0.835
15. ln 931 ln 32
16. ln (139 - 45)
137
North Carolina StudyText, Math BC, Volume 2
Lesson 8-7
The functions f (x) = ex and f ( x) = ln x are inverse functions.
NAME
DATE
8-7
Study Guide
PERIOD
SCS
(continued)
MBC.A.6.2
Base e and Natural Logarithms
Equations and Inequalities with e and ln All properties of logarithms from
earlier lessons can be used to solve equations and inequalities with natural logarithms.
Example
Solve each equation or inequality.
a. 3e2x + 2 = 10
3e2x + 2 = 10
3e2x = 8
8
e2x = −
3
8
ln e2x = ln −
3
8
2x = ln −
3
8
1
x=−
ln −
2
3
x ≈ 0.4904
Original equation
Subtract 2 from each side.
Divide each side by 3.
Property of Equality for Logarithms
Inverse Property of Exponents and Logarithms
1
Multiply each side by −
.
2
Use a calculator.
b. ln (4x - 1) < 2
ln (4x - 1) < 2
eln (4x - 1) < e2
0 < 4x - 1 < e2
Original inequality
Write each side using exponents and base e.
Inverse Property of Exponents and Logarithms
Addition Property of Inequalities
1
1 2
−
<x<−
(e + 1)
4
4
Multiplication Property of Inequalities
0.25 < x < 2.0973
Use a calculator.
Exercises
Solve each equation or inequality. Round to the nearest ten-thousandth.
1. e4x = 120
2. ex ≤ 25
3. ex - 2 + 4 = 21
4. ln 6x ≥ 4
5. ln (x + 3) - 5 = -2
6. e-8x ≤ 50
7. e4x - 1 - 3 = 12
8. ln (5x + 3) = 3.6
9. 2e3x + 5 = 2
10. 6 + 3ex + 1 = 21
Chapter 8
11. ln (2x - 5) = 8
138
12. ln 5x + ln 3x > 9
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1 < 4x < e2 + 1
NAME
8-7
DATE
PERIOD
Practice
SCS
MBC.A.6.2
Base e and Natural Logarithms
1. ln 50 = x
2. ln 36 = 2x
3. ln 6 ≈ 1.7918
4. ln 9.3 ≈ 2.2300
5. ex = 8
6. e5 = 10x
7. e-x = 4
8. e2 = x + 1
Solve each equation or inequality. Round to four decimal places.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
9. ex < 9
10. e-x = 31
11. ex = 1.1
12. ex = 5.8
13. 2ex - 3 = 1
14. 5ex + 1 ≥ 7
15. 4 + ex = 19
16. -3ex + 10 < 8
17. e3x = 8
18. e-4x = 5
19. e0.5x = 6
20. 2e5x = 24
21. e2x + 1 = 55
22. e3x - 5 = 32
23. 9 + e2x = 10
24. e-3x + 7 ≥ 15
25. ln 4x = 3
26. ln (-2x) = 7
27. ln 2.5x = 10
28. ln (x - 6) = 1
29. ln (x + 2) = 3
30. ln (x + 3) = 5
31. ln 3x + ln 2x = 9
32. ln 5x + ln x = 7
33. INVESTING Sarita deposits $1000 in an account paying 3.4% annual interest
compounded continuously. Use the formula for continuously compounded interest,
A = Pert, where P is the principal, r is the annual interest rate, and t is the time
in years.
a. What is the balance in Sarita’s account after 5 years?
b. How long will it take the balance in Sarita’s account to reach $2000?
34. RADIOACTIVE DECAY The amount of a radioactive substance y that remains after
t years is given by the equation y = aekt, where a is the initial amount present and k is
the decay constant for the radioactive substance. If a = 100, y = 50, and k = -0.035,
find t.
Chapter 8
139
North Carolina StudyText, Math BC, Volume 2
Lesson 8-7
Write an equivalent exponential or logarithmic equation.
NAME
8-7
DATE
Word Problem Practice
PERIOD
SCS
MBC.A.6.2
Base e and Natural Logarithms
1. INTEREST Horatio opens a bank account
that pays 2.3% annual interest
compounded continuously. He makes an
initial deposit of 10,000. What will be the
balance of the account in 10 years?
Assume that he makes no additional
deposits and no withdrawals.
a. If Linda can invest the money for
5 years only, which account would
give her the higher return on her
investment? How much more money
would she make by choosing the
higher paying account?
2. INTEREST Janie’s bank pays 2.8%
annual interest compounded
continuously on savings accounts.
She placed $2000 in the account. How
long will it take for her initial deposit
to double in value? Assume that she
makes no additional deposits and no
withdrawals. Round your answer to the
nearest quarter year.
b. If Linda can invest the money for
10 years only, which account would
give her the higher return on her
investment? How much more money
would she make by choosing the
higher paying account?
c. If Linda can invest the money for
20 years only, which account would
give her the higher return on her
investment? How much more money
would she make by choosing the
higher paying account?
4. BACTERIA A bacterial population grows
exponentially, doubling every 72 hours.
x
2x
4x
8x
time
0
72
144
216
Let P be the initial population size and
let t be time in hours. Use the equation
from Exercise 3 to write a formula using
the natural base exponential function
that gives the size of the population y as
a function of P and t.
Chapter 8
140
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3. POPULATION The equation A = A0ert
describes the growth of the world’s
population where A is the population
at time t, A0 is the population at t = 0,
and r is the annual growth rate. The
world’s population at the start of 2008
was estimated at 6,641,000,000. If the
annual growth rate is 1.2%, when will
the world population reach 9 billion?
bacteria
5. MONEY MANAGEMENT Linda wants
to invest $20,000. She is looking at two
possible accounts. Account A is a
standard savings account that pays 3.4%
annual interest compounded
continuously. Account B would pay her
a fixed amount of $200 every quarter.
NAME
DATE
8-8
PERIOD
Study Guide
SCS
MBC.A.7.4
Using Exponential and Logarithmic Functions
Exponential Growth and Decay
Exponential
Growth
f(x) = aekt where a is the initial value of y, t is time in years, and k is a constant
representing the rate of continuous growth.
Exponential
Decay
f(x) = ae–kt where a is the initial value of y, t is time in years, and k is a constant
representing the rate of continuous decay.
Example
POPULATION In 2000, the world population was estimated to be
6.124 billion people. In 2005, it was 6.515 billion.
a. Determine the value of k, the world’s relative rate of growth.
y = ae kt
6.515
−
= e5k
6.124
y = 6.515, a = 6.124, and t = 2005 - 2000 or 5
Divide each side by 6.124.
6.515
ln −
= ln e5k
Property of Equality for Logarithmic Functions
6.515
ln −
= 5k
ln ex = x
0.01238 ≈ k
Divide each side by 5 and use a calculator.
6.124
6.124
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The world’s relative rate of growth is about 0.01238 or 1.2%
b. When will the world’s population reach 7.5 billion people?
7.5 ≈ 6.124e 0.01238t
7.5
−
≈ e 0.01238t
6.124
7.5
ln −
≈ ln e 0.01238t
6.124
y = 7.5, a = 6.124, and k = 0.01238
Divide each side by 6.124.
Property of Equality for Logarithmic Functions
7.5
ln −
≈ 0.01238t
ln ex = x
16.3722 ≈ t
Divide each side by 0.01238 and use a calculator.
6.124
The world’s population will reach 7.5 billion in 2016.
Exercise
1. CARBON DATING Use the formula y = ae-0.00012t, where a is the initial amount of
carbon 14, t is the number of years ago the animal lived, and y is the remaining
amount after t years.
a. How old is a fossil that has lost 95% of its Carbon-14?
b. How old is a skeleton that has 95% of its Carbon-14 remaining?
Chapter 8
141
North Carolina StudyText, Math BC, Volume 2
Lesson 8-8
6.515 = 6.124e
Formula for continuous growth
k(5)
NAME
DATE
8-8
Study Guide
PERIOD
SCS
(continued)
MBC.A.7.4
Using Exponential and Logarithmic Functions
Logistic Growth
A logistic function models the S-curve of growth of some set λ. The
initial stage of growth is approximately exponential; then, as saturation begins, the growth
slows, and at some point, growth stops.
Example
The population of a certain species of fish in a lake after t years is
1880
given by P(t) = −
.
-0.037t
1 + 1.42e
a. Graph the function.
2500
P(t)
2250
2000
1750
1500
1250
1000
750
500
250
0
t
50
100
150
200
250
b. Find the horizontal asymptote. What does it represent in the situation?
The horizontal asymptote is P(t) = 1880. The population of fish will reach a
ceiling of 1880.
c. When will the population reach 1875?
1880
1875 = −
-0.037t
Replace P(t) with 1875.
1875(1 + 1.42e-0.037t) = 1880
Multiply each side by (1 + 1.42e-0.037t).
2662.5e-0.037t = 5
Simplify and subtract 1875 from each side.
5
e
=−
2662.5
5
-0.037t = ln −
2662.5
5
t = ln −
÷ (-0.037)
2662.5
-0.037t
(
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1 + 1.42e
)
Divide each side by 2662.5.
Take the natural logarithm of each side.
Divide each side by -0.037.
t ≈ 169.66
Use a calculator.
The population will reach 1875 in about 170 years.
Exercise
1. Assume the population of gnats in a specific habitat follows the function
17,000
(1 + 15e
.
P(t) = −
-0.0082t
)
14,000
12,000
10,000
8000
6000
4000
2000
a. Graph the function for t ≥ 0.
b. What is the horizontal asymptote?
c. What is the maximum population?
P (t)
t
0
50
150
250
350
450
550
d. When does the population reach 15,000?
Chapter 8
142
North Carolina StudyText, Math BC, Volume 2
NAME
8-8
DATE
Practice
PERIOD
SCS
MBC.A.7.4
Using Exponential and Logarithmic Functions
1. BACTERIA How many hours will it take a culture of bacteria to increase from 20 to
2000? Use k = 0.614.
2. RADIOACTIVE DECAY A radioactive substance has a half-life of 32 years. Find the
constant k in the decay formula for the substance.
3. RADIOACTIVE DECAY Cobalt, an element used to make alloys, has several isotopes.
One of these, cobalt 60, is radioactive and has a half-life of 5.7 years. Cobalt 60 is used
to trace the path of nonradioactive substances in a system. What is the value of k for
cobalt 60?
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5. POPULATION The population of rabbits in an area is modeled by the growth equation
P(t) = 8e0.26t, where P is in thousands and t is in years. How long will it take for the
population to reach 25,000?
6. RADIOACTIVE DECAY A radioactive element decays exponentially. The decay model is
given by the formula A = A0e-0.04463t. A is the amount present after t days and A0 is the
amount present initially. Assume you are starting with 50g. How much of the element
remains after 10 days? 30 days?
7. POPULATION A population is growing continuously at a rate of 3%. If the population is
now 5 million, what will it be in 17 years’ time?
8. BACTERIA A certain bacteria is growing exponentially according to the model y = 80ekt.
Using k = 0.071, find how many hours it will take for the bacteria reach a population of
10,000 cells?
9. LOGISTIC GROWTH The population of a certain habitat follows the function
16,300
.
P(t) = −−
(1 + 17.5e-0.065t)
a. What is the maximum population?
b. When does the population reach 16,200?
Chapter 8
143
North Carolina StudyText, Math BC, Volume 2
Lesson 8-8
4. WHALES Modern whales appeared 5-10 million years ago. The vertebrae of a whale
discovered by paleontologists contain roughly 0.25% as much carbon-14 as they would
have contained when the whale was alive. How long ago did the whale die? Use
k = 0.00012.
NAME
8-8
DATE
PERIOD
Word Problem Practice
SCS
MBC.A.7.4
Using Exponential and Logarithmic Functions
1. PROGRAMMING For reasons having
to do with speed, a computer
programmer wishes to model population
size using a natural base exponential
function. However, the programmer
is told that the users of the program
will be thinking in terms of the annual
percentage increase. Let r be the
percentage that the population increases
each year. Find the value for k in terms
of r so that
e k = 1 + r.
5. CONSUMER AWARENESS Jason wants
to buy a brand new high-definition (HD)
television. He could buy one now because
he has $7000 to spend, but he thinks
that if he waits, the quality of HD
televisions will improve. His $7000 earns
2.5% interest annually compounded
continuously. The television he wants to
buy costs $5000 now, but the cost
increases each year by 7%.
a. Write a natural base exponential
function that gives the value of
Jason’s account as a function of time t.
2. CARBON DATING Archeologists
uncover an ancient wooden tool. They
analyze the tool and find that it has
22% as much carbon 14 compared to
the likely amount that it contained when
it was made. Given that the half-life of
carbon 14 is about 5730 years, about how
old is the artifact? Round your answer to
the nearest 100 years.
c. In how many years will the cost of the
television exceed the value of the
money in Jason’s account? In other
words, how much time does Jason
have to decide whether he wants to
buy the television? Round your
answer to the nearest tenth of a year.
6. LOGISTIC GROWTH The population of a
bacteria can be modeled by
4. POPULATION Louisa read that the
population of her town has increased
steadily each year. Today, the population
of her town has grown to 68,735. One
year ago, the population was 67,387.
Based on this information, what was the
population of her town 100 years ago?
22,000
1 + 1.2e
P(t) = −
where t is time in
-kt
hours and k is a constant.
a. After 1 hour the bacteria population is
10,532, what is the value of k?
b. When does the population reach
21,900?
Chapter 8
144
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3. POPULATION The doubling time of
a population is d years. The population
size y can be modeled by an exponential
equation of the form y = aekt, where a is
the initial population size and t is time.
What is k in terms of d?
b. Write a natural base exponential
function that gives the cost of the
television Jason wants as a function
of time t.
NAME
DATE
9-1
PERIOD
Study Guide
SCS
MBC.A.4.1, MBC.A.4.2
Multiplying and Dividing Rational Expressions
Simplify Rational Expressions
A ratio of two polynomial expressions is a rational
expression. To simplify a rational expression, divide both the numerator and the
denominator by their greatest common factor (GCF).
Multiplying Rational Expressions
c
c a .−
a
ac
For all rational expressions −
and −
,−
=−
, if b ≠ 0 and d ≠ 0.
Dividing Rational Expressions
c a
c
ad
a
For all rational expressions −
and −
,−÷−
=−
, if b ≠ 0, c ≠ 0, and d ≠ 0.
d b
b
d
bd
d
bc
Lesson 9-1
Example
d b
b
Simplify each expression.
24a 5b 2
a. −
4
(2ab)
1
1
1
1
1 1
1
1
1
1
1
1 1
1
1
1
24a 5b 2
2 · 2 · 2 · 3 · a · a ·a · a · a · b · b
3a
−
= −−
= −
2 · 2 · 2 · 2 · a · a ·a · a · b · b · b · b
2b 2
(2ab) 4
1
1
3
3r 2n · 20t 2
−
b. −
4
3
5t
9r n
1
1
1
1
1
1
1
3r 2n 3 · 20t 2
· r · r · n · n · n · 2 · 2 · 5 · t · t
· 2 · n · n
4n 2
−
−
= 3−−
= 2−
= −
4
3
5·t·t·t·t·3·3·r·r·r·n
3·r·t·t
5t
9r n
3rt 2
1
2
1
1
1
1
1
1
2
x + 8x + 16
2x - 2
x + 2x - 8
x-1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
c. − ÷ −
x 2 + 8x + 16
x 2 + 2x - 8
x 2 + 8x + 16
x-1
−÷ − = − · −
2x - 2
x-1
2x - 2
x 2 + 2x - 8
1
=
1
(x + 4) (x + 4) (x - 1)
−−
2(x - 1) (x - 2) (x + 4)
1
x+4
2(x - 2)
= −
1
Exercises
Simplify each expression.
(-2ab 2) 3
20ab
3m 3 - 3m
4m 5
4. −
· −
4
6m
x2 + x - 6
x - 6x - 27
4x2 - 12x + 9
9 - 6x
2. −
1. −
4
m+1
3. −
2
c 2 - 3c
5. −
2
c - 25
·
c 2 + 4c - 5
−
c 2 - 4c + 3
4
(m - 3) 2
m 3 - 9m
· −
m - 6m + 9
m2 - 9
6xy
18xz 2
7. −3 ÷ −
16p 2 - 8p + 1
14p
2m - 1
4m 2 - 1
9. −
÷ −
2
6. −
2
4p 2 + 7p - 2
7p
8. −
÷ −
4
5
Chapter 9
25z
5y
m - 3m - 10
145
4m + 8
North Carolina StudyText, Math BC, Volume 2
NAME
DATE
9-1
Study Guide
PERIOD
SCS
(continued)
MBC.A.4.1, MBC.A.4.2
Multiplying and Dividing Rational Expressions
Simplify Complex Fractions A complex fraction is a rational expression with a
numerator and/or denominator that is also a rational expression. To simplify a complex
fraction, first rewrite it as a division problem.
3n - 1
−
Example
n
Simplify −
.
2
3n + 8n - 3
−
n4
3n - 1
−
n
3n 2 + 8n - 3
3n - 1
−
−
= −
÷
2
n
3n + 8n - 3
n4
Express as a division problem.
−
4
n
3n - 1
=−
n
4
·
1
=
n
−
2
Multiply by the reciprocal of the divisor.
3n + 8n - 3
n3
(3n - 1) n4
−
n(3n - 1)(n + 3)
Factor and eliminate.
1
3
n
= −
Simplify.
n+3
Exercises
Simplify each expression.
2
3
a bc
−
2 2
2
b -1
−
xy
3b + 2
b+1
−
3b 2 - b - 2
2. −
2
3. −
ab
−
c 4x 2y
2
4.
b - 100
−
2
b
−
3b 2 - 31b + 10
−
2b
x-4
−
2
x + 6x + 9
x - 2x - 8
−
3+x
5. −
2
7.
2x 2 + 9x + 9
−
x+1
−
10x 2 + 19x + 6
−
5x 2 + 7x + 2
9.
x2 - x - 2
−
x2 + x - 6
−
x+1
−
x+3
2
a - 16
−
a+2
a + 3a - 4
−
a2 + a - 2
6. −
2
8.
b+2
b - 6b + 8
−
b2 + b - 2
−
b 2 - 16
−
2
Chapter 9
146
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1.
x 3y 2z
ab
−
a 3x 2y
−
b2
−
2 2
NAME
DATE
9-1
PERIOD
Practice
SCS
MBC.A.4.1, MBC.A.4.2
Multiplying and Dividing Rational Expressions
Simplify each expression.
2. −
5 4
27a b c
3. −
2
k 2 - k - 15
4. 2−
2
k -9
25 - v
5. −
2
2
x 4 + x 3 - 2x 2
x -x
25x
· −
7. −
5
2 2
a+y
n
n - 6n
· −
9. −
8
3v - 13v - 10
-2u 3y
15xz
6. −
4
3
a-y
2
2
10x - 2
( )
2xy
w
3
14. −
2
n
6x + 2x
25x - 1
−
2
x - 10x + 25
2
·
5x
−
8-x
5 3
ay
a 3w 2
13. −7 ÷ −
5 2
wy
wy
x2 - y2
3
x+y
6
24x 2
÷ −
5
3x + 6
x -9
2
2
- 5x - 24
11. x−
2
2
·
14u y
n-6
w -n
· −
10. −
y-a
w+n
x-5
12. −
3
5
4
8. − · −
y+a
6
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
10y 2 + 15y
35y - 5y
(2m 3n 2) 3
-18m n
3
Lesson 9-1
2
9a b
1. −
4 4
15. − ÷ −
w
6x 2 + 12x
4x + 12
16. −
÷−
2
9 - a2
2a - 6
18. −
÷−
2
5a + 10
a + 5a + 6
s 2 - 10s + 25
s 2 - 7s - 15
17. 2−
÷ −
2
2x + 1
−
x
19. −
4-x
−
x
2
x -9
−
4
20. −
21.
3-x
−
s+4
(s + 4)
8
x3 + 23
x - 2x
−
(x + 2) 3
−
x 2 + 4x + 4
−
2
22. GEOMETRY A right triangle with an area of x2 - 4 square units has a leg that
measures 2x + 4 units. Determine the length of the other leg of the triangle.
x 2 + 3x - 10
2x
23. GEOMETRY A rectangular pyramid has a base area of − square centimeters
2
x - 3x
centimeters. Write a rational expression to describe the
and a height of −
2
x - 5x + 6
volume of the rectangular pyramid.
Chapter 9
147
North Carolina StudyText, Math BC, Volume 2
NAME
9-1
DATE
Word Problem Practice
PERIOD
SCS
MBC.A.4.1, MBC.A.4.2
Multiplying and Dividing Rational Expressions
1. JELLY BEANS A large vat contains G
green jelly beans and R red jelly beans.
A bag of 100 red and 100 green jelly
beans is added to the vat. What is the
new ratio of red to green jelly beans in
the vat?
4. OIL SLICKS David was moving a drum of
oil around his circular outdoor pool when
the drum cracked, and oil spilled into the
pool. The oil spread itself evenly over the
surface of the pool. Let V denote the
volume of oil spilled and let r be the
radius of the pool. Write an equation for
the thickness of the oil layer.
2. MILEAGE Beth drives a hybrid car that
gets 45 miles per gallon in the city and
48 miles per gallon on the highway. Beth
uses C gallons of gas in the city and H
gallons of gas on the highway. Write an
expression for the average number of
miles per gallon that Beth gets with her
car in terms of C and H.
5. RUNNING Harold runs to the local food
mart to buy a gallon of soy milk. Because
he is weighed down on his return trip, he
runs slower on the way back. He travels
S1 feet per second on the way to the food
mart and S2 feet per second on the way
back. Let d be the distance he has to run
to get to the food mart. Remember:
distance = rate × time.
b. What speed would Harold have to run
if he wanted to maintain a constant
speed for the entire trip yet take the
same amount of time running?
x
What is the height of the triangle in
terms of x?
Chapter 9
148
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3. HEIGHT The front face of a Nordic house
is triangular. The surface area
of the face is x2 + 3x + 10 where x is
the base of the triangle.
a. Write an equation that gives the total
time Harold spent running for this
errand.
NAME
DATE
9-2
Study Guide
PERIOD
SCS
MBC.A.4.1, MBC.A.4.2
Adding and Subtracting Rational Expressions
LCM of Polynomials
To find the least common multiple of two or more polynomials,
factor each expression. The LCM contains each factor the greatest number of times it
appears as a factor.
Example 1
Example 2
Find the LCM of 16p2q3r,
40pq4r2, and 15p3r4.
Find the LCM of
3m - 3m - 6 and 4m2 + 12m - 40.
16p2q3r
40pq4r2
15p3r4
LCM
3m2 - 3m - 6 = 3(m + 1)(m - 2)
4m2 + 12m - 40 = 4(m - 2)(m + 5)
LCM = 12(m + 1)(m - 2)(m + 5)
2
= 24 · p2 · q3 · r
= 23 · 5 · p · q4 · r2
= 3 · 5 · p3 · r4
= 24 · 3 · 5 · p3 · q4 · r4
= 240p3q4r4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find the LCM of each set of polynomials.
1. 14ab2, 42bc3, 18a2c
2. 8cdf 3, 28c2f, 35d4f 2
3. 65x 4y, 10x2y2, 26y4
4. 11mn5, 18m2n3, 20mn4
5. 15a4b, 50a2b2, 40b8
6. 24p7q, 30p2q2, 45pq3
7. 39b2c2, 52b4c, 12c3
8. 12xy4, 42x2y, 30x2y3
9. 56stv2, 24s2v2, 70t3v3
10. x2 + 3x, 10x2 + 25x - 15
11. 9x2 - 12x + 4, 3x2 + 10x - 8
12. 22x2 + 66x - 220, 4x2 - 16
13. 8x2 - 36x - 20, 2x2 + 2x - 60
14. 5x2 - 125, 5x2 + 24x - 5
15. 3x2 - 18x + 27, 2x3 - 4x2 - 6x
16. 45x2 - 6x - 3, 45x2 - 5
17. x3 + 4x2 - x - 4, x2 + 2x - 3
18. 54x3 - 24x, 12x2 - 26x + 12
Chapter 9
149
North Carolina StudyText, Math BC, Volume 2
Lesson 9-2
Exercises
NAME
DATE
9-2
Study Guide
PERIOD
SCS
(continued)
MBC.A.4.1, MBC.A.4.2
Adding and Subtracting Rational Expressions
Add and Subtract Rational Expressions
To add or subtract rational expressions,
follow these steps.
Step
Step
Step
Step
Step
1
2
3
4
5
Find the least common denominator (LCD). Rewrite each expression with the LCD.
Add or subtract the numerators.
Combine any like terms in the numerator.
Factor if possible.
Simplify if possible.
6
2
Simplify −
- −
.
2
2
Example
2x + 2x - 12
x -4
6
2
−
- −
2
2
2x + 2x - 12
x -4
6
2
- −
= −
2(x + 3)(x - 2)
Factor the denominators.
(x - 2)(x + 2)
6(x + 2)
2(x + 3)(x - 2)(x + 2)
2 · 2(x + 3)
2(x + 3)(x - 2)(x + 2)
= −− - −−
The LCD is 2(x + 3)(x - 2)(x + 2).
Subtract the numerators.
= −−
6x + 12 - 4x - 12
2(x + 3)(x - 2)(x + 2)
Distribute.
2x
= −−
Combine like terms.
x
= −−
Simplify.
2(x + 3)(x - 2)(x + 2)
(x + 3)(x - 2)(x + 2)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
6(x + 2) - 4(x + 3)
2(x + 3)(x - 2)(x + 2)
= −−
Exercises
Simplify each expression.
-7xy
3x
4y 2
2y
1. − + −
2
1
2. −
- −
4a
15b
- −
3. −
3
4. −
+ −
3bc
x-3
5ac
x+2
3x + 3
x-1
+ −
5. −
2
2
x + 2x + 1
Chapter 9
x -1
x-1
4x + 5
3x + 6
5x
4
6. −
- −
2
2
4x - 4x + 1
150
20x - 5
North Carolina StudyText, Math BC, Volume 2
NAME
DATE
9-2
Practice
PERIOD
SCS
MBC.A.4.1, MBC.A.4.2
Adding and Subtracting Rational Expressions
Find the LCM of each set of polynomials.
1. x2y, xy3
2. a2b3c, abc4
3. x + 1, x + 3
4. g - 1, g2 + 3g - 4
5. 2r + 2, r2 + r, r + 1
6. 3, 4w + 2, 4w2 - 1
7. x2 + 2x - 8, x + 4
8. x2 - x - 6, x2 + 6x + 8
9. d2 + 6d + 9, 2(d2 - 9)
Simplify each expression.
5
7
10. −
-−
5
1
11. −
-−
4
2 3
3
1
12. −
+−
2
3
4m
13. −
+2
x-8
14. 2x - 5 - −
9
4
15. −
+−
16
2
+−
16. −
2
2 - 5m
4m - 5
17. −
+−
18. −
+−
2
2
5
20
- −
19. −
2
5
20. −
-−
2
2
8a
12x y
3mn
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
x - 16
2x - 12
5x y
6c d
x+4
x+4
x - 4x - 12
2a
2a
36
-−
+ −
22. −
a-3
a+3
a2 - 9
m-9
9-m
2p - 3
p - 5p + 6
p -9
4cd
a-3
a-5
y-5
y - 3y - 10
y
y +y-2
3
7
1
21. −
-−
+−
5n
4
10n
r+6
2
1
−
x-y +−
x+y
1
−
r -−
r+2
r + 4r + 3
−
r 2 + 2r
23. −
24. −
2
1
−
x-y
5x 20
10
25. GEOMETRY The expressions −
, − , and −
represent the lengths of the sides of a
2
x+4
x-4
triangle. Write a simplified expression for the perimeter of the triangle.
26. KAYAKING Mai is kayaking on a river that has a current of 2 miles per hour. If r
represents her rate in calm water, then r + 2 represents her rate with the current, and
r - 2 represents her rate against the current. Mai kayaks 2 miles downstream and then
d
back to her starting point. Use the formula for time, t = −
r , where d is the distance, to
write a simplified expression for the total time it takes Mai to complete the trip.
Chapter 9
151
North Carolina StudyText, Math BC, Volume 2
Lesson 9-2
6ab
NAME
DATE
9-2
PERIOD
Word Problem Practice
SCS
MBC.A.4.1, MBC.A.4.2
Adding and Subtracting Rational Expressions
1. SQUARES Susan’s favorite perfect
square is s2 and Travis’ is t2, where s and
t are whole numbers. What perfect
square is guaranteed to be divisible by
both Susan’s and Travis’ favorite perfect
squares regardless of their specific value?
4. FRACTIONS In the seventeenth century,
Lord Brouncker wrote down a most
peculiar mathematical equation:
4
−
π =1+
12
32
2+
2+
52
2
2 + .7
..
This is an example of a continued
fraction. Simplify the continued fraction
1
n+−
.
2. ELECTRIC POTENTIAL The electrical
potential function between two electrons
is given by a formula that has the form
1
1
−
+−
. Simplify this expression.
r
1
n+−
n
1-r
5. RELAY RACE Mark, Connell, Zack, and
Moses run the 400 meter relay together.
Each of them runs 100 meters. Their
average speeds were s, s + 0.5, s - 0.5,
and s - 1 meters per second,
respectively.
3. TRAPEZOIDS The cross section of a
stand consists of two trapezoids stacked
one on top of the other.
a. What were their individual times for
their own legs of the race?
x
x+2
The total area of the cross section is x2
square units. Assuming the trapezoids
have the same height, write an
expression for the height of the stand in
terms of x. Put your answer in simplest
form. (Recall that the area of a trapezoid
with height h and bases b1 and b2 is
1
given by −
h(b1+ b2).)
2
Chapter 9
152
b. Write an expression for their time as
a team. Write your answer as a ratio
of two polynomials.
c. The world record for the 100 meter
relay is 37.4 seconds. What will s
equal if the team ties the world
record?
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
x+4
NAME
DATE
9-3
Study Guide
SCS
PERIOD
MBC.A.7.3, MBC.A.8.1, MBC.A.8.2, MBC.A.8.3,
MBC.A.8.4, MBC.A.8.5, MBC.A.10.1, MBC.A.10.2
Graphing Reciprocal Functions
Vertical and Horizontal Asymptotes
Parent Function of Reciprocal Functions
Parent Function
1
y= −
x
Type of Graph
hyperbola
Domain
all nonzero real numbers
Range
all nonzero real numbers
Symmetry
over the x- and y-axes
Intercepts
none
Asymptotes
the x- and y-axes
Example
Identify the asymptotes, domain, and range of the function
3
.
f (x) = −
x+2
4
Identify x values for which f(x) is undefined.
2
x + 2 = 0, so x = -2. f(x) is not defined when x = -2,
so there is an asymptote at x = -2.
-4
From x = -2, as x-values decrease, f(x) approaches 0.
As x-values increase, f(x) approaches 0. So there is an
asymptote at f(x) = 0.
-2
0
2
4x
-2
-4
Lesson 9-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
f(x)
The domain is all real numbers not equal to -2, and
the range is all real numbers not equal to 0.
Exercises
Identify the asymptotes, domain, and range of each function.
-3
2. f (x) = −
1
1. f (x) = −
x
4
3. f(x) = −
+2
x-1
4
f (x)
4
2
x+1
f (x)
4
f (x)
2
2
x
-4
Chapter 9
-2
0
2
4x
-4
0
-2
2
4
-4
-2
0
-2
-2
-2
-4
-4
-4
153
2
4x
North Carolina StudyText, Math BC, Volume 2
NAME
DATE
9-3
Study Guide
SCS
(continued)
Graphing Reciprocal Functions
PERIOD
MBC.A.7.3, MBC.A.8.1, MBC.A.8.2,
MBC.A.8.3, MBC.A.8.4, MBC.A.8.5,
MBC.A.10.1, MBC.A.10.2
Transformations of Reciprocal Functions
Equation Form
a
f(x) = −
+k
Horizontal Translation
The vertical asymptote moves to x = h.
Vertical Translation
The horizontal asymptote moves to y = k.
Reflection
The graph is reflected across the x-axis when a < 0.
Compression and Expansion
The graph is compressed vertically when |a| < 1 and expanded
vertically when |a| > 1.
x-h
-1
−
Example
2
Graph f(x) = −
- 3. State the domain and range.
x+1
a < 0: The graph is reflected over the x-axis.
0 < |a| < 1: The graph is compressed vertically.
h = -1: The vertical asymptote is at x = -1.
k = -3: The horizontal asymptote is at f(x) = -3.
D = {x | x ≠ -1}; R = { f(x) | f(x) ≠ -3}
2
-6
-4
-2
f(x)
0
2x
-2
-4
Exercises
-6
Graph each function. State the domain and range.
-2
2. f(x) = −
-1
3. f(x) = −
x-2
x-3
f (x)
f (x)
f(x)
x
x
x
0
1 +3
4. f(x) = −
Chapter 9
0
1
6. f(x) = −
+4
-2
5. f(x) = −
+2
x-1
x+5
0
0
f (x)
f(x)
x
0
x-3
f(x)
x
154
0
x
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1
1. f(x) = −
x +1
NAME
DATE
9-3
Practice
SCS
PERIOD
MBC.A.7.3, MBC.A.8.1, MBC.A.8.2, MBC.A.8.3,
MBC.A.8.4, MBC.A.8.5, MBC.A.10.1, MBC.A.10.2
Graphing Reciprocal Functions
Identify the asymptotes, domain, and range of each function.
1
1. f(x) = −
-3
-3
3. f(x) = −
+5
1
2. f(x) = −
+3
x-1
x+1
f(x)
x-2
f(x)
f (x)
6
6
-2
4
4
-4
2
2
-2
2
0
4
x
-6
-4
-2
0
2
x
-2
0
2
4
6x
Graph each function. State the domain and range.
x-3
f (x)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
0
3
6. f(x) = −
+4
-1
5. f(x) = −
-4
x+1
x-2
f(x)
x
f(x)
x
0
0
x
7. RACE Kate enters a 120-mile bicycle race. Her basic rate is 10 miles per hour, but Kate
will average x miles per hour faster than that. Write and graph an equation relating x
(Kate’s speed beyond 10 miles per hour) to the time it would take to complete the race.
If she wanted to finish the race in 4 hours instead of 5 hours, how much faster should
she travel?
t
14
12
10
8
6
4
2
0
Chapter 9
2
4 6
8 10 12 14 x
155
North Carolina StudyText, Math BC, Volume 2
Lesson 9-3
1
4. f(x) = −
-5
NAME
DATE
9-3
PERIOD
Word Problem Practice
SCS
Graphing Reciprocal Functions
1. VACATION The Porter family takes a
trip and rents a car. The rental costs
$125 plus $0.30 per mile.
MBC.A.7.3, MBC.A.8.1, MBC.A.8.2,
MBC.A.8.3, MBC.A.8.4, MBC.A.8.5,
MBC.A.10.1, MBC.A.10.2
3. BIOLOGY A rabbit population follows
40
+ 10, with P(t)
the function P(t) = −
t+2
equal to the rabbit population after t
months. Eventually, what happens to
the rabbit population?
a. Write the equation that relates the
cost per mile to the number of
miles traveled.
b. Explain any limitations to the range
or domain in this situation.
4. COMPUTERS To make computers, a
company must pay $5000 for rent and
overhead and $435 per computer
for parts.
b. Graph the function you found in
part a.
C
700
600
500
400
300
r
700
200
100
600
500
0
10 20 30 40 50 60 70 n
400
300
c. What is the minimum number of
computers the company needs to
make so that the average cost is less
than $685?
200
100
0
Chapter 9
1
2 3
4
5
6
7 t
156
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. PLANES A plane is scheduled to leave
Dallas for an 800-mile flight to Chicago’s
O’Hare airport at time t = 0. The
departure is delayed for two hours.
Write two equations that represent the
planes’ speed, r, on the vertical axis as a
function of travel time, t, on the
horizontal axis. Graph the equations
below. How do the two curves relate?
a. Write the equation relating average
cost to make a computer to how many
computers are being made.
NAME
DATE
9-4
Study Guide
SCS
PERIOD
MBC.A.1.2, MBC.A.7.3, MBC.A.8.3, MBC.A.8.5
Graphing Rational Functions
Vertical and Horizontal Asymptotes
p(x)
Rational Function
A function with an equation of the form f(x) = −, where p(x) and q(x) are polynomial
q(x)
expressions and q(x) ≠ 0
Domain
The domain of a rational function is limited to values for which the function is defined.
Vertical Asymptote
An asymptote is a line that the graph of a function approaches. If the simplified form of the
related rational expression is undefined for x = a, then x = a is a vertical asymptote.
Horizontal
Asymptote
Often a horizontal asymptote occurs in the graph of a rational function where a value is
excluded from the range.
x2 + x - 6
x+1
Example
Graph f(x) = − .
(x + 3)(x - 2)
x2 + x - 6
− = −
x+1
x+1
Therefore the graph of f(x) has zeroes at x = -3 and x = 2 and a vertical asymptote at
x = –1. Because the degree of x2 + x - 6 is greater than x + 1, there is no horizontal
asymptote. Make a table of values. Plot the points and draw the graph.
x
-4
-2
-3
0
0
-6
-2
4
4
1
-2
2
0
3
1.5
4
2.8
f (x)
2
-4
-2
0
2
4x
-2
-4
Exercises
Lesson 9-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
f(x)
-5
-3.5
Graph each function.
4
1. f(x) = −
2
x + 3x - 10
x 2 - 2x + 1
x + 2x + 1
f(x)
0
3. f (x) = −
2
f(x)
f(x)
x
0
0
Chapter 9
2x + 9
2x - x - 3
2. f (x) = −
2
157
x
x
North Carolina StudyText, Math BC, Volume 2
NAME
DATE
9-4
Study Guide
PERIOD
SCS
(continued)
MBC.A.1.2, MBC.A.7.3,
MBC.A.8.3, MBC.A.8.5
Graphing Rational Functions
Oblique Asymptotes and Point Discontinuity An oblique asymptote is an
asymptote that is neither horizontal nor vertical. In some cases, graphs of rational functions
may have point discontinuity, which looks like a hole in the graph. That is because the
function is undefined at that point.
a(x)
b(x)
Oblique Asymptotes
If f(x) = −, a(x) and b(x) are polynomial functions with no common factors other than
1 and b(x) ≠ 0, then f(x) has an oblique asymptote if the degree of a(x) minus the
degree of b(x) equals 1.
a(x)
b(x)
Point Discontinuity
If f(x) = −, b(x) ≠ 0, and x - c is a factor of both a(x) and b(x), then there is a point
discontinuity at x = c.
x-1
Graph f(x) = −
.
2
Example
x + 2x - 3
x-1
x-1
1
−
= −
or −
2
(x - 1)(x + 3)
x + 2x - 3
x+3
4
Therefore the graph of f(x) has an asymptote at x = -3
and a point discontinuity at x = 1.
f (x)
2
Make a table of values. Plot the points and draw the graph.
-6
-4
2x
O
-2
-2
-2.5 -2 -1 -3.5 -4
f(x)
2
1
0.5
-2
-5
-4
-1 -0.5
Exercises
Graph each function.
x2 + 5x + 4
x+3
3. f(x) = −
2
x-3
f(x)
0
x 2 - 6x + 8
x -x-2
2
-x-6
2. f(x) = x−
1. f(x) = −
f(x)
f(x)
x
O
O
Chapter 9
158
x
x
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
x
NAME
DATE
9-4
Practice
SCS
PERIOD
MBC.A.1.2, MBC.A.7.3, MBC.A.8.3, MBC.A.8.5
Graphing Rational Functions
Graph each function.
x-3
2. f(x) = −
x-2
3x
3. f(x) = −
2
x-2
f (x)
(x + 3)
f (x)
f (x)
O
x
O
x
O
2x2 + 5
6x - 4
x2 + 2x - 8
x-2
4. f(x) = −
x2 - 7x + 12
x-3
5. f(x) = −
f (x)
6. f(x) = −
f (x)
f (x)
x
0
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
0
x
x
0
x
7. PAINTING Working alone, Tawa can give the shed a coat of paint
in 6 hours. It takes her father x hours working alone to give the
6+x
shed a coat of paint. The equation f(x) = − describes the
6x
portion of the job Tawa and her father working together can
6+x
complete in 1 hour. Graph f(x) = − for x > 0, f(x) > 0. If Tawa’s
6x
father can complete the job in 4 hours alone, what portion of the
job can they complete together in 1 hour? What domain and range
values are meaningful in the context of the problem?
8. LIGHT The relationship between the illumination an object
receives from a light source of I foot-candles and the square of
the distance d in feet of the object from the source can be
f (x)
x
O
Lesson 9-4
-4
1. f(x) = −
II
4500
4500
. Graph the function I(d) = −
for
modeled by I(d) = −
2
2
d
d
0 < I ≤ 80 and 0 < d ≤ 80. What is the illumination in
foot-candles that the object receives at a distance of 20 feet
from the light source? What domain and range values are
meaningful in the context of the problem?
Chapter 9
159
O
d
North Carolina StudyText, Math BC, Volume 2
NAME
9-4
DATE
Word Problem Practice
PERIOD
SCS
MBC.A.1.2, MBC.A.7.3,
MBC.A.8.3, MBC.A.8.5
Graphing Rational Expressions
1. ROAD TRIP Robert and Sarah start
off on a road trip from the same house.
During the trip, Robert’s and Sarah’s
cars remain separated by a constant
distance. The graph shows the ratio of
the distance Sarah has traveled to the
distance Robert has traveled. The dotted
line shows how this graph would be
extended to hypothetical negative values
of x. What does the x-coordinate of the
vertical asymptote represent?
3. NEWTON Sir Isaac Newton studied
the rational function
ax 3 + bx 2 + cx + d
.
f(x) = −−
x
Assuming that d ≠ 0, where will there
be a vertical asymptote to the graph of
this function?
4. BATTING AVERAGES Alex Rodriguez
had a lifetime batting average of .305 at
the begining of the 2007 season. He had
2067 hits out of 6767 at bats. During the
2007 season, he had 183 hits.
y
a. Write an equation describing
Rodriguez’s batting average at the end
of the 2007 season using x for the
number of at bats he had during the
season.
x
O
2. GRAPHS Alma graphed the function
x 2 - 4x
f(x) = −
below.
x-4
6
c. What is the meaning of the horizontal
asymptote for the graph of this
equation?
f (x)
4
2
O
-2
2
4
6x
-2
There is a problem with her graph.
Explain how to correct it.
Chapter 9
160
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
b. Determine where the horizontal and
vertical asymptotes for the graph of
the equation would be.
NAME
DATE
9-6
PERIOD
Study Guide
SCS
MBC.A.7.2, MBC.A.7.3
Solve Rational Equations A rational equation contains one or more rational
expressions. To solve a rational equation, first multiply each side by the least common
denominator of all of the denominators. Be sure to exclude any solution that would produce
a denominator of zero.
Example
9
2
2
Solve −
+−
=−
. Check your solution.
10
x+1
2
2
9
−
+−
=−
10
x+1
5
2
2
9
10(x + 1) − + − = 10(x + 1) −
x+1
5
10
(
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Check
Original equation
()
)
9(x + 1) + 2(10)
9x + 9 + 20
5x
x
5
=
=
=
=
4(x + 1)
4x + 4
-25
-5
2
2
9
+−
=−
−
x+1
5
10
2
2
9
+−
−
−
-5 + 1
5
10
10
2
18
−-−−
20
5
20
2
2 =−
−
5
5
Multiply each side by 10(x + 1).
Multiply.
Distribute.
Subtract 4x and 29 from each side.
Divide each side by 5.
Original equation
x=
-5
Simplify.
Exercises
Solve each equation. Check your solution.
2y
3
y+3
6
1. − - − = 2
4t - 3
4 - 2t
2. −
-−
=1
3m + 2
2m - 1
=4
4. − + −
4
5. −
=−
5m
2m
5
x-1
3
x+1
12
2x + 1
x-5
1
3. − - −
=−
3
4
2
x
4
6. −
+−
= 10
x-2
x-2
7. NAVIGATION The current in a river is 6 miles per hour. In her motorboat Marissa can
travel 12 miles upstream or 16 miles downstream in the same amount of time. What is
the speed of her motorboat in still water? Is this a reasonable answer? Explain.
8. WORK Adam, Bethany, and Carlos own a painting company. To paint a particular
1
house alone, Adam estimates that it would take him 4 days, Bethany estimates 5−
days,
2
and Carlos 6 days. If these estimates are accurate, how long should it take the three of
them to paint the house if they work together? Is this a reasonable answer?
Chapter 9
161
North Carolina StudyText, Math BC, Volume 2
Lesson 9-6
Solving Rational Equations and Inequalities
NAME
DATE
9-6
Study Guide
PERIOD
SCS
(continued)
MBC.A.7.2, MBC.A.7.3
Solving Rational Equations and Inequalities
Solve Rational Inequalities
Step 1
Step 2
Step 3
To solve a rational inequality, complete the following steps.
State the excluded values.
Solve the related equation.
Use the values from steps 1 and 2 to divide the number line into regions. Test a value in each region to
see which regions satisfy the original inequality.
2
4
2
Solve −
+−
≤−
.
Example
3n
5n
3
Step 1 The value of 0 is excluded since this value would result in a denominator of 0.
Step 2 Solve the related equation.
2
4
2
−
+−
=−
Related equation
3n
5n
3
4
2
2
15n − + − = 15n −
5n
3n
3
(
)
( )
10 + 12 = 10n
22 = 10n
2.2 = n
Multiply each side by 15n.
Simplify.
Add.
Divide each side by 10.
Step 3 Draw a number with vertical lines at the
excluded value and the solution to the equation.
0
-3 -2 -1
1
2.2
2 3
Test n = 1.
Test n = 3.
2
2
4
+ -−
≤−
is true.
-−
3
3
5
2
4
2
−
+−
≤−
is not true.
3
5
3
2
4
2
−
+−
≤−
is true.
( )
9
15
3
The solution is n < 0 or n ≥ 2.2.
Exercises
Solve each inequality. Check your solutions.
3
1. −
≥3
a+1
3
2
1
-−
>−
4. −
2x
Chapter 9
x
4
1
2. −
x ≥ 4x
1
4
2
3. −
+−
>−
5
4
5. −
+−
x <2
x-1
3
2
6. −
+1>−
2
2p
162
x -1
5p
3
x-1
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Test n = -1.
NAME
DATE
9-6
PERIOD
Practice
SCS
MBC.A.7.2, MBC.A.7.3
Solve each equation or inequality. Check your solutions.
3
3
12
1. −
x +−=−
4
p + 10
3. −
=
p2 - 2
x
x
2. −
-1=−
2
x-1
5s + 8
s+2
s
4. −
+s=−
4
−
p
s+2
y
y-5
5
=−-1
5. −
5
1
6. −
+−
x =0
3x - 2
5
9
<−
7. −
5
3
1
8. −
+−
=−
y-5
t
2t + 1
2h
w-2
h
h-1
3
7
10. 5 - −
a <−
a
4
-1
=−
9. −
w+3
3
4
1
11. −
+−
<−
3
19
12. 8 + −
y >−
y
4
1
1
13. −
p +−<−
6
4
2
14. −
=−
+−
5x
10
2x
3p
5
g
2
15. g + − = −
g-2
g-2
x-1
x-2
x+1
2b
b-3
16. b + −
=1-−
b-1
c+1
c-3
b-1
3
1
1
+−
=−
17. −
2
12
18. − = 4 - −
2
3
25
4
+−
=−
19. −
2
4v
5v
2
20. −
-−
=−
2
n+2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
n-2
n -4
k-3
k-4
k - 7k + 12
y
7
14
=−
21. − + −
y+2
y-5
y 2 - 3y - 10
2
r + 16
r - 16
r
4
+−
=−
23. −
2
r+4
r-4
v-1
c - 2c - 3
v-2
v - 3v + 2
2
x +4
x -4
x
2
22. −
+−
=−
2
2-x
x+2
6a - 1
22
24. 3 = −
+−
2a + 7
a+5
27. BASKETBALL Kiana has made 9 of 19 free throws so far this season. Her goal is to
make 60% of her free throws. If Kiana makes her next x free throws in a row, the
9+x
function f(x) = − represents Kiana’s new ratio of free throws made. How many
19 + x
successful free throws in a row will raise Kiana’s percent made to 60%? Is this a
reasonable answer? Explain.
1
1
1
28. OPTICS The lens equation −
relates the distance p of an object from a
p +−
q =−
f
lens, the distance q of the image of the object from the lens, and the focal length f of
the lens. What is the distance of an object from a lens if the image of the object is
5 centimeters from the lens and the focal length of the lens is 4 centimeters? Is this
a reasonable answer? Explain.
Chapter 9
163
North Carolina StudyText, Math BC, Volume 2
Lesson 9-6
Solving Rational Equations and Inequalities
NAME
9-6
DATE
Word Problem Practice
PERIOD
SCS
MBC.A.7.2, MBC.A.7.3
Solving Rational Equations and Inequalities
1. HEIGHT Serena can be described as
being 8 inches shorter than her sister
Malia, or as being 12.5% shorter than
4. PROJECTILES A projectile target
is launched into the air. A rocket
interceptor is fired at the target. The
ratio of the altitude of the rocket to the
altitude of the projectile t seconds after
the launch of the rocket is given by the
8
1
Malia. In other words, −
=−
,
H+8
8
where H is Serena’s height in inches.
How tall is Serena?
333t
formula −−
. At what time
2
-32t + 420t + 27
are the target and interceptor at the
same altitude?
2. CRANES For a wedding, Paula wants to
fold 1000 origami cranes.
5. FLIGHT TIME The distance between
John F. Kennedy International Airport
and Los Angeles International Airport is
about 2500 miles. Let S be the airspeed
of a jet. The wind speed is 100 miles per
hour. Because of the wind, it takes longer
to fly one way than the other.
She does not want to make anyone fold
more than 15 cranes. In other words, if
N is the number of people enlisted to
1000
fold cranes, Paula wants −
≤ 15.
N
What is the minimum number of people
that will satisfy this inequality?
3. RENTAL Carlos and his friends rent a
car. They split the $200 rental fee evenly.
Carlos, together with just two of his
friends, decide to rent a portable DVD
player as well, and split the $30 rental
fee for the DVD player evenly among
themselves. Carlos ends up spending $50
for these rentals. Write an equation
involving N, the number of friends Carlos
has, using this information. Solve the
equation for N.
Chapter 9
164
b. Solve the equation you wrote in
part a for S.
c. Write an equation and find how much
longer it would take to fly between
New York and Los Angeles if the wind
speed increases to 150 miles per hour
and the airspeed of the jet is 525 miles
per hour.
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
a. Write an equation for S if it takes
2 hours and 5 minutes longer to fly
between New York and Los Angeles
against the wind versus flying with
the wind.
NAME
DATE
10-3
PERIOD
Study Guide
SCS
MBC.G.5.1
Circles
Equations of Circles
2
2
The equation of a circle with center (h, k) and radius r units is
2
(x - h) + (y - k) = r .
A line is tangent to a circle when it touches the circle at only one point.
Example
Write an equation for a circle if the endpoints of a diameter are at
(-4, 5) and (6, -3).
Use the midpoint formula to find the center of the circle.
( x +2 x y +2 y )
-4 + 6 5 + (- 3)
= (−
, −)
2
2
1
2
1
2
, −
(h, k) = −
(2 2)
2 2
= −
, − or (1, 1)
Midpoint formula
(x1, y1) = (-4, 5), (x2, y2) = (6, -3)
Simplify.
Use the coordinates of the center and one endpoint of the diameter to find the radius.
r=
(x2 - x1)2 + ( y2 - y1)2
√
Distance formula
= √(-4
- 1)2 + (5 - 1)2
(x1, y1) = (1, 1), (x2, y2) = (-4, 5)
2
= √(-5)
+ 42 = √
41
Simplify.
Exercises
Write an equation for the circle that satisfies each set of conditions.
1. center (8, -3), radius 6
2. center (5, -6), radius 4
3. center (-5, 2), passes through (-9, 6)
4. center (3, 6), tangent to the x-axis
5. center (-4, -7), tangent to x = 2
6. center (-2, 8), tangent to y = -4
7. center (7, 7), passes through (12, 9)
Write an equation for each circle given the end points of a diameter.
8. (6, 6) and (10, 12)
9. (-4, -2) and (8, 4)
10. (-4, 3) and (6, -8)
Chapter 10
165
North Carolina StudyText, Math BC, Volume 2
Lesson 10-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The radius of the circle is √
41 , so r2 = 41.
An equation of the circle is (x - 1)2 + ( y - 1)2 = 41.
NAME
DATE
10-3
Study Guide
PERIOD
SCS
(continued)
MBC.G.5.1
Circles
Graph Circles
To graph a circle, write the given equation in the standard form of the
equation of a circle, (x - h)2 + (y - k)2 = r2.
Plot the center (h, k) of the circle. Then use r to calculate and plot the four points (h + r, k),
(h - r, k), (h, k + r), and (h, k - r), which are all points on the circle. Sketch the circle that
goes through those four points.
Example
Find the center and radius of the circle
whose equation is x2 + 2x + y2 + 4y = 11. Then graph
the circle.
2
y
x2 + 2x + y2 + 4y = 11
2
2
x + 2x + y + 4y = 11
x + 2x + + y2 + 4y + = 11 + + x2 + 2x + 1 + y2 + 4y + 4 = 11 + 1 + 4
(x + 1)2 + ( y + 2)2 = 16
Therefore, the circle has its center at (-1, -2) and a radius of
√
16 = 4. Four points on the circle are (3, -2), (-5, -2), (-1, 2),
and (-1, -6).
-4
2
2
O
-2
4
x
-2
-4
-6
Exercises
Find the center and radius of each circle. Then graph the circle.
1. (x - 3)2 + y2 = 9
2. x2 + (y + 5)2 = 4
y
-4
2
-2
4x
2
O
-2
6x
4
2
O
y
-4
4 x
2
O
-2
-2
-2
-4
-4
-6
-6
-2
-4
4. (x - 2)2 + (y + 4)2 = 16
5. x2 + y2 - 10x + 8y + 16 = 0 6. x2 + y2 - 4x + 6y = 12
y
-2
y
2
O
4
6
x
y
2
O
-2
-2
-4
-4
-6
-6
-8
-8
4
6
8
x
-2
2
O
4
6
x
-2
-4
-6
-8
Chapter 10
166
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
y
4
3. (x - 1)2 + (y + 3)2 = 9
NAME
DATE
10-3
PERIOD
Practice
SCS
MBC.G.5.1
Circles
Write an equation for the circle that satisfies each set of conditions.
1. center (-4, 2), radius 8 units
(
2. center (0, 0), radius 4 units
)
1
, - √
3 , radius 5 √
2 units
3. center - −
4
4. center (2.5, 4.2), radius 0.9 units
5. endpoints of a diameter at (-2, -9) and (0, -5)
6. center at (-9, -12), passes through (-4, -5)
7. center at (-6, 5), tangent to x-axis
Find the center and radius of each circle. Then graph the circle.
8. (x + 3)2 + y2 = 16
-2
-4
y
4
y
4
2
2
x
O
-2
-4
10. x2 + y2 + 2x + 6y = 26
y
4
4x
2
O
-2
-2
-4
-4
–8
–4
4
O
8x
–4
–8
11. (x - 1)2 + y2 + 4y = 12
12. x2 - 6x + y2 = 0
y
2
-2
y
4
2
O
-2
13. x2 + y2 + 2x + 6y = -1
4
y
-4
2
x
-2
2
O
4x
-2
2
O
-4
-2
-6
-4
4
6
x
-4
-6
14. WEATHER On average, the circular eye of a hurricane is about 15 miles in diameter.
Gale winds can affect an area up to 300 miles from the storm’s center. A satellite photo
of a hurricane’s landfall showed the center of its eye on one coordinate system could be
approximated by the point (80, 26).
a. Write an equation to represent a possible boundary of the hurricane’s eye.
b. Write an equation to represent a possible boundary of the area affected by gale winds.
Chapter 10
167
North Carolina StudyText, Math BC, Volume 2
Lesson 10-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
-6
9. 3x2 + 3y2 = 12
NAME
10-3
DATE
PERIOD
Word Problem Practice
SCS
MBC.G.5.1
Circles
1. RADAR A scout plane is equipped
with radar. The boundary of the
radar’s range is given by the circle
(x - 4)2 + (y - 6)2 = 4900. Each unit
corresponds to one mile. What is the
maximum distance that an object can be
from the plane and still be detected by
its radar?
4. POOLS The pool on an architectural
blueprint is given by the equation
x2 + 6x + y2 + 8y = 0. What point on
the edge of the pool is farthest from
the origin?
5. TREASURE A mathematically inclined
y
pirate decided to hide the location of a
treasure by marking it as the center of
a circle given by an equation in nonstandard form.
Scout plane
5
O
5
x
y
5
-5
O
5
x
-5
The circle can be represented by:
x2 + y2 - 2x + 14y + 49 = 0.
a. Rewrite the equation of the circle in
standard form.
3. FERRIS WHEEL The Texas Star, the
largest Ferris wheel in North America,
is located in Dallas, Texas. It weighs
678,554 pounds and can hold 264 riders
in its 44 gondolas. The Texas Star
has a diameter of 212 feet. Use the
rectangular coordinate system with
the origin on the ground directly below
the center of the wheel and write the
equation of the circle that models the
Texas Star.
Chapter 10
b. Draw the circle on the map. Where is
the treasure?
168
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. STORAGE An engineer uses a
coordinate plane to show the layout
of a side view of a storage building.
The y-axis represents a wall and the
x-axis represents the floor. A 10-meter
diameter cylinder rests on its side flush
against the wall. On the side view, the
cylinder is represented by a circle in the
first quadrant that is tangent to both
axes. Each unit represents 1 meter.
What is the equation of this circle?
NAME
10-7
DATE
Study Guide
PERIOD
SCS
MBC.A.2.2
Solving Linear-Nonlinear Systems
Systems of Equations
Like systems of linear equations, systems of linear-nonlinear
equations can be solved by substitution and elimination. If the graphs are a conic section
and a line, the system will have 0, 1, or 2 solutions. If the graphs are two conic sections, the
system will have 0, 1, 2, 3, or 4 solutions.
Solve the system of equations. y = x 2 - 2x - 15
x + y = -3
Rewrite the second equation as y = -x - 3 and substitute it into the first equation.
-x - 3 = x2 - 2x - 15
0 = x2 - x - 12
Add x + 3 to each side.
0 = (x - 4)(x + 3)
Factor.
Use the Zero Product property to get
x = 4 or x = -3.
Substitute these values for x in x + y = -3:
4 + y = -3 or -3 + y = -3
y = -7
y=0
The solutions are (4, -7) and (-3, 0).
Exercises
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Solve each system of equations.
1. y = x2 - 5
y=x-3
2. x2 + ( y - 5)2 = 25
y = -x2
3. x2 + ( y - 5)2 = 25
y = x2
4. x2 + y2 = 9
x2 + y = 3
5. x2 - y2 = 1
x2 + y2 = 16
6. y = x - 3
x = y2 - 4
Chapter 10
169
North Carolina StudyText, Math BC, Volume 2
Lesson 10-7
Example
NAME
DATE
10-7
Study Guide
PERIOD
SCS
(continued)
MBC.A.2.2
Solving Linear-Nonlinear Systems
Systems of Inequalities
Systems of linear-nonlinear inequalities can be solved
by graphing.
Example 1
2
Solve the system of inequalities by graphing.
y
2
x + y ≤ 25
2
(x - −25 )
4
25
+ y2 ≥ −
2
4
The graph of x2 + y2 ≤ 25 consists of all points on or inside
the circle with center (0, 0) and radius 5. The graph of
-4
-2
2
4
x
2
4
x
-2
2
)
(
O
5
25
x-−
+ y2 ≥ −
consists of all points on or outside the
2
4
5
5
circle with center −
, 0 and radius −
. The solution of the
2
2
-4
( )
system is the set of points in both regions.
Example 2
2
Solve the system of inequalities by graphing.
y
2
x + y ≤ 25
4
y2
x2
−-−
>1
4
9
2
2
2
The graph of x + y ≤ 25 consists of all points on or inside
the circle with center (0, 0) and radius 5. The graph of
2
4
9
the hyperbola shown. The solution of the system is the set of
points in both regions.
-2
O
-2
-4
Exercises
Solve each system of inequalities by graphing.
y2
4
2
x
1. −
+− ≤1
16
2. x2 + y2 ≤ 169
3. y ≥ (x - 2)2
x2 + 9y2 ≥ 225
1
y>−
x-2
(x + 1)2 + ( y + 1)2 ≤ 16
2
y
O
Chapter 10
y
y
x
O
170
x
O
x
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
y
x2
−-−
> 1 are the points “inside” but not on the branches of
-4
NAME
DATE
10-7
PERIOD
Practice
SCS
MBC.A.2.2
Solving Linear-Nonlinear Systems
Solve each system of equations.
2. x = 2( y + 1)2 - 6
x+y=3
3. y2 - 3x2 = 6
y = 2x - 1
4. x2 + 2y2 = 1
y = -x + 1
5. 4y2 - 9x2 = 36
4x2 - 9y2 = 36
6. y = x2 - 3
x 2 + y2 = 9
7. x2 + y2 = 25
4y = 3x
8. y2 = 10 - 6x2
4y2 = 40 - 2x2
9. x2 + y2 = 25
x = 3y - 5
14. x2 + y2 = 4
2
2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
y2
7
x
16. −
+−=1
17. x + 2y = 3
3x2 - y2 = 9
x 2 + y2 = 9
7
15. x2 - y2 = 3
y 2 - x2 = 3
y2
x
−
+−=1
4
8
2
5
x = -−
y2
2
x
11. x = -( y - 3)2 + 2 12. −
-−=1
9
16
x = ( y - 3)2 + 3
2
x + y2 = 9
10. 4x2 + 9y2 = 36
2x2 - 9y2 = 18
13. 25x2 + 4y2 = 100
Lesson 10-7
1. (x - 2)2 + y2 = 5
x-y=1
18. x2 + y2 = 64
x 2 - y2 = 8
Solve each system of inequalities by graphing.
19. y ≥ x2
y > -x + 2
y
(x + 2) 2
4
21. − + − ≤ 1
(x + 1)2 + ( y - 2)2 ≤ 4
y
O
O
(y - 3) 2
16
20. x2 + y2 < 36
x2 + y2 ≥ 16
y
x
x
O
22. GEOMETRY The top of an iron gate is shaped like half an
ellipse with two congruent segments from the center of the
ellipse to the ellipse as shown. Assume that the center of
the ellipse is at (0, 0). If the ellipse can be modeled by the
equation x2 + 4y2 = 4 for y ≥ 0 and the two congruent
√3
2
x
B
A
(0, 0)
√3
2
segments can be modeled by y = − x and y = - − x,
what are the coordinates of points A and B?
Chapter 10
171
North Carolina StudyText, Math BC, Volume 2
NAME
DATE
10-7
PERIOD
Word Problem Practice
SCS
MBC.A.2.2
Solving Linear-Nonlinear Systems
1. GRAPHIC DESIGN A graphic designer
is drawing an ellipse and a line. The
ellipse is drawn so that it appears on top
of the line. In order to determine if the
line is partially covered by the ellipse,
the program solves for simultaneous
solutions of the equations of the line and
the ellipse. Complete the following table.
No. of Intersections
Covered? Y/N
4. COLLISION AVOIDANCE An object
is traveling along a hyperbola given by
y2
x2
- − = 1. A probe is
the equation −
9
36
launched from the origin along a
straight-line path. Mission planners
want the probe to get closer and closer
to the object, but never hit it. There are
two straight lines that meet their
criteria. What are they?
0
1
2
2. ORBITS Objects in the solar system
travel in elliptical orbits where the Sun
is one focal point. Earth’s orbit is an
ellipse. What is the maximum number
of times per orbit that an asteroid also
in an elliptical orbit can cross Earth’s
orbit?
5. TANGENTS An architect wants a
straight path to run from the origin of a
coordinate plane to the edge of an
elliptically shaped patio so that the
pathway forms a tangent to the ellipse.
The ellipse is given by the equation
(x - 6)2
12
y2
96
−+−=1
b. Solve for m in the equation you found
for part a when x = 4.
Chapter 10
172
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3. CIRCLES An artist is commissioned to
complete a painting of only circles. She
wants to include all possible ways circles
can relate. What are the possible
numbers of intersection points between
two circles? For each case, sketch two
distinct circles that intersect with the
corresponding number of points. Explain
why more intersections are not possible.
a. Using the equation y = mx to describe
the path, substitute into the equation
for the ellipse to get a quadratic
equation in x.
NAME
DATE
13-1
PERIOD
Study Guide
SCS
MBC.A.9.3
Trigonometric Functions in Right Triangles
Trigonometric Functions for Acute Angles
Trigonometry is the study of
relationships among the angles and sides of a right triangle. A trigonometric function
has a rule given by a trigonometric ratio, which is a ratio that compares the side lengths
of a right triangle.
hyp
A
opp
θ
C
adj
adj
hyp
hyp
sec θ = −
adj
opp
hyp
hyp
csc θ = −
opp
sin θ = −
cos θ = −
opp
adj
adj
cot θ = −
opp
tan θ = −
3
In a right triangle, ∠B is acute and cos B = −
. Find the value of
Example
tan B.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
If θ is the measure of an acute angle of a right triangle, opp is the
measure of the leg opposite θ, adj is the measure of the leg adjacent
to θ, and hyp is the measure of the hypotenuse, then the following
are true.
7
Step 1
Draw a right triangle and label one acute angle B. Label the
adjacent side 3 and the hypotenuse 7.
Step 2
Use the Pythagorean Theorem to find b.
a 2 + b2 = c2
Pythagorean Theorem
32 + b2 = 72
a = 3 and c = 7
2
9 + b = 49
Simplify.
2
b = 40
Subtract 9 from each side.
b = √
40 = 2 √
10 Take the positive square root of each side.
Find tan B.
Step 3
opp
adj
2 √10
tan B = −
3
tan B = −
3
#
7
$
b
"
Tangent function
Replace opp with 2 √
10 and adj with 3.
Exercises
Find the values of the six trigonometric functions for angle θ.
2.
1.
5
θ
13
3.
17
θ
16
12
θ
8
In a right triangle, ∠ A and ∠ B are acute.
7
, what is
4. If tan A = −
12
cos A?
Chapter 13
1
5. If cos A = −
, what is
2
tan A?
173
3
6. If sin B = −
, what is
8
tan B?
North Carolina StudyText, Math BC, Volume 2
Lesson 13-1
Trigonometric Functions
in Right Triangles
B
NAME
DATE
13-1
Study Guide
PERIOD
SCS
(continued)
MBC.A.9.3
Trigonometric Functions in Right Triangles
Use Trigonometric Functions You can use trigonometric functions to find missing
side lengths and missing angle measures of right triangles. You can find the measure of the
missing angle by using the inverse of sine, cosine, or tangent.
Example
Find the measure of ∠C. Round to the nearest tenth if necessary.
You know the measure of the side opposite ∠C and the measure of the
hypotenuse. Use the sine function.
opp
hyp
8
sin C = −
10
8
sin-1 −
= m∠C
10
sin C = −
"
Sine function
10
8
Replace opp with 8 and hyp with 10.
#
Inverse sine
53.1° ≈ m∠C
$
Use a calculator.
Exercises
Use a trigonometric function to find each value of x. Round to the nearest tenth
if necessary.
1.
2.
14.5
x
4
10
x
20°
x
4.
5.
5
x
6.
8
x
x 70°
32°
45°
9
Find x. Round to the nearest tenth if necessary.
7.
8.
7
x°
33
x°
9.
13
x°
10
4
4
Chapter 13
174
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
38°
3.
63°
NAME
DATE
13-1
PERIOD
Practice
SCS
MBC.A.9.3
Trigonometric Functions in Right Triangles
Find the values of the six trigonometric functions for angle θ.
1.
2.
5
45
3 3
3.
θ
3
11
θ
Lesson 13-1
24
In a right triangle, ∠ A and ∠ B are acute.
8
11
4. If tan B = 2, what is cos B? 5. If tan A = −
, what is sin A? 6. If sin B = −
, what is cos B?
17
15
Use a trigonometric function to find each value of x. Round to the nearest tenth
if necessary.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
7.
8.
9.
x
49°
x
32
30°
x
17
20°
7
Use trigonometric functions to find the values of x and y. Round to the nearest
tenth if necessary.
10.
11.
x
y
12.
y°
19.2
y°
x°
41°
28
7
15.3
17
x°
13. SURVEYING John stands 150 meters from a water tower and sights the top at an angle of
elevation of 36°. If John’s eyes are 2 meters above the ground, how tall is the tower? Round
to the nearest meter.
Chapter 13
175
North Carolina StudyText, Math BC, Volume 2
NAME
DATE
13-1
PERIOD
Word Problem Practice
SCS
MBC.A.9.3
Trigonometric Functions in Right Triangles
1. ROOFS The roof on a house is built with
a pitch of 10/12, meaning that the roof
rises 10 feet for every 12 feet of
horizontal run. The side view of the
roof is shown in the figure below.
y°
3. SCALE DRAWING The collection pool
for a fountain is in the shape of a right
triangle. A scale drawing shows that the
angles of the triangle are 40°, 50°, and
90°. If the hypotenuse of the actual
fountain will be 30 feet, what are
the lengths of the other two sides of
the fountain?
10 ft
x°
12 ft
4. GEOMETRY A regular hexagon is
inscribed in a circle with a diameter of
8 inches.
a. What is the angle x at the base of
the roof ?
8 in.
c. What is the length ℓ of the roof ?
d. If the width of the roof is 26 feet,
what is the total area of the roof ?
a. What is the perimeter of the hexagon?
2. BUILDINGS Jessica stands 150 feet from
the base of a tall building. She measures
the angle from her eye to the top of the
building to be 84°. If her eye level is
5 feet above the ground, how tall is the
building?
Chapter 13
176
b. What is the area of the hexagon?
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
b. What is the angle y at the peak of
the roof ?
NAME
DATE
13-3
PERIOD
Study Guide
SCS
MBC.A.9.3
Trigonometric Functions of General Angles
Trigonometric Functions for General Angles
Trigonometric Functions,
θ in Standard Position
y
P(x, y)
y
y
r
x
Let θ be an angle in standard position and let P(x, y) be a point on
the terminal side of θ. By the Pythagorean Theorem, the distance r
x2 + y2 . The trigonometric functions
from the origin is given by r = √
of an angle in standard position may be defined as follows.
sin θ = −r
θ
O
x
cos θ = −
r
y
tan θ = −
x, x ≠ 0
r
r
x
csc θ = −
y , y ≠ 0 sec θ = −
x , x ≠ 0 cot θ = −
y, y ≠ 0
x
Example
Find the exact values of the six trigonometric functions of θ if the
terminal side of θ in standard position contains the point (-5, 5 √
2 ).
You know that x = -5 and y = 5. You need to find r.
x2 + y2
r = √
=
Pythagorean Theorem
√
(-5)2 + (5 √
2 )2
Replace x with -5 and y with 5 √
2.
= √
75 or 5 √
3
2 , and r = 5 √
3 to write the six trigonometric ratios.
Now use x = -5, y = 5 √
5 √2
√6
x
-5
cos θ = −
= -−
r = −
3
5 √3
5 √3
√6
r
sec θ = −
= - √
3
x = −
-5
r
csc θ = −
=−
y =−
2
5 √2
√3
5 √3
y
5 √2
tan θ = −
= - √
2
x =−
-5
√2
x
-5
cot θ = −
= -−
y =−
2
5 √2
Exercises
The terminal side of θ in standard position contains each point. Find the exact
values of the six trigonometric functions of θ.
1. (8, 4)
2. (4, 4)
3. (0, 4)
4. (6, 2)
Chapter 13
177
North Carolina StudyText, Math BC, Volume 2
Lesson 13-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
y
sin θ = −
= −
r =−
3
5 √3
NAME
DATE
13-3
Study Guide
PERIOD
SCS
(continued)
MBC.A.9.3
Trigonometric Functions of General Angles
Trigonometric Functions with Reference Angles
If θ is a nonquadrantal angle
in standard position, its reference angle θ ' is defined as the acute angle formed by the
terminal side of θ and the x-axis.
y Quadrant I
Reference
Angle Rule
Quadrant II y
θ
x
O
y
O
θ
x
Quadrant IV
θ = θ - 180°
(θ = θ - π)
θ = 360° - θ
(θ = 2π - θ)
Example 2
Example 1
Use a reference angle
3π
to find the exact value of cos −
.
Sketch an angle of
measure 205°. Then find its reference
angle.
4
3π
Because the terminal side of − lies in
4
Because the terminal side of 205° lies in
Quadrant III, the reference angle θ ' is
205° - 180° or 25°.
Quadrant II, the reference angle θ ' is
3π
π
π- −
or −
.
4
y
4
The cosine function is negative in
Quadrant II.
θ = 205°
√2
3π
π
= -cos −
= -−.
cos −
4
x
O
4
2
Exercises
Sketch each angle. Then find its reference angle.
1. 155°
4π
3. −
2. 230°
y
O
π
4. - −
3
6
y
x
O
y
y
x
O
O
x
x
Find the exact value of each trigonometric function.
5. tan 330°
Chapter 13
11π
6. cos −
4
7. cot 30°
178
π
8. csc −
4
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
θ
θ
Quadrant III
θ = 180° - θ
(θ = π - θ)
θ = θ
y
O x
θ
θ
θ
x
O
θ
NAME
DATE
13-3
PERIOD
Practice
SCS
MBC.A.9.3
Trigonometric Functions of General Angles
The terminal side of θ in standard position contains each point. Find the exact
values of the six trigonometric functions of θ.
1. (6, 8)
2. (-20, 21)
3. (-2, -5)
Sketch each angle. Then find its reference angle.
7π
6. - −
5. -210°
8
4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
O
y
y
y
O
x
O
x
x
Lesson 13-3
13π
4. −
Find the exact value of each trigonometric function.
7. tan 135°
5π
11. tan −
3
8. cot 210°
(
3π
12. csc - −
4
9. cot (-90°)
)
10. cos 405°
13π
14. tan −
13. cot 2π
6
15. LIGHT Light rays that “bounce off” a surface are reflected
by the surface. If the surface is partially transparent, some
of the light rays are bent or refracted as they pass from the
air through the material. The angles of reflection θ1 and of
refraction θ2 in the diagram at the right are related by the
equation sin θ1 = n sin θ2. If θ1 = 60° and n = √
3 , find the
measure of θ2.
air
θ1
θ1
surface
θ2
800 N
16. FORCE A cable running from the top of a utility pole to the
ground exerts a horizontal pull of 800 Newtons and a vertical
pull of 800 √
3 Newtons. What is the sine of the angle θ between the
cable and the ground? What is the measure of this angle?
Chapter 13
179
800
3N
North Carolina StudyText, Math BC, Volume 2
NAME
13-3
DATE
PERIOD
Word Problem Practice
SCS
MBC.A.9.3
Trigonometric Functions of General Angles
1. RADIOS Two correspondence radios are
located 2 kilometers away from a base
camp. The angle formed between the
first radio, the base camp, and the
second radio is 120°. If the first radio has
coordinates (2, 0) relative to the base
camp, what is the position of the second
radio relative to the base camp?
4. SOCCER Alice kicks a soccer ball
towards a wall. The ball is deflected off
the wall at an angle of 40°, and it travels
6 meters. How far is the soccer ball from
the wall when it stops rolling?
Alice
6m
Radio 2
2 km
40°
120°
5. PAPER AIRPLANES The formula
2 km
Base
Camp
V 2 sin 2θ
Radio 1
0
+ 15 cos θ gives the
R=−
32
distance traveled by a paper airplane
that is thrown with an initial velocity of
V0 feet per second at an angle of θ with
the ground.
(2
)
time in seconds after leaving the bottom
of the swing. Determine the measure of
the angles in radians for t = 0, 0.5, 1,
1.5, 2, 2.5, and 3 seconds.
a. If the airplane is thrown with an
initial velocity of 15 feet per second at
an angle of 25°, how far will the
airplane travel?
b. Two airplanes are thrown with
an initial velocity of 10 feet per
second. One airplane is thrown at an
angle of 15° to the ground, and the
other airplane is thrown at an angle
of 45° to the ground. Which will
travel farther?
3. FERRIS WHEELS Janice rides a Ferris
wheel in Japan called the Sky Dream
Fukuoka, which has a radius of about
60 m and is 5 m off the ground. After she
enters the bottom car, the wheel rotates
210.5° counterclockwise before stopping.
How high above the ground is Janice
when the car has stopped?
Chapter 13
180
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. CLOCKS The pendulum of a grandfather
clock swings back and forth through an
arc. The angle θ of the pendulum is given
π
by θ = 0.3 cos −
+ 5t where t is the
NAME
DATE
13-6
PERIOD
Study Guide
SCS
MBC.A.9.3
Circular Functions
If the terminal side of an angle θ in standard position
intersects the unit circle at P(x, y), then cos θ = x and
sin θ = y. Therefore, the coordinates of P can be
written as P(cos θ, sin θ).
Definition of
Sine and Cosine
(0,1) y
P(cos θ, sin θ)
θ
(-1,0)
(1,0)
x
O
(0,-1)
Example
The terminal side of angle θ in standard position intersects the unit
)
(
P(- − , −) = P(cos θ, sin θ), so cos θ = - − and sin θ = −.
√
11
6
6
5
, − . Find cos θ and sin θ.
circle at P - −
5
6
√
11
6
√
11
6
5
6
Exercises
The terminal side of angle θ in standard position intersects the unit circle at each
point P. Find cos θ and sin θ.
(
)
2. P(0, -1)
)
3
4
4. P - −
, -−
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
√3
1
1. P - − , −
(
2
2
√5
3
2
,−
3. P - −
3
(6
(
)
5
5
)
( 4 4)
√
35
1
, -−
5. P −
√7
3
6. P −, −
7. P is on the terminal side of θ = 45°.
8. P is on the terminal side of θ = 120°.
9. P is on the terminal side of θ = 240°.
10. P is on the terminal side of θ = 330°.
Chapter 13
6
181
North Carolina StudyText, Math BC, Volume 2
Lesson
Lesson 13-6
X-X
Circular Functions
NAME
DATE
13-6
Study Guide
PERIOD
SCS
(continued)
MBC.A.9.3
Circular Functions
Periodic Functions
A periodic function has y-values that repeat at regular intervals. One complete pattern is
called a cycle, and the horizontal length of one cycle is called a period.
The sine and cosine functions are periodic; each has a period of 360° or 2π radians.
Example 1
Determine the period of the function.
The pattern of the function repeats every 10 units,
so the period of the function is 10.
y
1
O
5
-1
10
15
Example 2
20
25
30
35
θ
Find the exact value of each function.
(6)
31π
7π
cos (−
= cos (−
+ 4π)
6 )
6
31π
b. cos −
a. sin 855°
sin 855° = sin (135° + 720°)
√2
2
= sin 135° or −
√3
7π
= cos −
or - −
2
Exercises
Determine the period of each function.
1.
1
y
O
2.
π
2π
3π
4π
y
0
θ
2
-1
6
4
8
10
x
Find the exact value of each function.
(
2
(
4
)
3. sin (-510°)
4. sin 495°
5π
5. cos - −
5π
6. sin −
11π
7. cos −
3π
8. sin - −
3
Chapter 13
4
182
)
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
6
NAME
DATE
13-6
PERIOD
Practice
SCS
MBC.A.9.3
The terminal side of angle θ in standard position intersects the unit circle at each
point P. Find cos θ and sin θ.
(
√3
2
)
( 29
)
1
1. P - −
,−
20
21
2. P −
, -−
4. P(0, -1)
5. P - −, - −
2
(
29
√2
2
√2
2
3. P(0.8, 0.6)
( 2 2)
)
√3
1
6. P −, −
Determine the period of each function.
7.
1
y
O
-1
1
2
3
4
5
6
7
8
10 θ
9
-2
8.
y
1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
O
-1
π
2π
3π
4π
5π
6π
θ
-2
Find the exact value of each function.
7π
9. cos −
(
)
10. sin (-30°)
2π
11. sin - −
13. cos 600°
9π
14. sin −
15. cos 7π
17. sin (-225°)
18. sin 585°
10π
19. cos - −
4
2
(
3
12. cos (-330°)
(
11π
16. cos - −
3
)
4
)
20. sin 840°
21. FERRIS WHEELS A Ferris wheel with a diameter of 100 feet completes 2.5 revolutions
per minute. What is the period of the function that describes the height of a seat on the
outside edge of the Ferris wheel as a function of time?
Chapter 13
183
North Carolina StudyText, Math BC, Volume 2
Lesson 13-6
Circular Functions
NAME
13-6
DATE
PERIOD
Word Problem Practice
SCS
MBC.A.9.3
Circular Functions
1. TIRES A point on the edge of a car tire is
marked with paint. As the car moves
slowly, the marked point on the tire
varies in distance from the surface of the
road. The height in inches of the point is
given by the expression h = -8 cos t + 8,
where t is the time in seconds.
2. GEOMETRY The temperature T in
degrees Fahrenheit of a city t months
into the year is approximated
π
t.
by the formula T = 42 + 30 sin −
(6 )
a. What is the highest monthly
temperature for the city?
b. In what month does the highest
temperature occur?
c. What is the lowest monthly
temperature for the city?
a. What is the maximum height above
ground that the point on the tire
reaches?
d. In what month does the lowest
temperature occur?
3. THE MOON The Moon’s period of
revolution is the number of days it takes
for the Moon to revolve around Earth.
The period can be determined by
graphing the percentage of sunlight
reflected by the Moon each day, as seen
by an observer on Earth. Use the graph
to determine the Moon’s period of
revolution.
c. How many rotations does the tire
make per second?
Moon’s Orbit
Sunlight Reflected (%)
d. How far does the marked point travel
in 30 seconds? How far does the
marked point travel in one hour?
100
80
60
40
20
0
7 14 21 28 35
Days
Chapter 13
184
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
b. What is the minimum height above
ground that the point on the tire
reaches?
NAME
DATE
13-7
Study Guide
PERIOD
SCS
MBC.A.8.1, MBC.A.8.2, MBC.A.9.2,
MBC.A.9.4, MBC.A.10.1
Graphing Trigonometric Functions
Sine,
Cosine, and
Tangent
Functions
Parent
Function
y = sin θ
y = cos θ
y = tan θ
Domain
{all real numbers}
{all real numbers}
{θ | θ ≠ 90 + 180n, n is an integer}
Range
{y | -1 ≤ y ≤ 1}
{y | -1 ≤ y ≤ 1}
{all real numbers}
Amplitude
1
1
undefined
Period
360°
360°
180°
Example
Find the amplitude and period of each function. Then graph
the function.
θ
a. y = 4 cos −
1
b. y = −
tan 2θ
3
2
First, find the amplitude.
| a | = | 4 |, so the amplitude is 4.
Next find the period.
The amplitude is not defined, and the
period is 90°.
− = 1080°
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1
−
2
3
O
Use the amplitude and period to help
graph the function.
4
45°
90° 135° 180° θ
-2
y
-4
y = 4 cos θ–3
2
O
y
4
360°
180° 360° 540° 720° 900° 1080°
θ
-2
-4
Exercises
Find the amplitude, if it exists, and period of each function. Then graph the
function.
1. y = -4 sin θ
θ
2. y = 2 tan −
y
y
2
2
O
2
90° 180° 270° 360°
O
θ
-2
Chapter 13
90° 180° 270° 360° 450° 540°
θ
-2
185
North Carolina StudyText, Math BC, Volume 2
Lesson 13-7
Sine, Cosine, and Tangent Functions Trigonometric functions can be graphed on
the coordinate plane. Graphs of periodic functions have repeating patterns, or cycles; the
horizontal length of each cycle is the period. The amplitude of the graph of a sine or cosine
function equals half the difference between the maximum and minimum values of the
function. Tangent is a trigonometric function that has asymptotes when graphed.
NAME
DATE
13-7
Study Guide
(continued)
PERIOD
SCS
MBC.A.8.1, MBC.A.8.2, MBC.A.9.2,
MBC.A.9.4, MBC.A.10.1
Graphing Trigonometric Functions
Graphs of Other Trigonometric Functions
The graphs of the cosecant, secant,
and cotangent functions are related to the graphs of the sine, cosine, and tangent functions.
Cosecant, Secant,
and Cotangent
Functions
Example
Parent Function
y = csc θ
y = sec θ
y = cot θ
Domain
{ θ | θ ≠ 180n, n is
an integer}
{ θ | θ ≠ 90 + 180n, n
is an integer}
{ θ | θ ≠ 180n, n is
an integer}
Range
{y | -1 > y or y > 1}
{y | -1 > y or y > 1}
{all real numbers}
Amplitude
undefined
undefined
undefined
Period
360°
360°
180°
1
Find the period of y = −
csc θ. Then graph the function.
2
1
1
Since − csc θ is a reciprocal of − sin θ, the graphs
2
2
have the same period, 360°. The vertical
1
asymptotes occur at the points where −
sin θ = 0.
4
y
y = 1 csc θ
2
2
2
So, the asymptotes are at θ = 0°, θ = 180°,
1
sin θ and use it to graph
and θ = 360°. Sketch y = −
2
1
csc θ.
y=−
0
90°
180° 270° 360°
θ
-2
2
-4
Exercises
Find the period of each function. Then graph the function.
4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. y = cot 2θ
y
2
0
90°
180° 270° 360°
θ
180° 270° 360°
θ
-2
-4
2. y = sec 3θ
4
y
2
0
90°
-2
-4
Chapter 13
186
North Carolina StudyText, Math BC, Volume 2
NAME
DATE
13-7
Practice
PERIOD
SCS
MBC.A.8.1, MBC.A.8.2, MBC.A.9.2,
MBC.A.9.4, MBC.A.10.1
Graphing Trigonometric Functions
Find the amplitude, if it exists, and period of each function. Then graph
the function.
1
2. y = cot −
θ
4
y
1
y
2
O
90° 180° 270° 360°
O
θ
-2
-2
-4
-4
3
4. y = csc −
θ
4
120° 240° 360° 480°
θ
O
45°
90° 135° 180°
θ
1
6. y = −
sin θ
2
y
1.0
2
2
O
-1
2
y
O
θ
90° 180° 270° 360°
1
5. y = 2 tan −
θ
4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
y
4
2
4
3. y = cos 5θ
2
y
0.5
180° 360° 540° 720°
θ
O
-2
-2
-0.5
-4
-4
-1.0
90° 180° 270° 360°
7. FORCE An anchoring cable exerts a force of 500 Newtons on a pole.
The force has the horizontal and vertical components Fx and Fy. (A force
of one Newton (N), is the force that gives an acceleration of 1 m/sec2
to a mass of 1 kg.)
a. The function Fx = 500 cos θ describes the relationship between the
angle θ and the horizontal force. What are the amplitude and period
of this function?
500 N
θ
Fy
θ
Fx
b. The function Fy = 500 sin θ describes the relationship between the angle θ and the
vertical force. What are the amplitude and period of this function?
π
t, where t is in months and t = 0 corresponds
8. WEATHER The function y = 60 + 25 sin −
6
to April 15, models the average high temperature in degrees Fahrenheit in Centerville.
a. Determine the period of this function. What does this period represent?
b. What is the maximum high temperature and when does this occur?
Chapter 13
187
North Carolina StudyText, Math BC, Volume 2
Lesson 13-7
1. y = -4 sin θ
NAME
13-7
DATE
PERIOD
Word Problem Practice
SCS
Graphing Trigonometric Functions
1. PHYSICS The following chart gives
functions which model the wave patterns
of different colors of light emitted from a
particular source, where y is the height
of the wave in nanometers and t is the
length from the start of the wave
in nanometers.
Color
y
π
= 300 sin (−
t
350 )
Orange
y
π
= 125 sin (−
t
305 )
Yellow
y
π
= 460 sin (−
t
290 )
Green
2. SWIMMING As Charles swims a
25 meter sprint, the position of his right
hand relative to the water surface can be
modeled by the graph below, where g is
the height of the hand in inches from the
water level and t is the time in seconds
past the start of the sprint. What
function describes this graph?
Function
Red
y
=
8
y
6
4
2
π
200 sin −
t
260
(
MBC.A.8.1, MBC.A.8.2,
MBC.A.9.2, MBC.A.9.4,
MBC.A.10.1
)
O
Blue
y
π
= 40 sin (−
t
235 )
-2
Violet
y
π
= 80 sin (−
t
210 )
-4
1.25
2.5
3.75
5
6.25
7.5
8.75
t
10
-6
a. What are the amplitude and period of
the function describing green light
waves?
-8
c. The color of light depends on the
period of the wave. Which color has
the shortest period? The longest
period?
3. ENVIRONMENT In a certain forest, the
leaf density can be modeled by the
π
(t - 3)
equation y = 20 + 15 sin −
(6
)
where y represents the number of leaves
per square foot and t represents the
month where January = 1.
a. Determine the period of this function.
What does this period represent?
b. What is the maximum leaf density
that occurs in this forest and when
does this occur?
Chapter 13
188
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
b. The intensity of a light wave
corresponds directly to its amplitude.
Which color emitted from the source is
the most intense?
NAME
DATE
13-8
PERIOD
Study Guide
SCS
MBC.A.9.4, MBC.A.10.2
Translations of Trigonometric Graphs
Horizontal Translations
When a constant is subtracted from the angle measure in a
trigonometric function, a phase shift of the graph results.
The phase shift of the graphs of the functions y = a sin b(θ - h), y = a cos b(θ - h),
and y = a tan b(θ - h) is h, where b > 0.
If h > 0, the shift is h units to the right.
If h < 0, the shift is ⎪h⎥ units to the left.
Phase Shift
Example
State the amplitude, period, and
π
1
cos 3 θ - −
. Then graph
phase shift for y = −
2
2
the function.
(
)
1.0
1
1
Amplitude: a = −
or −
2
2
2π
2π
2π
=−
or −
Period: −
3
| b|
| 3|
y
0.5
π
6
π
Phase Shift: h = −
π
3
2π
3
π
2
5π
6
π
θ
-1.0
2
π
> 0.
The phase shift is to the right since −
2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
State the amplitude, period, and phase shift for each function. Then graph
the function.
π
2. y = tan θ - −
(
1. y = 2 sin (θ + 60°)
)
y
y
2
2
O
-90°
-2
90° 180° 270° 360°
O
θ
π
2
-2
π
3π
2
2π
θ
π
1
4. y = −
sin 3 θ - −
3. y = 3 cos (θ - 45°)
(
2
y
1.0
2
O
2
3
)
y
0.5
90° 180° 270° 360° 450°
O
-0.5
θ
-2
π
6
π
3
π
2
2π
3
5π
6
π
θ
-1.0
Chapter 13
189
North Carolina StudyText, Math BC, Volume 2
Lesson 13-8
O
-0.5
NAME
DATE
13-8
Study Guide
PERIOD
SCS
(continued)
MBC.A.9.4, MBC.A.10.2
Translations of Trigonometric Graphs
Vertical Translations
When a constant is added to a trigonometric function, the graph
is shifted vertically.
Vertical Shift
The vertical shift of the graphs of the functions y = a sin b(θ - h) + k, y = a cos b(θ - h) + k,
and y = a tan b(θ - h) + k is k.
If k > 0, the shift is k units up.
If k < 0, the shift is ⎪k⎥ units down.
The midline of a vertical shift is y = k.
Step 1
Step 2
Graphing
Trigonometric
Functions
Step 3
Step 4
Determine the vertical shift, and graph the midline.
Determine the amplitude, if it exists. Use dashed lines to indicate the maximum and
minimum values of the function.
Determine the period of the function and graph the appropriate function.
Determine the phase shift and translate the graph accordingly.
Example
State the amplitude, period, vertical shift, and equation of the
midline for y = cos 2θ - 3. Then graph the function.
Amplitude: | a | = | 1 | or 1
2
1
2π
2π
=−
or π
Period: −
| 2|
O
-1
π
2
π
3π
2
2π
θ
Vertical Shift: k = -3, so the vertical shift is 3 units down.
The equation of the midline is y = -3.
Since the amplitude of the function is 1, draw dashed lines
parallel to the midline that are 1 unit above and below the midline.
Then draw the cosine curve, adjusted to have a period of π.
Exercises
State the amplitude, period, vertical shift, and equation of the midline for each
function. Then graph the function.
1
1. y = −
cos θ + 2
2. y = 3 sin θ - 2
2
3
2
1
1
y
O
-1
-2
Chapter 13
π
2
π
3π
2
2π
O
-1
-2
-3
-4
-5
-6
θ
190
y
π
2
π
3π
2
2π
θ
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
| b|
y
NAME
DATE
13-8
PERIOD
Practice
SCS
MBC.A.9.4, MBC.A.10.2
Translations of Trigonometric Graphs
State the amplitude, period, phase shift, and vertical shift for each function. Then
graph the function.
(
2
)
2
2. y = 2 cos (θ + 30°) + 3
y
4
π
2
-2
π
3π
2
2π
O
2
θ
180° 360° 540° 720°
(
4
)
y
0
-1
-2
-3
-4
-5
π
2
π
3π
2
2π
θ
-12
5. y = 3 cos 2(θ + 45°) + 1
4
6. y = -1 + 4 tan (θ + π)
y
4
2
0
θ
-8
θ
-2
π
4. y = -3 + 2 sin 2 θ + −
90° 180° 270° 360°
-4
O
-4
y
4
4
O
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
y
6
2
3. y = 3 csc (2θ + 60°) - 2.5
Lesson 13-8
π
1
1. y = −
tan θ - −
y
2
90° 180° 270° 360° 450° 540°
θ
0
-2
-2
-4
-4
π
2
π
3π
2
2π
θ
7. ECOLOGY The population of an insect species in a stand of trees follows the growth
cycle of a particular tree species. The insect population can be modeled by the function
y = 40 + 30 sin 6t, where t is the number of years since the stand was first cut in
November, 1920.
a. How often does the insect population reach its maximum level?
b. When did the population last reach its maximum?
c. What condition in the stand do you think corresponds with a minimum
insect population?
Chapter 13
191
North Carolina StudyText, Math BC, Volume 2
NAME
DATE
13-8
Word Problem Practice
PERIOD
SCS
MBC.A.9.4, MBC.A.10.2
Translations of Trigonometric Graphs
1. CLOCKS A town hall has a tower with a
clock on its face. The center of the
clock is 40 feet above street level. The
minute hand of the clock has a length
of four feet.
2. ANIMAL POPULATION The population
of predators and prey in a closed
ecological system tends to vary
periodically over time. In a certain
system, the population of snakes
S can be represented by
π
S = 100 + 20 sin −
t , where t is the
(5 )
number of years since January 1, 2000.
In that same system, the population of
rats can be represented by
π
π
R = 200 + 75 sin −
t+−
.
(5
40 ft
10
)
a. What is the maximum snake
population?
a. What is the maximum height of the tip
of the minute hand above street level?
b. When is this population first reached?
c. Write a sine function that represents
the height above street level of the
tip of the minute hand for t minutes
after midnight.
c. What is the minimum rat population?
d. Graph the function from your answer
to part c.
48
44
40
36
32
28
24
20
16
12
8
4
O
Chapter 13
y
d. When is this population first reached?
x
5 10 15 20 25 30 35 40 45 50 55 60
192
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
b. What is the minimum height of the tip
of the minute hand above street level?
NAME
DATE
13-9
PERIOD
Study Guide
SCS
MBC.A.8.1
Inverse Trigonometric Functions
Inverse Trigonometric Functions If you know the value of a trigonometric
function for an angle, you can use the inverse to find the angle. If you restrict the
function’s domain, then the inverse is a function. The values in this restricted domain
are called principal values.
π
π
y = Sin x if and only if y = sin x and - −
≤x≤−
.
Principal Values
of Sine, Cosine,
and Tangent
2
2
y = Cos x if and only if y = cos x and 0 ≤ x ≤ π.
π
π
≤x≤−
.
y = Tan x if and only if y = tan x and - −
2
2
Inverse Sine,
Cosine, and
Tangent
Given y = Sin x, the inverse sine function is defined by y = Sin-1 x or y = Arcsin x.
Given y = Cos x, the inverse cosine function is defined by y = Cos-1 x or y = Arccos x.
Given y = Tan x, the inverse tangent function is given by y = Tan-1 x or y = Arctan x.
Example 1
3
Find the value of Sin-1 −
. Write angle measures in degrees
(√ )
2
and radians.
√3
π
π
≤θ≤−
that has a sine value of − .
Find the angle θ for - −
2
2
2
√3
π
or 60°.
Using a unit circle, the point on the circle that has y-coordinate of − is −
-1
So, Sin
2
( )
3
√3
π
− =−
or 60°.
2
3
(
)
1
Find tan Sin-1 −
. Round to the nearest hundredth.
Example 2
2
π
π
π
1
1
Let θ = Sin −
. Then Sin θ = −
with - −
<θ<−
. The value θ = −
satisfies both
2
2
2
6
2
√3
√3
π
1
conditions. tan −
= − so tan Sin-1 −
= −.
6
3
2
3
(
)
Exercises
Find each value. Write angle measures in degrees and radians.
(√ )
(
3
2
1. Cos-1 −
( 2)
1
3. Arccos - −
(
√2
2
√3
2
)
2. Sin-1 - −
4. Arctan √
3
)
6. Tan-1 (-1)
5. Arccos - −
Find each value. Round to the nearest hundredth if necessary.
)
(
⎡
√2
⎤
7. cos ⎢ Sin-1 - − 2 ⎦
⎣
(
5
9. sin Tan-1 −
12
)
11. cos (Arctan 5)
Chapter 13
⎡
5 ⎤
8. tan ⎢ Arcsin - −
7 ⎦
⎣
( )
10. Cos [Arcsin (-0.7)]
12. sin (Cos-1 0.3)
193
North Carolina StudyText, Math BC, Volume 2
Lesson 13-9
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
-1
NAME
13-9
DATE
Study Guide
PERIOD
SCS
(continued)
MBC.A.8.1
Inverse Trigonometric Functions
Solve Equations by Using Inverses
You can rewrite trigonometric equations to
solve for the measure of an angle.
Example
Solve the equation Sin θ = -0.25. Round to the nearest tenth
if necessary.
The sine of angle θ is -0.25. This can be written as Arcsin (-0.25) = θ.
Use a calculator to solve.
KEYSTROKES:
2nd
[SIN -1]
(–)
.25
ENTER
–14.47751219
So, θ ≈ -14.5°
Exercises
Solve each equation. Round to the nearest tenth if necessary.
2. Tan θ = 4.5
3. Cos θ = 0.5
4. Cos θ = -0.95
5. Sin θ = -0.1
6. Tan θ = -1
7. Cos θ = 0.52
8. Cos θ = -0.2
9. Sin θ = 0.35
Chapter 13
10. Tan θ = 8
194
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. Sin θ = 0.8
NAME
DATE
13-9
PERIOD
Practice
SCS
MBC.A.8.1
Inverse Trigonometric Functions
Find each value. Write angle measures in degrees and radians.
( -3 3 )
( - 2 2 )
√
√
1. Arcsin 1
√2
2
4. Arccos −
2. Cos-1 −
3. Tan-1 −
5. Arctan (- √
3)
1
6. Sin-1 - −
( 2)
Find each value. Round to the nearest hundredth if necessary.
(
2)
(
13 )
1
7. tan Cos-1 −
( )
(
√3
3
)
11. sin Arctan −
9. cos [Arctan (-1)]
(
)
3
12. cos Arctan −
4
Solve each equation. Round to the nearest tenth if necessary.
13. Tan θ = 10
14. Sin θ = 0.7
15. Sin θ = -0.5
16. Cos θ = 0.05
17. Tan θ = 0.22
18. Sin θ = -0.03
19. PULLEYS The equation cos θ = 0.95 describes the angle through which pulley A moves,
and cos θ = 0.17 describes the angle through which pulley B moves. Which pulley moves
through a greater angle?
20. FLYWHEELS The equation Tan θ = 1 describes the counterclockwise angle through
which a flywheel rotates in 1 millisecond. Through how many degrees has the flywheel
rotated after 25 milliseconds?
Chapter 13
195
North Carolina StudyText, Math BC, Volume 2
Lesson 13-9
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
12
10. tan Sin-1 −
⎡
3 ⎤
8. cos ⎢ Sin-1 - −
5 ⎦
⎣
NAME
13-9
DATE
Word Problem Practice
PERIOD
SCS
MBC.A.8.1
Inverse Trigonometric Functions
1. DOORS The exit from a restaurant
kitchen has a pair of swinging doors that
meet in the middle of the doorway. Each
door is three feet wide. A waiter needs to
take a cart of plates into the dining area
from the kitchen. The cart is two feet
wide.
3
3
2. SURVEYING In ancient times, it was
known that a triangle with side lengths
of 3, 4, and 5 units was a right triangle.
Surveyors used ropes with knots at each
unit of length to make sure that an angle
was a right angle. Such a rope was
placed on the ground so that one leg of
the triangle had three knots and the
other had four. This guaranteed that the
triangle formed was a right triangle,
meaning that the surveyor had formed a
right angle.
2
θ
θ
To the nearest degree, what are the
angle measures in a triangle formed in
this way?
b. If only one of the two doors could be
opened, what is the minimum angle θ
through which the door must be
opened to prevent the cart from hitting
the door?
c. If the pair of swinging doors were
replaced by a single door the full width
of the opening, what is the minimum
angle θ through which the door must
be opened to prevent the cart from
hitting the door?
Chapter 13
196
3. TRAVEL Beth is riding her bike to her
friend Marco’s house. She can only ride
on the streets, which run north-south or
east-west.
a. Beth rides two miles east and four
miles south to get to Marco’s. If Beth
could have traveled directly from her
house to Marco’s, in what direction
would she have traveled?
b. Beth then rides three miles west and
one mile north to get to the grocery
store. If Beth could have traveled
directly from Marco’s house to the
store, in what direction would she
have traveled?
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
a. What is the minimum angle θ through
which the doors must each be opened
to prevent the cart from hitting either
door?
NAME
DATE
PERIOD
CSB 9 Study Guide
SCS
MBC.S.2.3
Appropriateness of Linear Models
Sum of the Squared Errors The sum of the squared errors (SSE) is used to
evaluate the appropriateness of a linear model for a set of data. The sum of the squared
errors is the sum of the squared residuals. A residual is the difference between an actual
y-value and the predicted y-value on the regression line.
The table below shows the number of customers serviced by a lawn
Example
company for several years. Calculate the sum of the squared errors.
Year
2002
2003
2004
2005
2006
2007
2008
2009
Customers
13
24
47
68
81
88
109
126
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Step 1 Make a scatter plot of the data and calculate
the least squares regression line. Determine
if the linear model could be appropriate.
The least squares regression equation is
about y = 16.1x - 19.0, where x represents
the number of years since 2000 and y is the
[0, 10] scl: 1 by [0, 150] scl: 15
number of customers. The scatter plot
resembles a straight line, so this model could
be appropriate.
Step 2 Set up a table of values to calculate SSE. The y-value that is predicted by the
regression equation is represented by yp.
(y – yp)2
x
y
yp
y – yp
2
13
13.2
-0.2
0.04
3
24
29.3
-5.3
28.09
4
47
45.4
1.6
2.56
5
68
61.5
6.5
42.25
6
81
77.6
3.4
11.56
7
88
93.7
-5.7
32.49
8
109
109.8
-0.8
0.64
9
126
125.9
0.1
0.01
SSE = 117.64
The sum of the squared errors of using the least squares regression equation to
predict the y-values for the data is 117.64.
Exercise
The table below shows the number of students that failed the
state test on the first attempt each year. Make a scatter plot
of the data and calculate the regression equation. Then
calculate the sum of the squared errors. Let x represent
the number of years since 2000.
Year
2003
2004
2005
2006
2007
2008
2009
Students
112
93
76
51
43
29
18
Concepts and Skills Bank
197
North Carolina StudyText, Math BC, Volume 2
NAME
DATE
PERIOD
CSB 9 Study Guide
SCS
MBC.S.2.3
Appropriateness of Linear Models
Coefficient of Determination As a percentage, the coefficient of determination
provides the likelihood that the related regression line will make an accurate prediction of
SSE
the data. The formula for the coefficient of determination is r2 = 1 - −
, where SSE is the
SST
sum of squared errors and SST is the total sum of squares. The total sum of squares (SST)
is the sum of the squared differences between each y-value and the average y-value.
Example
An ice cream vendor compared the daily sales with the high
temperature. Calculate the coefficient of determination and determine if a linear
model is appropriate.
Temperature(˚F)
Sales ($)
84
88
86
83
89
91
1225
1417
1394
1260
1603
1598
Step 1 Make a scatter plot of the data and calculate the least
squares regression line. Determine if the linear model
could be appropriate. The least squares regression
equation is about y = 49.58x - 2889.35. The scatter plot
resembles a straight line, so this model could be
appropriate.
[80, 100] scl: 2 by [1100, 1700] scl: 50
Step 2 Set up a table of values to calculate SSE and SST. The predicted y-value is yp and
the average of the actual y-values is −
y. Calculate and analyze r2.
SSE
(y – yp)
−
y
SST
y–−
y
(y – −
y)2
x
y
yp
84
1225
1275.37
-50.37
2537.14
1416.17
-191.17
36,545.97
88
1417
1473.69
-56.69
3213.76
1416.17
0.83
0.69
86
1394
1374.53
19.47
379.08
1416.17
-22.17
491.51
83
1260
1225.79
34.21
1170.32
1416.17
-156.17
24,389.07
89
1603
1523.27
79.73
6356.87
1416.17
186.83
34,905.45
91
1598
1622.43
-24.43
596.82
1416.17
181.83
33,062.15
SSE =14,253.99
SST =129,394.84
14,253.99
129,394.84
r2 = 1 - − or about 0.89
The coefficient of determination is about 0.89, so the regression equation is about
89% likely to accurately predict the data. Thus the linear model is appropriate.
Exercise
The adjusted gross domestic product in billions of dollars is shown in the table.
The amounts are compared to those in 2000. Calculate the coefficient of
determination, and determine if a linear model is appropriate.
Year
2004
2005
2006
2007
2008
2009
GDP ($)
42,809
43,969
48,536
53,016
53,249
51,955
Concepts and Skills Bank
198
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
y – yp
2
NAME
DATE
PERIOD
CSB 9 Practice
SCS
MBC.S.2.3
Appropriateness of Linear Models
For each exercise, analyze the appropriateness of the linear model.
a. Make a scatter plot of the data and determine whether the relationship between the
x-values and y-values could be linear.
b. Identify the least squares regression line, rounding to the nearest thousandth.
c. Find the sum of the squared errors.
d. Find the total sum of squares.
e. Calculate the coefficient of determination.
f. Determine whether the linear model is appropriate. Explain your reasoning.
1. EXERCISE Marisa began exercising with a rowing machine. She kept track of her
progress for several weeks.
Average Rowing Time Per Session
Week
1
2
3
4
5
6
7
8
Rowing Time (minutes)
4.3
5.1
5.7
6.4
6.8
7.3
7.4
7.9
2. RETAIL The table below gives the sales of jeans at a department
store chain since 2004. Let x represent the number of years since 2000.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Jeans Sales By Year
Year
2004
2005
2006
2007
2008
2009
Sales (millions of dollars)
6.8
7.6
10.9
15.4
17.6
21.2
3. MARATHON The Boston Marathon has been run each year since 1897. The number of
entrants in several years are shown. Let x represent the number of years since 1975.
Boston Marathon Entrants
Year
1975
1980
1985
1990
1995
2000
2005
2009
Entrants
2395
5417
5594
9412
9416
17,813
20,453
26,331
4. MAID SERVICE The manager of a maid service kept track of the
average amount of time it took her employees to clean a house.
Average Cleaning Times
2
Size of House (ft )
900
Time (minutes)
63
Concepts and Skills Bank
1200 1500 1800 2100 2400 2700 3000
78
94
106
122
199
141
158
172
North Carolina StudyText, Math BC, Volume 2
NAME
DATE
PERIOD
CSB 9 Word Problem Practice
SCS
MBC.S.2.3
Appropriateness of Linear Models
1. BIRTHS The table below shows the total
number of births in the United States
for selected years.
2. EDUCATION The table below shows the
total expenditures by state governments
on education.
Year
Births
(thousands)
Year
Births
(thousands)
Year
1985
3761
2002
4022
1990
1990
4158
2003
4090
1995
1995
3900
2004
4112
1999
1999
3959
2005
4140
2000
4059
2006
4317
2001
4026
2007
4265
Amount
($ million)
Year
Amount
($ million)
7253
2002
16,589
10,042
2003
17,727
12,294
2004
19,632
2000
14,077
2005
20,632
2001
14,936
2006
20,623
Source: Statistical Abstract of the United States
Source: Statistical Abstract of the United States
a. Make a scatter plot of the data and
determine whether the relationship
between the x-values and y-values
could be linear. Let x represent
the number of years since 1990.
a. Make a scatter plot of the data and
determine whether the relationship
between the x-values and y-values
could be linear. Let x represent the
number of years since 1985.
b. Calculate SSE and SST.
c. Calculate the coefficient of
determination. Determine whether
the linear model is appropriate, and
explain your reasoning.
c. Calculate the coefficient of
determination. Determine whether
the linear model is appropriate, and
explain your reasoning.
Concepts and Skills Bank
200
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
b. Calculate SSE and SST.
Name
Date
Diagnostic Test
1
Which is an equation of a circle with
center at (4, -2) and a radius of 6?
3
Which of the following numbers is
a real number?
(x - 4)2 + (y + 2)2 = 36
A
(-2) 3
B
(x + 4)2 + (y - 2)2 = 36
B
(-3) 2
C
(x - 4)2 + (y + 2)2 = 6
C
1
-_
D (x + 4)2 + (y - 2)2 = 6
D
2
__1
A
Tamara graphs the function
f(x) = log2 x as shown below.
4
y
8
6 G(x) = log2 x
4
2
−8−6−4−20
__1
( 3)
(-_12 )
__1
2
__1
4
Parallel lines j and k are cut by a
transversal in the figure below.
1
2 4 6 8x
2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
−4
−6
−8
3
4
She then reflects the graph of f(x) over
the line y = x. What is the equation of
the reflected graph?
j
k
Which statement must be true?
__
A
g(x) = √x
A
∠1 and ∠3 are congruent.
B
g(x) =2x
B
∠2 and ∠4 are supplementary.
C
g(x) =log2 x
C
∠1 and ∠4 are congruent.
D ∠3 and ∠4 are supplementary.
1
D g(x) = −
log2 x
5
Solve log x - log 10x3 = -7 for x.
A
x = 0.18
B
x = 3.5
C
x = 316
D x = 1000
A1
North Carolina StudyText, Math BC, Volume 2
Name
Date
Diagnostic Test
6
(continued)
Tomi created the graph shown below.
8
y
O
x
Residents of North Carolina can buy
lifetime coastal fishing licenses, with
prices determined based on the age
of the applicant. The table below
shows the number of lifetime coastal
fishing licenses sold over a three-day
period.
Age (years)
Which set of steps best describes the
transformation of the graph of f(x) = x2
into the graph that Tomi created?
A
C
a vertical shrink by a factor of 3, a
shift 1 unit right and 5 units down
65 plus
Day 1
8
17
5
Day 2
14
22
8
Day 3
18
25
6
⎡ 8 17 5 ⎤ ⎡x ⎤ ⎡ 5600 ⎤
14 22 8 y = 7840
⎣ 18 25 6 ⎦ ⎣z ⎦ ⎣ 9130 ⎦
⎢
⎢ ⎢
Using the matrix equation, what is the
price of a lifetime coastal fishing
license for an 18 year old applicant?
D a vertical stretch by a factor of 3, a
shift 1 unit right and 5 units down
A
$30
B
$64
C
$150
D $250
7
QRS ~ TUV, QR = 10, RS = 6,
QS = 14, and UV = 4. What is the
perimeter of TUV ?
9
1
−
Simplify (32b5) 4 .
A
45
B
30
A
8b
20
B
D 12
C
2b(2b) 4
2b √2b
C
1
−
1
D 16b(2b)−4
A2
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
a reflection over the x-axis, a
vertical stretch by a factor of 3, a
shift 1 unit right and 5 units down
12 to 64
The following matrix equation can be
used to calculate the price in dollars
of a license for each age range.
a reflection over the x-axis, a
vertical shrink by a factor of 3, a
shift of 1 unit right and 5 units
down
B
1 to 11
Name
Date
Diagnostic Test
10
Which set of statements illustrates the
structure of inductive reasoning?
A
B
C
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
(continued)
12
Thomas Wolfe was born in
Asheville. Asheville is located in
western North Carolina. Therefore,
Thomas Wolfe was born in western
North Carolina.
Many gardeners living in North
Carolina grow the native plant,
foamflower. Tim is a gardener who
lives in North Carolina. Therefore,
Tim grows foamflower in his
garden.
A container shaped like a cone holds
245 cubic centimeters of water. How
many cubic centimeters of water can a
cylindrical container with the same
radius and height hold?
A
81.7 cm3
B
122.5 cm3
C
490 cm3
D 735 cm3
13
The 17th president of the United
States was born in North Carolina.
Andrew Johnson was the 17th
president. Therefore Andrew
Johnson was born in North
Carolina.
D Many tourists visit Cape Hatteras
Lighthouse located in Buxton,
North Carolina. Sally visits Cape
Hatteras Lighthouse. Therefore,
Sally is in Buxton.
The length of a football field is
100 yards. Audrey starts at one end of
the field and walks back and forth in a
straight line from one end of the field
to another. The situation is modeled
by the graph below, where x
represents time in minutes and y
represents distance in yards from her
starting point.
y
80
60
40
20
11
Which equation represents the
relationship between f(x) = 5x - 20
1
and g(x) = −x + 4?
0
f(g(x)) = -f(g(x))
B
f(g(x)) = g(f(x))
C
f(g(x)) = -g(f(x))
D
1
f(g(x)) = −
f(g(x))
2
3
4
x
At what rate is Audrey walking?
5
A
1
A
20 yards per minute
B
50 yards per minute
C
100 yards per minute
D 500 yards per minute
A3
North Carolina StudyText, Math BC, Volume 2
Name
Date
Diagnostic Test
14
(continued)
Janice draws isosceles triangle QRS
−−
with vertex angle R and median RT as
shown below.
15
3
2
5
A store sells three models of a motor
scooter at a discount during June and
July. Matrix N shows the number of
each model sold during each month.
Matrix S shows the regular price and
the sale price for each model.
Model X Model Y Model Z
⎡61
June
45
39⎤
⎢
N=
⎣38
July
29
40⎦
4
Which argument can Janice use to
−−
prove that RT is also an altitude of
QRS?
Regular
Price
($)
A
Model X ⎡1699
S = Model Y 1895
Model Z ⎣2195
⎢
Matrix P is the product of N and S, as
shown below.
⎡P
P12 ⎤
11
P=N×S=⎢
⎣P21 P22 ⎦
Which of the following strategies can
be used to find the total amount of
discounts on all models for the month
of June?
A
Subtract P11 from P12.
B
Subtract P12 from P11.
C
Subtract P21 from P11.
D Subtract P22 from P12.
A4
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Base angles and legs are
congruent, so ∠Q ∠S and
−− −− −− −−
QR RS. RT RT by the reflexive
property. QTR STR by SSA.
Corresponding parts of congruent
triangles are congruent, so
∠RTQ ∠RTS. Supplementary
−− −−
angles are congruent, so RT ⊥ QS.
−−
Therefore RT is an altitude.
−−
B Since the median bisects QS at a
−− −−
right angle, RT ⊥ QS.
−−
Therefore RT is an altitude.
−− −−
C QT TS by the definition of a
median. QTR STR by SSS.
Corresponding parts of congruent
triangles are congruent, so ∠RTQ
∠RTS. Supplementary angles
−− −−
are congruent, so RT ⊥ QS.
−−
Therefore RT is an altitude.
−− −−
D QR RS since legs are congruent.
∠RTQ ∠RTS since they are
opposite congruent sides.
Supplementary angles are
−− −−
congruent, so RT ⊥ QS.
−−
Therefore RT is an altitude.
Sale
Price
($)
999⎤
1199
1399⎦
Name
Date
Diagnostic Test
16
(continued)
The table below shows the
relationship between the number of
pages in each chapter of a book and
the number of spelling errors in the
chapter.
Pages
2
23
22
24
23
17
24
12
Errors
27
45
59
23
28
57
59
43
18
1
x
Add − + −
.
2
A
B
x ≠ -2, x ≠ 0, and x ≠ 2
C
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Using the sum of the squared errors
from the least-squares regression line
for the data, which statement best
describes the appropriateness of a
linear model for this relationship?
A
The sum of the squared errors is
105.37, so a linear model is
appropriate.
B
The sum of the squared errors is
105.37, so a linear model is not
appropriate.
C
The sum of the squared errors is
1757.83, so a linear model is
appropriate.
D
19
4x = −
B
1
=4
(−
16 )
C
4 = 16x
x2 + 4x - 4
−
;
4x(x2 - 4)
Troy buys an ice cream cone at the
concession stand. He can select
vanilla, chocolate, or strawberry, and
he can choose either 1, 2, or 3
toppings. Which sample space
represents Troy’s choices?
A
(V, C), (V, S), (C, S), (1, 2), (1, 3),
(2, 1), (2, 3), (3, 1), (3, 2)
B
(V, 1), (V, 2), (V, 3), (C, 1), (C, 2),
(C, 3), (S, 1), (S, 2), (S, 3)
C
(V, 1), (C, 2), (S, 3), (V, 1), (C, 2),
(S, 3), (V, 1), (C, 2), (S, 3)
D (V, C), (V, S), (S, C), (V, 1), (V, 2),
(V, 3), (C, 1), (C, 2), (C, 3), (S, 1),
(S, 2), (S, 3)
1
16
A
4x(x - 4)
x ≠ -2, x ≠ 0, and x ≠ 2
Which equation is equivalent to
1
log4 − = x?
16
2
5x - 4
−
;
2
x ≠ -2, x ≠ 0, and x ≠ 2
D The sum of the squared errors is
1757.83, so a linear model is not
appropriate.
17
4x
x -4
5
−; x ≠ -2, x ≠ 0, and x ≠ 2
4x
1+x
−
;
x2 + 4x - 4
x
1
−
D 4 16 = x
A5
North Carolina StudyText, Math BC, Volume 2
Name
Date
Diagnostic Test
20
(continued)
Triangle EFG is transformed into
triangle E´F´G´ as shown below.
22
y
&'
'
('
x
0
(
''
&
The coordinate matrix for EFG is
multiplied by matrix T, resulting in the
coordinate matrix for E´F´G´. What is
matrix T?
⎡ 0⎤
A T = ⎢1 ⎣0 1⎦
B
⎡-1 0⎤
T=⎢
⎣ 0 1⎦
C
⎡ -1 0 ⎤
T=⎢
⎣ 0 -1 ⎦
A
2 feet 2 inches
B
4 feet 2 inches
C
6 feet 2 inches
D 8 feet 2 inches
Quadrilateral FGHJ below is a
parallelogram.
)
(
-
'
A metronome’s pendulum completes
a cycle every 2 seconds. It has a
center point of zero and swings a total
distance of 18 centimeters. At t = 0,
the pendulum is at equilibrium and is
starting to swing to the right. Which
equation describes the motion of the
pendulum cycle?
A
y = 9 sin πt
B
y = 9 sin −t
C
y = 18 sin πt
π
2
π
D y = 18 sin −
t
2
+
What must be true to prove that
FGHJ is a rectangle?
−− −−
−− −−−
A FJ GH and FG HJ
−− −−
B FH GJ
−− −−
−− −−
C GL LJ and FL LH
D ∠HGJ ∠FJG
A6
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
23
⎡0 1⎤
D T=⎢
⎣1 0⎦
21
In 1903, the Wright brothers took the
first controlled-power flight at Kitty
Hawk. Two propellers pushed their
plane as they completed their
12-second, 120-foot flight. Each
propeller was 8 feet in diameter and
rotated by a chain-and-sprocket
transmission system. If the plane was
not moving and the edge of the
propeller was 2 inches above the
ground at its lowest point, how far
above the ground would that same
edge be, after the propeller rotated
990° counterclockwise?
Name
Date
Diagnostic Test
24
(continued)
Which statement supports the
Pythagorean Theorem?
26
Felix graphed a function as shown
below.
8
6
4
2
Y
A
B
[
−8−6−4−20
Z
−4
−6
−8
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2 4 6 8x
Which type of function did Felix
graph?
1
The area of the triangle is −xy.
2
The area of the smallest square
1
is − the area of the largest square.
A
linear
B
quadratic
The area of the largest square is
equal to the sum of the areas of
the smaller two squares.
C
radical
3
C
y
D absolute value
D The area of the largest square is
greater than the sum of the areas
of the smaller two squares.
27
and RS
are secants to
Given: QR
= 87°, and mTU
= 23°.
circle O, mQS
2
25
A ball is released from the top of a
ramp. Which of the following variables
would most likely have a strong
negative correlation with the time t
after the ball is released?
3
5
6
A
the speed s of the ball
What is m∠QRS?
B
the height h of the ball above the
ground
A
26°
B
32°
C
the distance d traveled by the ball
C
55°
D the weight w of the ball
0
4
D 64°
A7
North Carolina StudyText, Math BC, Volume 2
Name
Date
Diagnostic Test
28
(continued)
Colin finds the median-fit line for the
data in the table below.
x
3
7
12
10
9
16
5
14
8
y
23
35
53
40
27
56
17
32
19
30
Which ordered pairs most likely
represents the three median points
he used to find the median-fit line?
A
(5, 17), (9, 27), and (14, 32)
B
(5, 19), (9, 32), and (14, 53)
C
(5, 23), (9, 27), and (14, 53)
A
B
(f - g)(x) = 2(x2 - 2x - 1)
C
(f - g)(x) = -2(x2 - 2x - 1)
(f - g)(x) = 4(x2 - 2x + 2)
D (f - g)(x) = 2(x2 - 4x - 1)
D (7, 35), (9, 27), and (14, 32)
31
The graph below shows the feasible
region for the production of ballasts
and wickets.
In the figure below, h represents the
height of the tree, and sin x = 0.5.
Wickets
29
Given:
f(x) = 3(x - 1)2
g(x) = x2 - 2x + 5
Which equation represents (f - g)(x)?
h
h
0
C
60 ft
)
x
10 20 30 40 50 60 70 80 90
The profit on a ballast is $7. The profit
on a wicket is $3. Which point
represents the maximum profit, given
the constraints?
How tall is the tree?
16 ft
(
Ballasts
x°
A
B
'
30 ft
D 64 ft
A
B
E
C
G
F
D H
A8
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
32 ft
y
90
&
80
70
60
50
40
30
20
10
Name
Date
Diagnostic Test
32
(continued)
Bethany won the Geography Bee
trophy. The globe has a diameter of
12 inches, and the dimensions of the
base are shown in the diagram below.
34
The graph of f (x) = 3(x - 1)2 - 3 is
shifted up 2 units. Which graph
represents the transformation?
y
A
12 in.
x
O
4 in.
18 in.
y
B
12 in.
What is the approximate volume of
the trophy?
905 in3
C
1015 in3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
y
C
D 1769 in3
33
x
O
864 in3
A
B
x
O
Which expression is equivalent to
x-3
x
x
−
+ −
- −
?
2
x-2
A
x - 5x + 6
x-3
3
x-2
-−;
y
D
x ≠ 2 and x ≠ 3
B
-x + 9
x - 5x + 6
−
;
2
O
x
x ≠ 2 and x ≠ 3
C
x-3
−
;
2
x - 5x + 6
x ≠ 2 and x ≠ 3
D
3x + 9
x - 5x + 6
−
;
2
x ≠ 2 and x ≠ 3
A9
North Carolina StudyText, Math BC, Volume 2
Name
Date
Diagnostic Test
35
(continued)
Randy shades several squares in his
grid paper as shown below.
37
If x = 12 in the right triangle below,
then what is the value of y?
x
60°
What is the probability that a
randomly selected point on the paper
lies in a shaded square?
A
B
0.15
C
0.6
A
B
C
y
4
4 √
3
6 √2
D 24
0.4
D 0.75
38
Max dropped an object from a
window 100 feet above ground level.
The equation -16t2 + 100 = 0 can be
used to determine the time in seconds
it will take the object to hit the
ground. About how long will the
object be in the air before it hits the
ground?
A
B
2.5 seconds
C
6.25 seconds
A
a shift 1 unit up and a shift 4 units
right
B
a shift 3 units right
C
a shift 1 unit up and a shift 4 units
left
D a shift 4 units up and a shift 1 unit
right
4 seconds
D 9.2 seconds
A10
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
36
Which of the following phrases best
describes the translation of the graph
of f(x) = (x + 2)2 + 1 to the graph of
f(x) = (x - 2)2 + 2 in the coordinate
plane?
Name
Date
Diagnostic Test
39
(continued)
Yvonne is calculating the total area A
for a garden with a walkway. The
walkway will surround the garden on
all sides. The garden will be 5 meters
in width and n meters in length. The
walkway will be 1 meter in width. The
table below shows the relationship
between the length of the garden n
and the total area A, including garden
and walkway.
41
Simplify the equation below. What
restrictions must be placed on x?
2x2 + 6x
x -9
y=−
2
A
2x
x+3
y=−;
x ≠ -3 and x ≠ 3
B
2x
x+3
y=−;
x ≠ -3, x ≠ 0, and x ≠ 3
Length, n
(m)
10
15
20
25
30
Area, A
(m2)
84
119
154
189
224
C
x ≠ -3 and x ≠ 3
2x
D y=−
;
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Which function best describes the
relationship between length n and
total area A?
A
A = 5n + 34
B
A = 6n + 24
C
A = 7n + 14
x-3
x ≠ -3, x ≠ 0, and x ≠ 3
42
D A = n2 - 16
40
What is the domain and range of
f(x) = sin x?
A
2x
x-3
y=−;
Which of the following is sufficient to
guarantee congruence of
quadrilaterals?
A
AAAS
B
SSSS
C
SASA
D SASAS
Domain = all real numbers
Range = all real numbers
B
Domain = -1 ≤ x ≤ 1
Range = all real numbers
C
Domain = all real numbers
Range = -1 ≤ f (x) ≤ 1
D Domain = -1 ≤ x ≤ 1
Range = -1 ≤ f (x) ≤ 1
A11
North Carolina StudyText, Math BC, Volume 2
Name
Date
Diagnostic Test
43
(continued)
If g(x) = x2 - 4x + 7 and
h(x) = 5x2 + x - 1,
what is g(x) + h(x)?
45
A circle is graphed on the coordinate
plane as shown below.
y
A
(g + h)(x) = 5x - 3x + 6
B
(g + h)(x) = 6x2 - 5x + 6
C
(g + h)(x) = 6x2 - 3x + 6
2
D (g + h)(x) = 6x2 - 3x + 8
44
0
What is the equation of a congruent
circle that is a reflection across the
y-axis?
Which statement was used to create
the truth table below?
4 math
credits
Yes
4 English
credits
Requirements
met?
A
(x - 3)2 + (y - 2)2 = 16
Yes
Yes
B
(x - 3)2 + (y + 2)2 = 16
Yes
No
No
C
(x - 3)2 + (y - 2)2 = 4
No
Yes
No
D (x + 3)2 + (y - 2)2 = 16
No
No
No
High school students must earn
4 math credits or 4 English credits
to meet the course of study
requirements for admission to the
University of North Carolina.
B
High school students must earn
4 math credits and 4 English
credits to meet the course of study
requirements for admission to the
University of North Carolina.
46
If high school students complete
4 math credits, then they will
complete 4 English credits.
The vertices for triangle RST are
R(2, -3), S(3, 4), and T(-3, 1). If RST is
represented by the vertex matrix M,
which of the following transformations
will result in a vertex matrix for a
triangle that is congruent to RST?
A
⎡-3 -3 -3⎤
2M + ⎢
⎣ 5
5
5⎦
B
M+M
C
⎡ 2
M-⎢
⎣-5
⎡
D M + ⎢1
⎣2
D If high school students complete
4 math credits and 4 English
credits, then they will attend the
University of North Carolina.
A12
2
2⎤
-5 -5⎦
2
4
3 ⎤
6⎦
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A
C
x
Name
Date
Diagnostic Test
47
(continued)
In the figure shown below, which
given information would be sufficient
to prove that XYZ WZY by
ASA?
49
Given: Polygon EFGHJ ∼ KLMNP.
11 ft
&
A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
50
x +1
x
−
÷ −
?
x2 + 5x
x2 + 3x - 10
C
D
x+1 x-2
−
x
x ·−
B
4
−
ft
B
3
16
−
ft
3
C
12 ft
D 24 ft
Which expression is equivalent to
x+1
x(x + 5) (x - 2)(x + 5)
x+1 x-2
−
·−
x
x2
x+1 x+2
−
x
x ·−
(
If LM = 27 feet, what is the length
−−−
of MN ?
;
A
18 ft
'
8
−− −−−
−− −−−
A XY ZW and XZ YW
−−− −−
−− −−−
B YW XZ and XY WZ
−−− −−
−− −−−
C YW XZ and XZ YW
−− −−−
D ∠X ∠W and XY ZW
48
8 ft
9 ft
:
9
+ 2 ft
)
Subtract
⎡6
⎢
⎣8
-3
-10
⎡-12 -3 -7⎤
-7⎤
-2⎢
.
⎣-4
5 -8⎦
-8⎦
A
⎡-18
⎢
⎣ 16
3 7⎤
0 8⎦
B
⎡30
⎢
⎣ 0
C
⎡-18
⎢
⎣ 0
1
−· −
-9 -21 ⎤
-20 -24 ⎦
-9
0
-21⎤
-24⎦
⎡30
3 7⎤
D ⎢
⎣16 -20 8⎦
A13
North Carolina StudyText, Math BC, Volume 2
Name
Date
Diagnostic Test
51
(continued)
Which of the following expressions is
3
equivalent to −−1 ?
53
1
The graph of f(x) = -−
(x - 3)2 + 2 is
2
shown below.
(27x 3) 2
A
B
C
3
2
1
√
3x
−
3x2
√
3x
_
O
3x|x|
1 2 3 4 5 6 7x
−2
−3
−4
−5
1
_
x|x|
√
3x
3x
Use the graph. What are all the zeros
of the function?
D −
52
y
Quadrilateral EFGH below is a
parallelogram.
A
1 and 5
B
1, 3, and 5
C
-2.5, 1, and 5
D -2.5, 1, 3, and 5
&
'
8
)
54
8
(
Which argument proves that EFGH is
also a square?
A
B
C
Since all the sides are congruent,
EFGH is a square.
−− −−
Since EG FH, EFGH is a square.
Since the diagonals are congruent
−− −−−
and EF HG, EFGH is a square.
D Since the diagonals are congruent
−− −−
and EF FG, EFGH is a square.
There are 4 yellow stickers and 3 blue
stickers in a bag. Evan pulls out a
yellow sticker and keeps it. Then
Karen pulls out a blue sticker. In which
situation would the chances be the
greatest for the next student after
Karen to pick a blue sticker?
A
Karen keeps the blue sticker.
B
Karen returns the blue sticker to
the bag.
C
Karen returns the blue sticker to
the bag. Then she pulls out a
different blue sticker and keeps it.
D Karen returns the blue sticker to
the bag. Then she pulls out a
yellow sticker and keeps it.
A14
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
8
8
+
Name
Date
Diagnostic Test
55
(continued)
A 120-foot bridge support column is
stabilized by four support cables using
a total of 1290 feet of cable.
57
Which are logically equivalent?
A
conditional and inverse
B
contrapositive and conditional
C
converse and inverse
D inverse and contrapositive
58
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Approximately how far from the base
of the column is each support cable
planted? Assume that all cables are
the same length.
A
299.3 ft
B
344.1 ft
C
1197.4 ft
D 1284.4 ft
56
Researchers at Duke University in
Durham, North Carolina, are
developing techniques to make
objects invisible. The techniques
involve refraction, which is the
bending of light as it moves through
transparent materials such as water or
glass. The index of refraction is the
ratio of the speed of light through a
material and the speed of light
through a vacuum. The index is given
x
by the function f(x) = −8 , where x
3 × 10
is the speed of light through the
material and 3 × 10 8 is the speed of
light through the vacuum, both are
measured in meters per second. If
there is another function, g, such that,
g(f(x)) = x, what is g?
Simplify log 6 6x m.
A
g(x) = (3 × 10 8)(x)
A
m log6 x
B
3 × 10
g(x) = −
x
B
1 + m log6 x
C
C
6 + m log6 x
D
1
(3 × 10 )(x)
x
g(x) = - −
3 × 108
D m + m log6 x
A15
8
g(x) = −
8
North Carolina StudyText, Math BC, Volume 2
Name
Date
Diagnostic Test
59
(continued)
Emily draws a point and a line on a
piece of paper. She folds the paper at
the point so that her original line
coincides with itself as shown in the
diagram below.
60
Which graph represents the function
f(x) = 2x 2 - 3?
y
A
x
0
y
B
x
0
y
C
x
0
A
If consecutive interior angles are
supplementary, then lines are
parallel.
B
If alternate interior angles are
congruent, then lines are parallel.
C
Given a line and a point not on
that line, there is exactly one line
perpendicular to the given line
which contains the given point.
y
D
0
x
D There are an infinite number of
perpendicular lines to a given line.
A16
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Which statement represents the
geometric relationship illustrated by
this fold?
Name
Date
Diagnostic Test
61
(continued)
A store sells red, blue, black, orange,
and white T-shirts in small, medium,
and large sizes. The information is
organized in a matrix, with each
element representing the number of
T-shirts sold for a different color and
size combination. Which of the
following statements best describes
the matrix?
A
The matrix has 8 elements.
B
The matrix has 5 rows and
3 columns.
C
63
12 in.
x
The matrix has 3 more rows than
columns.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4.0 feet
B
8.8 feet
C
15.0 feet
x
x
x
x
x
x
What is the value of x if the base of
the bin has an area of 32 square
inches?
In 1979, one of the world’s largest
windmills was built on top of Howard’s
Knob, a mountain in Boone, North
Carolina. Although the experimental
windmill was designed to power 300
to 500 average-sized homes, it had to
be shut down in 1983 because it was
too loud. The windmill consisted of
two blades, each of which measured
30 meters from the center of the
windmill to the tip. If the tip of one
blade rotated 30° counterclockwise
from a straight upward position, what
was the approximate vertical distance
from the initial position of the blade
to its finishing position?
A
x
8 in.
D The matrix has 15 columns and
1 row.
62
Lakisha is making a bin out of a flat
rectangular piece of cardboard. The
cardboard is 12 inches by 8 inches.
She cuts four squares of length x, in
inches, out from the corners. Then she
folds the cardboard up along the
dotted lines to make the sides of
the bin.
A
1
B
2
C
4
D 8
64
Max dropped an object from a
window 100 feet above ground level.
The function f(t) = -16t 2 + 100 can
be used to determine the height of
the object after t seconds.
Approximately how long will the
object be in the air before it hits the
ground?
A
0.4 second
B
1.25 seconds
C
2.5 seconds
D 6.25 seconds
D 26.0 feet
A17
North Carolina StudyText, Math BC, Volume 2
Name
Date
Diagnostic Test
65
(continued)
Rosa wants to organize a bake sale to
raise money for her environmental
care club. She makes a list of all the
necessary tasks, approximately how
many days each task will take to
complete, and what tasks must
precede other tasks.
66
Pilar creates the graph below to
display the least-squares regression
line for the eight data points shown.
45
40
35
30
25
20
15
10
5
Bake Sale
Task
Code
1
Task
Time
(days)
Prior
Task
y
x
Pick recipes and
date of bake sale
2
2
Assign recipes to
volunteer bakers
7
1
3
Rent tables
1
1
4
Design posters
and flyers
5
1
5
Print posters and
flyers
2
4
She calculates the slope m of the
least-squares regression line and the
correlation coefficient r for the data.
Which of the following statements
best describes how m and r would
change if the point with coordinates
(40, 49) is added to the data?
6
Advertise
14
5
A
Both m and r get closer to 0.
7
Collect finished
baked goods and
wrap in individual
serving sizes
B
Both m and r get closer to -1.
2
2
C
While m gets closer to 0, r gets
closer to -1.
Set up table and
lay out baked
goods
1
6,7
5 10 15 20 25 30 35 40 45
D While m gets closer to -1, r gets
closer to 0.
Rosa has enough help that many tasks
can be in progress at one time. What
is the minimum number of days
needed for Rosa’s bake sale?
67
What is the solution to the equation
4 x = 12?
A
17 days
B
20 days
C
24 days
B
D 34 days
C
A
log 12
log 4
log 4
x=−
log 12
x=−
x=3
1
D x=−
3
A18
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
8
0
Name
Date
Diagnostic Test
68
(continued)
Given:
71
∠WXZ and ∠ZXY form a linear pair.
Using the given statement, which
conclusion can be inferred?
A
∠WXZ ∠ZXY
B
m∠WXZ + m∠ZXY = 90
C
Points W, X, and Y are collinear.
Laura plays softball, soccer, and
basketball. She has one game every
Saturday at the possible fields listed
below.
.BTPO'JFME
4PGUCBMM
1SPTQFDU'JFME
%SFX'JFME
"SMJOHUPO'JFME
4PDDFS
D ∠WXZ and ∠WXY are
supplementary.
3FBEJOH'JFME
.BTPO'JFME
"SMJOHUPO'JFME
#BTLFUCBMM
-PDLT'JFME
1SPTQFDU'JFME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
69
What is the solution to the system of
equations shown below?
⎧ 2x - 3y + 4z = 3
⎨
x+y+z=6
4x
8y + 4z = 12
⎩
If all options are equally likely, what is
the probability that she will have a
game at Mason Field on a random
Saturday?
A
2
−
A
(0, 4, 2)
B
2
−
B
(7, 1, -2)
C
no solutions
C
1
−
6
9
Which function is the inverse of
1
f(x) = − x + 4?
2
9
1
D −
D infinitely many solutions
70
3
72
1
2
Which function has a graph for which
f(x)→ ∞ as x → -∞ and as x → ∞?
A
f -1(x) = - −x - 4
A
f(x) = x 5 - x 4 + x 3
B
f -1(x) = 2x - 8
B
f(x) = x 6 - x 5 + x 4
C
f -1(x) = 2x - 4
C
f(x) = - x 5 - x 4 + x 3
D f -1(x) = 2x + 8
D f(x) = - x 6 - x 5 + x 4
A19
North Carolina StudyText, Math BC, Volume 2
Name
Date
Diagnostic Test
73
(continued)
Which graph represents the system of
inequalities below?
74
2x - y + 2 < 0
x+2≤0
y
A
What are the x- and y-intercepts of
the graph of g(x) = x 3 + 8?
A
x-intercept = (0, 8)
y-intercept = (2, 0)
B
x-intercept = (2, 0)
y-intercept = (0, 8)
C
x-intercept = (-2, 0)
y-intercept = (0, 8)
x
0
D x-intercept = (-2, 0)
y-intercept = (0, -8)
y
B
75
x
0
In Greenville, North Carolina, the
Ground Cloud is a 12-foot circular
fountain illuminated at night. When
Maria looks at the fountain, only the
portion shown below is illuminated.
12 ft
x
0
If the edge of the lighted portion of
the fountain is 4 feet long, what is the
arc measure of the lighted portion?
y
D
0
A
9.4˚
B
12.7˚
C
19.1˚
D 38.2˚
x
A20
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
y
C
Name
Date
Diagnostic Test
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
76
(continued)
North Carolina is the largest producer
of sweet potatoes in the United States.
The table below shows sweet potato
production for the years 2000 to 2008.
77
Which table of values best represents
a radical function?
A
x
f(x)
1
-1
Years since
2000
0
Production
(millions of lb)
4
-4
555
9
-9
1
558
16
-16
2
481
25
-25
3
588
4
688
x
f(x)
5
595
1
1
6
4
2
702
7
9
3
666.5
8
16
4
874
25
5
x
f(x)
1
2
4
8
B
Using the least-squares regression line
for the data, predict the approximate
production of sweet potatoes in North
Carolina for the year 2020.
C
A
684 million pounds
9
18
B
839 million pounds
16
32
C
1009 million pounds
25
50
x
f(x)
1
1
4
16
9
81
16
256
25
625
D 1181 million pounds
D
A21
North Carolina StudyText, Math BC, Volume 2
Name
Date
Diagnostic Test
78
(continued)
A pizza delivery driver needs to
deliver a pizza to Gary’s house.
Possible routes with the travel times in
minutes are shown in the vertex-edge
graph below.
Post
Office
Pizza
Shop
6
3
9
5
8
Mall
3 Pharmacy
Given: Triangle LMN is an isosceles
triangle with vertex angle M.
Prove: Angles 1 and 2 are congruent.
.
Library
5
Cinema
2
80
4
5
Park
10
2
-
Gary’s
Home
1
2
/
Which statement would be made in an
indirect proof?
School
What is the earliest time Gary can
expect his pizza if the driver leaves
the pizza shop at 7:10 PM?
A
Triangle LMN is not an isosceles
triangle.
B
Angle M is not a vertex angle.
A
7:11 PM
C
Angles 1 and 2 are not congruent.
B
7:21 PM
C
7:23 PM
D Points L, M, and N do not form a
triangle.
79
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
D 7:31 PM
What are all the values of x for which
x 2 - 5x + 4
x + 3x - 4
is discontinuous?
f(x) = −
2
A
-4, 1, 4
B
1, 4
C
-4, 1
D -4
A22
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice By Standard
Clarifying Objective MBC.A.4.1
1
x+1
5
Add _ + _
.
2
x-2
A
B
C
D
2
x2 + x - 6
2
x_
+ 9x + 6
; x ≠ 2 and x ≠ -3
x2 + x - 6
x+6
_
; x ≠ 2 and x ≠ -3
x-2
C
D
between their average speeds. Which
expression is equivalent to
x_
+ 4x + 8
; x ≠ 2 and x ≠ -3
x2 + x - 6
110
110
_
_
x - x+2 ?
2x
2
-x
- 3x + 2
__
;x≠0
2x 2
x+2
_
;x≠0
2x 2
x 2 + 3x + 2
__
;x≠0
2x 2
5
x -4
1
Divide _
÷_
.
2
3x + 3
x -x-2
2
B
x - 2; x ≠ -1 and x ≠ 2
C
3x + 6; x ≠ -1 and x ≠ 2
110
_
; x ≠ 0 and x ≠ - 2
B
110
_
; x ≠ 0 and x ≠ - 2
C
220
_
; x ≠ 0 and x ≠ - 2
(x + 2)
(x)(x + 2)
(x + 2)
(x)(x + 2)
2x 2
x + 2; x ≠ -1 and x ≠ 2
A
220
D _
; x ≠ 0 and x ≠ - 2
1 - 3x 2
_
;x≠0
A
Janelle and Rick are both riding in a
110-mile bike race from Charlotte,
North Carolina, to Chapel Hill, North
Carolina. Janelle completes the course
in x hours and Rick completes the
course in x + 2 hours. The expression
110
110
_
_
x - x + 2 models the difference
2
x
B
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
x+8
_
; x ≠ 2 and x ≠ -3
x+3
1
Subtract _2 - _.
A
3
4
x +x-6
x 2 - 3x
x2 - 4
Multiply _ · __
.
3
2
x+2
x - 5x + 6x
A
1; x ≠ 0, x ≠ 2, x ≠ -2, and x ≠ 3
B
x; x ≠ 2, x ≠ -2, and x ≠ 3
C
x + 2; x ≠ 2, and x ≠ -2
D
x-3
_
; x ≠ 2, and x ≠ -2
x+2
D 3x - 6; x ≠ -1 and x ≠ 2
A23
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice By Standard
Clarifying Objective MBC.A.4.2
1
Celia is explaining how to simplify the
3
x 2 - 3x - 4
rational expression _
. She
x 2 - 2x - 8
Which expression is the simplest form
x-3
x-3
?
of _ ÷ _
2
x+2
says that the first step is to factor
both the numerator and the
denominator. Which expression is
A
x 2 - 3x - 4
equivalent to __
?
2
2
(x + 1)(x + 4)
__
B
(x - 1)(x + 4)
__
C
(x + 1)(x - 4)
__
D
(x - 1)(x - 4)
__
B
(x + 4)(x + 2)
C
(x + 4)(x - 2)
(x - 3)(x 2 - 4);
x ≠ -2, x ≠ 2, and x ≠ 3
(x
- 3)(x - 2)
__
;
(x - 3)(x + 2)
x ≠ -2, x ≠ 2, and x ≠ 3
D x - 2;
x ≠ -2, x ≠ 2, and x ≠ 3
(x - 4)(x + 2)
(x - 4)(x - 2)
4
Which expression is equivalent to
x 2 - 2x
2x - 10
_
·_
?
1+ _
x-2
x 2 - 4x - 5
x-2
A _
; x ≠ 1 and x ≠ 2
x-1
A
x-2
B _
; x ≠ 1 and x ≠ 2
3x
3x
x 2 - 2x
_
· _;
x 2 + 4x - 5 2x - 10
x ≠ -1, x ≠ 0, and x ≠ 5
2
B
1
C _
; x ≠ 1 and x ≠ 2
x-1
2
x-2 _
_
· ;
x+1 3
x ≠ -1, x ≠ 0, and x ≠ 5
x-2
_
; x ≠ 1 and x ≠ 2
C
x-3
1
2x - 10
_
·_
;
x-3
3
x ≠ -1, x ≠ 0, and x ≠ 5
1
1
D _
· _;
x + 1 3x
x ≠ -1, x ≠ 0, and x ≠ 5
A24
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1
What is the simplest form of _
?
1
D
(x
- 3)(x 2 - 4)
__
;
(x + 2)(x - 3)
x ≠ -2, x ≠ 2, and x ≠ 3
x - 2x - 8
A
x -4
Name
Date
Practice By Standard
Clarifying Objective MBC.A.5.1
1
If f (x) = 4x 2 - 5x + 2 and
g (x) = x 3 + 2x, what is f (x) - g (x)?
A
f (x) - g (x) = 3x 2 - 7x + 2
B
f (x) - g (x) = -x 3 + 4x 2 - 7x + 2
C
f (x) - g (x) = x 3 + 4x 2 - 7x + 2
4
What is the product of the functions
x +3
x2 - 9
?
f (x) = _ and g (x) = _
2
x+2
A
B
D f (x) - g (x) = -x 3 + 4x 2 - 3x + 2
C
x + 5x + 6
x+3
(f · g)(x) = _ ; x ≠ 3
x-3
x_
-3
(f · g)(x) =
; x ≠ -3
x+3
x2 - 9
(f · g)(x) = _
;
2
x + 4x + 4
x ≠ -2 and x ≠ -3
2
x 2 + 6x + 9
D (f · g)(x) = _
;
2
If g (x) = 3x - 4 and h (x) = x 2 - 1,
what is g (x) + h (x)?
A
g (x) + h (x) = 3x 3 + 4
B
g (x) + h (x) = 4x 2 - 5
C
g (x) + h (x) = -x 2 - 3x + 5
x -4
x ≠ -2 and x ≠ 2
5
D g (x) + h (x) = x 2 + 3x - 5
If h (x) = - 4x and j (x) = 6x, what is
h (x) ÷ j (x)?
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A
3
_2
C
3
2
B -_
A cell phone service charges access
fees and late fees. Both fees are
percentages of the basic monthly
service cost. Suppose x represents the
cost of basic monthly service,
f (x) = 0.03x represents the access fee,
and g (x) = 0.05x represents the late
fee. Which function can be used for
the total of both fees, f + g?
3
6
_2 x
3
2
D -_
x
3
Which function represents
f (x) · g (x) + f (x) if f (x) = x + 3
and g (x) = x - 5?
A
f (x) · g (x) + f (x) = 2x 2 + 4x - 6
A
(f + g)(x) = 0.08x
B
f (x) · g (x) + f (x) = x 2 - 2x - 15
B
(f + g)(x) = 1.08x
C
f (x) · g (x) + f (x) = x 2 - x - 12
C
(f + g)(x) = x + 0.08
D f (x) · g (x) + f (x) = 3x + 1
D (f + g)(x) = x + 1.08
A25
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice By Standard
Clarifying Objective MBC.A.5.2
1
Which function is the inverse of
f (x) = 3x - 1?
A
f -1 (x) = 3x + 1
B
f -1 (x) = - 3x - 1
C
x-1
f -1 (x) = _
3
The graph of a function is shown
below.
4
3
2
1
3
2
Which function could be the inverse of
the function in the graph?
2
A f (x) = _
x-1
B
D
A
y=_x+2
B
y=-_x+4
C
y = 2x + 2
1
2
1
2
D y = - 2x + 4
1
_3
4
2
Which table represents a function that
has an inverse that is also a function?
A
1
B
C
D
A26
x
1
2
3
4
f(x)
2
4
3
4
x
1
2
3
4
f(x)
4
3
2
1
x
2
4
6
8
f(x)
1
1
1
1
x
2
4
6
8
f(x)
6
2
8
6
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
C
1 2 3 4x
-2
-3
-4
Which functions are inverses of each
other?
3
3
3
g (x) = _x + _
2
2
2
f (x) = _x - 1
3
2
g (x) = - _x +
3
3
3
f (x) = _x + _
2
2
3
g (x) = - _x 2
3
3
f (x) = _x + _
2
2
2
_
g (x) = - x 3
O
-4-3-2
x+1
D f (x) = _
3
-1
y
Name
Date
Practice By Standard
Clarifying Objective MBC.A.5.3
1
The functions g (x) and g -1(x) are
inverses of each other. What is the
value of g -1(g(x))?
A
0
B
1
C
4
If the functions f (x) and f -1(x) are
inverses of each other, what
generalization can be made about
f (f -1(x)) and f -1(f (x))?
A
B
-1
C
D x
D
2
The highest temperature on record in
North Carolina is 110°F, recorded in
Fayetteville in 1983. The function
5
f (x) = _ (x - 32) converts degrees
9
Fahrenheit to degrees Celsius and the
5
function f -1(x) = _ x + 32 converts
5
degrees Celsius to degrees Fahrenheit.
What is the value of f -1(f (110))?
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
9
A
43°F
B
78°F
C
110°F
D 144°F
3
f (f -1(x)) = f
-1
-1
(f (x))
-1
f (f (x)) + f (f (x)) = 0
f (f -1(x)) = _
-1
1
f (f(x))
1
-1
f (f (x)) = _
f (f -1(x))
Suppose f (x) = 4x + 2 and
1
1
g (x) = _x - _. If f (g(x)) = x,
4
2
which statement best describes the
relationship between the functions
f and g?
A
The functions f and g are
equivalent.
B
The functions f and g are inverses
of each other.
C
For any value of x, f (x) + g (x) = 0.
D For any value of x, f (x) · g (x) = 1.
If the functions h (x) and h -1(x) are
inverses of each other, what is the
value of h (h -1(1))?
A
0
B
1
C
-1
D x
A27
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice By Standard
Clarifying Objective MBC.A.5.4
1
The table below displays several
points of an exponential function.
x
f(x)
1
6
2
36
3
216
4
1296
5
7776
4
A
f
B
f
-1
C
f
-1
(x) = 6 log6 x
D f
-1
(x) = log6 x 6
(x) = log6 x
(x) = log6 x 2
A
2
B
4
C
16
D 64
Which logarithmic function is an
inverse of the function in the table?
-1
If h (x) = log8 x and h -1(x) = 8x,
what is h -1(h(2))?
5
The function f (x) = 2x is shown in the
graph below.
8
6
4
2
−8−6−4−20
2
B
x
C
3x
Which statement best describes the
relationship between the graph of
f (x) = 2x and the graph of
g (x) = log2 x?
D log3 x
3
What is the inverse of g (x) = log10 x?
A
g -1(x) = 10
B
g -1(x) = 10x
C
g -1(x) = x10
2 4 6 8x
A
The graphs are reflections of each
other over the x-axis.
B
The graphs are reflections of each
other over the y-axis.
C
The graphs are reflections of each
other over the line y = x.
D The graphs are reflections of each
other over the line y = -x.
D g -1(x) = 10x
A28
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3
G(x) = 2x
−4
−6
−8
If f (x) = 3x and g (x) = log3 x, what
is g (f (x))?
A
y
Name
Date
Practice By Standard
Clarifying Objective MBC.A.6.1
1
Which equation is equivalent to
log2 16 = x?
2
A
16 = x
B
16 2 = x
C
2 x = 16
4
1
Solve log 9 y = _.
2
A
y = 18
C
B
y = 4_
1
D y=_
2
1
2
y=3
512
D 2 16 = x
5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
In 2007, plants were dying at a
nursery in Greene County, North
Carolina. Tests done by the North
Carolina Department of Agriculture
and Consumer Services showed the
problem was due to well water with
an abnormally acidic pH of 3.0. The
pH of a water sample is given by the
formula pH = -log 10 [H +], where [H+]
is the hydrogen ion concentration in
moles per liter. What was the
hydrogen ion concentration of the
well water at the nursery?
A
0.001 moles per liter
B
0.003 moles per liter
C
0.300 moles per liter
6
log 3 4 = 81
B
log 3 81 = 4
C
log 4 3 = 81
log 10 x = 1
C
B
log 10 e = x
D e 10 = x
Norah wrote an exponential equation
1 = x.
that is equivalent to log 5 _
25
7
Which equation is equivalent to
81 = 3 4 ?
A
e x = 10
A
Which equation did she write?
D 0.477 moles per liter
3
Which equation is equivalent to
ln x = 10 ?
x
1
_
=5
A
( 25 )
B
5x = _
1
25
C
1
_
5 25 = x
D 5 = 25 x
Which equation is equivalent to
x 3 = 216?
A
log 3 x = 216
B
log 3 216 = x
C
log x 3 = 216
D log x 216 = 3
D log 4 81 = 3
A29
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice By Standard
Clarifying Objective MBC.A.6.2
1
In photography, an f-stop is a number
that represents the amount of light
exposure. Exposure values are
numbers that refer to certain
combinations of f-stops and shutter
speeds. Exposure value (EV) is given
4
Which expression is equivalent to
3x
log 5 _ ?
2y
N
by the equation EV = log 2 _
t , where
2
A
log 5 3x - log 5 2y
B
log 5 3x + log 5 2y
C
(log 5 3x)(log 5 2y)
D log 5 (6xy)
N is the f-stop number and t is the
shutter speed in seconds. What is the
exposure value when the f-stop is 8.0
1
and the shutter speed is _ second?
2
5
Which expression is equivalent to
A
5
B
6
36x
log 6 _
?
4
C
7
A
2 log 6 x - 4 log 6 y
D 8
B
2 + log 6 x - 4 log 6 y
C
8 + 4 log 6 x - 4 log 6 y
y
D 8 + 8 log 6 x - 8 log 6 y
2
A
3
B
4
C
64
A
2
C
D 81
B
5
D 25
6
Simplify log 8 x 9 y 6.
A
54 log 8 xy
B
15 log 8 xy
C
15 + log 8 x + log 8 y
7
What is the value of y if 5 log
5
(2y - 6)
= y?
6
Sanjay is using the properties of
1
__
logarithms to simplify 9 log (3x - 1) . Which
equivalent expression can he write?
9
D 9 log 8 x + 6 log 8 y
A30
A
-9 log (3x - 1) C
B
3x - 1
D
9
_
log (3x - 1)
1
_
3x - 1
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3
What is the value of x if log 4 4 3 = x ?
Name
Date
Practice By Standard
Clarifying Objective MBC.A.7.1
1
The Koury Natatorium, at the
University of North Carolina, has a
diving platform that is 5 meters high.
If a diver jumps off the platform with
an upward velocity of 2.25 meters per
second, the dive can be modeled by
the equation h = -5t 2 + 2t + 5,
where h is the height above the water
and t is the time in seconds of the
dive. Approximately how long will it
take for the diver to hit the water?
A 0.8 second
B
1.2 seconds
C
2.4 seconds
4
Johanna is solving 2x 2 - 3x - 5 = 0.
Which of the following shows Johanna
correctly using the quadratic formula?
A
- 3 ± √
(-3) - 4(2)(-5)
x = __
2
B
2
3 ± √(-3)
- 4(2)(-5)
__
x=
2(2)
C
2
-(3) ± √(-3)
- 4(2)(-5)
___
x=
2
2
-(- 3) ± √
(-3) - 4(2)(-5)
D x = ___
2
2(2)
D 2.8 seconds
A
x = 2; x = 5
Catherine is solving
15x 2 + 14x - 8 = 0. Which of the
following shows Catherine correctly
factoring the equation?
A 5x 2 - 14x = - 8
B 15x (x - 1) = 8
B
x = 2; x = -5
C
C
x = -2; x = 5
D (3x + 4)(5x - 2) = 0
5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
What are the solutions to the equation
x 2 - 7x + 10 = 0?
(3x + 4)(5x - 2) = 8
D x = -2; x = -5
6
3
If Pilar graphs the quadratic function
y = 2x 2 + 2x - 12, what are the
x-intercepts of her graph?
A
2 and 3
B
-2 and 3
C
2 and -3
What are the roots of the equation
x 2 - 6x = 0?
A
x = -3; x = 2
B
x = 0; x = 6
C
x = 1; x = 6
D x = 3; x = 2
D -2 and -3
A31
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice By Standard
Clarifying Objective MBC.A.7.2
1
x2 - 4
Steve rewrites the equation y = _
4
x+2
as y = x - 2. What restrictions must
he identify for the variable x?
A
x≠2
B
x ≠ -2
C
Simplify the equation below. What
restrictions must be placed on x?
x 2 - 16
y=_
2
x + 6x + 8
A
x+4
y = _;
x+2
x ≠ -4 and x ≠ 4
x ≠ 2 and x ≠ -2
B
D There are no restrictions.
x-4
y = _;
x-2
x ≠ -2 and x ≠ 2
C
2
3x + 8
Simplify _ .
15x + 40
x+4
D y = _;
x-2
10
x ≠ -4, x ≠ -2 and x ≠ 2
B
8
1
_
; x ≠ -_
C
_1 ; x ≠ 0
3
10
5
5
8
1
D _
; x ≠ -_
x 2 + 4x
The equation y = _
can
2
x + 3x - 4
x
also be written as y = _ . What
x-1
B
x ≠ 1 and x ≠ -4
C
x ≠ 0, x ≠ 1, and x ≠ -4
A
x ≠ -3
B
x≠1
C
x≠5
D x ≠ 1 and x ≠ 5
restrictions must be placed on x when
the fraction is in its simplest form?
x≠1
2x + 6
a quadratic numerator and a linear
denominator. What restrictions must
be placed on x?
3
A
2x 2 - 12x + 10
The equation y = __ has
D There are no restrictions.
A32
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3
x+2
x ≠ -4 and x ≠ -2
1
A _
;x≠0
5
x-4
y = _;
Name
Date
Practice By Standard
Clarifying Objective MBC.A.7.3
1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
3
North Carolina’s state bird, the
northern cardinal, can live up to 15
years in the wild. The cardinal’s life
span is affected by its basal metabolic
rate (BMR), which is given by the
3
_
formula BMR = 5.914(m) 4 , where m is
the body mass in kilograms and BMR
is measured in watts. If the northern
cardinal’s body mass is 39.9 grams,
what is its basal metabolic rate?
5
900
800
700
600
500
400
300
200
100
y
x
1
0
2
3
4
A
0.339 watt
B
0.528 watt
Use the graph. Which could be the
solution to the equation 3 2x = 729?
C
60.206 watts
A
x = 1000
C
D 93.888 watts
B
x = 729
D x = 1.5
Solve 4 = log 4 x.
6
A
x=1
C
B
x = 16
D x = 256
x = 64
C
x=_
1
8
1
x=_
2
B
x=3
A certain radioactive element decays
over time according to the equation
t
1 _
y = A _ 300 , where A is the number
2
of grams present initially and t is time
in years. If 100 grams were initially
present, how many grams will remain
after 600 years?
()
If 2 5x - 8 = 16 x, what is the value of x?
A
x=2
A
0.8 gram
C
B
25 grams
D 100 grams
50 grams
D x=8
7
4
The graph of 3 2x is shown below.
Solve ln 6x - ln x 2 = 1.
Solve 2 x = 12.
A
A
x = log 60
B
B
x = log 12 - log 2
C
x=6
x=_
e
x = √6e
6
C
x=6
D x = 3 ± 2 √
2
log 3
log 2
D x=_+2
A33
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice By Standard
Clarifying Objective MBC.A.8.1
1
What is the domain of the function
3
x2 - 1
f(x) = _
?
2
x + 2x - 3
A
all x
B
all x not equal to 1 or −1
C
all x not equal to 1 or −3
D all x not equal to 1, −1, or −3
2
The function f(x) = -x4 + 2 is shown
in the graph below.
8
6
4
2
−8−6−4−20
In 2007, civil engineering students at
North Carolina State University
participated in a contest to create a
concrete cube that meets strict weight
and strength requirements. The
volume of a cube can be determined
by the function f(x) = x3, where x is
the length of one side. The theoretical
domain of f(x) = x3 is all real numbers.
Which of the following descriptions
best represents the practical domain
of f(x) = x3 in the given situation?
The practical domain is all real
numbers because the theoretical
and practical domains are the
same.
B
The practical domain is all whole
numbers because the length of a
side is a whole number.
C
The practical domain is all real
numbers greater than or equal to
zero because the length of a side
is a nonnegative number.
y
2 4 6 8x
−4
−6
−8
What are the domain and range of
f(x) = -x4 + 2?
A
Domain = all real numbers
D The practical domain is all real
numbers greater than zero
because the length of a side is a
positive number.
Range = all real numbers
B
Domain = all real numbers
Range = y ≤ 2
C
Domain = x ≤ 2
Range = all real numbers
4
D Domain = y ≤ 2
Range = x ≤ 2
What is the domain of the radical
function f(x) = √
-x + 4 ?
A
x≥4
B
x≤4
C
x≥2
D x≤2
A34
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A
Name
Date
Practice By Standard
Clarifying Objective MBC.A.8.2
1
Which point is an x-intercept of
4
2x - 2
?
h(x) = _
x2 - 4
A
(−2, 0)
C
B
(−1, 0)
D (2, 0)
(1, 0)
What is an x-intercept of the graph of
the function f(x) = 3x + 4?
A
(0, 3)
B
(0, 4)
C
(0, 9)
D The graph has no x-intercept.
2
At what coordinate is there a
minimum of the quadratic function
f(x) = 2(x + 3)2 - 5?
A
(3, 5)
What are the zeros of the function
g(t) = t2 + 3t + 2?
B
(−3, 5)
A
−1 and −2
C
(3, −5)
B
1 and 2
C
√_23 and - √_23
5
D (−3, −5)
√
√
2
2
D _ and −_
3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3
3
Which function is graphed on the
coordinate grid shown below?
3
2
1
−3−2
O
y
6
1 2 3 4 5x
−2
−3
−4
−5
7
What is the y-intercept of
y = -2 cos x + 5?
A
(0, 5)
C
B
(0, 1)
D (0, −5)
(0, 3)
What is the maximum value of
f(x) = -2x2 + 8x + 1?
A
f(x) = -x2 - 2x + 1
B
f(x) = -x2 + 4x - 3
C
f(x) = x2 - 4x + 3
A
9
D f(x) = x2 + 2x - 1
B
8
C
1
D 0
A35
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice By Standard
Clarifying Objective MBC.A.8.3
1
At which value does the graph of
4
x2 - 4
f(x) = _ have a point of
x-2
What is the horizontal asymptote of
the graph of f(x) = 3(x - 2)?
discontinuity?
A
y=3
A
x=4
B
y=2
B
x=2
C
y=0
C
x=0
D y = -2
D x = -2
2x + 1
graph of f(x) = _ ?
Mitch graphs the function
f(x) = -2x2 + 8x - 4 on a coordinate
plane. What are all the values of x for
which the function is decreasing?
A
x=0
A
x>2
B
x=-_
B
x<2
C
x=1
C
x>4
D
x = -1
5
2
What is the vertical asymptote of the
x+1
D x<4
6
For which values of x is the function
f(x) = x2 + 4x - 25 increasing?
Which of the following functions has a
graph with a slant asymptote of y = x?
A
f(x) = x + _
x
x > -2
B
x+1
f(x) = √
-2 < x < 2
C
f(x) = x3 + x
A
x < -2
B
C
1
x -1
D f(x) = _
2
D 2 ≥ x ≥ -2
x+1
A36
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3
1
2
Name
Date
Practice By Standard
Clarifying Objective MBC.A.8.4
1
Which function has a graph that
extends down at each end?
4
A
f(x) = 2x4 - x2 + 5
B
f(x) = 4x5 + 2x3 - 6x + 1
A
Both ends extend up.
C
f(x) = -3x4 - 2x3 + x - 5
B
Both ends extend down.
C
The end on the left extends up and
the end on the right extends down.
D f(x) = x5 - 3x2 - 3
2
D The end on the left extends down
and the end on the right extends
up.
Which term can be used to determine
the end behavior of the polynomial
function f(x) = x3 + 4x2 + 3x + 6?
A
x3
5
2
B
4x
C
3x
D 6
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Which of the following descriptions
best represents the end behavior of
the graph of f(x) = -2x3 + 4?
Which function has a graph with one
end that extends up and one end that
extends down?
A
f(x) = -2x6 - 3x4 + x2
B
f(x) = 5x5 + 4x4 - 3x3
C
f(x) = x2 + 2x + 1
D f(x) = x2 - 2x + 1
3
Rosa graphs function f on a coordinate
plane. She observes that as the value
of x becomes increasingly large or
increasingly small, the value of f(x)
becomes closer and closer to zero.
Which of the following functions has
this type of end behavior?
6
Which of the following descriptions
best represents the end behavior of
10x + 2
the graph of f(x) = _
?
2
-5x - 15
A
The left end extends down and
the right end extends up.
A
f(x) = x2
B
f(x) = x3
B
Both ends extend down.
C
f(x) =
C
Both ends extend toward a
horizontal asymptote described
by y = −2.
√
x
1
D f(x) = _
x
D Both ends extend toward a
horizontal asymptote described
by y = 0.
A37
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice By Standard
Clarifying Objective MBC.A.8.5
1
At a height of 220 feet, the Laurel
Creek Bridge is the second tallest
bridge in North Carolina. An object is
dropped from the bridge into the
water below. The table represents the
object’s height h, in feet, above the
water after 1, 2, and 3 seconds.
3
Bacteria in a culture are growing as
shown in the table below.
Bacteria Growth
Day
Number of Bacteria
0
25
1
625
2
15,625
Height of Object
Time t
(seconds)
Height h
(feet above water)
0
220
1
204
2
156
3
76
Which statement best describes the
growth rate of the bacteria?
The number of bacteria is
increasing at a linear rate.
B
Which function represents the object’s
height h in feet above the water after
t seconds?
The number of bacteria is
increasing at an exponential rate.
C
The number of bacteria is
decreasing at a linear rate.
A
h = 220 - 16t
B
h = -16t 2 + 220
D The number of bacteria is
decreasing at an exponential rate.
C
h = 220 - 16t
1
D h = 220 - _
t
16
4
Which function could be represented
by the graph shown below?
y
2
Which function has a graph that
intersects the y-axis at (0, 3)?
A
y = x2 + 3
B
y=2 +3
C
y=_
x +3
40
20
0
x
2
4
6
x
1
A
y = 2x
B
y = x2 - 2x + 1
C
y=_
3
D y=x
x3
5
D y=_
2
3
x -5
A38
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A
Name
Date
Practice By Standard
Clarifying Objective MBC.A.9.1
1
A clock has a circular face with the
numerals 1 through 12 positioned at
equal intervals around the outer edge.
If the minute hand on the clock
rotates 630° clockwise, which
statement best describes the position
of the minute hand before and after
the rotation?
A
3
Which situation could best be
represented by the graph below?
y
The minute hand starts at 6 and
ends at 9.
B
The minute hand starts at 6 and
ends at 12.
C
The minute hand starts at 3 and
ends at 9.
x
A
the distance x, in miles, traveled
by a person walking y miles per
hour
B
the position of a rock dropped
from a height of y meters after
x seconds
C
the air y, in liters, in a person’s
lungs after breathing for x seconds
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
D The minute hand starts at 3 and
ends at 12.
2
D average speed after walking
x miles in y hours
The spinner for a carnival game is a
wheel with numbers painted along the
edge. The wheel is mounted vertically
on the wall. Before Jason spins the
wheel, the number he chooses is
exactly at the bottom of the wheel.
After Jason spins the wheel, that same
number is exactly at the top of the
wheel. Which description best
represents the rotation of the wheel
when Jason spins it?
A
1050° counterclockwise rotation
B
1080° clockwise rotation
C
1110° clockwise rotation
4
D 1260° counterclockwise rotation
A duck bobs up and down as it floats
on the water, moving from its highest
position to its lowest position every
6 seconds. The distance between the
duck’s highest and lowest position is
8 inches. If x represents time, in
seconds, and the equilibrium point is
y = 0, which function could model the
movement of the duck?
A
y = 3x + 4
B
y = 3x2 + 4x
C
π
y = 4 cos _x
(3 )
D y = 3x2 - 4x
A39
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice By Standard
Clarifying Objective MBC.A.9.2
1
Jolene is pacing back and forth across
a rug with a length of 3.5 meters and a
width of 1.5 meters, as shown below.
3
Which graph best represents average
monthly precipitation that fluctuates
between 8 and 3 inches over time?
A
1.5 cm
It takes Jolene a total of 35 seconds
to walk across the length of the rug
20 times. If she continues pacing at
the same rate, how many times can
she walk across the width of the rug in
30 seconds?
10
C
B
25
D 45
6
4
2
x
0
6
B
18
24
40
y
9
6
3
x
0
6
-3
12
18
24
Months
The pendulum shown in the picture is
0.25 meter long.
Precipitation
(inches)
C
0.25 m
y
9
6
3
x
0
6
12
18
24
Months
can be
The equation T = 2 √0.25
used to find the approximate time T in
seconds for one full swing of the
pendulum. About how many full swings
will the pendulum make in 10 seconds?
5
B
10
C
20
6
Precipitation
(inches)
A
D
y
3
x
0
-3
-6
6
12
18
24
Months
D 40
A40
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
12
Months
Precipitation
(inches)
A
y
8
Precipitation
(inches)
3.5 cm
Name
Date
Practice By Standard
Clarifying Objective MBC.A.9.3
1
Points P and Q lie on the unit circle
shown below.
3
y
1
1
135°
-1
0
A carousel at a carnival has horses
arranged in a circle with a radius of
10 feet. Jenna gets onto a horse
directly to the right of the center at
point H, and Tyler gets onto a horse
directly above the center at point G.
(
2
1
x
10 ft
10 ft
-1
What is the horizontal distance
between P and Q?
When the ride is over, Jenna and Tyler
π
are at a _ counterclockwise rotation
√
3
A _
from their starting positions. What is
the horizontal distance between Tyler
and Jenna at the end of the ride?
3
2
B
√
2
2
1+_
√
3
C 1+_
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
)
2
A
5 ft
B
5 + 5 √
3 ft
10 ft
ft
D 10 √3
C
D 2
4
2
When Sarah gets into the seat at the
bottom of a Ferris wheel at the
Cleveland County Fair, she is 5 feet
above the ground. If the seats are
30 feet from the center of the wheel,
how far above the ground is Sarah
after the Ferris wheel rotates 660°
counterclockwise?
A
20 feet
B
C
A vertical gear in a machine is in the
shape of a circle with a radius of
8 inches. A blue dot has been painted
on the edge of the gear. Before the
machine starts up, the blue dot is in
the bottom position, 3 inches above
the floor. When the machine starts,
the gear rotates until the blue dot is
7 inches above the floor. Which
statement best describes the rotation
of the gear?
A
The gear rotated 750° clockwise.
25 feet
B
The gear rotated 765° clockwise.
35 feet
C
The gear rotated 780° clockwise.
D 65 feet
D The gear rotated 790° clockwise.
A41
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice By Standard
Clarifying Objective MBC.A.9.4
1
What is the period for the function
θ
?
y = cos _
A
π
_
B
π
C
2π
4
What is the amplitude for the function
1
y = _ sin 2θ ?
2
2
A
2
B
_1
4
_1
2
C
1
D 2
D 4π
5
2
The musical note A above middle C
has a frequency of 440 hertz. A sine
function that represents the behavior
of the note A is y = 0.2 sin (880πt).
What is the amplitude of the note A
modeled by this function?
The average high monthly
temperatures for Raleigh, North
Carolina, are given below.
Average High Temperature
Raleigh, North Carolina
Month
January
49
53
0.1
March
61
B
0.2
April
71
May
78
C
176
June
84
July
88
August
86
September
80
October
70
November
61
December
52
Which periodic function is equivalent
to 2 cos x + 3 ?
A
B
C
D
π
y = 2 cos x - _
+3
( 2)
π
+3
y = 2 cos (x + _
2 )
π
+3
y = 2 sin (x - _
2 )
π
+3
y = 2 sin (x + _
2 )
If t = 1 represents January, which of
the following functions best models
this data?
A
B
C
D
A42
π
f(t) = -19.5 cos _
t - 0.5 + 68.5
(6
)
π
t - 0.5) + 19.5
f(t) = -68.5 cos (_
6
π
t - 0.5) + 68.5
f(t) = -38 cos (_
6
π
f(t) = -68.5 cos (_
t - 0.5) + 39
6
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
February
A
D 4400
3
Temperature (°F)
Name
Date
Practice By Standard
Clarifying Objective MBC.A.10.1
1
Karen invests $10,000 in a savings
plan that guarantees $200 per year for
ten years. Juan invests $10,000 in a
savings plan that guarantees 2%
interest compounded annually. Which
description best represents the
situation after ten years?
A
Karen has about $200 more than
Juan.
B
Juan has about $200 more than
Karen.
C
Karen has about $2000 more than
Juan.
3
Which type of function could be
represented by the graph below?
y
x
0
A
periodic
C
B
exponential
D absolute value
logarithmic
D Juan has about $2000 more than
Karen.
4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
A
What function is shown in the graph
below?
8
6
4
2
−8−6−4−20
B
1
f(x) = _
x
B
f(x) = x
C
f(x) = x2
f(x) = x
2
f(x) = x
y
5
2 4 6 8x
−4
−6
−8
A
Which function has a vertical
asymptote of x = 0?
6
D f(x) = x3
A43
C
f(x) = |x|
1
D f(x) = _
x
Which function does not have a
y-intercept of −2?
1
y=_
x -2
A
y=x-2
C
B
y = x2 - 2
D y = |x| - 2
Which type of function repeats its
values in regular intervals?
A
quadratic
C
B
periodic
D rational
radical
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice By Standard
Clarifying Objective MBC.A.10.2
1
The graph of f(x) = log x is shifted
2 units down and shrunk horizontally
by a factor of 4. What is the equation
of the new graph?
A
f(x) = log 2(x - 4)
B
f(x) = log 2x - 4
C
f(x) = log 4(x - 2)
4
The graph of f(x) = x2 is reflected over
the x-axis, shifted up 2 units, and
shifted left 2 units. Which is the graph
of the new parabola?
A
D f(x) = log 4x - 2
5
4
3
2
1
−4 −3−2
y
1 2 3x
O
−2
−3
2
_
Consider f(x) = _
x and g(x) = x - 3 .
Which translation will transform the
graph of f(x) into the graph of g(x)?
1
A
1
B
−4−3−2
a shift 3 units right
B
a shift 3 units left
C
a shift 3 units up
y
3
2
1
1 2 3x
O
−2
−3
−4
−5
D a shift 3 units down
3
If the graph of the function f(x) = 2x is
shifted 1 unit down and stretched by a
factor of 0.5, which function would
represent the new graph?
A
f(x) = 0.5(1 - 2x)
B
f(x) = 20.5x - 1
C
f(x) = (2 - 1)0.5x
3
2
1
−4 −3−2
O
y
1 2 3x
−2
−3
−4
−5
D
3
2
1
−4−3−2
D f(x) = 0.5(2x - 1)
O
y
1 2 3 4x
−2
−3
−4
−5
A44
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
C
Name
Date
Practice By Standard
Clarifying Objective MBC.G.4.1
1
The vertices of a quadrilateral are
represented using a 2 × 4 matrix. If
matrix multiplication is used to reflect
the quadrilateral across the x-axis,
how does the new matrix compare to
the original matrix?
A
B
C
3
⎡ -3 -3 -3
⎢
⎣ 1
1
1
The first row is the same. In the
second row, the entries are each
multiplied by -1.
A
The second row is the same. In the
first row, the entries are each
multiplied by -1.
Translate a quadrilateral 1 unit up
and 3 units left.
B
Translate a quadrilateral 3 units
down and 1 unit left.
Every entry is multiplied by -1.
C
Translate a pentagon 1 unit up
and 3 units left.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
D Translate a pentagon 3 units down
and 1 unit left.
4
The vertices of triangle RST are
R (3, 5), S (2, -4), and T (0, 4). A
transformation of RST is represented
by the matrix equation shown below.
⎡ 0
⎢
⎣ -1
1 ⎤⎡ 3
⎢
0 ⎦⎣ 5
2
-4
0⎤ ⎡ 5
=⎢
4 ⎦ ⎣ -3
-4
-2
Triangle EFG is transformed into
triangle EFG as shown below.
&(−2, 3)
4⎤
0⎦
((−3, −1)
&'(–3, –2)
Which statement best describes the
transformation of RST?
A
RST is reflected across the x-axis.
B
RST is rotated clockwise 90° about
the origin.
C
RST is rotated clockwise 180°
about the origin.
'(−1, −4)
y
x
0
''(4, –1)
('(1, –3)
The coordinate matrix for EFG is
multiplied by matrix T resulting in the
coordinate matrix for EFG. What is
matrix T?
⎡
⎤
⎡
⎤
A T = ⎢ 1 0 C T = ⎢ -1 0 ⎣0 1⎦
⎣ 0 1⎦
D RST is rotated clockwise 270°
about the origin.
Chapter 1
-3 ⎤
1⎦
-3
1
For which transformation would
addition of the matrix most likely be
used?
D The entries in each row are
interchanged with the entries in
other rows.
2
Consider the transformation matrix
shown below.
B
A45
⎡ 0 1⎤
⎡ 0 -1 ⎤
T=⎢
D T=⎢
⎣ -1 0 ⎦
⎣1
0⎦
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice By Standard
Clarifying Objective MBC.G.4.2
1
Which transformation results in an
image that is similar, but not
congruent?
A
180° rotation about the origin
B
reflection across the line x = 2
C
dilation by a factor of 2
4
⎡ 0
P=⎢
⎣ -4
-1
3
-2
6
0⎤
2⎦
0⎤
4⎦
Which statement is true?
The coordinate matrix for a triangle is
multiplied by a transformation matrix
T. The resulting triangle is congruent
to the original. What is matrix T?
⎡2 0⎤
⎡1 0⎤
A T=⎢
C T=
1
⎣0 1⎦
0 _
2 ⎦
⎣
⎡1 1⎤
⎡2 0⎤
B T=⎢
D T=⎢
⎣1 1⎦
⎣0 2⎦
⎢
3
1
⎡ 0 6
I=⎢
⎣ -8 2
D translation 2 units up and 7 units
right
2
Pre-image matrix P and image matrix I
for a quadrilateral are shown below.
A
The transformation is a translation.
B
The image is similar and
congruent to the pre-image.
C
The image is congruent to the
pre-image.
D The image is similar but not
congruent to the pre-image.
5
y
3
The elements of the vertex matrix for
a polygon are multiplied by 3.5. Which
of the following statements best
describes the relationship between
the pre-image and image?
A
The image and pre-image are similar
because the transformation is rigid.
B
The image and pre-image are
congruent because the
transformation is rigid.
C
0
x
If the map is represented as a vertex
matrix, which statement best
describes the result of multiplying the
entries of the matrix by 2?
A
The size of the map is halved.
B The size of the map is doubled.
C The map is translated 2 units to
the right.
D The map is translated 2 units to
the left.
The image and pre-image are
similar because the transformation
is a dilation.
D The image and pre-image are
congruent because the
transformation is a dilation.
A46
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A sketch of North Carolina is placed
on a coordinate grid as shown.
Name
Date
Practice By Standard
Clarifying Objective MBC.G.5.1
1
What is the equation of the graph of a
circle with radius 4 and center (-1, 5)?
A
(x - 1) 2 + (y + 5) 2 = 16
B
(x + 1) 2 + (y - 5) 2 = 4
C
(x + 1) 2 + (y - 5) 2 = 16
4
A circle is graphed on the coordinate
plane as shown below.
y
D (x - 1) 2 + (y + 5) 2 = 4
0
2
What is the equation of the circle
when it is reflected across the x-axis?
The diameter of a circle has endpoints
at (5, 6) and (-3, 12). What is the
equation of the circle?
A
(x + 1) 2 + (y - 9) 2 = 25
2
x
2
A
(x - 3) 2 + (y + 4) 2 = 9
B
(x + 3) 2 + (y - 4) 2 = 9
(x - 3) 2 + (y - 4) 2 = 9
B
(x + 1) + (y + 9) = 25
C
C
(x - 1) 2 + (y - 9) 2 = 25
D (x + 4) 2 + (y + 3) 2 = 9
2
2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
D (x - 1) + (y + 9) = 25
5
3
The distance by air from Kings
Mountain, North Carolina, to Tarboro,
North Carolina, is 351.68 miles.
Suppose each city is represented as a
point on the coordinate plane, with
Kings Mountain at the origin and
Tarboro at (351.68, 0). If these points
are endpoints of the diameter of a
circle, what is the equation of the
circle?
A
x 2 + (y - 175.84) 2 = 175.84 2
2
2
2
B
x + y = 175.84
C
(x - 175.84) 2 + y 2 = 351.68 2
A
Circle E intersects circle F exactly
once.
B
Circle E intersects circle F twice.
C
Circle E is completely inside
circle F.
D Circle F is completely inside
circle E.
D (x - 175.84) 2 + y 2 = 175.84 2
Chapter 1
Circle E is represented by the
equation (x - 3) 2 + (y - 1) 2 = 16, and
circle F is represented by the equation
(x - 2) 2 + (y + 1) 2 = 4. Which of the
following statements best describes
the relationship between circle E and
circle F?
A47
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice By Standard
Clarifying Objective MBC.S.2.1
1
Population estimates for Union
County, North Carolina, are shown in
the table below.
Years since 1970
0
3
y
Population
54,714
5
63,000
10
70,436
15
76,712
20
84,210
25
100,437
It is nearly the same as the
population given in the table.
B
It is about 1100 less than the
population given in the table.
C
It is about 4200 greater than the
population given in the table.
Which of the following numbers is the
approximate value of the x-intercept
for the least-squares regression line?
4
D It is about 4200 less than the
population given in the table.
1
2
4
7
10
13
17
19
25
y
8
15
13
16
20
25
25
29
33
12.5
C
B
15
D 20
17.5
Several samples of apples are
weighed, and the data is given in the
table.
Number of
Apples
3
5
4
4
3
5
2
3
Weight (oz)
32
54
41
44
29
52
19
31
A median-fit line is used to model the
data in the table. According to the
linear model, how many ounces will a
sample of 10 apples weigh? Round to
the nearest ounce.
A median-fit line is used to model the
data in the table below.
x
A
A
100 ounces
B
107 ounces
C
111 ounces
If the slope and y-intercept of the line
are rounded to the nearest hundredth,
which equation best fits the data?
A
y = 10.00 + 0.96x
D 115 ounces
B
y = 10.94 + 0.94x
C
y = 9.07 + 1.22x
D y = 23.86 + 1.04x
A48
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A
x
0
A least-squares regression line is used
to model the data in the table. Which
statement best compares the 1995
population predicted by the model to
the 1995 population in the table?
2
A least-squares regression line is
determined for the points plotted on
the graph below.
Name
Date
Practice By Standard
Clarifying Objective MBC.S.2.2
1
A rubber ball is dropped from various
heights. For each drop height, the
rebound height is recorded. Which
statement best describes the
relationship between the height of the
drop and the rebound height?
A
There is a positive correlation
between drop height and rebound
height.
B
There is a negative correlation
between drop height and rebound
height.
C
4
Edward calculated the correlation
coefficient for the graph below and
found a strong linear correlation
between x and y.
y
x
0
After adding three points to the graph,
he found almost no linear correlation
between x and y. Which three points
did Edward add to the graph?
There is no correlation between
drop height and rebound height.
D The correlation between drop
height and rebound height may
be positive or negative.
A
(6, 1), (7, 0), and (8, -1)
B
(6, 2), (7, 4), and (8, 3)
C
(6, 6), (7, 7), and (8, 8)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
D (6, 1), (7, 1), and (8, 1)
2
Which of the following would most
likely have the least correlation strength
to the posted speed limit of a road?
A
the time it takes to drive a mile
B
the average speed of the cars
C
the fraction of drivers exceeding
the speed limit
5
The table below shows the rating
system for mountain biking trails in
Western North Carolina.
D the revolutions per minute on the
cars’ tires
Rating
Difficulty
1
easy
2
moderate
3
more difficult
4
most difficult
Which of the following measurements
most likely has the strongest
correlation with how a trail is rated?
3
Which r-value indicates the weakest
strength of correlation?
A
B
trail length
elevation change (climb)
A
C
1
_
r = 0.36
B r = 0.19
C r = -0.16
D r = -0.89
Chapter 1
D
A49
trail length
elevation change (climb)
__
trail length
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice By Standard
Clarifying Objective MBC.S.2.3
1
What is the mean absolute deviation
of the following data?
4
27, 15, 18, 35, 17, 19, 8, 21, 27, 16
A
0
B
5.76
C
7.62
The correlation coefficient r of the
relationship between two data sets is
0.14. Which statement best describes
the relationship between the data sets?
A
There is a strong positive linear
relationship between the data sets.
B
There is a weak positive linear
relationship between the data sets.
C
There is a strong negative linear
relationship between the data sets.
D 20.3
2
x
5.5
6.7
3.2
4.8
9
6.6
4.5
y
55
75
50
35
80
50
20
A
15
C
B
81
D 1254
5
283
18
32
26
38
16
34
14
36
y
29
57
45
69
25
61
21
65
Which statement best describes what
the correlation coefficient shows about
the fit of a linear model for this data?
A
B
C
SAT Math
700
430
580
450
720 490
SAT Verbal
550
420
630
390
740 640
Using the value of the correlation
coefficient, which statement best
describes the relationship between
the SAT math scores and the SAT
verbal scores?
Juanita finds the correlation
coefficient for the following data.
x
Alex calculates the correlation
coefficient r for the SAT math scores
and SAT verbal scores shown in the
table below.
A linear model fits the data
perfectly.
A linear model fits the data fairly
well.
A
Since r = 0.70, the relationship is
moderately linear.
B
Since r = 0.70, the relationship is
weakly linear.
C
Since r = 0.88, the relationship is
moderately linear.
D Since r = 0.88, the relationship is
strongly linear.
A linear model does not fit the
data well.
D A linear model is inappropriate for
this data.
A50
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3
D There is a weak negative linear
relationship between the data sets.
What is the sum of the squared
deviations from the line of best fit for
the following data? Round to the
nearest whole number.
Name
Date
Practice By Standard
Clarifying Objective MBC.S.2.4
1
The Carolina Hurricanes play in the
National Hockey League. The table
and graph below show the number of
goals and assists for 10 team players
during the 2008–2009 season.
53
Staal
40
35
Ruutu
26
28
Brind’ Amour
16
35
Samsonov
16
32
Cullen
22
21
Corvo
14
24
Babchuk
16
19
Pitkanen
7
26
LaRose
19
12
80
70
60
50
40
30
20
10
0
The graph below shows the leastsquares regression line for data
represented by the plotted points.
y
Goals Assists
24
Assists
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Player
Whitney
2
0
If the value of the correlation
coefficient is 0.02, which statement
best describes the fit of the regression
line?
y
A
Given an x-value, the line is a
perfect predictor of the related
y-value.
B
Given an x-value, the line is a very
strong predictor of the related
y-value.
C
Given an x-value, the line is a
moderate predictor of the related
y-value.
.
x
10 20 30 40 50 60 70 80
Goals
Which line is the least squares
regression line? What is its strength as
a predictor for the number of assists if
a player has 35 goals?
A
Line L; strong predictor
B
Line L; weak predictor
C
Line M; strong predictor
x
D Given an x-value, the line is a very
weak predictor of the related
y-value.
D Line M; weak predictor
Chapter 1
A51
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice By Standard
Clarifying Objective MBC.D.2.1
1
Which linear function maximizes 4x + y
for the feasible region shown below?
A
the line through point E
A manufacturer makes two products.
The deluxe model requires 5 hours of
development time and 3 hours of
finishing time, which results in a $45
profit. The economy model requires
3.5 hours of development time and
1.5 hours of finishing time, which
results in a $30 profit. Time for
development is limited to 50 hours,
and time for finishing is limited to
35 hours. How many of each type of
product should be developed to
maximize profit?
B
the line through point F
A
C
the line through point G
5 deluxe, 6 economy
B
8 deluxe, 0 economy
C
10 deluxe, 3 economy
3
y
&
'
(
0
)
x
D the line through point H
D 14 deluxe, 0 economy
2
Consider the constraints below.
4
Which ordered pair is a vertex of the
feasible region?
A
(1, 0)
B
(1, 3)
C
(3, 0)
D (5, 2)
A health food store makes two types of
trail mix using a mixture of almonds
and cashews. A batch of Mix A contains
4 pounds of almonds and 2 pounds of
cashews. A batch of Mix B contains
3 pounds of almonds and 5 pounds of
cashews. The store only has 62 pounds
of almonds and 80 pounds of cashews.
The profit is $4 on each batch of Mix A
and $5 on each batch of Mix B. How
many batches of each type of mix
should be made to maximize profit?
What is the maximum profit?
A
16 batches of Mix A and no
batches of Mix B; $64
B
10 batches of Mix A and 8 batches
of Mix B; $72
C
5 batches of Mix A and 14 batches
of Mix B; $90
D 12 batches of Mix A and
10 batches of Mix B; $98
A52
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
x≥1
y≤4
2x + y ≤ 10
x - 2y ≥ -5
Name
Date
Practice By Standard
Clarifying Objective MBC.N.1.1
1
Which expression is equivalent
13
y?
to _ √
3 1 _13
A y
C _
y
3 _1
_1
B y3
D 3y 3
3
Which process best demonstrates how
1
_
to simplify (-32) 5 ?
1
_
1
A (-32) 5 = _
(-32) = -6.4
5
1
_
5
B (-32) 5 = √
-32 = -2
C
1
_
-5
(-32) 5 = √
-32 = 2
1
_
2
D (-32) 5 = √32 5 ≈ -5793
A map of Randolph County, North
Carolina, is shown below.
4
Randleman
1 4
If f(x) = - _
x x , what is f (3)?
( )
Archdale
4
Asheboro
Franklinville
A
√
3
-_
B
1
-_
C
R ANDOL PH
C OU NT Y
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1
_
Seagrove
3
5
√
3
1
-_
243
D -27
RANDOLPH COUNTY
Area = 790 mi2
Randolph County is considered to be
a rough square. The distance from the
southern border of the county to the
1
_
northern border is about (2.2)790 2
kilometers. Which expression
represents the same distance?
A
2.2(790) km
√
B
4.84 √
790 km
km
√4.84(790)
C
D
5
Which of the following is equivalent to
1
_
1
_
?
the expression _
1
_
82 · 82
A
8
22
8 √
2
2
C _
√
2
D 4 √
2
B
√
2.2 √
790 km
A53
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice By Standard
Clarifying Objective MBC.N.1.2
1
Which expression is equivalent to
1
_
(27x 3) 2 for all values of x?
A
B
5
What is the simplest form of
9x √
3x
D 9|x| √3x
C
2
_
2
2
_
(a _c _) (a _b 2c _) _
__
?
r3 - z3
A
1
2
1
_
1
_
r -z
3
3 4
4
1
3
4
3
3
2
(2a 2bc) 2
1
_
3
1
B _
1
1
_
_
A
1
_
1
_
B
1
_
1
_
r3 + z3
C
1
_
Solve 4 6x + 1 = 16 4x .
1
A x = -_
2
1
B x = -_
4
C x=0
1
D x=_
5
3x √
3x
3|x| √3x
r3 - z3
Simplify _
.
1
1
_
_
1
_
4
r3 - z3
D r3 + z3
C
c4
_
4a 3
c3
_
4a 3b
bc 3
_
_3
4a 2
D
3
2
⎡
⎤ _1
Simplify ⎢243(x 15y 10) 2 5 .
⎦
⎣
A 9x 6y 2
B
C
1
_
3 2
9x y
6
2
-_
8a -2b 3
4a -1b 3
Simplify _ ÷ _.
12x -2y -3
30 20
3x y
3x -4y -1
2 4
D 3x 6y 4
A
9x y
_
B
b 3y
_
8a b 2
3
_1
2
2ax 2
_1
2
8b 3 y
C _
2
9ax
2
D
A54
by
_
2ax 2
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
_1
a 2 b 2c 5
_
Name
Date
Practice By Standard
Clarifying Objective MBC.N.2.1
1
In Asheville, North Carolina, the
average high temperatures for
December, January, and February are
50°, 46°, and 50°. For the same
months in Cape Hatteras, North
Carolina, the average high
temperatures are 57°, 53°, and 54°.
Which is a possible matrix for the high
temperatures of these two locations
for the given months?
A
⎡ 50 46
⎢
⎣ 44 40
⎡ 50
B
57
⎣ 46
⎡ 50
C ⎢
⎣ 57
⎡ 50
D ⎢
⎣ 53
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
⎢
2
3
y
50 ⎤
41 ⎦
Which of the following matrices could
be used to display ordered pairs from
the line?
⎡
2
4⎤
A ⎢4
⎣ 0 1 -2 ⎦
50 ⎤
54 ⎦
46 ⎤
54 ⎦
2 clothing stores and 4 shirt
brands
B
8 clothing stores and 8 shirt
brands
C
8 clothing stores and 1 shirt brand
B
⎡ -2 7
⎢
⎣ 0 4
C
⎡0
⎢
⎣4
⎡
D ⎢ -1
⎣ 5.5
Terrell is shopping for shirts at
different clothing stores. He creates
the matrix of prices below.
⎡ 37.95 31.50
29 39.99⎤
⎢
⎣ 19.98 27.49 21.50 27.97⎦
Which type of data is most likely
represented by the matrix?
A
x
0
53 ⎤
50
54 ⎦
46
53
57
50
Consider the line graphed below.
4
-1 5.5 ⎤
1 2.5 ⎦
3
-0.5
1
2.5
4⎤
2⎦
3 ⎤
-0.5 ⎦
Which of the following could be true
of a matrix that contains 12 elements?
A
The matrix has 6 rows and
6 columns.
B
The matrix has 5 rows.
C
The matrix has 7 columns.
D The matrix has 3 rows and
4 columns.
D 1 clothing store and 8 shirt brands
A55
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice By Standard
Clarifying Objective MBC.N.2.2
1
The North Carolina state sales tax is
4.5%. If a matrix contains item prices,
what matrix operation would result in
the prices with state sales tax
included?
A
adding a matrix where each
element is 0.045
B
multiplying the matrix by a scalar
of 0.045
C
multiplying the matrix by a scalar
of 1.045
4
A
1
11 -24 ⎤
32
2 -14
⎣ -5 -15
1⎦
⎢
B
9 -21 ⎤
4
-7 -21
28
⎣ 0
-1 ⎦
0
C
8
-3
3⎤
-23 -21 -4
⎣ 12
36 -5 ⎦
D
⎡ -8
23
⎣-12
⎢
⎡
⎡ 1
1⎤
D ⎢
⎣ 7.6 1.4 ⎦
5
3
⎢
Matrix G has 36 elements. Which of
the following ensures that 2G - H
exists for some matrix H?
⎢
⎢
3 -3 ⎤
4
21
5⎦
-36
For two matrices I and J, the sum of
⎡
3⎤
. The
the matrices I + J is ⎢ 7
⎣ 6 -4 ⎦
⎡ 1
5⎤
. What
difference of I - J is ⎢
⎣
⎦
-4
10
is Matrix J?
A
Matrix H has 36 elements.
B
Matrix H has 72 elements.
A
⎡ 8 -2 ⎤
⎢
⎣ 10
6⎦
C
C
Matrix H has the same number of
rows and columns as matrix G.
B
⎡ 6 -2 ⎤
⎢
⎣ 10 -6 ⎦
⎡ 3 -1 ⎤
D ⎢
⎣ 5 -7 ⎦
D Matrix H has the same number of
rows as the number of columns in
matrix G.
A56
⎡3
⎢
⎣1
4⎤
-7 ⎦
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
⎡
3 5.1 ⎤
⎢
⎣-1.5
5⎦
⎡
⎡ 2 5.1 ⎤
Consider the matrices K = ⎢
⎣ 1 3.2 ⎦
⎡
0 ⎤. What is K + L?
and L = ⎢ -1
2.5
-1.8
⎦
⎣
⎡ 1 5.1 ⎤
⎡ 3 5.1 ⎤
A ⎢
C ⎢
⎣ 3.5 1.4 ⎦
⎣ 3.5 -5 ⎦
B
⎢
⎡
D multiplying the matrix by a scalar
of 4.5
2
Matrices E and F are shown below.
⎡ 1 -3
6⎤
E = -4
0 -8
⎣ 3
9 -1 ⎦
⎡ -2 -1
3⎤
F=
5
7 -4
⎣ -2 -6
1⎦
What is -3F - 2E?
Name
Date
Practice By Standard
Clarifying Objective MBC.N.2.3
1
Suppose M is a 5 × 3 matrix, N is a
4 × 3 matrix, and P is a 4 × 4 matrix.
Which matrix product exists?
A
MN
C
B
MP
D PN
3
Given the matrices:
⎡ 3 2⎤
⎡-1
Q = -1 0 R = ⎢
⎣ 2
⎣ 4 1⎦
What is QR?
⎡ 1 -4 -4 ⎤
A
0
2
1
⎣ -2 -7 -2 ⎦
0⎤
⎡ 1 -10
B
0
2
1
⎣ -5 -10 -2 ⎦
⎢
NP
⎢
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
The Duke Blue Devils played the
North Carolina Tar Heels in an NCAA
basketball game in March of 2008. In
the matrices below, LF represents
successful long field goals, SF
represents successful short field goals,
FT represents successful free throws,
and PT is the number of points for
each type of shot.
0⎤
-2 ⎦
⎢
⎡ -1 -2 ⎤
⎢
⎣ -3
2⎦
⎡
⎤
D ⎢ -7 0 9
-6
⎣
⎦
C
PT
⎡
LF 3⎤
SF 2
FT ⎣1⎦
What is the product matrix? What is
its real-world meaning?
LF SF FT
Duke ⎡ 10 15 8 ⎤
⎢
N.C. ⎣ 5 26 9 ⎦
-2
1
4
⎢
A
[35 108 43]; the number of
points from each type of
successful shot
⎡ 68 ⎤
B ⎢
; the number of points
⎣ 76 ⎦
scored by all successful shots by
Duke and North Carolina,
respectively
⎡ 67 60 ⎤
C ⎢
; the number of points
8 ⎦
⎣ 9
from field goals and the number
of points from free throws
⎡
⎤
D ⎢ 33 ; the combined number of
⎣ 40 ⎦
successful goals and free throws
for Duke and North Carolina,
respectively
Felicia sells lunches at three different
food carts around town. Each cart
offers lunch in small, medium, and
large portions. The matrices below
show her prices for each lunch and
the number of lunches she sold
yesterday.
Sm
Med Lg
Prices ($): [2.75 3.50 4.00]
Number Sold:
cart cart cart
A B
C
Sm ⎡12 15 16 ⎤
Med 10 24 18
Lg ⎣ 8 20 12 ⎦
Felicia multiplies the matrices to find
matrix S, her total sales in dollars at
each cart. What is S 13?
⎢
A57
A
$100
C
B
$140
D $205.25
$155
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice By Standard
Clarifying Objective MBC.A.1.1
1
An open garage door starts at a
height of 7 feet and closes at a rate of
7 inches per second. A sensor will
reverse the direction of the door if an
object is encountered while the door
is moving.
3
The function y = x represents the
greatest integer function, where x
returns the nearest integer that is less
than or equal to x. Which is the graph
of y = x?
y
A
Height (ft)
Garage Door Height
7
6
5
4
3
2
1
y
0
x
0
x
1 2 3 4 5 6 7 8
Which equation results in the graph
shown for the domain {x | 0 ≤ x ≤ 12}?
A
y = |x| + 1
B
y = |7 - x| + 1
y
B
Time (s)
x
0
C
y
x
0
2
Which equation is graphed on the
coordinate grid shown below?
3
2
1
−3−2
O
y
y
D
1 2 3 4 5x
−2
−3
−4
−5
0
x
y = -x 2 - 2x + 1
A
B
y = -x 2 + 4x - 3
C
y = -x 2 - 4x + 3
D y = x 2 + 2x + 1
A58
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
y = |x - 6| + 1
⎧
D y = ⎨ 7 - x if x < 7
if x ≥ 7
⎩x
C
Name
Date
Practice By Standard
Clarifying Objective MBC.A.1.2
1
Juanita wants to deposit her savings
in an account that compounds
interest. If she leaves the money in the
account for 2 years, the final value of
her deposit is a function of the
interest rate. The relationship is shown
in the table.
r
F(r)
2%
$260.10
3%
$265.23
4%
$270.40
5%
$275.63
3
F(r) = 250(1 + r)2
B
F(r) = 250(1 - r)2
C
F(r) = 250r2
x
0
1
2
3
4
5
y
0
1
8
27
64
125
Which function describes the
difference d(x) between y and the
next value of y in the table?
Which function does the table
represent?
A
The cubes of whole numbers are
given in the table.
A
d(x) = x3
B
d(x) = 6x - 5
C
d(x) = x3 - x
D d(x) = 3x2 + 3x + 1
4
What function is graphed below?
2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
D F(r) = 250 + (1 + r)
2
y
The number of handshakes h in a
room with n people is given in the
table.
n
1
2
3
4
5
h
0
1
3
6
10
0
Which function best describes the
relationship given in the table?
A
h=n-1
B
h = n(n - 1)
1
h = _(n2 - n)
2
n2
D h=_
-n
2
A
y=
B
y=
C
y=
C
D y=
A59
{
{
{
{
x
1 + x2 if x < 1
(x - 1)2 if x ≥ 1
1 - x2 if x < 1
x2 - 1 if x ≥ 1
1 - x2 if x < 1
2x - 1 if x ≥ 1
1 + x2 if x < 1
x2 if x ≥ 1
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice By Standard
Clarifying Objective MBC.A.2.1
1
What matrix equation can be used to
solve the system of equations below?
4
Which matrix equation is the solution
to the system of equations?
4x = 5y
-2x - 3y = 8
2
A
⎡ 4
⎢
⎣ -2
B
⎡4
⎢
⎣2
C
⎡ 4
⎢
⎣-2
-5 ⎤ ⎡ x ⎤ ⎡ 0 ⎤
⎢ =⎢ -3 ⎦ ⎣ y ⎦ ⎣ 8 ⎦
⎡ 4
D ⎢
⎣-2
0⎤ ⎡x⎤ ⎡5⎤
⎢ =⎢ -3 ⎦ ⎣ y ⎦ ⎣ 8 ⎦
x - 3y = 7
2x - y = 5
3⎤ ⎡ 7 ⎤ ⎡ x ⎤
⎢ =⎢ -1⎦ ⎣ 5 ⎦ ⎣ y ⎦
5⎤ ⎡x⎤ ⎡5⎤
⎢ =⎢ 3⎦ ⎣y⎦ ⎣8⎦
A
⎡1
⎢
⎣2
5⎤ ⎡x⎤ ⎡0⎤
⎢ =⎢ -3 ⎦ ⎣ y ⎦ ⎣ 8 ⎦
B
_1 ⎡⎢-1
C
1 ⎡⎢-1
-_
5 ⎣-3
D
_1 ⎡⎢-1
A
Multiply the coefficient matrix by
the variable matrix.
B
Multiply the inverse of the coefficient
matrix by the variable matrix.
C
Multiply the constant matrix by the
variable matrix.
5
3⎤ ⎡ 7 ⎤ ⎡ x ⎤
⎢ =⎢ 5 ⎣-2 1⎦ ⎣ 5 ⎦ ⎣ y ⎦
The number of adult, discount, and
infant admissions tickets sold over
three days, to the Asheville Art
Museum, and the total amount of
ticket sales for those days are shown
in the matrix equation below.
Adult
⎡
Day 1 342
Day 2 421
Day 3 ⎣377
⎢
D Multiply the inverse of the coefficient
matrix by the constant matrix.
-2⎤ ⎡ 7 ⎤ ⎡ x ⎤
⎢ =⎢ 1⎦ ⎣ 5 ⎦ ⎣ y ⎦
Disc. Inf.
316
380
408
Total
⎤⎡ ⎤ ⎡
⎤
24 x
3632
16 y = 4426
18⎦ ⎣ z⎦ ⎣4302⎦
⎢ ⎢ What was the price of admission for
each type of ticket?
3
⎡-2
Solve ⎢
⎣ 3
5⎤ ⎡ x ⎤ ⎡-21⎤
⎢ =⎢
.
7⎦ ⎣ y ⎦ ⎣-12⎦
B
12
21
x = _, y = _
29
29
x = 3, y = -3
C
x = -3, y = 3
A
A
adult: $7, discount: $4, infant: $1
B
adult: $8, discount: $6, infant: free
C
adult: $6, discount: $5, infant: free
D adult: $6, discount: $3.50, infant: $1
111 , y = _
129
D x = -_
29
29
A60
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Which process can be used to solve a
matrix equation?
-2⎤ ⎡ 7 ⎤ ⎡ x ⎤
⎢ =⎢ 5 ⎣-3
1⎦ ⎣ 5 ⎦ ⎣ y ⎦
Name
Date
Practice By Standard
Clarifying Objective MBC.A.2.2
1
Solve
A
B
C
2a - 3b = -2
-2a + b = -6.
4
a = 5; b = 4
1
1
a = _; b = _
5
4
infinitely many solutions
The state vegetable of North Carolina
is the sweet potato. The table below,
on the left, shows the amount of each
energy source of three different
serving sizes of sweet potato. The
table below, on the right, shows the
total energy in each serving size.
D no solution
2
Which system of inequalities has no
solution?
A
B
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
C
D
{
{
{
{
Fat
(g)
Protein
(g)
Energy
(Cal.)
23.6
0.2
2.3
114
47.2
0.4
4.6
228
70.8
0.6
6.9
342
Can a system of equations from the
data be used to find the Calories
contributed by each energy source?
2x + y > 0
3x + y ≤ 0
A
Yes, you can set up 3 equations for
the 3 unknown values and solve.
B No, the system is inconsistent.
C No, the system is dependent.
D No, the system would need to be
a system of inequalities.
2y - x > 2
2y - x < 6
3x + y > -1
x - 3y > 13
4y - 2x > 12
2y - x < 6
5
3
Carb.
(g)
What is the solution to the system of
equations shown below?
What system of inequalities best
represents the graph shown below?
3
2
1
9x + 3y + 3z = -12
27x + 9y + 9z = -36
-3x - y - z = 4
A
(0, 0, _14 )
B
(0, 4, 0)
C
infinitely many solutions
−3−2 O
y
1 2 3 4 5x
−2
−3
−4
−5
D no solution
A
2x + 3y < 0 and x - y ≥ 5
B
2x + 3y ≥ 0 and x - y < 5
C
2x + 3y < 0 and x - y ≤ 5
D 2x + 3y > 0 and x - y ≤ 5
A61
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice By Standard
Clarifying Objective MBC.A.3.1
1
Which best describes the
transformation of the parent function
f(x) = |x| to g(x) = - |x - 1|?
A
B
C
4
The graph is shifted 1 unit right
and reflected across the x-axis.
The graph is shifted 1 unit left and
reflected across the x-axis.
The graph is shifted 1 unit right
and reflected across the y-axis.
In the general transformation of the
parent quadratic function f(x) = x 2 to
g(x) = a(x - h) 2 + k, which value(s) will
affect the location of the vertex?
A
h and k
B
a and h
C
a only
D h only
D The graph is shifted 1 unit left and
reflected across the y-axis.
5
2
Which translation of the graph of
f(x) = 2|x| shares at least two points
with function f ?
Which description most accurately
describes the translation of the graph
f(x) = 2(x - 1) 2 - 5 to the graph of
g(x) = 2(x - 2) 2 - 3?
A
2 units right
B
2 units up and 1 unit right
2 units up and 1 unit left
g(x) = |x|
C
B
g(x) = |x - 1| - 2
D 1 unit up and 2 units right
C
g(x) = 2|x - 1| + 2
D g(x) = 2|x| + 1
6
3
Which transformations
of the function
x
⎛_
1⎞
f(x) = ⎪ ⎥ are equivalent?
⎝2⎠
x
x-1
⎛ 1⎞
⎛_
1⎞
A g(x) = 2⎪_
⎥ and h(x) = ⎪ ⎥
⎝2 ⎠
⎝2 ⎠
x
B
C
⎛ 1⎞
g(x) = 2⎪_⎥ and h(x) =
⎝2 ⎠
g(x) = 2
x +1
⎛ 1⎞
D g(x) = 2⎪_
⎥
⎝2 ⎠
x+1
⎛_
1⎞
⎪ ⎥
⎝2 ⎠
⎛ 1⎞
and h(x) = ⎪_⎥
⎝2 ⎠
x+1
x-1
⎛ 1⎞
and h(x) = ⎪_⎥
⎝2 ⎠
Which of the following transformations
shifts the function f (x), 3 units left and
2 units down?
A
f(x) = (x - 1) 2 and
g(x) = 2 - (x + 2) 2
B
f(x) = (x + 1) 2 and
g(x) = (x - 2) 2 - 2
C
f(x) = (x - 1) 2 and
g(x) = (x + 2) 2 - 2
D f(x) = x 2 and
g(x) = (x + 3) 2 - 2
x-1
A62
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A
Name
Date
Practice By Standard
Clarifying Objective MBC.A.3.2
1
Camila reflected and translated
f(x) = (x + 4) 2 + 1 to make a new
function. The graph below shows
f and its transformation.
3
A
y
8
6 G(x) = (x + 4)2 + 1
4
2
−8−6−4−20
If the graph of the equation
f(x) = - (x + 1) 2 + 2 is shifted 1 unit
left, which would be the graph of
the transformed parabola?
2 4 6 8x
5
4
3
2
1
−4 −3−2
−4
−6
−8
B
A
g(x) = (-x) 2 + 1
B
g(x) = -x 2 - 1
C
g(x) = (x - 4) 2 - 1
3
2
1
−4−3−2
Which transformation of f(x) = x 2
results in the widest curve?
g(x) = 2(x + 10) 2 + 3
B
g(x) = - _ (x - 2) 2 - 4
C
g(x) = _ (x - 5) 2 + 20
1 2 3 4x
O
C
A
y
−2
−3
−4
−5
D g(x) = (-x + 4) 2 + 5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1 2 3x
O
−2
−3
Which equation represents the
transformed function?
2
y
3
2
1
−4 −3−2
O
y
1 2 3x
−2
−3
−4
−5
5
2
7
3
D
3
2
1
2
(x - 1)
D g(x) = _
5
−4−3−2
O
y
1 2 3 4x
−2
−3
−4
−5
A63
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice Test
1
For their school trip this year,
Mrs. Callahan’s students decided to
visit one state park and one historical
battleground. Their state park choices
are Fort Macon, Goose Creek, and
Fort Fisher. The battleground choices
are Almanac Battleground and
Bettonville Battlefield. Which is a
diagram of the sample space for the
class field trip?
A
2
⎢
3
9⎤
⎡-10
4 12 14
⎣-12
6⎦
0
⎡-4 -2 ⎤
B ⎢
⎣ 18 12 ⎦
⎡-10
0⎤
C ⎢
⎣ 0 12 ⎦
⎡-12
0 5⎤
D ⎢
⎣ 0 12 4 ⎦
A
'PSU.BDPO
#FUUPOWJMMF
(PPTF$SFFL
'PSU'JTIFS
"MNBOBD
#FUUPOWJMMF
B
Consider the matrices below.
⎡ 5 1⎤
⎡-2 0 1⎤
⎢
T = -2 4
S=
⎣ 0 3 4⎦
⎣ 6 0⎦
What is the product of S and T ?
'PSU.BDPO
⎢
(PPTF$SFFL
'PSU'JTIFS
"MNBOBD
#FUUPOWJMMF
3
C
What are the x-intercepts of the graph
of the equation f(x) = 2x2 - 4x + 1?
+ √
2
2_
2 - √
2
and _
2
2
2 - √
6
2 + √
6
B _ and _
2
2
C 1+ 2 √
2 and 1 - 2 √
2
and -1 - 2 √
D -1+ 2 √2
2
"MNBOBD
A
'PSU.BDPO
#FUUPOWJMMF
"MNBOBD
(PPTF$SFFL
#FUUPOWJMMF
"MNBOBD
'PSU'JTIFS
#FUUPOWJMMF
D
'PSU'JTIFS
'PSU.BDPO
"MNBOBD
'PSU'JTIFS
(PPTF$SFFL
"MNBOBD
'PSU'JTIFS
#FUUPOWJMMF
"MNBOBD
A64
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
(PPTF$SFFL
Name
Date
Practice Test
4
(continued)
Reggie uses a compass and a
straightedge to draw a line containing
as
point P which is parallel to MN
shown below.
1
6
2
The center of a circle is located at
(4, 2). If A(3, 5) is a point on the circle,
which equation represents the circle?
A
(x - 3)2 + (y - 5)2 = 100
B
(x - 4)2 + (y - 2)2 = 100
C
(x - 3)2 + (y - 5)2 = 10
D (x - 4)2 + (y - 2)2 = 10
.
/
7
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Which statement best supports
Reggie’s construction?
A
If consecutive interior angles are
supplementary, then lines are
parallel.
B
If alternate interior angles are
congruent, then lines are parallel.
C
Parallel lines have the same slope.
D Through any point not on a line,
there is exactly one line parallel to
the given line.
Don is making a pet food mixture by
combining ingredient H with
ingredient J. Don wants to make at
least 5 pounds of pet food that
contains at least 60 grams of protein.
Ingredient H has 8 grams of protein
per pound, and Ingredient J has
16 grams of protein per pound. If
ingredient H costs $4 per pound and
ingredient J costs $3 per pound, what
is the minimum cost of the pet food?
A
$14.00
B
$17.50
C
$20.00
D $22.50
5
If f (x) = 4x2 - x and g(x) = 3x2, which
is an equivalent form of g(x) - f (x)?
A
g(x) - f(x) = -x2 - x
B
g(x) - f(x) = -x2 + x
C
g(x) - f(x) = x2 - x
8
D g(x) - f(x) = x2 + x
Which function has a maximum at
x = 2?
A
f(x) = -4x2 + 8x - 3
B
f(x) = -2x2 + 8x + 1
C
f(x) = x2 - 4x - 7
D f(x) = 3x2 + 12x + 10
A65
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice Test
9
(continued)
Lines q and r are parallel in the figure
below.
q
9
3
6
5
13
r
2
1
10
14
12
7
11
15
4
8
12
s
B
∠1 ∠7
C
∠5 ∠11
8 log4 xy
B
15 log4 xy
C
3 log4 x + 5 log4 y
t
16
13
∠4 ∠16
A
D 8 + log4 x + log4 y
If ∠3 ∠9, which relationship is true?
A
Simplify log4 x3 y5.
Which matrix is equivalent to
⎡ 4 0⎤
⎡4 0⎤
?
-2 ⎢
⎢
⎣-3 1⎦
⎣3 1⎦
⎡ 4
0⎤
⎢
⎣-3 -1⎦
B
⎡4
0⎤
⎢
⎣9 -1⎦
C
⎡-4
0⎤
⎢
⎣-3 -1⎦
D
⎡-4
0⎤
⎢
⎣ 9 -1⎦
D ∠4 ∠15
10
The graph of f (x) = x is translated
3 units up and 2 units left. Which
equation is represented by the
translated graph?
A
g(x) = x + 2 + 3
B
g(x) = x - 2 + 3
C
g(x) = x + 2 - 3
D g(x) = x - 2 - 3
11
14
For which values of x is the function
f (x) = x2 - 8x + 15 decreasing?
A
x>0
C
B
x>4
D x<4
x<0
Brett finds the median-fit line for a
set of data. The three medians are
(8, 24), (17, 19), and (26, 12). What is
the equation of the median-fit line?
A
y = 29.33 - 0.67x
B
y = 29.5 - 0.67x
C
y = 29.67 - 0.67x
D y = 30 - 0.67x
A66
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A
Name
Date
Practice Test
15
(continued)
Which is a pair of similar figures?
17
A
B
The height in feet of a kicked football
is represented by the function
f (t) = -16t2 + 64t + 2, where t is the
time in seconds. What is the maximum
height reached by the football?
A
32 feet
B
50 feet
C
64 feet
D 66 feet
C
18
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
D
North Carolina’s Research Triangle is
an area known for its high technology
businesses, many of which have
benefited from exponential
improvements in computer hardware.
Moore’s law states that the number of
transistors that can inexpensively be
placed on an integrated circuit
doubles every two years. Since the
number of transistors on a circuit was
2300 in 1971, ythe equation
_
16
n = 2300 × 2 2 can be used to predict
the number of transistors n on a
circuit y years after 1971. Which
logarithmic equation is equivalent
What is the vertical shift when
transforming y = cos x into
π
y = cos x - _ + 3 ?
4
π
A _
unit up
4
π
B _
unit down
4
C 3 units up
(
_y
)
to n = 2300 × 2 2 ?
A
B
C
D 3 units down
y = logn 2300
1
y = _ logn 2300
2
n
y = log2 _
2300
(
D y = 2 log2
A67
)
n
(_
2300 )
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice Test
19
(continued)
Which is a valid conclusion drawn
from the true conditional statement in
the box below?
21 What is the simplest form of
1
_
x-3
_
?
1
1+_
If Katia is in the capital of
North Carolina, then she is in
Raleigh.
A
If Katia is in North Carolina, then
she is in the state capital.
B
If Katia is in North Carolina, then
she is not in Raleigh.
C
If Katia is not in Raleigh, then she
is not in North Carolina.
x-3
A
1
x-4
B _
(x - 3)2
1
C _
x-4
1
D _
x-2
D If Katia is not in the state capital of
North Carolina, then she is not in
Raleigh.
Which matrix equation represents the
system of equations below?
⎧ -2y + z = 4
⎨ -x + 3y = 0
⎩ 2x + 5z = 8
A
⎡x⎤ ⎡4⎤
⎡-2 1 ⎤
y = 0
⎢
⎣-1 3 ⎦
⎣z⎦ ⎣8⎦
⎢ ⎢ B
⎡-2 1 0 ⎤ ⎡ x ⎤ ⎡ 4 ⎤
y = 0
-1 3 0
⎣ 0 0 0⎦⎣z⎦ ⎣8⎦
C
⎡0 -2 1 ⎤ ⎡ x ⎤ ⎡ 4 ⎤
y = 0
0 -1 3
⎣2
5 0⎦⎣z⎦ ⎣8⎦
⎢
⎢
⎢ ⎢ Adam is one of 7 students whose
names are in a bag. Two names will be
selected from the bag at random to
win a prize. The first student chosen is
Kelly. Which statement is true about
Adam’s chances of winning?
A
Adam has the same chance of
winning a prize whether Kelly’s
name is returned to the bag or
not.
B
Adam has a better chance of
winning if Kelly’s name is returned
to the bag.
C
Adam has a better chance of
winning if Kelly’s name is kept out
of the bag.
D Adam’s chances of winning are
greater than those of the other
remaining students.
⎢ ⎢ ⎡ 0 -2 1 ⎤ ⎡ x ⎤ ⎡ 4 ⎤
y = 0
D -1
3 0
⎣ 2
0 5⎦ ⎣z⎦ ⎣8⎦
⎢
⎢ ⎢ A68
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
20
22
Name
Date
Practice Test
23
Kai wants to simplify
C
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3x + 6
x2 - 1 _
_
·
25
4x
x2 - 4
before multiplying. Which expression
3x + 6
x2 - 1 _
·
is equivalent to _
?
2
4x
x -4
1 _
A _
· 3;
2 4x
x ≠ -2, x ≠ 0, and x ≠ 2
B
24
(continued)
The sum of the interior angles of a
polygon with 3 sides equals 180°.
II. The sum of the interior angles of a
polygon with 4 sides equals 360°.
III. The sum of the interior angles of a
polygon with 5 sides equals 540°.
IV. The sum of the interior angles of a
polygon with 6 sides equals 720°.
I.
(x - 1) _
_
· 3;
(x - 2) 4x
x ≠ -2, x ≠ 0, and x ≠ 2
(x + 1)(x - 1) _
__
· 3;
(x - 2)
4x
x ≠ -2, x ≠ 0, and x ≠ 2
What conclusion can be inferred
based on the given information?
A
When a polygon has n sides, the
sum of the interior angles will be
360n°.
B When a polygon has n sides, the
sum of the interior angles will be
180n°.
C When a polygon has n sides, the
sum of the interior angles will be
360(n - 2)°.
D When a polygon has n sides, the
sum of the interior angles will be
180(n - 2)°.
(x + 1)(x - 1) _
3(x + 2)
__
·
;
(x - 2)
4x
x ≠ -2, x ≠ 0, and x ≠ 2
A circle is graphed on the coordinate
plane. The diameter of the circle has
endpoints at (-4, 5) and (3, -7).
What is the equation of the circle?
2
1 + (y - 1) 2 = 48.25
A x-_
2
2
1 + (y + 1) 2 = 48.25
B x+_
2
(
(
C
Given:
)
)
26
2
(x - 1)2 + (y - 2) = 56
D (x + 1)2 + (y + 2) 2 = 56
Which transformation results in a
congruent figure?
a dilation by a factor of 3 and a 90°
clockwise rotation about the origin
B a reflection across the line y = -1
and a translation up 3 units
C a translation down 2 units and a
dilation by a factor of 2
D a 270° rotation about the origin
and a dilation by a factor of 3
A
A69
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice Test
(continued)
27 Kalisha is proving that the angle
bisectors of LMN meet at a single
point Z. She begins by constructing
−−
−−−
angle bisectors NX and MY. Next,
she constructs perpendicular
−− −−
−−
segments ZQ, ZR, and ZS. Her work is
shown below.
28
What function is represented by the
graph below?
8
6
4
2
−8−6−4−20
2 4 6 8x
−4
−6
−8
-
4
9
2
A
⎧ _
x if x < 0
y = ⎨ 2
⎩ x - 2 if x ≥ 0
B
⎧ _
1 if x < 0
y = ⎨ 2x
2x - 2 if x ≥ 0
⎩
:
;
.
3
/
Which reasoning could Kalisha use to
−−
show that ZL is also an angle bisector?
QZM RZM and RZN −− −− −−
SZN by SAS, so ZQ ZR ZS.
Angles opposite congruent sides
are congruent, so ∠QLZ ∠SLZ.
−−
Therefore, LZ is an angle bisector.
B
XZM MZN and MZN −− −−
YZN by AAS, so ZX ZY. Angles
opposite congruent sides are
congruent, so ∠XLZ ∠YLZ.
−−
Therefore, LZ is an angle bisector.
⎧ _
x - 2 if x < 0
2
C y = ⎨
2(x - 2) if x ≥ 0
⎩
⎧ _
1 x - 2 if x < 0
D y = ⎨ 2
⎩ 2(x - 1) if x ≥ 0
29
XZM MZN and MZN −− −−
YZN by AAS, so ZX ZY. XLZ
YLZ by HL congruence, so
−−
∠XLZ ∠YLZ. Therefore, LZ is an
angle bisector.
Which is the best example of an
undefined term?
A
theorem
B
postulate
C
point
D angle
D QZM RZM and RZN −− −− −−
SZN by AAS, so ZQ ZR ZS.
QLZ SLZ by HL congruence,
−−
so ∠QLZ ∠SLZ. Therefore, LZ is
an angle bisector.
A70
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A
C
y
Name
Date
Practice Test
30
(continued)
= 132° and QT
ST
Given: mQT
32
The figure below is from a proof of
the Pythagorean Theorem.
4
a
b
c
2
5
What is m∠QTS ?
A
24°
B
48°
Which statement could be used in the
proof of the Pythagorean Theorem?
C
66°
A
The area of the inner square is
equal to half the area of the larger
square.
B
The diagonals of the square are
congruent.
C
The area of all four right triangles
sums to 2ab.
D 114°
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
31
Which function could be represented
by the graph below?
1
D The area of a triangle is _
ac.
y
0
2
x
33
What is the amplitude and period for
1
the function y = 3.5 sin _θ ?
(3 )
A
amplitude: 3.5
period: 3π
A
y=
B
y=_
x
1
B
amplitude: 3.5
period: 6π
C
y = sin x
C
amplitude: 7.0
period: 3π
√x
4
D y=x
D amplitude: 7.0
period: 6π
A71
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice Test
34
(continued)
Which system of inequalities is
represented by the graph below?
4
3
2
1
−4 −3−2
36
y
The number of North Carolina votes
for the winning United States
presidential candidate are shown in
the table below.
Years since
1980
1 2 3 4x
O
−2
−3
−4
35
⎧x - 2y < 3
⎨
⎩ 3x + y ≥ 4
A
⎧x - 2y < 3
⎨
⎩ 3x + y ≤ 4
C
B
⎧x - 2y > 3
⎨
⎩ 3x + y ≥ 4
⎧x - 2y > 3
D ⎨
⎩ 3x + y ≤ 4
Votes
0
915,018
4
1,346,481
8
1,237,258
12
1,114,042
16
1,107,849
20
1,631,163
24
1,961,166
28
2,142,651
Using the least-squares regression line
for the data, what will be the
approximate number of North
Carolina votes for the winning
candidate in the 2012 presidential
election?
A
1,970,000 votes
B
2,120,000 votes
C
2,270,000 votes
D 2,420,000 votes
45 ft
Approximately how far is the laser
measurement device from the
monument?
Which function is the inverse of
f (x) = 7x -10 ?
1
A f -1(x) = _
x + 10
7
1
10
B f -1(x) = _
x+_
7
7
C f -1(x) = -7x - 10
A
30.7 ft
C
D f -1(x) = 7x + 10
B
31.5 ft
D 35.4 ft
37
33.5 ft
A72
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Arrowhead Monument in Old Fort,
North Carolina, is 30 feet tall.
A laser measurement device placed
on the ground forms the right triangle
shown below.
Name
Date
Practice Test
38
(continued)
Tracy and Orazio have two similar
wooden blocks as shown below.
40
Which of the following expressions is
1
_
1
_
32 2 · 2 2
equivalent to _ ?
1
_
A
5 cm Tracy
10 cm
90 cm3
B
180 cm3
C
270 cm3
Orazio
D 4 √
2
41
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
D 360 cm3
39
The table below displays several
points of a logarithmic function.
x
f(x)
3
1
9
2
_4
9
2
8 √
B _
3
2
4 √
C _
3
If the volume of Tracy’s block is
45 cubic centimeters, what is the
volume of Orazio’s block?
A
Ray RZ bisects ∠QRS. What statement
can be inferred based on the given
information?
A
∠QRZ ∠QRS
B
m∠QRS = 90°
C
m∠QRZ < m∠QRS
D ∠QRZ and ∠ZRS are acute angles.
3
_
42
What is 81 4 in simplest form?
27
3
81
4
A
243
5
B
27
4
√
27
C
78
Which exponential function is an
inverse of the function in the table?
A
f -1(x) = x3
B
f -1(x) = 3
C
f -1(x) = (x + 3) 3
18 2
D 108
x
D f -1(x) = 3 x + 1
A73
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice Test
43
(continued)
In Euclidean geometry, which diagram
could be used to provide a
counterexample to the conjecture
below?
45
Lola records survey responses in the
pie chart shown below. The pie chart
has a diameter of 40 inches, and it is
divided into 3 sections.
Preferred Foods
Through any three points, there
exists exactly one plane.
A
1J[[B
2
4UFBL
=°
4
$IJDLFO
=°
3
B
2
What is the approximate length of
the arc intercepted by the Pizza
section of the chart?
4
C
3
2
3
C
B
34.9 in.
D 122.2 in.
61.1 in.
4
3
46
4
2
44
29.7 in.
What is the domain of the function
x2 - x - 6 ?
f (x) = __
x2 + 6x + 8
North Carolina’s population has been
increasing exponentially since 2000.
The growth of the population P can
be modeled according to the
t
equation P = 8,046,500(1.0146) ,
where t represents the number of
years since 2000. Using the equation,
predict how many years it will take for
the population of North Carolina to
reach 10,000,000.
A
2
A
all x
B
7
B
all x not equal to -4 or -2
C
15
C
all x not equal to -2 or 3
D 86
D all x not equal to -4, -2, or 3
A74
North Carolina StudyText, Math BC, Volume 2
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D
A
Name
Date
Practice Test
47
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
48
(continued)
The radius of the tire on Maggie’s
bike is 12 inches. A point on the edge
of the tire is at the bottom, touching
the ground. After she rides a few
feet, that same spot is 6 √
2 inches
above the ground. Which description
best represents the rotation of the
tire?
50
Maria records her January heating
costs for ten years in the table below.
January Heating Costs,
1998–2008
Years Since 1998
Cost ($)
0
163.00
1
166.60
2
172.20
3
177.10
A
675° clockwise rotation
B
690° clockwise rotation
4
179.88
C
750° clockwise rotation
5
183.96
D 780° clockwise rotation
6
188.90
7
195.30
8
201.60
9
207.34
10
215.30
Two functions, g and h, are inverses of
each other. If x is a real number, for
what values of x is the equation
g (h(x)) = x true?
A
all values of x
She finds the least-squares regression
line for the data in the table. For
which of the following years is the
absolute deviation from the leastsquares regression line the greatest?
B
only some values of x
A
C
no values of x
1998
B
2003
C
2004
D impossible to determine
D 2008
49
Which function has a graph with ends
that both extend up?
51
x+4
A
f (x) = 3
B
f (x) = 4 x + 3
C
f (x) = 4x3
D f (x) = 3x4
A cylinder has a volume of
225 cubic inches. What is the volume
of a sphere with the same height and
diameter as the cylinder?
A
75 in3
B
112.5 in3
C
150 in3
D 337.5 in3
A75
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice Test
52
(continued)
Suppose WXYZ is graphed on the
coordinate plane. Which argument
could be used to prove WXYZ is a
rectangle?
A
54
.
Use the distance formula to verify
both pairs of opposite sides are
congruent.
B
Use the slope formula to verify
both pairs of opposite sides are
parallel.
C
Use the midpoint formula to verify
the diagonals bisect each other.
Given: LMN ~ HJK
x-2
)
-
16
+
24
28
32
,
/
What is the value of x?
D Use the distance formula to verify
the diagonals are congruent.
A
10
B
12
C
14
D 16
53
55
x-3
A y=_
;
x+5
x ≠ -5 and x ≠ 5
x-3
B y=_
;
x+5
x ≠ -5, x ≠ 3, and x ≠ 5
x-3
C y=_
;
x-5
x ≠ -5 and x ≠ 5
x-3
D y=_
;
x-5
x ≠ -5, x ≠ 3, and x ≠ 5
Subtract
y-2
4y - 3
_
- _.
6y
A
3y + 1
-_
24y2
B
2
y__
- 8y + 9
6y2
C
2
2y
- 16y - 9
__
12y2
4y2
2
D
A76
2y - 16y + 9
__
12y2
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Simplify the equation below. What
restrictions must be placed on x?
x2 + 2x - 15
y = __
x2 - 25
Name
Date
Practice Test
56
(continued)
In the triangle below, which equation
could be used to find the value of x?
58
In the figure shown below, what
additional information is needed to
prove the triangles are similar?
65°
x
-
18
2 in.
A
9
x = 18 cos 65°
5 in.
;
+
B
x = 18 sin 65°
18
C x=_
cos 65°
18
D x=_
sin 65°
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
57
59
The matrix has 5 rows and 5
columns, with 25 nonzero entries.
B
The matrix has 5 rows and 5
columns, with 20 nonzero entries.
C
The matrix has either 5 rows and 2
columns, or 2 rows and 5 columns,
with 10 nonzero entries.
A
KL = 4 in.
B
KL = 25 in.
C
JK = 4 in.
D JK = 7 in.
A matrix is used to represent the
distances between each of the
following cities: Asheville, Raleigh,
Charlotte, Greensboro, and Durham.
Which statement best describes the
matrix?
A
10 in.
,
:
What is the x-intercept of the graph of
f (x) = √
x - 1?
A
(1, 0)
B
(0, 0)
C
(-1, 0)
D (0, -1)
60
D The matrix has either 5 rows and 4
columns, or 4 rows and 5 columns,
with 20 nonzero entries.
If log10 x = 5, what is the value of x?
1
A x=_
100,000
1
B x=_
50
C x = 50
D x = 100,000
A77
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice Test
61
(continued)
In the figure below, the base areas
and heights of the solids are equal.
The volume of the pyramid is
562 cubic feet.
63
The vertex-edge graph below shows the
approximate number of miles between
rest stops along several bike paths.
16
East exit
Almost
Peddler’s Perch There
15
12
8
2 4
Pond View
Hill 18 Biker’s Bend
10
West exit
4
Top
3
Peaceful View
7
Bird Watch
What is the volume of the prism?
A
187 ft3
B
281 ft3
What is the distance of the shortest
route between the East and West exits?
A
41 miles
3
B
42 miles
D 1686 ft3
C
43 miles
C
1124 ft
D 45 miles
The table below shows data for the
heights of six students and the
number of minutes they exercise each
week.
64
Student Exercise Time
Height
(inches)
55
Time
130
(minutes)
61
20
70
160
63
240
73
100
68
The function f (x) = 2(x + 1)2 - 5 is
transformed to g(x) = -2(x + 1) 2 + 5
in the coordinate plane. Which
statement best describes the
transformation?
A
The graph of f (x) = 2(x + 1)2 - 5 is
reflected across the x-axis.
B
The graph of f (x) = 2(x + 1)2 - 5 is
reflected across the x-axis and
translated 10 units down.
C
The graph of f (x) = 2(x + 1)2 - 5 is
reflected across the x-axis and
translated 10 units up.
90
Using the correlation coefficient r
which statement best describes the
relationship between height and
exercise time?
A
r ≈ 0.0, there is no relationship
B
r ≈ 0.25, there is a weak
relationship
C
r ≈ 0.5, there is a moderate
relationship
D The graph of f (x) = 2(x + 1)2 - 5 is
reflected across the y-axis and
translated 10 units right.
D r ≈ 0.75, there is a strong
relationship
A78
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
62
Name
Date
Practice Test
65
(continued)
The graph below shows the leastsquares regression line for data points
relating a person’s weight and the
number of months the person has
been on a restricted diet.
67
−−
In the figure below, H bisects GY
−−
and JX.
(
9
175
)
Weight (pounds)
170
165
+
160
155
:
150
145
Laura writes the argument below to
prove ∠JGH ∠XYH. What is the
missing step in Laura’s argument?
−−
−−− −−
−−
i. H bisects GY and JX, so GH HY
−− −−
and JH HX.
140
0
1
2
3
4
5
6
7
Time (months)
Using the mean absolute deviation
from the least-squares regression line,
which statement best describes the fit
of the linear model for the data?
ii.
iii. ?
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A
The mean absolute deviation is
about 1 pound, so the linear
model fits the data.
B The mean absolute deviation is
about 3.8 pounds, so the linear
model fits the data.
C The mean absolute deviation is
about 14.9 pounds, so the linear
model does not fit the data.
D The mean absolute deviation is
about 27 pounds, so the linear
model does not fit the data.
66
∠GHJ is vertical to ∠YHX, so
∠GHJ ∠YHX.
iv. Therefore, ∠JGH ∠XYH.
−− −−
GJ XY, so ∠JGH ∠XYH are
alternate interior angles.
−− −− −−−
−−
−− −−
B GJ XY, GH, HY and JH HX,
so GHJ YHX by SSS.
−− −−
C GY ⊥ JX, so GHJ YHX by LL.
−−− −−
D ∠GHJ ∠YHX, GH HY,
−− −−
and JH HX, so GHJ YHX
by SAS.
A
What is the horizontal asymptote of
the graph of f(x) = -2 x + 3 + 4?
A
y=4
B
y=3
C
y=0
D y = -2
A79
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice Test
68
(continued)
If a point is randomly chosen inside
the square, what is the probability that
it is outside of the gray circle?
70
24
Joshua plans to invest $500 in a
savings plan. Plan A guarantees $20
of interest per year. Plan B guarantees
4% interest compounded annually.
Which functions represent the amount
of money he would have on each plan
after x years?
A
Plan A: f (x) = 500 + 20x
Plan B: f (x) = 500 + x4
A
0.21
B
0.32
C
0.68
B
Plan A: f (x) = 500 + 20x
Plan B: f (x) = 500(1 + 0.04)x
C
Plan A: f (x) = 500 + 20x2
Plan B: f (x) = 500 + x4
D 0.79
D Plan A: f (x) = 500 + 20x2
Plan B: f (x) = 500(1 + 0.04)x
69
Triangle EFG is transformed into
triangle E´F´G´ as shown below.
&'(-3, 1)
71
('(2, 0)
0
&(-3, -1)
((2, 0)
x
'(-1, -4)
The coordinate matrix for EFG is
multiplied by matrix T, resulting in the
coordinate matrix for E´F´G´. What is
matrix T ?
A
⎡1
T=⎢
⎣0
B
⎡ 0
T=⎢
⎣-1
0⎤
1⎦
1⎤
0⎦
⎡1
T= ⎢
⎣0
0⎤
-1⎦
⎡1
D T= ⎢
⎣0
-1⎤
0⎦
C
A80
Regina rolls a 6-sided die and
randomly picks a token from a bag.
There are 2 red tokens and 4 blue
tokens in the bag. What is the
probability of rolling an odd number
and picking a red token?
1
A _
2
1
B _
3
1
C _
4
1
D _
6
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
y
''(-1, 4)
Name
Date
Practice Test
72
(continued)
Although blood pressure devices no
longer use mercury, blood pressure
values are still reported in millimeters
of mercury. For example, if blood
pressure oscillates between 110 and
90 millimeters, the person’s blood
pressure is 110 over 90. Rajiv’s blood
pressure is modeled by the graph
below, where m is millimeters and t is
time in seconds.
160
140
120
100
80
60
40
20
0
74
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Passes
written test
Passes
driving test
Yes
Yes
Yes
No
No
Yes
No
No
m
License?
How many times should Jack write
“Yes” in the License column?
1
2
3
4 t
A
100 over 40
B
100 over 80
C
120 over 40
A
1
B
2
C
3
D 4
What is Rajiv’s blood pressure?
75
D 120 over 80
73
Jack reads the following statement:
“The student will earn his driver’s
license if he passes the written test
and the driving test.” Then, he creates
the truth table shown below.
Which translation will transform the
graph of y = x2 into the graph of
y = (x - 1)2 + 2 ?
The price of a first-class United States
postage stamp is graphed on a
coordinate plane. The x-axis
represents the number of years since
1932, and the y-axis represents the
price of the stamp. What type of
function is most likely represented by
the graph?
A
an absolute value function
A
2 units right
B
a quadratic function
B
2 units up and 1 unit right
C
an exponential function
C
2 units up and 1 unit left
D a step function
D 1 unit up and 2 units right
A81
North Carolina StudyText, Math BC, Volume 2
Name
Date
Practice Test
76
(continued)
Elaine has a scarf in the shape of the
triangle shown below.
78
x in.
x in.
16 in.
What is the value of x?
A
8
B
8 √
2
C
16
Junior Senior
Steak ⎡162 143⎤
S = Chicken 143 121
Fish ⎣ 96
77⎦
⎢
D 16 √
2
77
⎡ P11 P12⎤
P=C×S=⎢
⎣ P21 P22⎦
110°
.
2 140°
7
2
;
120°
Which element in matrix P represents
the total cost of meals for juniors at
Restaurant 2?
8
/
:
8
9
80°
C
100°
P11
B
P12
C
P21
D P22
What is m∠W?
A 20°
B
A
D 120°
A82
North Carolina StudyText, Math BC, Volume 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
-
Matrix P is the product of C and S, as
shown below.
Given: Polygon LMNPQ ˜ VWXYZ.
1
Juniors and seniors at a high school
will choose one of two restaurants for
a school banquet. Matrix C shows the
costs of meals at each restaurant.
Matrix S shows the number of
students that select each type of
meal.
Steak Chicken Fish
Restaurant 1 ⎡8.50 9.25 7.70⎤
⎢
C=
Restaurant 1 ⎣ 11 12.50 10.25⎦
Name
Date
Practice Test
79
(continued)
The height of the tide in Cape
Hatteras, North Carolina, is modeled
in the graph below, where h
represents height in feet and t
represents time in hours.
80
Barry drew the weighted graph below
to represent a training program that
involves 11 courses. The edges are
labeled with the time, in hours, it will
take to complete each course.
h
8
7
6
5
4
3
2
1
0
&
15
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
B
C
15
+
(
'
)
96
28
3 6 9 12 15 18 21 24 t
65
120
,
45
Which statement best describes the
height of the tide as modeled by the
graph?
A
15
15
8
-
/
.
The tide peaks about every
3 hours and ranges from
0 to 3 feet.
2
55
37
72
0
1
22
The tide peaks about every
3 hours and ranges from
0.1 to 3.5 feet.
Training
Complete
The tide peaks about every
12.5 hours and ranges from
0 to 3 feet.
What is the critical path?
D The tide peaks about every
12.5 hours and ranges from
0.1 to 3.5 feet.
A
E, G, L, P
B
E, H, L, P
C
E, F, K, M, P
D E, F, K, N, O, P
A83
North Carolina StudyText, Math BC, Volume 2
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