Additional Vocabulary Support

Additional Vocabulary Support
Name
4-1
Class
Date
Additional Vocabulary Support
Quadratic Functions and Transformations
Concept List
Choose the concept from the list below that best represents the item in each box.
axis of symmetry
parabola
translation
maximum value
parent quadratic function
vertex form
minimum value
quadratic function
vertex of the parabola
1. y 5 ax2 1 bx 1 c
quadratic function
2. a line that divides a
y
3.
parabola into two
mirror images
axis of symmetry
x
parabola
4. (h, k), where
5. the y-value of the vertex
y 5 a(x 2 h)2 1 k
vertex of the parabola
when the parabola
opens up
6. y 5 x2
parent quadratic
function
minimum value
7. the y-value of the vertex
when the parabola
opens down
8. y 5 a(x 2 h)2 1 k
vertex form
9. a shift of the graph
horizontally or vertically
translation
maximum value
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1
Name
Class
4-1
Date
Think About a Plan
Quadratic Functions and Transformations
Write a quadratic function to represent the areas of all rectangles with a perimeter of
36 ft. Graph the function and describe the rectangle that has the largest area.
1. Write an equation that represents the area of a rectangle with a perimeter of 36 ft.
Let x 5 width and y 5 length.
2x 1 2y 5 36
2. Solve your equation for y.
y 5 18 2 x
3. Write a quadratic function for the area of the rectangle.
z
z z
z
x
18 2 x z R
5 z z ? Q z 2 x2 z
5 z 18x
x
y
A 5 ? y
4. Graph the quadratic function you wrote.
80
5. What point on the graph has a coordinate that
60
represents the largest area?
40
the maximum of the graph: the vertex
6. How can you find the coordinates of this point? What
20
0
are the coordinates?
x
0
4
8
12
16
Answers may vary. Sample: Read the x-coordinate of the vertex from the graph and then
substitute that value into the quadratic function to get the y-value; (9, 81)
7. Describe the rectangle that has the largest area. What is its area?
The rectangle that has the largest area is a square with length 9 ft; 81 ft2
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Name
Class
4-1
Date
Practice
Form G
Quadratic Functions and Transformations
Graph each function.
1. y 5 3x2
2. f (x) 5 25x2
y
4
3
2
1
5
4. f (x) 5 26x2
⫺2⫺1 O
⫺1
⫺2
⫺3
⫺4
x
⫺2⫺1 O
1 2
y
x 5.
⫺2⫺1 O
⫺1
⫺2
⫺3
⫺4
x 3.
y
87
f (x) 5 10x2
4
3
2
1
1 2
⫺2⫺1 O
8
y 5 3 x2
4
3
2
1
1 2
y
x
⫺2⫺1 O
4
6. f (x) 5 5x2
y
4
3
2
1
x
1 2
y
⫺2⫺1 O
1 2
x
1 2
Graph each function. Describe how it was translated from f (x) 5 x2 .
7. f (x) 5 x2 1 4
up 4 units
5
4
3
2
1
8. f (x) 5 (x 2 3)2
right 3 units
y
x
4
3
2
1
y
x
O
1 2 3 4
Identify the vertex, axis of symmetry, the maximum or minimum value, and
the domain and the range of each function.
9. y 5 (x 2 2)2 1 3
vertex: (2, 3); axis of symmetry: x 5 2;
minimum value: 3; domain: all real
numbers; range: all real numbers # 3
10. f (x) 5 20.2(x 1 3)2 1 2
vertex: (–3, 2); axis of symmetry: x 5 23;
maximum value: 2; domain: all real
numbers; range: all real numbers " 2
Graph each function. Identify the axis of symmetry.
11. y 5 (x 1 2)2 2 1
x 5 22
3
2
1
⫺3⫺2⫺1 O
⫺1
12. y 5 24(x 2 3)2 1 2
x53
y
x
y
2
1
x
O
⫺1
⫺2
1
1 2 3 4
Write a quadratic function to model each graph.
13.
7
6
5
4
3
2
1
⫺1 O
y 5 (x 2 2)2 1 3 14.
y
x
6
5
4
3
2
1
⫺7⫺6⫺5⫺4⫺3⫺2⫺1 O
⫺1
1 2 3 4 5 6
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y
x
y 5 2(x 1 3)2 2 1
Name
Class
Date
Practice (continued)
4-1
Form G
Quadratic Functions and Transformations
Describe how to transform the parent function y 5 x2 to the graph of each
function below. Graph both functions on the same axes.
15. y 5 3Ax 1 2B 2
4
3
2
1
Translate 2 units
to the left; stretch
vertically by the
factor 3.
⫺3⫺2⫺1 O
1
17. y 5 2 Ax 1 4B 2 2 2
Translate 4 units
to the left; shrink
vertically by the
factor 12 ; translate
2 units down.
16. y 5 2Ax 1 5B 2 1 1
y
Translate 5 units
to the left; reflect
across the x-axis;
translate 1 unit up.
x
⫺6⫺4⫺2 O
18. y 5 20.08Ax 2 0.04B 2 1 1.2
8
6
4
2
Translate 0.04 units to
the right; shrink vertically
by a factor of 0.08;
reflect across the x-axis;
translate 1.2 units up.
2 x
x
⫺6⫺4⫺2 O
⫺2
⫺4
1
y
8
6
4
2
y
4
2
⫺4⫺2 O
y
x
2 4
Write the equation of each parabola in vertex form.
1
20. vertex Q 2, 1 R , point (2, 28)
19. vertex (3, 22), point (2, 3)
y 5 5(x 2 3)2 2 2
y 5 24 Q x 2 12 R 2 1 1
21. vertex (24, 224), point (25, 225)
y 5 2(x 1
4)2
22. vertex (212.5, 35.5), point (1, 400)
y 5 2(x 1 12.5)2 1 35.5
2 24
23. The amount of cloth used to make four curtains is given by the function
A 5 24x2 1 40x, where x is the width of one curtain in feet and A is the total
area in square feet. Find the width that maximizes the area of the curtains.
What is the maximum area?
5 ft; 100 ft2
24. The diagram shows the path of a model rocket launched from
384 ft
the ground. It reaches a maximum altitude of 384 ft when it
is above a location 16 ft from the launch site. What quadratic
function models the height of the rocket?
f(x) 5 21.5(x 2 16)2 1 384
Launch
16 ft
25. To make an enclosure for chickens, a rectangular
area will be fenced next to a house. Only three sides
will need to be fenced. There is 120 ft of fencing material.
x
a. What quadratic function represents the area of the
rectangular enclosure, where x is the distance from the house? A 5 22x2 1 120x
b. What dimensions will maximize the area of the enclosure? 30 ft 3 60 ft
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2
32 ft
Name
Class
Date
Practice
4-1
Form K
Quadratic Functions and Transformations
Graph each function.
1. y 5 4x2
y
4
3. y 5 22 x2
8 y
8 y
4
x
⫺8 ⫺4 O
⫺4
1
2. f(x) 5 23x2
⫺8 ⫺4 O
8
4
x
4
x
⫺8 ⫺4 O
⫺4
8
4
8
⫺8
⫺8
Graph each function. How is each graph a translation of f(x) 5 x2 ?
4. f(x) 5 x2 1 4
5. f(x) 5 (x 1 3)2
8 y
8 y
4
⫺8 ⫺4 O
⫺4
6. f(x) 5 x2 2 2
8 y
x
4
4
x
⫺8 ⫺4 O
⫺4
8
⫺8
4
⫺8 ⫺4 O
⫺4
8
⫺8
The vertex moved to (0, 4)
8. f(x) 5 x2 1 6
9. f(x) 5 (x 1 1)2
y
6
4
⫺8 ⫺4 O
⫺4
8
The vertex moved to
(0, 22)
y
y
4
⫺8
The vertex moved to (23, 0)
7. f(x) 5 (x 2 5)2
x
4
4
x
4
⫺8 ⫺4 O
⫺8
The vertex moved to (5, 0)
4
x
8
The vertex moved to (0, 6)
2
⫺4 ⫺2 O
x
2
The vertex moved to (21, 0)
What are the vertex, the axis of symmetry, the maximum or minimum value,
the domain, and the range of each function?
10. f(x) 5 2(x 2 4)2 1 3
11. f(x) 5 2(x 1 3)2 2 2.5
vertex: (4, 3); axis of sym:
x 5 4; min: 3; domain: all
real numbers; range: all real
numbers L 3
vertex: (23, 22.5); axis of
sym: x 5 23; max value:
22.5; domain: all real
numbers; range: all real
numbers K 22.5
12. f(x) 5 22(x 2 6)2
vertex: (6, 0); axis
of sym: x 5 6; max
value: 0; domain: all real
numbers; range: all real
numbers K 0
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Name
Class
Date
Practice (continued)
4-1
Form K
Quadratic Functions and Transformations
What is the graph of each function? Identify the axis of symmetry.
13. f(x) 5 6(x 2 1)2 2 4
x51
4 y
2
⫺2 O
14. f(x) 5 2(x 1 5)2 1 2
y x 5 25
2
x
2
15. f(x) 5 (x 1 2)2 2 7
y
x 5 22
x
4
⫺8⫺6 ⫺4 ⫺2 O
4
⫺8 ⫺4 O
⫺4
⫺4
⫺6
⫺4
x
4
8
⫺8
What quadratic function models each graph?
16.
17.
y
8
6
4
2
⫺4 ⫺2 O
y
⫺6⫺4⫺2
⫺2
⫺4
x
18.
x
O2
⫺8
2 4
f(x) 5 4(x 2 1)2 1 5
8
6
4
2
⫺6⫺4⫺2
f(x) 5 234(x 1 3)2 2 2
y
x
O2
f(x) 5 (x 1 4)2 1 1
19. Error Analysis A classmate said that the vertex of y 5 25(x 1 2)2 2 6 is
(2, 6). What mistake did your classmate make? What is the correct vertex?
Your classmate forgot to change the sign for h and not to change the sign for k.
The correct vertex is (22, 26).
20. Open-Ended Write a quadratic function that has a maximum value.
any quadratic function with a negative a value
Write the equation of each parabola in vertex form.
21. vertex (23, 7), point (22, 25)
f(x) 5 212(x 1
3)2
22. vertex (4, 0), point (26, 23)
f(x) 5 20.03(x 2 4)2
17
23. vertex (22, 25), point (26, 0)
f(x) 5
5
16 (x
1
2)2
24. vertex (1, 3), point (2, 5)
f(x) 5 2(x 2 1)2 1 3
25
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Name
Class
4-1
Date
Standardized Test Prep
Quadratic Functions and Transformations
Multiple Choice
For Exercises 1−4, choose the correct letter.
1. What is the vertex of the function y 5 3(x 2 7)2 1 4? D
(27, 24)
(27, 4)
(7, 24)
(7, 4)
2. Which is the graph of the function f (x) 5 22(x 1 3)2 1 5? F
y
6
6
4
4
2
⫺6 ⫺4 ⫺2 O
y
⫺6 ⫺4 ⫺2 O
⫺2
y
2
x
⫺6 ⫺4 ⫺2 O
x
x
8 y
6
⫺4
4
⫺6
2
O
⫺8
x
2
3. Which of the following best describes how to transform y 5 x2 to the graph of
y 5 4(x 2 2.5)2 2 3? C
Translate 2.5 units left, stretch by a factor of 4, translate 3 units down.
Translate 3 units right and 2.5 units down, stretch by a factor of 4.
Translate 2.5 units right, stretch by a factor of 4, translate 3 units down.
Stretch by a factor of 4, translate 2.5 units left and 3 units down.
4. What is the equation of the parabola with vertex (24, 6) passing through the
point (22, 22)? I
y 5 22(x 1 4)2 2 6
y 5 2(x 1 4)2 1 6
y 5 2(x 2 4)2 2 6
y 5 22(x 1 4)2 1 6
Short Response
5. A baseball is hit so that its height above ground is given by the equation
h 5 216t2 1 96t 1 4, where h is the height in feet and t is the time in
seconds after it is hit. Show your work.
a. How long does it take the baseball to reach its highest point?
b. How high will it go?
[2] a. 3 s; b. 148 ft
[1] incorrect time to highest point OR incorrect final height OR correct time and
distance, but no work shown
[0] incorrect answers and no work shown OR no answers given
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4
6
Name
Class
4-1
Date
Enrichment
Quadratic Functions and Transformations
Parabolas in Other Coordinate Systems
1. What are the coordinates of the vertex of the parabola y 5 ax2 ? (0, 0)
2. How do you determine whether the vertex is a maximum or
a minimum? If a S 0, vertex is a minimum; if a R 0, vertex is a maximum.
3. What is the equation of the axis of symmetry? x 5 0
Suppose you choose any point (h, k). Through (h, k), draw two
lines, one parallel to the x-axis and one parallel to the y-axis.
Let the line parallel to the x-axis be called the u-axis, and let
the line parallel to the y-axis be called the v-axis. You have now
established a new coordinate system—the u-v system.
y
y ⴝ ax2
u
(h, k)
(0, 0)
4. In the x-y system, what are the coordinates of the origin
of the u-v system? (h, k)
5. In the u-v system, what are the coordinates of the vertex of the parabola
y 5 ax2 ? What is the equation of its axis of symmetry? (−h, −k); u 5 2h
Suppose point P has coordinates (x, y) in the x-y system and coordinates (u, v)
in the u-v system.
6. Write an equation expressing the relationship between u and x. x 5 u 1 h
7. Write an equation expressing the relationship between v and y. y 5 v 1 k
8. Use these relationships to write an equation of the parabola y 5 ax2 in terms
of u and v. v 1 k 5 a(u 1 h)2
9. Expand and simplify your equation to express v as a quadratic function of u.
v 5 au2 1 2ahu 1 ah2 2 k
If we let b 5 2ah and c 5 ah2 2 k, the parabola represented by the quadratic
equation v 5 au2 1 bu 1 c in the u-v system is equivalent to the parabola
y 5 ax2 in the x-y system.
10. In the u-v system, express the coordinates of the vertex of this parabola in
terms of a, b, and c. What is the equation of its axis of symmetry?
b
b2
b
Q 2 2a , c 2 4a R ; u 5 2 2a
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v
x
Name
Class
4-1
Date
Reteaching
Quadratic Functions and Transformations
Parent Quadratic Function
The parent quadratic function is y 5 x2 .
y
Axis of symmetry
Substitute 0 for x in the function to get y 5 0. The vertex of the
parent quadratic function is (0, 0).
xⴝ0
A few points near the vertex are:
x
y
23 22 21
9
4
1
1
2
3
1
4
9
y ⴝ x2
x
Vertex (0, 0)
The graph is symmetrical about the line x 5 0 . This line is the axis
of symmetry.
y
Vertex Form of a Quadratic Function
y ⴝ a(xⴚh)2ⴙk
h)2
The vertex form of a quadratic function is y 5 a(x 2
1 k.
The graph of this function is a transformation of the graph of the
parent quadratic function y 5 x2. The vertex of the graph is
(h, k). If a 5 1, you can graph the function by sliding the graph of
the parent function h units along the x-axis and k units along the
y-axis.
y ⴝ x2
Slide
k units
x
Slide h units
Problem
What is the graph of y 5 (x 1 3)2 1 2? What are the vertex and axis of symmetry
of the function?
Step 1
Write the function in vertex form: y 5 1fx 2 (23)g 2 1 2
Step 2
Find the vertex: h 5 23, k 5 2. The vertex is (23, 2).
Step 3
Find the axis of symmetry. Since the vertex is (23, 2), the
graph is symmetrical about the line x 5 23. The axis of
symmetry is x 5 23.
Step 4
Because a 5 1, you can graph this function by sliding
the graph of the parent function 23 units along the x-axis
and 2 units along the y-axis. Plot a few points near the
vertex to help you sketch the graph.
x
y
y ⴝ (xⴙ3)2 ⴙ 2
Slide
2 units
Slide ⴚ3 units
25 24 23 22 21
6
3
2
3
6
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y
y ⴝ x2
x
Name
Class
Reteaching
4-1
Date
(continued)
Quadratic Functions and Transformations
If a 2 1, the graph is a stretch or compression of the parent function by a factor of u a u .
0 , uau , 1
The graph is a vertical
compression of the parent function.
uau . 1
The graph is a vertical
stretch of the parent function
y
y
Vertical
y ⴝ ax2
y ⴝ x2 compression
y ⴝ ax2
y ⴝ x2
Vertical
stretch
x
x
Problem
What is the graph of y 5 2(x 1 3)2 1 2?
Step 1
Write the function in vertex form: y 5 2fx 2 (23)g 2 1 2
Step 2
Step 3
Step 4
The vertex is (23, 2).
The axis of symmetry is x 5 23.
Because a 5 2, the graph of this function is a vertical
stretch by 2 of the parent function. In addition to sliding
the graph of the parent function 3 units left and
2 units up, you must change the shape of the graph. Plot
a few points near the vertex to help you sketch the graph.
x
y
y ⴝ 2(xⴙ3)2 ⴙ 2
y
Stretch
vertically
y ⴝ x2
Slide
2 units
Slide ⴚ3 units
x
25 24 23 22 21
10
4
2
4
10
Exercises
Graph each function. Identify the vertex and axis of symmetry.
1. y 5 (x 2 1)2 1 3
(1, 3);
y
8
x51
4
2
6
4
2
x
⫺2 O
2 4 6
⫺2 O
y
(1, 3);
x51
x
2 4 6
3. y 5 (x 1 2)2 1 1
(22, 1);
y
8
x 5 22
6
4
x
⫺8 ⫺6 ⫺4 ⫺2 O
⫺2
4. y 5 2(x 2 1)2 1 3
8
6
4
2
2. y 5 (x 1 4)2 2 2
y (24, 22);
6
x 5 24
⫺6 ⫺4 ⫺2 O
1
5. y 5 2(x 1 4)2 2 2
(24, 22);
y
6
x 5 24
4
2
6. y 5 0.9(x 1 2)2 1 1
y
8
6
4
2
x
⫺8 ⫺6 ⫺4 ⫺2 O
⫺2
⫺6 ⫺4 ⫺2 O
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2 x
(22, 1);
x 5 22
x
2
Name
Class
4-2
Date
Additional Vocabulary Support
Standard Form of a Quadratic Function
Standard Form of a Quadratic
The standard form of a quadratic is f (x) 5 ax2 1 bx 1 c, where a 2 0.
If a . 0, then the parabola opens upward. If a , 0, then the parabola opens
downward.
The y-intercept is (0, c).
b
The x-coordinate of the vertex is 22a
.
1. Which function is written in standard form? f (x) 5 4(x 2 5)2 1 1
f (x) 5 3x2 2 x 1 4
2. Which has an x-coordinate of 25?
(25, 2)
(2, 25)
3. Which quadratic has a y-intercept of 2?
f (x) 5 2(x 2 5)2 1 2
f (x) 5 2x2 1 5x 1 2
4. Which parabola opens downward?
f (x) 5 223(x 1 1)2 1 5
f (x) 5 4x2 1 1
Vertex Form of a Quadratic
The vertex form of a quadratic is f (x) 5 a(x 2 h)2 1 k, where a 2 0.
If a . 0, then the parabola opens upward. If a , 0, then the parabola opens
downward.
The vertex is (h, k).
5. Which function is written in vertex form?
f (x) 5 22(x 1 7)2 1 3
f (x) 5 2x2 1 2x 1 6
6. Which quadratic has a vertex of (22, 4)?
f (x) 5 x2 2 2x 1 4
f (x) 5 (x 1 2)2 1 4
7. Which parabola opens upward?
f (x) 5 25(x 2 7)2 1 1
f (x) 5 14(x 1 5)2 2 9
Write the vertex form of the quadratic written in standard form below.
8. f (x) 5 x2 2 6x 1 12
f(x) 5 (x 2 3)2 1 3
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Name
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Class
Date
Think About a Plan
Standard Form of a Quadratic Function
Landscaping A town is planning a playground.
It wants to fence in a rectangular space using an
existing wall. What is the greatest area it can fence
in using 100 ft of donated fencing?
Understanding the Problem
1. Write an expression for the width of the playground. Let l be the length of the
playground.
100 2 2l
2. Do you know the perimeter of the playground? Explain.
No; you do not know how much of the wall will be used.
3. What is the problem asking you to determine?
The largest area that can be enclosed by 100 ft of fence on three sides and an unknown
length of wall on the fourth side.
Planning the Solution
4. Write a quadratic equation to model the area of the playground.
A 5 100l 2 2l 2
5. What information can you get from the equation to find the maximum area?
Explain.
You can find the vertex. The x-coordinate of the vertex is the length that gives the
maximum area, which is the y-coordinate.
Getting an Answer
6. What is the value of l that produces the maximum area? 25 ft
7. What is the greatest area the town can fence in using 100 ft of fencing? 1250 ft2
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Name
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Class
Date
Practice
Form G
Standard Form of a Quadratic Function
Identify the vertex, the axis of symmetry, the maximum or minimum value, and
the range of each parabola.
1. y 5 x2 2 4x 1 1
2. y 5 2x2 1 2x 1 3
Vertex is the maximum: (1, 4); axis of
symmetry: x 5 1; range: y K 4
Vertex is the minimum: (2, 23); axis of
symmetry: x 5 2; range: y L 23
3. y 5 2x2 2 6x 2 10
4. y 5 3x2 1 18x 1 32
Vertex is the maximum: (23, 21); axis
of symmetry: x 5 23; range: y K 21
Vertex is the minimum: (23, 5); axis of
symmetry: x 5 23; range: y L 5
5. y 5 2x2 1 3x 2 5
6. y 5 23x2 1 4x
Vertex is the minimum: Q 2 34, 2 49
8 R ; axis
3
of symmetry: x 5 2 4 ; range: y L 2 49
8
Vertex is the maximum: Q 23, 43 R ; axis of
2
symmetry: x 5 3 ; range: y K 43
Graph each function.
7. y 5 x2 1 2x 2 5
8. y 5 2x2 1 3x 1 1
y
2
x
642 O
2
4
6
9. y 5 2x2 1 4x 2 4
3
2
1
1 2 3
1
10. y 5 22 x2 2 3x 1 3
y
2
8
6
4
2
2 x
8
2
42
2
4
6
x
1
642
2
4
11. y 5 3x2 2 8x
y
12. y 5 23x2 1 18x 2 27
y
x
x
42
y
2
4
6
8
2 4
y
x
2 4 6 8
Write each function in vertex form.
13. y 5 x2 2 8x 1 19 y 5 (x 2 4)2 1 3
14. y 5 x2 2 2x 2 6 y 5 (x 2 1)2 2 7
3 2
9
15. y 5 x2 1 3x y 5 Q x 1 2 R 2 4
16. y 5 2x2 1 x y 5 2 Q x 1 1 R 2 2 1
4
8
17. y 5 2x2 2 12x 1 11 y 5 2(x 2 3)2 2 7
18. y 5 22x2 2 4x 1 6 y 5 22(x 1 1)2 1 8
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Name
Class
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Practice (continued)
Form G
Standard Form of a Quadratic Function
19. A small independent motion picture company determines the profit P for
producing n DVD copies of a recent release is P 5 20.02n2 1 3.40n 2 16. P is
the profit in thousands of dollars and n is in thousands of units.
a. How many DVDs should the company produce to maximize the profit? 85,000
b. What will the maximize profit be? $128,500
Sketch each parabola using the given information.
20. vertex (4, 22), y–intercept 6
6
4
2
2
21. vertex (23, 12), point (21, 0)
y
12
8
4
x
2 4 6 8
y
x
8 642
4
For each function, the vertex of the function’s graph is given. Find the unknown
coefficients.
22. y 5 x2 1 bx 1 c; (24, 27) b: 8, c: 9
23. y 5 ax2 2 10x 1 c ; (25, 20) a: 21, c: 25
24. A local nursery sells a large number of ornamental trees every year. The
owners have determined the cost per tree C for buying and caring for each
tree before it is sold is C 5 0.001n2 2 0.3n 1 50. In this function, C is the cost
per tree in dollars and n is the number of trees in stock.
a. How many trees will minimize the cost per tree? 150
b. What will the minimum cost per tree be? $27.50
20 ft
25. To line an irrigation ditch, a farmer will use rectangular
metal sheets. Each side will be bent x feet from the edge
at an angle of 90° to form the trough. If the sheets are
20 ft wide, how far from the edge (x) should the farmer
bend them to maximize the area of a cross-section of the
trough. 5 ft
x
For each function, find the y-intercept.
26. y 5 (x 1 3)2 2 5 (0, 4)
27. y 5 22(x 2 2)2 1 6 (0, –2)
28. y 5 2(x 1 1)2 1 9 (0, 8)
1
29. y 5 2(x 1 4)2 2 15 (0, –7)
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x
Name
Class
Date
Practice
4-2
Form K
Standard Form of a Quadratic Function
What are the vertex, the axis of symmetry, the maximum or minimum value,
and the range of each parabola?
1. y 5 2x2 1 2x 2 5
vertex: (1, 24); axis of
sym: x 5 1; max: 24;
range: all real
numbers K 24
2. y 5 22x2 2 8x 1 3
vertex: (22, 11); axis of
sym: x 5 22; max: 11;
range: all real
numbers K 11
3. y 5 4x2 2 2x 1 1
vertex: Q 14, 34 R ; axis of
sym: x 5 14; min: 34 ;
range: all real
numbers L 34
What is the graph of each function?
4. y 5 2x2 2 6x 2 11
y
8 4 O
4
x
4
8
5. y 5 5x2 1 10x 1 8
12
y
8
8
12
4
8 4 O
7. y 5 22x2 1 4x 1 3
4
8 4
4
4
x
8
8. y 5 x2 1 4x 1 2
8 4 O
4
x
4
4
8
4
x
8
x
8
y
4
O
8 4
4
4
9. y 5 26x2 2 12x 1 5
y
O
11
y
8
8 y
1
6. y 5 2x2 2 3x 1 2
x
8 4 O
4
4
8
Sketch each parabola using the given information.
10. vertex (22, 1),
y-intercept 4
y
6
12. vertex (21, 27),
point (4, 1)
point (25, 1)
8
y
4
4
2
4 2 O
11. vertex (3, 25),
x
2
8
4
x
8 4 O
4
8
8
O
4
8
8
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y
x
4
8
Name
4-2
Class
Date
Practice (continued)
Form K
Standard Form of a Quadratic Function
What is the vertex form of the following quadratic functions?
13. y 5 4x2 1 16x 1 19
y 5 4(x 1 2)2 1 3
16. y 5 23x2 2 18x 2 29
y 5 23(x 1 3)2 2 2
14. y 5 2x2 1 2x 1 4
y 5 2(x 2 1)2 1 5
17. y 5 2x2 1 4x 1 9
y 5 2(x 1 1)2 1 7
1
15. y 5 2x2 2 6x 1 15
y 5 12(x 2 6)2 2 3
8
5
2
18. y 5 3x2 2 3x 1 3
y 5 23(x 2 2)2 2 1
19. Reasoning When is it better to have the quadratic function in vertex form
instead of standard form?
Answers may vary. Sample: When you need to quickly find information about
the graph of a quadratic function.
20. The Gateway Arch in St. Louis was built in 1965. It is the tallest monument
in the United States. The arch can be modeled with the function
y 5 20.00635x2 1 4x, where x and y are in feet.
a. How high above the ground is the tallest point of the arch? 630 ft
b. How far apart are the legs of the arch at their bases? 630 ft
21. The height of a batted ball is modeled by the function h 5 20.01x2 1 1.22x 1 3,
where x is the horizontal distance in feet from the point of impact with the bat,
and h is the height of the ball in feet.
a. What is the maximum height that the ball will reach? about 40 ft
b. At what distance from the batter will the ball hit the ground? about 124 ft
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Standardized Test Prep
Standard Form of a Quadratic Function
Multiple Choice
For Exercises 1−6, choose the correct letter.
1. What is the vertex of the parabola y 5 x2 1 8x 1 5? B
(4, 211)
(24, 211)
(24, 5)
(4, 5)
2. What is the maximum value of the function y 5 23x2 1 12x 2 8? F
4
28
8
2
2
1
3. Which function has the graph shown at the right? D
y 5 22x2 2 5x 1 1
y 5 2x2 1 5x 2 1
y 5 22x2 2 5x 2 1
y 5 22x2 1 5x 2 1
2 1
1
2
4. What is the vertex form of the function y 5 3x2 2 12x 1 17? F
y 5 3(x 2 2)2 1 5
y 5 3(x 2 2)2 1 17
y 5 3(x 2 2)2 1 11
y 5 3(x 1 2)2 1 5
5. What is the equation of the parabola with vertex (3, 220) and that passes
through the point (7, 12)? B
y 5 2x2 1 12x 2 2
y 5 2x2 2 12x 2 2
y 5 22x2 1 12x 2 38
y 5 2x2 2 12x 1 38
6. For the function y 5 25x2 2 10x 1 c, the vertex is (21, 8). What is c? H
213
3
23
13
Short Response
7. To increase revenue, a county wants to increase park fees. The overall income
will go up, but there will be expenses involved in collecting the fees. For a p%
increase in the fees, this cost C will be C 5 0.6p2 2 7.2p 1 48, in thousands
of dollars. What percent increase will minimize the cost to the county? Show
your work.
b
27.2
[2] 6%; the minimum is at the vertex, which is 2 2a
5 2 2(0.6)
5 6.
[1] computational error OR work not shown
[0] incorrect answer and no work shown OR no answer given
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y
x
1 2 3 4
Name
Class
4-2
Date
Enrichment
Standard Form of a Quadratic Function
The standard form of a quadratic function, y 5 ax2 1 bx 1 c , is useful, but it has
the disadvantage that only one of the three constants has a simple geometrical
interpretation.
1. Which of the constants in the equation y 5 ax2 1 bx 1 c can be interpreted
geometrically? c
2. What is its geometrical interpretation? the y-intercept
A more intuitive equation is expressed in terms of constants that have geometrical
interpretations. For instance, if I denotes the y-intercept of a parabola whose
vertex is (V, W), the equation y 5 I 2 2W (x 2 V)2 1 W describes a parabola with
V
an axis of symmetry parallel to the y-axis. Using this equation, write the equation
of the following parabolas in vertex form.
y-intercept
3.
22
4.
1
vertex
(1, 4) y 5 26(x 2 1)2 1 4
(23, 8) y 5 2 79 (x 1 3)2 1 8
I
The equation y 5 PQ(x 2 P)(x 2 Q) describes a parabola with an axis of
symmetry parallel to the y-axis in terms of its y-intercept I and its x-intercepts P
and Q, where P and Q are real numbers. Using this equation, write the equation of
the following parabolas.
y-intercept
x-intercepts
5.
24
1 and 5 y 5 2 45 (x 2 1)(x 2 5)
6.
6
24 and 22 y 5 34 (x 1 4)(x 1 2)
The equation y 5
I
(x 2 P)2 describes a parabola whose axis of symmetry is
P2
parallel to the y-axis with y-intercept I and exactly one x-intercept P.
7. Why is this a special case of the previous equation?
Answers may vary. Sample: It is the previous equation with P 5 Q.
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Name
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Reteaching
4-2
Standard Form of a Quadratic Function
• The graph of a quadratic function, y 5 ax2 1 bx 1 c, where a 2 0, is a parabola.
b
.
• The axis of symmetry is the line x 5 2 2a
b
. The y-coordinate of the vertex is
• The x-coordinate of the vertex is 2 2a
b
b
R , or the y-value when x 5 2 2a.
y 5 f Q 2 2a
• The y-intercept is (0, c).
Problem
What is the graph of y 5 2x2 2 8x 1 5?
x52
2(28)
b
8
5
5 52
2a
2(2)
4
Find the equation of the axis of symmetry.
b
2a
x-coordinate of vertex: 2
2
b
f Q 22a
R 5 f (2) 5 2(2)2 2 8(2) 1 5
Find the y-value when x 5 2.
5 8 2 16 1 5
5 23
y-coordinate of vertex: 23
The vertex is (2, 23).
y-intercept: (0, 5)
The y-intercept is at (0, c) 5 (0, 5).
y
6
5
(0, 5)
4
3
2
1
Because a is positive, the graph opens upward, and the vertex
is at the bottom of the graph. Plot the vertex and draw the axis
of symmetry. Plot (0, 5) and its corresponding point on the other
side of the axis of symmetry.
(4 , 5)
x
O
4 5
2
3
x2
(2, 3)
Exercises
Graph each parabola. Label the vertex and the axis of symmetry.
1. y 5 23x2 1 6x 2 9
(0, 9)
3. y 5 2x2 2 8x 1 1
2. y 5 2x2 2 8x 2 15
y x1
4 O
4 x
4 (1, 6)
8
(2, 9)
4 y
(0, 1)
4 O
4
8
4. y 5 22x2 2 12x 2 7
(4, 1)
4
x
8
(4, 1) 3 y x
9
3
x 4 6
(8, 15) 12 (0, 15)
18
(3, 11) 12 y
6 x = 3
O x
8 4
4
(6, 7) 6 (0, 7)
(2, 7)
x=2
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Name
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Reteaching (continued)
4-2
Standard Form of a Quadratic Function
• Standard form of a quadratic function is y 5 ax2 1 bx 1 c. Vertex form of a
quadratic function is y 5 a(x 2 h)2 1 k.
• For a parabola in vertex form, the coordinates of the vertex are (h, k).
Problem
What is the vertex form of y 5 3x2 2 24x 1 50?
y 5 ax2 1 bx 1 c
y 5 3x2 2 24x 1 50
Verify that the equation is in standard form.
b 5 224, a 5 3
Find b and a.
b
x-coordinate 5 22a
5 2224
For an equation in standard form, the x-coordinate of the vertex
b
can be found by using x 5 2 .
2a
Substitute.
54
Simplify.
2(3)
y-coordinate 5 3(4)2 2 24(4) 1 50 Substitute 4 into the standard form to find the y-coordinate.
52
y 5 3(x 2
4)2
Simplify.
12
Substitute 4 for h and 2 for k into the vertex form.
Once the conversion to vertex form is complete, check by multiplying.
y 5 3(x2 2 8x 1 16) 1 2
y 5 3x2 2 24x 1 50
The result is the standard form of the equation.
Exercises
Write each function in vertex form. Check your answers.
5. y 5 x2 2 2x 2 3
y 5 (x 2 1)2 2 4
8. y 5 x2 2 9x
y 5 Qx 2
9 2
2R
6. y 5 2x2 1 4x 1 6
2
11. y 5 4x2 1 8x 2 3
y 5 4(x 1 1)2 2 7
y 5 Q x 1 32 R 2 2 49
4
y 5 2(x 2 2)2 1 10
9. y 5 x2 1 x
81
4
7. y 5 x2 1 3x 2 10
y 5 Qx 1
1 2
2R
10. y 5 x2 1 5x 1 4
2
y 5 Q x 1 52 R 2 2 94
1
4
3
12. y 5 4x2 1 9x
y 5 34 (x 1 6)2 2 27
13. y 5 22x2 1 2x 1 1
y 5 22 Q x 2 12 R 2 1 32
Write each function in standard form.
14. y 5 (x 2 3)2 1 1
y 5 x2 2 6x 1 10
15. y 5 2(x 2 1)2 2 3
16. y 5 23(x 1 4)2 1 1
y 5 2x2 2 4x 2 1
y 5 23x2 2 24x 2 47
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Additional Vocabulary Support
Modeling With Quadratic Functions
A football player kicks a football and records the height of the ball at different
times. When kicked, at 0 seconds, the ball was 2 ft above the ground. One second
later the ball was 28 ft above the ground, and 2 seconds after being kicked the ball
was 20 ft above the ground. When will the ball hit the ground?
You wrote these steps to solve the problem on note cards, but they got
mixed up.
Substitute the x and y values
into the standard form of a
quadratic function.
Use the quadratic model to
determine when the ball hits
the ground.
Substitute the values of a, b,
and c into the standard form
of a quadratic function.
Solve the system of three
linear equations.
Use the note cards to write the steps in order.
1. First, substitute the x- and y-values into the standard form of a quadratic function
.
2. Second, solve the system of three linear equations
.
3. Next, substitute the values of a, b, and c into the standard form of a quadratic function
.
4. Finally, use the quadratic model to determine when the ball hits the ground
.
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Think About a Plan
Modeling With Quadratic Functions
a. Postal Rates Find a quadratic model for the data. Use 1981 as year 0.
Price of First-Class Stamp
Year
Price (cents)
1981
1991
1995
1999
2001
2006
2007
2008
18
29
32
33
34
39
41
42
b. Describe a reasonable domain and range for your model. (Hint: This is a
discrete, real situation.)
c. Estimation Estimate when first-class postage was 37 cents.
d. Use your model to predict when first-class postage will be 50 cents. Explain
why your prediction may not be valid.
1. How can you find the x-coordinates of the data points?
Subtract 1981 from each year.
2. What calculator function finds a quadratic model for data? QuadReg
z
z
z
z
z
0.930
18.586
3. Find a quadratic model for the data. y 5 20.0036
x2 1 x 1 4. What does the domain of your model represent? What set of numbers would
be a reasonable domain?
Years since 1981; answers may vary. Sample: positive integers
5. What does the range of your model represent? What set of numbers would be
a reasonable domain?
Cost of first-class stamp, in cents; answers may vary. Sample: positive integers
6. How can you find the x-value that produces a given y-value?
Answers may vary. Sample: Graph the function on a calculator and use the
TRACE function.
z
2002
7. Estimate the year when first-class postage was 37 cents. z
z
2021
8. Predict the year when first-class postage will be 50 cents. z
9. Why might your prediction not be valid?
Answers may vary. Sample: Many factors influence postal rates, such as
inflation, fuel costs, and demand for postal services.
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z
Name
Class
4-3
Date
Practice
Form G
Modeling With Quadratic Functions
Find an equation in standard form of the parabola passing through the points.
1. (1, 21), (2, 25), (3, 27)
y5
x2
2. (1, 24), (2, 23), (3, 24)
y 5 2x2 1 4x 2 7
2 7x 1 5
3. (2, 28), (3, 28), (6, 4)
x2
y5
4. (21, 212), (2, 26), (4, 212)
y 5 2x2 1 3x 2 8
2 5x 2 2
5. (21, 212), (0, 26), (3, 0)
6. (22, 24), (1, 21), (3, 11)
y 5 2x2 1 5x 2 6
y 5 x2 1 2x 2 4
7. (21, 26), (0, 0), (2, 6)
y5
9.
11.
2x2
8. (23, 2), (1, 26), (4, 9)
y 5 x2 2 7
1 5x
x
f(x)
21
10.
y 5 x2 2 x 1 5
x
f(x)
7
22
27
1
5
0
1
3
11
2
1
12.
y 5 x2 1 5x 2 2
x
f(x)
26
22
21
1
4
2
21
2
12
3
9
x
f(x)
21
y 5 2x2 1 2x 1 1
y 5 2x2 2 9
13. The table shows the number n of tickets to a school play
sold t days after the tickets went on sale, for several days.
a. Find a quadratic model for the data. n 5 22t2 1 24t 1 10
b. Use the model to find the number of tickets sold on day 7. 80
c. When was the greatest number of tickets sold? day 6
Day, t
Number of
Tickets Sold, n
1
32
3
64
4
74
14. The table gives the number of pairs of skis sold in a sporting
goods store for several months last year.
a. Find a quadratic model for the data, using
January as month 1, February as month 2, and
so on. s 5 2t2 2 28t 1 108
b. Use the model to predict the number of pairs of skis sold
in November. 42
c. In what month were the fewest skis sold? July
Month, t
Number of Pairs
of Skis Sold, s
Jan
82
Mar
42
May
18
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Name
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Date
Practice (continued)
Form G
Modeling With Quadratic Functions
Determine whether a quadratic model exists for each set of values. If so, write
the model.
15. f (21) 5 27, f (1) 5 1, f (3) 5 1
16. f (21) 5 13, f (0) 5 6, f (2) 5 28
no
y 5 2x2 1 4x 2 2
17. f (2) 5 2, f (24) 5 21, f (22) 5 0
18. f (2) 5 6, f (0) 5 24, f (22) 5 26
no
y 5 x2 1 3x 2 4
19. a. Complete the table. It shows the sum of the counting numbers from
1 through n.
Number, n
1
2
3
4
5
Sum, s
1
3
6
10
15
b. Write a quadratic model for the data. s 5 12n2 1 12n
c. Predict the sum of the first 50 counting numbers. 1275
20. On a suspension bridge, the roadway is hung from cables hanging between
support towers. The cable of one bridge is in the shape of the parabola
y 5 0.1x2 2 7x 1 150, where y is the height in feet of the cable above the
roadway at the distance x feet from a support tower.
a. What is the closest the cable comes to the roadway? 27.5 ft
b. How far from the support tower does this occur? 35 ft
21. The owner of a small motel has an unusual idea to increase revenue.
The motel has 20 rooms. He advertises that each night will cost a base
rate of $48 plus $8 times the number of empty rooms that night. For
example, if all rooms are occupied, he will have a total income of
20 3 $48 5 $960. But, if three rooms are empty, then his total income
will be (20 2 3) 3 ($48 1 $8 ? 3) 5 17 3 $72 5 $1224.
a. Write a linear expression to show how many rooms are occupied if n rooms
are empty. 20 2 n
b. Write a linear expression to show the price paid in dollars per room if
n rooms are empty. 48 1 8n
c. Multiply the expressions from parts (a) and (b) to obtain a quadratic model
for the data. Write the result in standard form. y 5 28n2 1 112n 1 960
d. What will the owner’s total income be if 10 rooms are empty? $1280
e. What is the number of empty rooms that results in the maximum income
for the owner? 7
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Name
Class
4-3
Date
Practice
Form K
Modeling With Quadratic Functions
Find an equation in standard form of the parabola passing through the points.
To start, substitute the (x, y) values into y 5 ax2 1 bx 1 c to write a system of
equations.
1. (2, 220), (22, 24), (0, 28)
2. (1, 23), (2, 0), (3, 9)
y 5 2x2 2 4x 2 8
y 5 3x2 2 6x 1 0
3. (3, 21), (2, 25), (4, 25)
4. (24, 3), (26, 7), (21, 12)
y 5 24x2 1 24x 2 37
y 5 x2 1 8x 1 19
5. (2, 1), (1, 21), (4, 27)
6. (21, 2), (22, 7), (0, 7)
y 5 22x2 1 8x 2 7
y 5 5x2 1 10x 1 7
7. A player hits a tennis ball across the court and records the
height of the ball at different times, as shown in the table.
1
11
a. Find a quadratic model for the data. y 5 22x2 1 x 1 2
b. Use the model to estimate the height of the ball
at 4 seconds. 1.5 ft
c. What is the ball’s maximum height? 6 ft
Time (s)
Height (ft)
0
1
2
5.5
6.0
5.5
3
4.0
8. Reasoning Explain why the quadratic model only works up to 4.5 seconds — that
height measurements made after 4.5 seconds are not valid. (Remember this
is a discrete, real situation.)
The ball hits the ground in 4.5 seconds. After it hits the ground, the ball cannot
go any lower.
9. The table at the right shows the height of the tides
measured at the Santa Monica Municipal Pier in
California. Hours are measured from 0.00 at midnight.
a. Find a quadratic model for this data using quadratic
regression. y 5 0.06x2 2 0.62x 1 4.1
b. Use the model to predict the lowest tide height. 2.5 ft
c. When does the lowest tide occur?
Time
Tide Height (ft)
0.33
3.9
3.30
2.7
11.11
4.6
SOURCE: www.tidesandcurrents.noaa.gov
about 5.17 h after midnight
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Practice (continued)
Form K
Modeling With Quadratic Functions
10. The table at the right shows average retail gasoline prices.
a. Find a quadratic model for the data using 1976
as year 0, 1986 as year 10, 1996 as year 20, and 2005
as year 29. y 5 0.11x2 1 1.87x 1 61.83
b. Use the model to estimate the average retail gasoline
price in 2000. 170.07 cents
Year
Price per
gallon (cents)
1976
61.4
1986
92.7
1996
141.3
2005
208.0
SOURCE: U.S. Dept. of Energy
Determine whether a quadratic model exists for each set of values. If so, write
the model.
11. f (0) 5 5, f (24) 5 13, f (2) 5 7
12. f (1) 5 1, f (23) 5 219, f (21) 5 29
no
yes; y 5 12x2 1 5
13. f (0) 5 0, f (1) 5 2, f (2) 5 4
14. f (25) 5 3, f (22) 5 6, f (0) 5 22
no
yes; y 5 2x2 2 6x 2 2
15. The table at the right shows in thousands
how many people in the U.S. subscribe to
a cellular telephone.
a. Find a quadratic model for the data.
Let x 5 the number of years since 1985.
b. Use the model to estimate the number
of subscribers in 1995.
c. Describe a reasonable domain and range
for this situation.
Year
U.S Cellular
Telephone Subscribership
(in thousands)
1985
340
1990
5283
2000
109,478
2004
182,140
SOURCE: CTIA Semi-Annual Wireless Industry
a. y 5 597.29x2 2 1724.76x 2 109.82
b. about 42,372 thousand subscribers
c. domain: all real numbers L 0 (or from 1985); range: all real numbers L 0
16. Error Analysis In Exercise 15 part (c), your friend said that the range was
equal to all real numbers. Why is this incorrect?
Because this is a real situation, you cannot have a negative number of
subscriptions. Therefore, the range must be greater than or equal to 0.
(It cannot be negative.)
17. Reasoning Explain how you know your answer to Exercise 15 part (b) is
reasonable.
The number of subscribers found in 1995 is reasonable because it is in
between the values for 1990 and 2000 from the table.
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Name
Class
4-3
Date
Standardized Test Prep
Modeling With Quadratic Functions
Multiple Choice
For Exercises 1−5, choose the correct letter.
1. Which parabola passes through the points (1, 22), (4, 1), and (5, 22)? B
y 5 2x2 1 x 2 3
y 5 x2 2 4x 1 1
y 5 2x2 1 6x 2 7
y 5 x2 2 4x 2 1
2. Which parabola passes through the points in the table at the right? I
y5
2x2
2x12
y5
y 5 12x2 2 52x 2 1
2x2
2 4x 2 4
y 5 x2 2 3x 2 2
3. A baseball coach records the height at every second of a ball thrown in
x
f(x)
21
2
2
24
4
2
the air. Some of the data appears in the table below.
Time (s)
0
1
3
Height (ft)
0
64
96
Which equation is a quadratic model for the data? A
h 5 216t2 1 80t
h 5 232t2 1 80t
h 5 248t2 1 112t
h 5 216t2 1 64t
4. Use the table in Exercise 3. What is the height of the ball at 2.5 s? H
80 ft
100 ft
88 ft
112 ft
5. Which of the following sets of values cannot be modeled with a quadratic function? C
(2, 3), (0, 21), (3, 2)
(2, 27), (21, 5), (3, 211)
f (2) 5 7, f (21) 5 22, f (0) 5 3
f (2) 5 26, f (0) 5 22, f (21) 5 3
Short Response
6. The accountant for a small company studied the amount spent on advertising
and the company’s profit for several years. He made the table below. What is a
quadratic model for the data? Show your work.
Advertising (Hundreds of Dollars)
Profit (Dollars)
1
2
3
269
386
501
[2] y 5 2x2 1 120x 1 150
[1] correct method but computational errors OR correct model but no work shown
[0] incorrect answers and no work shown OR no answers given
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Name
4-3
Class
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Enrichment
Modeling With Quadratic Functions
Baseballs in Flight
When baseballs are shot out of a cannon, their flight through the air
depends on both the angle at which the cannon is set and the initial velocity
of the baseball.
6
Height of baseball
The equation y 5 0.5x 2 0.01x2 represents the parabolic flight of
a certain baseball shot at an angle of 26° with the horizon and at
an initial velocity of 25 meters per second. In this equation, y is the
height of the baseball, in meters, and x is the horizontal distance
traveled, in meters. The graph of the equation is shown to the
right.
y
(10, 4)
4
(40, 4)
2
O
1. Given that the points (10, 4) and (40, 4) lie on the parabola, at
what x-coordinate must the vertex lie? x 5 25
(0, 0)
20
x
40
60
Distance traveled
2. Use the equation and your answer to question 1 to find the
maximum height of the baseball. 6.25 m
3. Use the point (0, 0) and the location of the vertex to find the
total horizontal distance that the baseball will travel. 50 m
4. What is the total horizontal distance that this baseball will
travel? 25 m
3
Height of baseball
When the angle of the cannon is decreased, the baseball will travel
in a different flight. The parabolic flight of the baseball is shown to
the right, with the vertex labeled.
y
2
(12.5, 1.25)
1
(0, 0)
20
O
x
40
60
Distance traveled
5. How far will the baseball travel horizontally before it reaches its
maximum height? 25 m
3
Height of baseball
Using the same angle, the initial velocity of the baseball is increased
to produce the graph of the flight shown to the right. The point
shown represents the total horizontal distance the baseball will
travel.
2
1
O
NOTE: SOURCES: Gustafson, R. David. Concepts of intermediate algebra: an early functions approach.
Pacific Grove, Calif.: Brooks/Cole Pub. Co., 1996.
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y
(0, 0) (50, 0)
20
40
Distance traveled
x
60
Name
4-3
Class
Date
Reteaching
Modeling With Quadratic Functions
Three non-collinear points, no two of which are in line vertically, are on the graph
of exactly one quadratic function.
Problem
A parabola contains the points (0, 22), (21, 5), and (2, 2). What is the equation of
this parabola in standard form?
If the parabola y 5 ax2 1 bx 1 c passes through the point (x, y), the coordinates
of the point must satisfy the equation of the parabola. Substitute the (x, y) values
into y 5 ax2 1 bx 1 c to write a system of equations.
First, use the point (0, 22).
Use the point (21, 5) next.
Finally, use the point (2, 2).
y 5 ax2 1 bx 1 c
Write the standard form.
22 5 a(0)2 1 b(0) 1 c
Substitute.
22 5 c
Simplify.
5 5 a(21)2 1 b(21) 1 c
Substitute.
55a2b1c
Simplify.
2 5 a(2)2 1 b(2) 1 c
Substitute.
2 5 4a 1 2b 1 c
Simplify.
Because c 5 22, the resulting system has two variables. Simplify the equations above.
a2b57
4a 1 2b 5 4
Use elimination to solve the system and obtain a 5 3, b 5 24, and c 5 22. Substitute
these values into the standard form y 5 ax2 1 bx 1 c.
The equation of the parabola that contains the given points is y 5 3x2 2 4x 2 2.
Exercises
Find an equation in standard form of the parabola passing through the given
points.
1. (0, 21), (1, 5), (21, 25) y 5 x2 1 5x 2 1
2. (0, 4), (21, 9), (2, 0) y 5 x2 2 4x 1 4
3. (0, 1), (1, 4), (3, 22) y 5 2x2 1 x 1 1
4. (1, 21), (22, 20), (2, 0) y 5 2x2 2 5x 1 2
5. (21, 25), (0, 21), (2, 1) y 5 2x2 1 3x 2 1
6. (1, 3), (22, 23), (21, 3) y 5 22x2 1 5
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Class
4-3
Date
Reteaching (continued)
Modeling With Quadratic Functions
Problem
A soccer player kicks a ball off the top of a building. His friend
records the height of the ball at each second. Some of her data
appears in the table.
a. What is a quadratic model for these data?
b. Use the model to complete the table.
Time (s)
Height (ft)
0
112
1
192
2
3
Use the points (0, 112), (1, 192), and (5, 192) to find the quadratic
model. Substitute the (t, h) values into h 5 at2 1 bt 1 c to write
a system of equations.
(0, 112) : 112 5 a(0)2 1 b(0) 1 c
c 5 112
(1, 192) : 192 5 a(1)2 1 b(1) 1 c
a 1 b 1 c 5 192
(5, 192) : 192 5 a(5)2 1 b(5) 1 c
25a 1 5b 1 c 5 192
4
5
192
6
7
Use c 5 112 and simplify the equations to obtain a system with just two variables.
a 1 b 5 80
25a 1 5b 5 80
Use elimination to solve the system. The quadratic model for the data is
h 5 216t2 1 96t 1 112
Time (s)
Height (ft)
Now use this equation to complete the table for the t-values 2, 3,
4, 6, and 7.
0
112
1
192
t 5 2: h 5 216(2)2 1 96(2) 1 112 5 264 1 192 1 112 5 240
2
240
3
256
4
240
t 5 3: h 5 216(3)2 1 96(3) 1 112 5 2144 1 288 1 112 5 256
t 5 4: h 5
216(4)2
1 96(4) 1 112 5 2256 1 384 1 112 5 240
5
192
t 5 6: h 5
216(6)2
1 96(6) 1 112 5 2576 1 576 1 112 5 112
6
112
7
0
t 5 7: h 5 216(7)2 1 96(7) 1 112 5 2784 1 672 1 112 5 0
Exercise
7. The number n of Brand X shoes in stock at the beginning of month t in a store
follows a quadratic model. In January (t 5 1), there are 36 pairs of shoes; in
March (t 5 3), there are 52 pairs; and in September, there are also 52 pairs.
a. What is the quadratic model for the number n of pairs of shoes at the
beginning of month t? n 5 2t2 1 12t 1 25
b. How many pairs are in stock in June? 61
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Name
Class
Date
Additional Vocabulary Support
4-4
Factoring Quadratic Expressions
Choose the word from the list that best matches each sentence.
factoring
greatest common
perfect square
difference of two
factor
trinomial
squares
1. the expression a2 2 b2
difference of two squares
2. rewriting an expression as a product of its factors
factoring
3. a trinomial that is the square of a binomial
perfect square trinomial
4. a common factor of each term in the expression
greatest common factor
Choose the word from the list that best matches each sentence.
factoring
5. 10 is the
greatest common
perfect square
difference of two
factor
trinomial
squares
greatest common factor
6. An example of a
7. When
of the expression 20x2 2 50.
perfect square trinomial
factoring
is x2 2 8x 1 16.
x2 1 8x 1 15, find numbers with product 15 and
sum 8.
8. The difference of two squares will always be a binomial.
Multiple Choice
9. Which of the following is a perfect square trinomial? B
2x 2 7
9x2 2 6x 1 1
4x2 2 25
9x2 2 4x
10. Which of the following is a difference of perfect squares? H
2x 2 7
9x2 2 6x 1 1
4x2 2 25
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9x2 2 4x
Name
4-4
Class
Date
Think About a Plan
Factoring Quadratic Expressions
Agriculture The area in square feet of a rectangular field is x2 2 120x 1 3500.
The width, in feet, is x 2 50. What is the length, in feet?
Know
z
z
z
z
length times the width .
1. The area of the field equals the 2
2. The area of the field is x 2 120x 1 3500 ft2 .
z
z
x 2 50 ft.
3. The width of the field is Need
4. To solve the problem I need to:
rewrite the expression for the area of the field as a product of its factors
.
Plan
z
z
x 2 50 .
5. One factor is 6. What is the coefficient of the first term of the other factor? 1
How do you know?
The coefficient of the trinomial is 1.
z
7. What is the sign of the second term of the other factor? negative
z
How do you know?
The second terms of the two factors must have the same sign because the third
term of the trinomial is positive. The second term of the first factor is negative.
z
z
70
8. The product of 50 and is 3500.
z
z
z
z
70
9. The sum of 50 and is 120.
x 2 70 .
10. The other factor is 11. What is the length of the rectangular field, in feet? x 2 70
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Name
4-4
Class
Date
Practice
Form G
Factoring Quadratic Expressions
Factor each expression.
1. x2 1 11x 1 28 (x 1 7)(x 1 4)
2. x2 1 11x 1 24 (x 1 8)(x 1 3)
3. s2 1 13s 1 42 (s 1 7)(s 1 6)
4. x2 2 10x 1 21 (x 2 7)(x 2 3)
5. y2 2 8y 1 15 ( y 2 5)( y 2 3)
6. x2 2 12x 1 32 (x 2 8)(x 2 4)
7. 2x2 1 9x 2 18 2(x 2 6)(x 2 3)
8. 2w2 1 12w 2 35 2(w 2 7)(w 2 5)
9. 2t2 2 3t 1 54 2(t 1 9)(t 2 6)
10. x2 2 7x 2 60 (x 2 12)(x 1 5)
Find the GCF of each expression. Then factor the expression.
11. 6x2 2 9 3(2x2 2 3)
12. 16m2 1 8m 8m(2m 1 1)
13. 2a2 1 22a 1 60 2(a 1 6)(a 1 5)
14. 5x2 1 25x 2 70 5(x 1 7)(x 2 2)
1
1
15. 3x2 1 3x 2 4 31 (x 2 3)(x 1 4)
16. 27x2 1 7x 1 14 27(x 1 1)(x 2 2)
Factor each expression.
17. 5x2 2 17x 1 6 (x 2 3)(5x 2 2)
18. 3x2 1 10x 1 8 (x 1 2)(3x 1 4)
19. 2b2 2 9b 2 5 (2b 1 1)(b 2 5)
20. z2 1 12z 1 36 (z 1 6)2
21. 9x2 2 6x 1 1 (3x 2 1)2
22. 4k2 1 12k 1 9 (2k 1 3)2
23. n2 2 49 (n 2 7)(n 1 7)
24. 2x2 2 50 2(x 2 5)(x 1 5)
25. The area of a rectangular field is x2 2 x 2 72 m2 . The length of the field is x 1 8 m.
What is the width of the field in meters? x 2 9
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4-4
Class
Date
Practice (continued)
Form G
Factoring Quadratic Expressions
26. The product of two integers is w2 2 3w 2 40, where w is a whole number. Write
expressions for each of the two integers in terms of w. w 2 8, w 1 5
27. John is j years old. The product of his younger brother’s and older sister’s
ages is j2 2 2j 2 15. How old are John’s brother and sister in terms of
John’s age? brother: j 2 5; sister: j 1 3
Factor each expression completely.
28. 2x2 1 9x 1 10 (2x 1 5)(x 1 2)
29. 6y2 2 5y 1 1 (2y 2 1)(3y 2 1)
30. 3x2 1 8x 2 3 (x 1 3)(3x 2 1)
31. 4x2 2 7x 2 15 (4x 1 5)(x 2 3)
32. 12t2 1 10t 2 12 2(2t 1 3)(3t 2 2)
33. 210x2 1 x 1 21 2(5x 1 7)(2x 2 3)
34. 24k2 1 2k 1 30 22(2k 1 5)(k 2 3)
1
1
35. 2x2 1 2x 2 10 12 (x 1 5)(x 2 4)
36. x2 2 16x 1 64 (x 2 8)2
37. m2 1 22m 1 121 (m 1 11)2
38. 16x2 2 40x 1 25 (4x 2 5)2
39. 36x2 1 12x 1 1 (6x 1 1)2
40. 22x2 2 32x 2 128 22(x 1 8)2
41. 225p2 1 30p 2 9 2(5p 2 3)2
42. r2 2 144 (r 1 12)(r 2 12)
1
1 1
43. 4x2 2 4 4(x 1 1)(x 2 1)
44. 27s2 1 175 27(s 1 5)(s 2 5)
1
1
45. 225z2 1 1 225(z 1 5)(z 2 5)
46. The radius of the outer circle in the illustration is R. The radius
of the inner circle is r.
a. Write an expression for the area of the outer circle. πR2
b. Write an expression for the area of the inner circle. πr2
c. Write an expression representing the area of the ring, the
shaded region in the illustration. Do not simplify. πR2 2 πr2
d. Factor the expression in part (c). π(R 1 r)(R 2 r)
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R
r
Name
Class
4-4
Date
Practice
Form K
Factoring Quadratic Expressions
Factor each expression.
1. x2 1 4x 2 5
(x 2 1)(x 1 5)
2. x2 1 13x 1 42
(x 1 6)(x 1 7)
3. 2x2 2 x 1 12
2(x 1 4)(x 2 3)
4. x2 2 8x 1 16
(x 2 4)(x 2 4)
5. 2x2 1 16x 2 55
2(x 2 11)(x 2 5)
6. x2 1 2x 2 48
(x 2 6)(x 1 8)
7. 2y2 1 17y 2 72
2(y 2 8)(y 2 9)
8. x2 1 7x 1 12
(x 1 4)(x 1 3)
9. x2 2 8x 1 12
(x 2 2)(x 2 6)
Find the GCF of each expression. Then factor the expression.
10. 3x2 1 15x 1 12
3(x 1 1)(x 1 4)
11. 29y2 1 6y
23y(3y 2 2)
12. 6x2 1 12x 2 48
6(x 2 2)(x 1 4)
13. 23x2 2 3x 1 60
23(x 2 4)(x 1 5)
14. 2x2 2 10x
2x(x 2 5)
15. 7x2 2 14x 2 56
7(x 1 2)(x 2 4)
16. 10x2 1 100x
10x(x 1 10)
17. 9x2 2 36x 1 27
9(x 2 3)(x 2 1)
18. 25xy2 2 30xy 2 25x
25x(y 1 1)(y 1 5)
19. Writing When you factor a quadratic expression, explain what it means when
c , 0 and b . 0.
When c R 0, one factor is positive and the other is negative and when b S 0, the
factor with the greater absolute value is positive.
20. Error Analysis You factored 2x2 1 10x 2 24 as 2(x 2 6)(x 2 4). Your friend factored
it as (x 1 12)(x 2 2). Which of you is correct? What mistake was made?
You are correct; your friend forgot to factor out 21.
21. Multiple Choice What is the factored form of 214a2 1 42ab? D
a(214a 1 42b)
7(22a2 1 6ab)
22a(7a 2 21b)
214a(a 2 3b)
22. Reasoning The area of a carpet is (x2 2 11x 1 28) ft2 . What are the length
and the width of the carpet? (x 2 7) ft and (x 2 4) ft
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Name
Class
4-4
Date
Practice (continued)
Form K
Factoring Quadratic Expressions
Factor each expression.
23. 2x2 1 7x 1 6
(2x 1 3)(x 1 2)
24. 3x2 2 14x 2 24
(3x 1 4)(x 2 6)
25. 5x2 2 22x 1 21
(5x 2 7)(x 2 3)
26. 4x2 1 18x 1 8
2(x 1 4)(2x 1 1)
27. 2x2 2 8x 1 6
2(x 2 3)(x 2 1)
28. 6x2 1 13x 2 28
(3x 2 4)(2x 1 7)
29. 4x2 2 4x 1 1
30. x2 1 6x 1 9
(2x 2 1)2
(x 1 3)2
31. 4x2 2 16
4(x 2 2)(x 1 2)
32. 9x2 2 4
33. 16x2 2 40x 1 25
(3x 1 2)(3x 2 2)
35. 9x2 2 36x 1 36
9(x 2 2)2
(4x 2 5)2
34. x2 2 25
(x 2 5)(x 1 5)
36. 25x2 2 9
(5x 2 3)(5x 1 3)
37. 4x2 1 24x 1 36
4(x 1 3)2
38. Error Analysis Which of the following examples is factored correctly? Explain. Example 1
The product of two terms with different
Example 1
Example 2
signs is negative, so the correct factorization
2
2
of 272 is 27 and 7, not 27 and 27.
4x 2 49
4x 2 49
(2x)2 2 72
(2x 2 7)(2x 1 7)
(2x)2 2 72
(2x 2 7)(2x 2 7)
39. You can represent the area of a square tabletop with the expression
16x2 1 24x 1 9. What is the side length of the tabletop in terms of x?
4x 1 3 units
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Date
Standardized Test Prep
Factoring Quadratic Expressions
Multiple Choice
For Exercises 1−6, choose the correct letter.
1. What is the complete factorization of 2x2 1 x 2 15? B
(x 2 5)(2x 1 3)
(x 2 3)(2x 1 5)
(x 1 3)(2x 2 5)
(x 1 5)(2x 2 3)
2. What is the complete factorization of 2x2 1 3x 1 28? I
(x 2 4)(x 2 7)
2(x 1 4)(x 1 7)
2(x 2 4)(x 1 7)
2(x 2 7)(x 1 4)
3. What is the complete factorization of 6x2 1 9x 2 6? A
3(2x 2 1)(x 1 2)
3(x 2 2)(2x 1 1)
(3x 1 2)(2x 2 3)
3(x 2 2)(2x 2 1)
4. What is the complete factorization of 16x2 2 56x 1 49? G
(4x 2 7)(4x 1 7)
(4x 1 7)2
(4x 2 7)2
16(x 2 7)2
5. What is the complete factorization of 5x2 2 20? C
(5x 2 4)(x 1 5)
5(x 1 2)(x 2 2)
5(x 1 4)(x 2 4)
5(x 2 2)2
6. What is the complete factorization of x2 2 14x 1 24? I
(x 2 8)(x 2 3)
(x 1 2)(x 2 12)
(x 2 4)(x 2 6)
(x 2 12)(x 2 2)
Short Response
7. The area in square meters of a rectangular parking lot is x2 2 95x 1 2100.
The width in meters is x 2 60. What is the length of the parking lot in meters?
Show your work.
[2] x 2 35
[1] correct method but an error in finding the length OR correct length
but no work shown
[0] incorrect answer and no work shown OR no answer given
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Enrichment
Factoring Quadratic Expressions
To factor a quadratic expression of the form ax2 1 bx 1 c, break the middle term of the
expression into two terms and use common factors to complete the factoring.
2x2 2 3x 2 5
2x2 1 2x 2 5x 2 5
Rewrite 23x as 2x 2 5x.
2x(x 1 1) 2 5(x 1 1)
Factor the first two terms. Then factor the third and
fourth terms.
(2x 2 5)(x 1 1)
Rewrite the expression using the Distributive Property.
This same method can be used to factor polynomials with more than three terms.
1. Rewrite x3 1 3x2 1 4x 1 12 by finding a common factor for the first two
terms and another for the last two terms. x2(x 1 3) 1 4(x 1 3)
2. Factor x3 1 3x2 1 4x 1 12 using the Distributive Property. (x2 1 4)(x 1 3)
Use this method to factor the polynomials below.
3. 3x2 1 xy 2 12x 2 4y (x 2 4)(3x 1 y)
4. a3 2 2a2 1 5a 2 10 (a2 1 5)(a 2 2)
5. x4 1 2x3 2 2x 2 4 (x3 2 2)(x 1 2)
6. b3 1 3b2 2 2b 2 6 (b2 2 2)(b 1 3)
7. m3 1 4m2 2 9m 2 36 (m 1 4)(m 2 3)(m 1 3)
8. c4 2 c2d2 1 c2d 2 d3 (c2 1 d )(c 2 d )(c 1 d )
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Reteaching
Factoring Quadratic Expressions
Problem
What is 6x2 2 5x 2 4 in factored form?
a 5 6, b 5 25, and c 5 24
Find a, b, and c; they are the coefficients of each term.
ac 5 224 and b 5 25
We are looking for factors with product ac and sum b.
Factors of 224 1, 224 21, 24 2, 212 22, 12 3, 28
Sum of factors
223
23
210
10
25
23, 8
4, 26
24, 6
5
22
2
The factors 3 and 28 are the combination whose sum is 25.
6x2 1 3x 2 8x 2 4
Rewrite the middle term using the factors you found.
3x(2x 1 1) 2 4(2x 1 1)
Find common factors by grouping the terms in pairs.
(3x 2 4)(2x 1 1)
Rewrite using the Distributive Property.
3
Check
3
(3x 2 4)(2x 1 1)
You can check your answer by multiplying the factors together.
6x2 1 3x 2 8x 2 4
6x2 2 5x 2 4
Remember that not all quadratic expressions are factorable.
Exercises
Factor each expression.
1. x2 1 6x 1 8 (x 1 4)(x 1 2)
2. x2 2 4x 1 3 (x 2 3)(x 2 1)
3. 2x2 2 6x 1 4 2(x 2 2)(x 2 1)
4. 2x2 2 11x 1 5 (2x 2 1)(x 2 5)
5. 2x2 2 7x 2 4 (2x 1 1)(x 2 4)
6. 4x2 1 16x 1 15 (2x 1 5)(2x 1 3)
7. x2 2 5x 2 14 (x 1 2)(x 2 7)
8. 7x2 2 19x 2 6 (7x 1 2)(x 2 3)
9. x2 2 x 2 72 (x 2 9)(x 1 8)
10. 2x2 1 9x 1 7 (2x 1 7)(x 1 1)
11. x2 1 12x 1 32 (x 1 4)(x 1 8)
12. 4x2 2 28x 1 49 (2x 2 7)(2x 2 7)
13. x2 2 3x 2 10 (x 2 5)(x 1 2)
14. 2x2 1 9x 1 4 (2x 1 1)(x 1 4)
15. 9x2 2 6x 1 1 (3x 2 1)(3x 2 1)
16. x2 2 10x 1 9 (x 2 1)(x 2 9)
17. x2 1 4x 2 12 (x 1 6)(x 2 2)
18. x2 1 7x 1 10 (x 1 5)(x 1 2)
19. x2 2 8x 1 12 (x 2 6)(x 2 2)
20. 2x2 2 5x 2 3 (2x 1 1)(x 2 3)
21. x2 2 6x 1 5 (x 2 1)(x 2 5)
22. 3x2 1 2x 2 8 (3x 2 4)(x 1 2)
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39
Name
Class
4-4
Date
Reteaching (continued)
Factoring Quadratic Expressions
• a2 1 2ab 1 b2 5 (a 1 b)2
Factoring perfect square trinomials
a2 2 2ab 1 b2 5 (a 2 b)2
• a2 2 b2 5 (a 1 b)(a 2 b)
Factoring a difference of two squares
Problem
What is 25x2 2 20x 1 4 in factored form?
There are three terms. Therefore, the expression may be a perfect square trinomial.
a2 5 25x2 and b2 5 4
Find a2 and b2.
a 5 5x and b 5 2
Take square roots to find a and b.
Check that the choice of a and b gives the correct middle term.
2ab 5 2 ? 5x ? 2 5 20x
Write the factored form.
a2 2 2ab 1 b2 5 (a 2 b)2
25x2 2 20x 1 4 5 (5x 2 2)2
(5x 2 2)2
Check
(5x 2 2)(5x 2 2)
25x2 2 10x 2 10x 1 4
25x2
2 20x 1 4
You can check your answer by multiplying the factors together.
Rewrite the square in expanded form.
Distribute.
Simplify.
Exercises
Factor each expression.
23. x2 2 12x 1 36
24. x2 1 30x 1 225
(x 2 6)2
25. 9x2 2 12x 1 4
(x 1 15)2
(3x 2 2)2
26. x2 2 64
(x 1 8)(x 2 8)
27. 9x2 2 42x 1 49
29. 27x2 2 12
3(3x 1 2)(3x 2 2)
30. 49x2 1 42x 1 9
32. 9x2 2 16
(3x 1 4)(3x 2 4)
33. 8x2 2 18
2(2x 1 3)(2x 2 3)
34. 81x2 1 126x 1 49
35. 125x2 2 100x 1 20
36. 2x2 1 196
37. 216x2 2 24x 2 9
5(5x 2 2)2
(3x 2 7)2
28. 25x2 2 1
(5x 1 1)(5x 2 1)
31. 16x2 2 32x 1 16
(7x 1 3)2
16(x 2 1)2
(9x 1 7)2
2(x 1 14)(x 2 14)
2(4x 1 3)2
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40
Name
Class
4-5
Date
Additional Vocabulary Support
Quadratic Equations
Problem
What are the solutions of the quadratic equation 2x2 2 4x 5 6?
Explain
Work
Justify
2x2 2 4x 5 6
First, write the equation.
Second, subtract 6 from
each side to set equal to 0.
2x2 2 4x 2 6 5 0
Next, factor out the GCF, 2.
2(x2 2 2x 2 3) 5 0
Then, factor the trinomial.
2(x 2 3)(x 1 1) 5 0
Then, use the Zero-Product
Property.
Finally, solve for x.
x 2 3 5 0 or x 1 1 5 0
x 5 3 or x 5 21
Original equation
Subtraction Property of Equality
Distributive Property
Factor the quadratic expression
Zero-Product Property
Addition Property of Equality
Additio
Solution
3 or 21
Exercise
What are the solutions of the quadratic equation 3x2 2 6x 5 23?
Explain
Work
First, write the equation.
Justify
3x2 2 6x 5 23
Original equation
Second, add 3 to each side
to set equal to 0.
3x2 2 6x 1 3 5 0
Addition Property of
Equality
Next, factor out the GCF, 3.
3(x2 2 2x 1 1) 5 0
Distributive Property
Then, factor the trinomial.
3(x 2 1)(x 2 1) 5 0
Then, use the Zero-Product
Property.
x 2 1 5 0 or x 2 1 5 0
Factor the quadratic
expression
Zero-Product Property
Finally, solve for x.
Addition Property of
Equality
x51
Solution
1
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41
Name
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Class
Date
Think About a Plan
Quadratic Equations
Landscaping Suppose you have an outdoor pool measuring 25 ft by 10 ft. You
want to add a cement walkway around the pool. If the walkway will be 1 ft thick
and you have 304 ft3 of cement, how wide should the walkway be?
Understanding the Problem
x
x
25 ft
walkway in feet.
2. If you lay the pieces of walkway end to end, what is the total length of
the walkway? 4x 1 70
3. What is the thickness of the walkway? 1 ft
4. What is the problem asking you to determine?
The width of a walkway that is (4x 1 70) ft long and 1 ft thick, that has volume 304 ft3.
Planning the Solution
5. Write a quadratic equation to model the volume of the walkway. V 5 4x2 1 70x
6. What method can you use to find the solutions of your quadratic equation?
Answers may vary. Sample: Substitute 304 for V in the equation. Subtract 304 from
each side to set it equal to zero. Graph y 5 4x2 1 70x 2 304 on a graphing calculator
and use the ZERO option in the CALC feature to solve for x.
Getting an Answer
7. How many solutions of your quadratic equation do you need to find? Explain.
One; only one solution is positive, and a negative solution does not make sense.
8. How wide should the walkway be? about 3.6 ft
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42
x
10 ft
1. Draw a diagram of the pool and the walkway. Let x 5 the width of the
x
Name
Class
4-5
Date
Practice
Form G
Quadratic Equations
Solve each equation by factoring. Check your answers.
1. x2 2 2x 2 24 5 0 24, 6
2. 3x2 5 x 1 4 21, 43
3. x2 2 6x 1 9 5 0 3
4. 3x2 1 45 5 24x 3, 5
3
5. 4x2 1 6x 5 0 22, 0
6. 7x2 5 21x 0, 3
7. (x 1 2)2 5 49 29, 5
8. x 1 3 5 24x2 21, 3
3 8
Solve each equation using tables. Give each answer to at most two decimal
places.
9. 5x2 1 7x 2 6 5 0 22, 0.6
10. x2 2 2x 5 1 20.41, 2.41
11. 2x2 2 x 5 5 21.35, 1.85
12. x2 2 4x 1 2 5 0 0.59, 3.41
13. 3x2 1 7x 5 1 22.47, 0.14
14. 2x2 2 3x 5 15 22.09, 3.59
Solve each equation by graphing. Give each answer to at most two decimal
places.
15. 10x2 5 4 2 3x 20.8, 0.5
16. 3x2 1 2x 5 2 21.22, 0.55
17. 4x2 2 x 5 6 21.11, 1.36
18. 4x2 1 3x 5 6 2 2x 22, 0.75
19. x2 1 4 5 6x 0.76, 5.24
1
20. 5 2 x 5 2x2 24.32, 2.32
21. A woman drops a front door key to her husband from their apartment window
several stories above the ground. The function h 5 216t2 1 64 gives the
height h of the key in feet, t seconds after she releases it.
a. How long does it take the key to reach the ground? 2 s
b. What are the reasonable domain and range for the function h?
0 K t K 2; 0 K h K 64
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Name
4-5
Class
Date
Practice (continued)
Form G
Quadratic Equations
22. The function C 5 75x 1 2600 gives the cost, in dollars, for a small company
to manufacture x items. The function R 5 225x 2 x2 gives the revenue, also
in dollars, for selling x items. How many items should the company produce
so that the cost and revenue are equal? 20 or 130
23. The function a 5 2.4t 2 0.1t2 gives the amount a, in micromilligrams (mmg),
of a drug in a patient’s bloodstream t hours after being ingested in tablet form.
When is the amount of the drug equal to 8 mmg? (Hint: Multiply the equation
you write by 10 before solving.) 4 h; 20 h
24. You use a rectangular piece of cardboard measuring 20 in. by 30 in. to
x
30 ft
construct a box. You cut squares with sides x in. from each corner of
the piece of cardboard and then fold up the sides to form the bottom.
a. Write a function A representing the area of the base of the box in
terms of x. A 5 (30 2 2x)(20 2 2x) 5 4x2 2 100x 1 600
b. What is a reasonable domain for the function A? 0 R x R 10
c. Write an equation if the area of the base must be 416 in.2. 4x2 2 100x 1 600 5 416
d. Solve the equation in part (c) for values of x in the reasonable domain. 2
e. What are the dimensions of the base of the box? 26 in. by 16 in.
Solve each equation by factoring, using tables, or by graphing. If necessary,
round your answer to the nearest hundredth.
25. 9x2 5 49 2 7, 7
3 3
2
27. 4x 1 1 5 8x 0.13, 1.87
26. x2 1 10x 1 17 5 0 27.83, 22.17
29. 4(x2 2 x) 5 19 21.74, 2.74
30. 25x2 1 20x 1 4 5 0 2 25
31. 3x2 5 4x 1 32 2 8, 4
3
32. x2 2 5x 2 12 5 0 21.77, 6.77
28. 5x2 2 2x 2 7 5 0 21, 75
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44
x
x
20 ft
x
Name
4-5
Class
Date
Practice
Form K
Quadratic Equations
Solve each equation by factoring. Check your answers. To start, factor the
quadratic expression.
1. x2 2 x 2 30 5 0
x 5 25, x 5 6
2. x2 2 10x 5 221
x 5 3, x 5 7
3. x2 5 210x 2 9
x 5 21, x 5 29
4. x2 2 5x 5 0
x 5 0, x 5 5
5. 10x 2 24 5 x2
x 5 4, x 5 6
6. x2 5 212x
x 5 0, x 5 212
Solve each equation using tables. Give each answer to at most two decimal
places. To start, enter the equation as Y1. Make a table and look for where the
y-values change sign.
7. x2 1 x 5 12
x 5 3, x 5 24
8. 10x2 1 26x 1 16 5 0
x 5 21, x 5 21.6
10. 2x2 2 13x 1 18 5 0
x 5 2, x 5 4.5
11. 2x2 5 10x
x 5 5, x 5 0
9. 2x2 1 11x 5 6
x 5 26, x 5 0.5
12. 0.5x2 2 8 5 0
x 5 24, x 5 4
Write a quadratic equation with the given solutions.
13. 4 and 25
x2 1 x 2 20 5 0
14. 26 and 0
x2 1 6x 5 0
15. 3 and 8
x2 2 11x 1 24 5 0
16. Writing Explain when you would prefer to use factoring to solve a quadratic
equation and when you would prefer to use tables.
When a 5 1 and c factors easily into integers, it is better to solve by factoring.
When c doesn’t factor easily, it is better to solve using tables.
17. A parabolic jogging path intersects both ends of a street. The path has the
equation x2 2 25x 5 0. If one end of the street is considered to be x 5 0 and
the street lies on the x-axis, where else does the path intersect the street?
at x 5 25
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45
Name
Class
4-5
Date
Practice (continued)
Form K
Quadratic Equations
Solve each equation by graphing. Give each answer to at most two decimal
places.
18. 2x2 2 x 2 10 5 0
x 5 22 and x 5 2.5
21. 4x2 2 5x 2 26 5 0
x 5 3.25 and x 5 22
19. 6x2 2 13x 5 28
x 5 21.33 and x 5 3.5
22. 6x2 2 23x 5 18
x 5 4.5 and x 5 20.67
20. 4x2 1 27x 5 12
x N 0.42 and x N 27.17
23. 4x2 2 9x 1 5 5 0
x 5 1.25 and x 5 1
24. The students in Mr. Wilson’s Physics class are making golf ball catapults. The
flight of group A’s ball is modeled by the equation y 5 20.014x2 1 0.68x,
where x is the ball’s distance from the catapult. The units are in feet.
a. How far did the ball fly? about 48.6 feet
b. How high above the ground did the ball fly? about 8.3 feet
c. What is a reasonable domain and range for this function?
Answers may vary. Sample: domain: 0 K x K 48.6; range: 0 K y K 8.3
25. A rectangular pool is 20 ft wide and 50 ft long. The pool is surrounded by a
walkway. The walkway is the same width all the way around the pool. The total
area of the walkway is 456 square ft. How wide is the walkway? 3 ft
26. Reasoning The equation used to solve Exercise 25 has two solutions. Why is
only one solution used to answer the question?
One of the solutions is negative, and the walkway cannot have a negative width.
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46
Name
Class
Date
Standardized Test Prep
4-5
Quadratic Equations
Gridded Response
Solve each exercise and enter your answer in the grid provided.
1. What is the positive solution of the equation x2 5 2x 1 35? Solve by
factoring.
2. What is the positive solution of the equation 5x2 1 2x 2 16 5 0? Solve by
factoring.
3. What is the positive solution of the equation x2 2 3x 5 1? Solve by using
a table or by graphing. If necessary, round your answer to the nearest
hundredth.
4. What is the positive solution of the equation 3x2 2 5x 2 7 5 0? Solve by
using a table or by graphing. If necessary, round your answer to the nearest
hundredth.
1
5. What is the positive solution of the equation 2x2 2 3x 5 5? Solve by using
a table or by graphing. If necessary, round your answer to the nearest
hundredth.
Answers
1.
–
2.
7
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
–
1 .6
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
3.
0
1
2
3
4
5
6
7
8
9
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
–
3 . 30
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
4.
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
–
2 .57
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
5.
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
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47
–
7 .3 6
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
Name
Class
4-5
Date
Enrichment
Quadratic Equations
Fencing Costs
Solve each of the following problems by using a quadratic equation.
1. The area of a rectangular field is 1200 m2 . Two parallel sides are fenced with
aluminum at $20/m. The remaining two sides are fenced with steel at $10/m.
The total cost of the fencing is $2200.
a. What is the length of each side fenced with aluminum? 40 m, or 15 m
b. What is the length of each side fenced with steel? 30 m, or 80 m, respectively
2. The perimeter of a rectangular field is 140 yards. The land sells for $10>yd2 and
the total cost of the land is $12,000. What are the dimensions of the field? 30 yd by 40 yd
3. The area of a rectangular field is 875 m2. Two adjacent sides of the field are
fenced with wood costing $5/m. The remaining two sides are fenced with steel
costing $10/m. The total cost of the fencing is $900. What are the dimensions
of the field? 25 m by 35 m
4. The area of a field shaped like a right triangle is 600 m2. The legs of the field are
fenced with steel at $10/m, while the hypotenuse is fenced with aluminum at
$20/m. The perimeter of the field is 120 m. The total cost of the fencing is $1700.
a. What is the length of each side fenced with steel? 30 m, 40 m
b. What is the length of the side fenced with aluminum? 50 m
5. The area of a field shaped like a right triangle is 750 yd2. One leg of the field is
fenced with wood costing $5/yd. The remainder of the perimeter of the field is
fenced with steel costing $10/yd. The perimeter of the field is 150 yd. The total
cost of the fencing is $1200.
a. What is the length of the leg fenced with wood? 60 yd
b. What is the length of the leg fenced with steel? 25 yd
6. The area of a rectangular field is 1000 yd2 . Two parallel sides are fenced with
aluminum at $15/yd. One of the remaining sides is fenced with steel at
$10/yd, and all but 10 yd of the remaining side is fenced with wood costing
$5/yd. The remaining 10 yd are left unfenced. The total cost of the fencing is $1525.
a. What is the length of the side fenced with steel? 25 yd or 80 yd
b. What is the length of each side fenced with aluminum? 40 yd, or 1212 yd, respectively.
7. Two rectangular fields with identical shapes and areas are to be fenced side by
side. The total area enclosed is 1200 yd2 . The shared side is fenced with wood
costing $5/yd. The remaining perimeter of the two joined fields is fenced with
aluminum at $15/yd. The total cost of the fencing is $2250.
a. What is the length of the shared side? 30 yd, or 3427 yd
b. What is the length of each side fenced with aluminum?
30 yd and 20 yd, or 3427 yd and 1712 yd, respectively
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Name
Class
Date
Reteaching
4-5
Quadratic Equations
There are several ways to solve quadratic equations. If you can factor the quadratic
expression in a quadratic equation written in standard form, you can use the
Zero-Product Property.
If ab 5 0 then a 5 0 or b 5 0.
Problem
What are the solutions of the quadratic equation 2x2 1 x 5 15?
2x2 1 x 5 15
2x2 1 x 2 15 5 0
(2x 2 5)(x 1 3) 5 0
2x 2 5 5 0 or x 1 3 5 0
2x 5 5 or x 5 23
5
x52
or
Write the equation.
Rewrite in standard form, ax2 1 bx 1 c 5 0.
Factor the quadratic expression (the nonzero side).
Use the Zero-Product Property.
Solve for x.
x 5 23
Check the solutions:
2
5
5
5
x 5 2: 2 Q 2 R 1 Q 2 R 0 15
25
5
2 1 2 0 15
x 5 23: 2(23)2 1 (23) 0 15
15 5 15
15 5 15
18 2 3 0 15
Both solutions check. The solutions are x 5 52 and x 5 23.
Exercises
Solve each equation by factoring. Check your answers.
1. x2 2 10x 1 16 5 0 2, 8
2. x2 1 2x 5 63 29, 7
3. x2 1 9x 5 22 211, 2
4. x2 2 24x 1 144 5 0 12
5. 2x2 5 7x 1 4 2 12, 4
6. 2x2 5 25x 1 12 24, 32
7. x2 2 7x 5 212 3, 4
8. 2x2 1 10x 5 0 25, 0
9. x2 1 x 5 2 22, 1
11. x2 5 25x 2 6 23, 22
12. x2 1 x 5 20 25, 4
10. 3x2 2 5x 1 2 5 0 23, 1
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Name
Class
4-5
Date
Reteaching (continued)
Quadratic Equations
Some quadratic equations are difficult or impossible to solve by factoring. You
can use a graphing calculator to find the points where the graph of a function
intersects the x-axis. At these points f (x) 5 0, so x is a zero of the function.
The values r1 and r2 are the zeros of the function y 5 (x 2 r1)(x 2 r2). The graph
of the function intersects the x-axis at x 5 r1 , or (r1, 0), and x 5 r2 , or (r2, 0).
Problem
What are the solutions of the quadratic equation 3x2 5 2x 1 7?
Step 1
Rewrite the equation in standard form, ax2 1 bx 1 c 5 0.
3x2 2 2x 2 7 5 0
Step 2
Enter the equation as Y1 in your calculator.
Step 3
Graph Y1. Choose the standard window and see if the
zeros of the function Y1 are visible on the screen. If
they are not visible, zoom out and determine a better
viewing window. In this case, the zeros are visible in the
standard window.
Step 4
Use the ZERO option in the CALC feature. For the first
zero, choose bounds of 22 and 21 and a guess of 21.5.
The screen display gives the first zero as x 5 21.230139.
Plot1 Plot2 Plot3
\Y153X2 2 2X 2 7
\Y25
\Y35
\Y45
\Y55
\Y65
\Y75
Zero
X 5 21.230139 Y 5 0
Similarly, the screen display gives the second zero as
x 5 1.8968053.
The solutions to two decimal places are x 5 21.23 and x 5 1.90.
Zero
X 5 1.8968053 Y 5 0
Exercises
Solve the equation by graphing. Give each answer to at most two decimal places.
13. x2 5 5 22.24, 2.24
14. x2 5 5x 1 1 20.19, 5.19
15. x2 1 7x 5 3 27.41, 0.41
16. x2 1 x 5 5 22.79, 1.79
17. x2 1 3x 1 1 5 0 22.62, –0.38
18. x2 5 2x 1 4 21.24, 3.24
19. 3x2 2 5x 1 9 5 8 0.23, 1.43
20. 4 5 2x2 1 3x 22.35, 0.85
21. x2 2 6x 5 27 1.59, 4.41
22. 2x2 5 8x 1 8 26.83, 21.17
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Name
Class
4-6
Date
Additional Vocabulary Support
Completing the Square
Problem
Solve 2x2 1 12x 2 2 5 0 by completing the square. Justify your steps.
2x2 1 12x 2 2 5 0
Write the original equation.
2x2 1 12x 5 2
Rewrite so the variable terms are on one side of the equation and
the constants are on the other side.
2x2
12x
2
2 1 2 52
Divide each side by 2 so the coefficient of x2 will be 1.
x2 1 6x 5 1
Simplify.
2
6 2
b 2
Q 2 R 5 Q 2 R 5 (3) 5 9
2
Find Q b2 R .
x2 1 6x 1 9 5 1 1 9
Add 9 to both sides.
(x 1 3)2 5 10
Factor the trinomial.
x 1 3 5 4 !10
x 5 23 4 !10
Find square roots.
Solve for x.
Exercise
Solve 3x2 2 24x 2 9 5 0 by completing the square. Justify your steps.
Write the original equation
3x2 2 24x 2 9 5 0
.
Rewrite so the variable terms are on one side of the
3x2 2 24x 5 9
equation and the constants are on the other side
3x2
24x
9
3 2 3 53
.
Divide each side by 3 so the coefficient of x2 will be 1 .
Simplify
x2 2 8x 5 3
.
b 2
Find Q 2 R
.
x2 2 8x 1 16 5 3 1 16
Add 16 to both sides
.
(x 2 4)2 5 19
Factor the trinomial
.
Find square roots
.
Solve for x
.
b 2
28 2
Q 2 R 5 Q 2 R 5 (24)2 5 16
x 2 4 5 4 !19
x 5 4 4 !19
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Name
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4-6
Date
Think About a Plan
Completing the Square
Geometry The table shows some possible dimensions of rectangles Width Length
with a perimeter of 100 units. Copy and complete the table.
1
49
a. Plot the points (width, area). Find a model for the data set.
2
48
b. What is another point in the data set? Use it to verify your
47
3
model.
46
4
c. What is a reasonable domain for this function? Explain.
45
5
d. Find the maximum possible area. What are its dimensions?
e. Find an equation for area in terms of width without using
y
the table. Do you get the same equation as in
200
part (a)? Explain.
Area
49
96
141
184
225
150
100
1. What points should you plot? Plot the points on the
graph.
50
(1, 49), (2, 96), (3, 141), (4, 184), (5, 225)
x
O
1
2
3
4
5
2. Use your graphing calculator to find a model for the
data set. A 5 2x2 1 50x
3. What is another point in the data set? Use it to verify your model.
?
Answers may vary. Sample: (6, 264); 264 5 2(6)2 1 50(6); 264 5 264
4. What does the domain of your function represent?
the width of the rectangle
5. The domain must be greater than
0
and less than 50 .
6. A reasonable domain is:
all real numbers greater than 0 and less than 50
7. Write the vertex form of your function. A 5 2(x 2 25)2 1 625
8. The maximum possible area is 625 units2 .
The dimensions of this rectangle are
25 units
9. If the width of the rectangle is x, then the length is
Area 5
length
times
width
5
by
.
25 units
50 2 x
(50 2 x)
?
.
x
5 2x2 1 50x
10. Is the equation in Exercise 9 the same as your model in Exercise 2? Explain.
Yes; the functions are the same.
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Name
4-6
Class
Date
Practice
Form G
Completing the Square
Solve each equation by finding square roots.
1. 3x2 5 75 w5
2. 5x2 2 45 5 0 w3
3. 4x2 2 49 5 0 w72
4. 6x2 5 216 w6
5. 2x2 5 14 w!7
6. 3x2 2 96 5 0 w4 !2
7. A box is 4 in. high. Its length is 1.5 times its width. The volume of the box is
1350 in.2. What are the width and length of the box? 15 in.; 22.5 in.
Solve each equation.
8. x2 1 12x 1 36 5 25 211, 21
9. x2 2 10x 1 25 5 144 27, 17
49 13 1
10. x2 1 6x 1 9 5 4 2 2 , 2
11. x2 2 22x 1 121 5 225 24, 26
12. 16x2 1 8x 1 1 5 16 254, 34
13. 25x2 2 30x 1 9 5 81 265, 12
5
Complete the square.
14. x2 1 22x 1 121
15. x2 2 30x 1 225
16. x2 1 5x 1 25
4
1
1
17. x2 2 2 x 1 16
18. 25x2 1 10x 1 1
19. 4x2 2 12x 1 9
Solve each quadratic equation by completing the square.
20. x2 1 10x 2 1 5 0 25 6 !26
21. x2 1 2x 2 7 5 0 21 6 2 !2
22. 2x2 1 6x 1 10 5 0 3 6 !19
23. x2 1 5x 5 3x 1 11 21 6 2 !3
24. 3x2 1 4x 5 2x2 1 3 22 6 !7
2 6 !7
3
25. x2 2 2x 2 4 5 0
2
26. 20.2x2 1 0.4x 1 0.8 5 0 1 6 !5
27. 4x2 1 20x 1 1 5 0
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25 6 2 !6
2
Name
Class
4-6
Date
Practice (continued)
Form G
Completing the Square
Rewrite each equation in vertex form.
28. y 5 x2 2 6x 1 4 y 5 (x 2 3)2 2 5
29. y 5 x2 1 14x 1 50 y 5 (x 1 7)2 1 1
30. y 5 3x2 1 8x 1 2 y 5 3 Q x 1 43 R 2 2 10
3
31. y 5 22x2 1 6x 2 2 y 5 22 Q x 2 32 R 2 1 52
Find the value of k that would make the left side of each equation a perfect
square trinomial.
32. x2 1 kx 1 196 5 0 28
33. 64x2 2 kx 1 1 5 0 16
34. x2 2 kx 1 16 5 0 8
35. 4x2 2 kx 1 9 5 0 12
36. 16x2 1 kx 1 9 5 0 24
1
1
37. 4x2 2 kx 1 25 5 0 15
38. The quadratic function d 5 2t2 1 4t 1 33 models the depth of water in a
flood channel after a rainstorm. The time in hours after it stops raining is t and
d is the depth of the water in feet.
a. Solve the equation 2t2 1 4t 1 33 5 0. 2 6 !37
b. Approximate the positive solution found in part (a) to two decimal places. 8.08
c. Interpret the answer to part (b) in terms of the problem.
The water level will be 0 ft after about 8 h.
39. While in orbit, a space scientist measures the pressure inside a container as it
is being heated and then cooled. She records the information and discovers
the pressure p, in pounds per square inch, is related to the time t in minutes
after the experiment begins according to the equation p 5 20.2t2 1 1.6t .
a. Complete the square in the expression 20.2t2 1 1.6t . 20.2t2 1 1.6t 2 3.2
b. Rewrite the equation for p in vertex form. p 5 20.2(t 2 4)2 1 3.2
c. What is a reasonable domain for this function? Explain. 0 K t K 8; other values of t result in
d. When does the maximum pressure occur? What is the maximum pressure? negative values for p.
4 min; 3.2 lbs/in.2
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Name
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Date
Practice
Form K
Completing the Square
Solve each equation by finding square roots. To start, remember to isolate x2.
1. x2 2 9 5 0
2. x2 1 4 5 20
3. x2 1 15 5 16
x2 5 9
x2 5 16
x2 5 1
x 5 w3
x 5 w4
x 5 w1
4. 2x2 2 64 5 0
x 5 w4 !2
5. 4x2 2 100 5 0
x 5 w5
6. 5x2 2 25 5 0
x 5 w!5
7. You are painting a large wall mural. The wall length is 3 times the height. The
area of the wall is 300 ft2.
a. What are the dimensions of the wall? height 5 10 ft; length 5 30 ft
b. If each can of paint covers 22 ft2, will 12 cans be enough to cover the wall?
No; 12 cans will only cover 264 ft2.
8. The lengths of the sides of a carpet have the ratio of 4.4 to 1. The area of the
carpet is 1154.7 ft2. What are the dimensions of the carpet?
16.2 ft, 71.3 ft
9. A packing box is 4 ft deep. One side of the box is 1.5 times longer than the
other. The volume of the box is 24 ft3. What are the dimensions of the box?
2 ft, 3 ft
Solve each equation. To start, factor the perfect square trinomial.
10. x2 2 14x 1 49 5 81
11. x2 1 6x 1 9 5 1
12. 9x2 2 12x 1 4 5 49
(x 2 7)2 5 81
(x 1 3)2 5 1
(3x 2 2)2 5 49
x 5 16, x 5 22
x 5 22, x 5 24
x 5 3, x 5 253
13. 4x2 1 36x 1 81 5 16
5
x 5 213
2 , x 5 22
14. x2 1 2x 1 1 5 36
x 5 27, x 5 5
15. x2 2 16x 1 64 5 9
x 5 11, x 5 5
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Practice (continued)
Form K
Completing the Square
Complete the following squares.
z
16. x2 1 8x 1 16 z
z
100 z
18. x2 2 14x 1 z
49 z
z
289 z
21. x2 2 46x 1 z
529 z
20 2
Q2R 5
8 2
Q 2 R 5 42 5 16
19. x2 2 24x 1 z
17. x2 1 20x 1 144 z
20. x2 1 34x 1 Solve the following equations by completing the square.
22. x2 2 8x 2 5 5 0
x2 2 8x 5 5
x2 2 8x 1 16 5 5 1 16
(x 2 4)2 5 21
23. x2 1 12x 1 9 5 0
x2 1 12x 5 29
x2 1 12x 1 36 5 29 1 36
24. x2 2 10x 5 211
x 5 5 6 "14
x 5 26 6 3 !3
x 2 4 5 4 !21
z
21
x 5 4 6 " z
25. 2x2 1 11x 2 23 5 2x 1 3 26. x2 2 18x 1 64 5 0
x 5 23 6 "22
x 5 9 6 "17
27. 3x2 2 42x 1 78 5 0
x 5 7 6 "23
Write the following equations in vertex form.
28. y 5 x2 1 10x 2 9
y 5 (x 1 5)2 2 34
29. y 5 x2 2 18x 1 13
y 5 (x 2 9)2 2 68
30. y 5 x2 1 32x 2 8
y 5 (x 1 16)2 2 264
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Standardized Test Prep
Completing the Square
Multiple Choice
For Exercises 1−6, choose the correct letter.
1. What are the solutions of the equation 36x2 2 12x 1 1 5 4? D
216 , 16
4, 8
212 , 16
216 , 12
2. What are the solutions of the equation 2x2 1 16x 1 28 5 0? G
24 4 !30
24 4 !2
4 4 !2
4 4 !30
9
24
3. Which value completes the square for x2 2 3x? A
9
4
3
2
9
49
4. Which value for k would make the left side of x2 1 kx 1 64 5 0 a perfect
square trinomial? H
7
2
7
7
4
7
8
5. What are the solutions of the equation x2 5 8x 2 1? C
24 4 !17
24 4 !15
4 4 !15
4 4 !17
6. Which equation is the vertex form of y 5 23x2 1 12x 2 7? G
y 5 23(x 2 2)2 2 5
y 5 23(x 1 2)2 2 5
y 5 23(x 2 2)2 1 5
y 5 23(x 1 2)2 1 5
Short Response
7. The equation p 5 2x2 1 8x 1 5 gives the price p, in dollars, for a product
when x million units are produced.
a. What are the solutions of the equation 2x2 1 8x 1 5 5 0?
b. What is the positive solution to part (a) rounded to two decimal places?
What does this solution mean in terms of this problem?
[2] a. 4 6 !21 b. 8.58; the price will be $0 when 8.58 million units are
produced.
[1] incorrect solution OR incorrect interpretation
[0] no answers given
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Enrichment
Completing the Square
You can approximate square roots with an interesting quadratic relationship.
2
b 2
1. Multiply Q a 1 2a R . a2 1 b 1 b 2
4a
2. What happens to the value of a fraction as the denominator gets larger?
As the denominator gets larger, the fraction becomes smaller.
2
If a is much larger than b, the value of the fraction b 2 is very small and has little
effect in the expression
write
a2
4a
b2
1 b 1 2 . So when a is much larger than b, you can
4a
b 2
Q a 1 2a R < a2 1 b
b
a 1 2a
< "a2 1 b
If you can write a number as a2 1 b where a . b, then an approximate value of
b
its square root is a 1 2a
.
For example, to approximate !17 , let a 5 4 and b 5 1.
b
1
< 4.125
!17 5 "42 1 1 so !17 < a 1 2a 5 4 1
2(4)
Evaluate this square root on a calculator.
!17 < 4.123
Use the formula to approximate each square root. Then find each square root
using your calculator. Round to the nearest thousandth.
3. !26 5.100; 5.099
4. !404 20.100; 20.100
5. !174 13.192; 13.191
6. !1773 42.107; 42.107
7. !963 31.032; 31.032
8. !83 9.111; 9.110
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Reteaching
4-6
Completing the Square
Completing a perfect square trinomial allows you to factor the completed
trinomial as the square of a binomial.
2
2
Start with the expression x2 1 bx. Add Q b2 R . Now the expression is x2 1 bx 1 Q b2 R ,
2
2
which can be factored into the square of a binomial: x2 1 bx 1 Q b2 R 5 Q x 1 b2 R .
To complete the square for an expression ax2 1 abx, first factor out a. Then find
the value that completes the square for the factored expression.
Problem
What value completes the square for 22x2 1 10x?
Think
Write
Write the expression in the
form a(x2 1 bx).
22x2 1 10x 5 22(x2 2 5x)
Find b2 .
5
b
25
2 5 2 5 22
2
2
Add Q b2 R to the inner expression
to complete the square.
22 cx2 2 5x 1 a22 b d 5 22ax2 2 5x 1 4 b
Factor the perfect square
trinomial.
22 Q x 2 2 R
Find the value that completes
the square.
22 Q 4 R 5 2 2
5
5 2
25
25
Exercises
What value completes the square for each expression?
1. x2 1 2x 1
2. x2 2 24x 144
4. x2 2 20x 100
5. x2 1 5x
7. 2x2 2 24x 72
8. 3x2 1 12x 12
10. 5x2 1 80x 320
11. 27x2 1 14x 27
25
4
3. x2 1 12x 36
81
6. x2 2 9x 4
9. 2x2 1 6x 29
75
12. 23x2 2 15x 2 4
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25
Name
Class
4-6
Date
Reteaching (continued)
Completing the Square
You can easily graph a quadratic function if you first write it in vertex form.
Complete the square to change a function in standard form into a function in
vertex form.
Problem
What is y 5 x2 2 6x 1 14 in vertex form?
Think
Write
Write an expression using the
terms that contain x.
x2 2 6x
Find b2 .
26
b
2 5 2 5 23
2
Add Q b2 R to the expression
to complete the square.
x2 2 6x 1 (23)2 5 x2 2 6x 1 9
Subtract 9 from the expression so
that the equation is unchanged.
y 5 x2 2 6x 1 9 1 14 2 9
Factor the perfect square
trinomial.
y 5 (x 2 3)2 1 14 2 9
Add the remaining constant
terms.
y 5 (x 2 3)2 1 5
Exercises
Rewrite each equation in vertex form.
13. y 5 x2 1 4x 1 3 (x 1 2)2 2 1
14. y 5 x2 2 6x 1 13 (x 2 3)2 1 4
15. y 5 2x2 1 4x 2 10 2(x 2 2)2 2 6
16. y 5 x2 2 2x 2 3 (x 2 1)2 2 4
17. y 5 x2 1 8x 1 13 (x 1 4)2 2 3
18. y 5 2x2 2 6x 2 4 2(x 1 3)2 1 5
19. y 5 2x2 1 10x 2 18 2(x 2 5)2 1 7
20. y 5 x2 1 2x 2 8 (x 1 1)2 2 9
21. y 5 2x2 1 4x 2 3 2(x 1 1)2 2 5
22. y 5 3x2 2 12x 1 8 3(x 2 2)2 2 4
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Additional Vocabulary Support
4-7
The Quadratic Formula
The column on the left shows the steps used to solve a problem using the
quadratic formula. Use the column on the left to answer each question in the
column on the right.
Problem
1. Read the title of the problem. What
Solve by using the
quadratic formula.
process are you going to use to solve the
problem?
4x2 1 x 5 2
Answers may vary. Sample: Solve by
using the quadratic formula.
Write in standard form.
2. What is the standard form of a quadratic
equation?
4x2 1 x 2 2 5 0
ax2 1 bx 1 c 5 0
Find the values of a, b, and c.
3. Explain how you know which value is a.
Answers may vary. Sample: It is the
a 5 4, b 5 1, c 5 22
leading coefficient.
Substitute the values into the quadratic formula.
x5
4. Write the quadratic formula.
x5
2(1) 4 "(1)2 2 4(4)(22)
2b 6 "b2 2 4ac
2a
2(4)
5. Explain what the symbol 4 means.
Simplify.
Answers may vary. Sample: The symbol
x5
5
21 4 "1 2 (232)
means there are two answers, one found
8
by adding and the other by subtracting.
21 4 "33
8
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Think About a Plan
The Quadratic Formula
Sports A diver dives from a 10 m springboard. The equation f (t) 5 24.9t2 1 4t 1 10
models her height above the pool at time t. At what time does she enter the water?
Understanding the Problem
1. What does the function represent?
the diver’s height above the water at time t in seconds
2. What is the problem asking you to determine?
at what time the diver’s height above the water equals 0
3. Do you need to use the height of the springboard to solve the problem?
Explain.
No; you don’t use the height of the springboard. You only need to find the zeros of the
function.
Planning the Solution
4. What are three possible methods for solving this problem?
Graph the equation, factor the equation; use the Quadratic Formula.
5. If a solution exists, which method will give an exact solution? Explain.
The Quadratic Formula; you can only estimate from a graph, and not all equations have
expressions that are factorable.
Getting an Answer
6. Is there more than one reasonable solution to the problem? Explain.
No; the equation has two zeros, but one of them is negative. Time moves forward and
cannot be negative.
7. At what time does the diver enter the water?
The diver enters the water about 1.89 s after starting her dive.
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Practice
Form G
The Quadratic Formula
Solve each equation using the Quadratic Formula.
1. x2 2 8x 1 15 5 0 3, 5
2. x2 1 12x 1 35 5 0 27, 25
3. 3x2 1 5x 5 2 22, 13
4. 2x2 1 3 5 7x 12, 3
5. x2 1 16 5 8x 4
6. x2 5 4x 2 1 2 6 "3
7. x(2x 2 5) 5 12 232, 4
8. 23x2 2 8x 1 16 5 0 24, 43
10. x2 1 10x 1 22 5 0 25 6 "3
9. x2 1 4x 5 3 22 6 "7
3
11. 4x(x 1 1) 5 7 21 6 2"2
12. x(2x 2 3) 5 9 22, 3
2
13. The principal at a high school is planning a concert to raise money for the
music programs. He determines the profit p from ticket sales depends on the
price t of a ticket according to the equation p 5 2200t2 1 3600t 2 6400.
All amounts are in dollars. If the goal is to raise $8500, what is the smallest
amount the school should charge for a ticket to the concert? $6.45
14. The equation y 5 x2 2 12x 1 45 models the number of books y sold in a
bookstore x days after an award-winning author appeared at an autographsigning reception. What was the first day that at least 100 copies of the book
were sold? day 16
15. The height of the tide measured at a seaside community varies according to
the number of hours t after midnight. If the height h, in feet, is currently given
by the equation h 5 212t2 1 6t 2 9, when will the tide first be at 6 ft?
about 3.55 h past midnight or about 3:33 A.M.
16. The height h, in feet, of a model rocket t seconds after launch is given by
h 5 256t 2 16t2 . As the rocket descends, it deploys a recovery parachute
when it reaches 200 ft above the ground. At what time does the
parachute deploy? about 15.2 s
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4-7
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Practice
Date
Form G
(continued)
The Quadratic Formula
Evaluate the discriminant for each equation. Determine the number of real
solutions.
17. x2 1 5x 1 8 5 0 27; none
18. x2 2 5x 1 4 5 0 9; two
19. 29x2 1 12x 2 4 5 0 0; one
20. 23x2 1 5x 2 4 5 0 223; none
21. 4x2 1 4x 5 21 0; one
22. 6x2 5 x 1 2 49; two
23. 5x 1 1 5 3x2 37; two
24. 4x2 2 x 1 3 5 0 247; none
25. 4x2 1 36x 1 81 5 0 0; one
26. 5x2 5 3x 2 2 231; none
27. 16x2 2 56x 1 49 5 0 0; one
28. 4x2 2 16x 1 11 5 0 80; two
29. In Exercise 16, the height of the rocket was given by h 5 256t 2 16t2 . Use the
discriminant to answer the following questions.
a. Will the rocket reach an altitude of 1000 ft? yes
b. Will the rocket reach an altitude of 1024 ft? yes
c. Will the rocket reach an altitude of 1048 ft? no
30. The number n of people using the elevator in an office building every hour is
given by n 5 t2 2 10t 1 40. In this equation, t is the number of hours after
the building opens in the morning, 0 # t # 12. Will the number of people
using the elevator ever be less than 15 in any one hour? Use the discriminant
to answer. no
Solve each equation using any method. When necessary, round real solutions
to the nearest hundredth.
3
31. 4x2 1 x 2 3 5 0 21, 4
32. 5x2 2 6x 2 2 5 0 20.27, 1.47
33. x2 2 5x 2 9 5 0 21.41, 6.41
1 1
34. 15x2 2 2x 2 1 5 0 25, 3
3
35. 2x2 5 5x 2 3 1, 2
36. 4x2 1 3x 5 5 21.55, 0.80
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Practice
Form K
The Quadratic Formula
Solve each equation using the Quadratic Formula. To start, find the values of a, b,
and c. Substitute those values into the Quadratic Formula. When necessary round
real solutions to the nearest hundredth.
1. x2 2 4x 1 3 5 0
2. 2x2 1 3x 2 4 5 0
3. 8x2 2 2x 2 5 5 0
a 5 1, b 5 24, c 5 3
a 5 2, b 5 3, c 5 24
a 5 8, b 5 22, c 5 25
2(24) 4 "(24)2 2 (4)(1)(3)
2(1)
x 5 3, x 5 1
2(3) 4 "(3)2 2 (4)(2)(24)
2(2)
x 5 0.85, x 5 22.35
2(22) 4 "(22)2 2 (4)(8)(25)
2(8)
x 5 0.93, x 5 20.68
4. x2 1 3x 5 3
5. 4x2 1 3 5 9x
x 5 0.79, x 5 23.79
x 5 0.41, x 5 1.84
6. 2x 2 5 5 2x2
x 5 1.45, x 5 23.45
7. Your school sells yearbooks every spring. The total profit p made depends on
the amount x the school charges for each yearbook. The profit is modeled by the
equation p 5 22x2 1 70x 1 520. What is the smallest amount in dollars the
school can charge for a yearbook and make a profit of at least $1000?
To start, substitute 1000 for p in the equation. 1000 5 22x2 1 70x 1 520
Then, write the equation in standard form. 2x2 2 70x 1 480 5 0 $9.36
8. Engineers can use the formula d 5 0.05s2 1 1.1s to estimate the minimum
stopping distance d in feet for a vehicle traveling s miles per hour.
a. If a car can stop after 65 feet, what is the fastest it could have been traveling
when the driver put on the brakes? 26.7 miles per hour
b. Reasoning Explain how you knew which of the two solutions from the
Quadratic Formula to use. (Hint: Remember this is a real situation.)
The other solution was negative. You cannot drive negative miles per hour.
9. Reasoning Explain why a quadratic equation has no real solutions if the
discriminant is less than zero.
The discriminant is the value under the square root sign. If it is less than zero,
you are trying to take the square root of a negative number. The square root of a
negative number is not a real number.
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Practice (continued)
Form K
The Quadratic Formula
Evaluate the discriminant for each equation. Determine the number of real
solutions.
10. 212x2 1 5x 1 2 5 0
11. x2 2 x 1 6 5 0
12. 2x 2 5 5 2x2
(5)2 2 4(212)(2)
(21)2 2 4(1)(6)
(2)2 2 4(1)(25)
121, 2 real solutions
223, 0 real solutions
24, 2 real solutions
13. 4x2 1 7 5 9x
231, 0 real solutions
14. x2 2 4x 5 24
0, 1 real solution
15. 3x 1 6 5 26x2
2135, 0 real solutions
Solve each equation using any method. When necessary, round real solutions
to the nearest hundredth.
16. 7x2 1 3x 5 12
17. x2 1 6x 2 7 5 0
18. 5x 5 23x2 1 2
7x2 1 3x 2 12 5 0
(x 1 7)(x 2 1) 5 0
23x2 2 5x 1 2 5 0
23 4 "(3)2 2 4(7)(212)
2(7)
x 5 1.11, x 5 21.54
x 5 27, x 5 1
5 4 "(5)2 2 4(23)(2)
2(23)
x 5 0.33, x 5 22
19. 212x 1 7 5 5 2 2x2
x 5 5.83, x 5 0.17
20. 9x2 2 6x 2 4 5 25
x 5 0.33
21. 2x 2 24 5 2x2
x 5 4, x 5 26
Without graphing, determine how many x-intercepts each function has.
22. y 5 2x2 2 3x 1 5
23. y 5 2x2 2 4x 1 1
24. y 5 x2 1 3x 1 3
(3)2 2 4(2)(5)
(24)2 2 4(2)(1)
(3)2 2 4(1)(3)
0 x-intercepts
2 x-intercepts
0 x-intercepts
25. y 5 9x2 2 12x 1 7
0 x-intercepts
26. y 5 25x2 1 8x 2 3
2 x-intercepts
27. y 5 x2 1 16x 1 64
1 x-intercept
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Standardized Test Prep
The Quadratic Formula
Multiple Choice
For Exercises 1−6, choose the correct letter.
1. What is the solution of 3x2 1 2x 2 5 5 0? Use the Quadratic Formula. A
253 , 1
21, 53
213 , 5
25, 13
2. What is the solution of 2x2 2 8x 1 3 5 0? Use the Quadratic Formula. H
24 4 !22
2
24 4 !10
2
4 4 !10
2
4 4 !22
2
3. What is the solution of x2 2 5x 5 5? Use the Quadratic Formula. D
25, 1
5 4 !5
2
21, 5
5 4 3 !5
2
4. What is the solution of x2 5 6x 2 3? Use the Quadratic Formula. I
23 4 !6
3
23
3 4 !6
5. What is the discriminant of the equation 3x2 2 7x 1 1 5 0? B
61
!37
37
219
6. What is the discriminant of the equation 4x2 1 28x 1 49 5 0? H
25472
0
2756
1568
Extended Response
7. The equation d 5 n2 2 12n 1 43 models the number of defective items d
produced in a manufacturing process when there are n workers in a restricted
area. Use the discriminant to answer the following questions. Show your work.
a. Will the number of defective items ever be 10?
b. Will the number of defective items ever be 7?
c. Will the number of defective items ever be 5?
[4] a. yes; b. yes; c. no; the answers are based on the value of the discriminant for each
case: (a) 12, (b) 0, and (c) –8.
[3] appropriate method with one computational error
[2] appropriate method with several computational errors
[1] correct answers but no work shown
[0] incorrect answers and no work shown OR no answers given
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Class
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Enrichment
The Quadratic Formula
The value of the discriminant of a quadratic equation tells you how many
solutions the quadratic equation will have and the type of solutions they will be.
The discriminant can also tell you if the quadratic equation is factorable.
Factor each of the following quadratic equations, if possible. Then determine
the value of the discriminant for each equation.
1. 6x2 2 11x 1 3 5 0 (3x 2 1)(2x 2 3); 49
2. 4x2 2 x 2 7 5 0 not factorable; 113
3. x2 1 7x 1 10 5 0 (x 1 5)(x 1 2); 9
4. 16x2 2 1 5 0 (4x 2 1)(4x 1 1); 64
5. x2 1 x 2 5 5 0 not factorable; 21
6. 2x2 1 5x 2 1 5 0 not factorable; 33
7. x2 2 11x 1 30 5 0 (x 2 5)(x 2 6); 1
8. x2 1 9x 1 2 5 0 not factorable; 73
9. x2 1 5x 2 8 5 0 not factorable; 57
10. 3x2 1 8x 1 4 5 0 (3x 1 2)(x 1 2); 16
11. x2 2 7x 2 5 5 0 not factorable; 69
12. x2 1 4x 1 3 5 0 (x 1 3)(x 1 1); 4
Copy and complete the table using your previous answers.
Discriminants of quadratics that are
factorable
49
9
64
1
16
4
Discriminants of quadratics that are
not factorable
113
21
33
73
57
69
13. What types of numbers did you get for the values of the discriminants for the
quadratics that were factorable? The numbers are all perfect squares.
14. Writing How can the value of the discriminant tell you if a quadratic equation
is factorable? Answers may vary. Sample: If the discriminant is a perfect
square, the equation will be factorable.
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Reteaching
4-7
The Quadratic Formula
You can solve some quadratic equations by factoring or completing the square.
You can solve any quadratic equation ax2 1 bx 1 c 5 0 by using the Quadratic
Formula:
x5
2b 4 "b2 2 4ac
2a
Notice the 4 symbol in the formula. Whenever b2 2 4ac is not zero, the
Quadratic Formula will result in two solutions.
Problem
What are the solutions for 2x2 1 3x 5 4? Use the Quadratic Formula.
x5
5
2x2 1 3x 2 4 5 0
Write the equation in standard form: ax2 1 bx 1 c 5 0
a 5 2 ; b 5 3 ; c 5 24
a is the coefficient of x2, b is the coefficient of x, c is the
constant term.
2b 4 "b2 2 4ac
Write the Quadratic Formula.
2a
2(3) 4 "(3)2 2 4(2)(24)
2(2)
Substitute 2 for a, 3 for b, and −4 for c.
5
23 4 !41
4
Simplify.
5
23 1 !41 23 2 !41
or
4
4
Write the solutions separately.
Check your results on your calculator.
Replace x in the original equation with
23 1 !41
23 2 !41
and
. Both values
4
4
for x give a result of 4. The solutions check.
(23 2 V (41)) / 4 X
22.350781059
2X2 1 3X
4
(23 1 V (41)) / 4 X
.8507810594
2X2 1 3X
4
Exercises
What are the solutions for each equation? Use the Quadratic Formula.
1. 2x2 1 7x 2 3 5 0
3. 2x2 5 4x 1 3
7 1 !37
2
2 1 !10
2
or
or
7 2 !37
2
2. x2 1 6x 5 10 23 1 !19 or 23 2 !19
2 2 !10
2
9
4. 4x2 1 81 5 36x 2
5. 2x2 1 1 5 5 2 7x 24 or 12
6. 6x2 2 10x 1 3 5 0
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5 1 !7
6
or
5 2 !7
6
Name
Class
4-7
Date
Reteaching (continued)
The Quadratic Formula
There are three possible outcomes when you take the square root of a real number n:
. 0 S two real values (one positive and one negative)
n • 5 0 S one real value (0)
, 0 S no real values
2b 4 "b2 2 4ac
. The value under the
Now consider the quadratic formula: x 5
2a
radical symbol determines the number of real solutions that exist for the equation
ax2 1 bx 1 c 5 0:
b2
. 0 S two real solutions
2 4ac • 5 0 S one real solution
, 0 S no real solutions
The value under the radical, b2 2 4ac,
is called the discriminant.
Problem
What is the number of real solutions of 23x2 1 7x 5 2?
23x2 1 7x 5 2
23x2 1 7x 2 2 5 0
Write in standard form.
a 5 23 , b 5 7 , c 5 22
b2
Find the values of a, b, and c.
2 4ac
Write the discriminant.
(7)2 2 4(23)(22)
Substitute for a, b, and c.
49 2 24
Simplify.
25
The discriminant, 25, is positive. The equation has two real roots.
Exercises
What is the value of the discriminant and what is the number of real solutions
for each equation?
7. x2 1 x 2 42 5 0
169; two
8. 2x2 1 13x 2 40 5 0
9; two
9. x2 1 2x 1 5 5 0
–16; none
1
10. x2 5 18x 2 81
0; one
11. 2x2 1 7x 1 44 5 0
225; two
12. 4x2 2 5x 1 25 5 0
0; one
13. 2x2 1 7 5 5x
–31; none
14. 4x2 1 25x 5 21
961; two
15. x2 1 5 5 3x
–11; none
1
16. 9x2 5 4x 2 36
0; one
1
17. 2x2 1 2x 1 3 5 0
–2; none
1
18. 6x2 5 2x 1 18
16; two
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Additional Vocabulary Support
Complex Numbers
Complete the vocabulary chart by filling in the missing information.
Word or
Word Phrase
Definition
Picture or Example
imaginary unit
The imaginary unit i is a complex
number whose square is 21.
i 5 !21
pure imaginary
number
A pure imaginary number is of the
form a 1 bi, where a 5 0 and b 2 0.
2. any number of the form
a 1 bi where a and b are
real numbers
complex number
complex number
plane
In the complex number plane, the
point (a, b) represents the complex
number a 1 bi. To graph, locate the
real part on the horizontal axis and
the imaginary part on the vertical
axis.
4. Answers may vary. Sample:
The absolute value of a
complex number is its distance
from the origin in the complex
number plane.
absolute value of a
complex number
complex conjugates
number pairs of the form a 1 bi
and a 2 bi
1. Answers may vary.
Sample: 5i
7 2 4i
3.
4i
2i
42
4i
Real
2 4
2 3i
u 8 1 6i u 5 "82 1 62 5 10
5. Answers may vary.
Sample: 23 1 4i
and 23 2 4i
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Imaginary
Name
Class
4-8
Date
Think About a Plan
Complex Numbers
A student wrote the numbers 1, 5, 1 1 3i, and 4 1 3i to represent the vertices of a
quadrilateral in the complex number plane. What type of quadrilateral has these vertices?
Know
1. The vertices of the quadrilateral are:
1, 5, 1 1 3i, and 4 1 3i
.
2. You can write the vertices in the form a 1 bi as:
1 1 0i, 5 1 0i, 1 1 3i, and 4 1 3i
.
Need
3. To solve the problem I need to:
graph the numbers in the complex plane
Plan
4. How do you find the coordinates that represent each complex number?
For the complex number a 1 bi, the real part a is the horizontal coordinate and
the imaginary part b is the vertical coordinate.
5. What are the points you need to graph?
(1, 0), (5, 0), (1, 3), (4, 3)
6. Graph your points in the complex plane. Connect the points with straight lines
to form a quadrilateral.
Imaginary axis
4i
2i
Real axis
O
2
4
6
2i
7. What type of quadrilateral did you draw? Explain how you know.
Trapezoid; the top and bottom sides are parallel, but the left and right sides are
not parallel.
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.
Name
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Class
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Practice
Form G
Complex Numbers
Simplify each number by using the imaginary number i.
1. !249 7i
2. !2144 12i
3. !27 i"7
4. !210 i"10
5. !28 2i"2
6. !248 4i"3
ⴚ4 ⴙ 8i
Plot each complex number and find its absolute value.
8. 6 2 4i 2"13
7. 23i 3
4i
2i
9. 24 1 8i 4"5
4 2
2i
Simplify each expression.
ⴚ3i
Imaginary
axis
Real axis
2 4
6 ⴚ 4i
10. (22 1 3i) 1 (5 2 2i) 3 1 i
11. (26 1 7i) 1 (6 2 7i) 0
12. (4 2 2i) 2 (21 1 3i) 5 2 5i
13. (25 1 3i) 2 (28 1 2i) 3 1 i
14. (4 2 3i)(25 1 4i) 28 1 31i
15. (2 2 i)(23 1 6i) 15i
16. (5 2 3i)(5 1 3i) 34
17. (21 1 3i)2 28 2 6i
18. (4 2 i)2 15 2 8i
19. (22i)(5i)(2i) 210i
20. A6 2 !216B 1 A24 1 !225B 2 1 i
21. A22 1 !29B 1 A21 2 !236B 23 2 3i
22. A25 1 !24B 2 A3 2 !216B 28 1 6i
23. A7 2 !21B 2 !281 7 2 10i
24. 3i(2 1 2i) 26 1 6i
25. 2(3 2 7i) 2 i(24 1 5i) 11 2 10i
26. A2 1 !24B A21 1 !29B 28 1 4i
27. A5 1 !21B A2 2 !236B 16 2 28i
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Practice (continued)
Form G
Complex Numbers
Write each quotient as a complex number.
28.
5 1 2i 1 5
2 2 4i
4i
29.
3 2 2i
1
30. 4 2 3i 18
25 1 25 i
3i
3
6
2 5i
22 1 i 5
7
14
31. 5 2 2i 35
29 1 29 i
Find the factors of each expression. Check your answer.
32. x2 1 36 ( x 1 6i)(x 2 6i)
33. 2x2 1 8 2( x 1 2i)(x 2 2i)
34. 5x2 1 5 5( x 1 i)(x 2 i)
1
35. x2 1 9 ( x 1 1 i)(x 2 1 i)
3
3
36. 16x2 1 25 (4 x 1 5i)(4 x 2 5i)
37. 24x2 2 49 2(2 x 1 7i)(2 x 2 7i)
Find all solutions to each quadratic equation.
38. x2 1 2x 1 5 5 0 21 6 2i
39. 2x2 1 2x 2 10 5 0 1 6 3i
"3
"31
3
41. 24x2 1 6x 2 3 5 0 4 6 4 i
3
40. 2x2 2 3x 1 5 5 0 4 6 4 i
"14
"13
1
42. 3x2 1 2x 1 5 5 0 23 6 3 i
1
43. 2x2 2 2x 1 7 5 0 2 6 2 i
44. a. Name the complex number represented by each point on the
graph at the right. A: 2 1 3i; B: 24 1 2i; C: 23 2 3i; D: 24i
A: 22 2 3i;
b. Find the additive inverse of each number. B: 4 2 2i; C: 3 1 3i;
D: 4i
c. Find the complex conjugate of each number.
A: 2 2 3i;
d. Find the absolute value of each number.
B: 24 2 2i;
A: !13; B: 2 !5; C: 3 !2; D: 4
C: 23 1 3i; D: 4i
Imaginary axis
B
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2i
4 2
2i
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4i
C 4i D
Real axis
2
4
Name
Class
Date
Practice
4-8
Form K
Complex Numbers
Simplify each number by using the imaginary number i.
1. !2100
2. !22
i!2
!21 ? 100
3. !248
4i!3
4. !236
6i
!21 ? !100
10i
Plot each complex number and find its absolute value.
5. 5i
6. 3 1 2i
8i imaginary axis
4i 5i
4i
real axis
8 4 O
4i
4
8 4 O
4i
8
8i
7. 7 2 1i
8i imaginary axis
3 2i
4i
8
8 4 O
4i
real axis
4
8i
4 9i
real axis
4
7 1i
!97
5 !2
(9 1 2) 1 (6i 2 i)
10. (212i) 2 (3 1 3i)
23 2 15i
11. (22i)(5 1 4i)
8 2 10i
11 1 5i
Write each quotient as a complex number.
12.
5 1 4i
7i
5 1 4i 27i
7i Q 27i R
4
7
21 1 5i
13. 3 2 2i
2 2 6i
14. 2 2 3i
22
13
21 1 i
6
2 13
i
2 57i
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4i
8i
Simplify each expression.
9. (9 1 6i) 1 (2 2 i)
8i imaginary axis
8 4 O
4i
8
8i
!13
5
8. 24 1 9i
8i imaginary axis
real axis
4
8
Name
Class
Date
Practice (continued)
4-8
Form K
Complex Numbers
Find the factors of each expression. Check your answer.
15. 2x2 1 32
2(x2 1 16)
16. x2 1 4
( x 1 2i)(x 2 2i)
17. 3x2 1 3
3( x 1 i)(x 2 i)
19. 9x2 1 49
(3 x 1 7i)(3 x 2 7i)
20. 4x2 1 25
(2 x 1 5i)(2 x 2 5i)
2(x2 1 42)
2( x 1 4i)(x 2 4i)
18. x2 1 64
( x 1 8i)(x 2 8i)
Find all solutions to each quadratic equation.
21. 2x2 2 3x 1 7 5 0
3 4 "(23)2 2 4(2)(7)
2(2)
3 4 !9 2 56
x5
4
x5
x5
22. 4x2 2 5x 1 6 5 0
23. x(x 2 3) 1 3 5 0
!3
x 5 32 6 2 i
5 4 "(25)2 2 4(4)(6)
x5
2(4)
!71
x 5 58 6 8 i
3 4 !247
4
3
x544
!47
4 i
24. Error Analysis Robert solved the equation 2x2 1 16 5 0. His solution was
x 5 4 !28 i. What errors did Robert make? What is the correct solution?
Robert made two errors. He left a negative number under the radical sign, and he did
not simplify !8. The correct solution is x 5 w2i!2.
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Standardized Test Prep
4-8
Complex Numbers
Multiple Choice
For Exercises 1−8, choose the correct letter.
1. What is the simplified form of (28 1 5i) 1 (3 2 2i)? B
214 1 31i
25 1 3i
211 1 7i
5 2 3i
2. What is the simplified form of (11 2 6i) 2 (24 1 12i)? I
7 1 6i
7 2 18i
15 1 6i
15 2 18i
3. What is the simplified form of (5 1 !236) 2 (24 2 !249)? D
9 2 13i
9 1 85i
12i
9 1 13i
15
15i
4. What is the simplified form of (25i)(23i)? G
215i
215
5. What is the simplified form of (23 1 2i)(1 2 4i)? C
22 2 2i
211 2 10i
5 1 14i
23 2 8i
55 2 48i
55 1 48i
5
3
4 2 2i
7
11
10 2 10i
6. What is the simplified form of (8 2 3i)2 ? H
73
16 2 6i
5 1 3i
7. What is 4 2 2i written as a complex number? A
7
13
11
1
10 1 10i
10 1 10i
8. What is the factored form of the expression 4x2 1 36? G
4(x 1 3i)2
4(x 1 6i)2
4(x 1 3i)(x 2 3i)
4(x 1 6i)(x 2 6i)
Short Response
9. What are the solutions of 2x2 1 3x 1 6 5 0? Show your work.
!39
[2] 234 6 4 i;
[1] quadratic formula properly used, but some computational
errors OR correct solutions without work shown
[0] incorrect answers and no work shown OR no answers given
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Enrichment
Complex Numbers
Mathematical quantities like distance, height, and area are represented by a single
real number that indicates the size or magnitude. Other quantities used in science
such as force, velocity, and acceleration have both magnitude and direction. These
quantities are called vectors. On a graph vectors are represented by a directed
line segment or arrow. Vector v, v , is shown on the complex number plane below.
Algebraically, vectors are represented by complex numbers. The vector shown
below is represented by k3, 5l because the arrow ends at the point represented by
the complex number 3 1 5i.
W
Imaginary
v
Real
The magnitude of the vector is the absolute value of the complex number.
Write the complex number represented by each vector. Then determine the
magnitude of the vector.
1. k22, 7l 22 1 7i; !53
2. k5, 1l 5 1 i; !26
3. k3, 24l 3 2 4i; 5
4. k210, 15l 210 1 15i; 5 !13
5. k213, 221l 213 2 21i; !610
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Reteaching
4-8
Complex Numbers
• A complex number consists of a real part and an imaginary part. It is written in
the form a 1 bi, where a and b are real numbers.
• i 5 !21 and i2 5 (!21)(!21) 5 21
• When adding or subtracting complex numbers, combine the real parts and
then combine the imaginary parts.
• When multiplying complex numbers, use the Distributive Property or FOIL.
Problem
What is (3 2 i) 1 (2 1 3i)?
(3 2 i) 1 (2 1 3i)
5 3 2 i 1 2 1 3i
Circle real parts. Put a square around imaginary parts.
5 (3 1 2) 1 (21 1 3)i
Combine.
5 5 1 2i
Simplify.
Problem
What is the product (7 2 3i)(24 1 9i)?
Use FOIL to multiply:
(7 2 3i)(24 1 9i) 5 7(24) 1 7(9i) 1 (23i)(24) 1 (23i)(9i)
(7 2 3i)(24 1 9i)
5 228 1 63i 1 12i 2 27i2
First 5 7(24)
5 228 1 75i 2 27i2
Outer 5 7(9i)
You can simplify the expression by substituting 21 for i2.
Inner 5 (23i)(24)
(7 2 3i)(24 1 9i) 5 228 1 75i 2 27(21)
Last 5 (23i)(9i)
5 21 1 75i
Exercises
Simplify each expression.
1. 2i 1 (24 2 2i)
24
2. (3 1 i)(2 1 i)
5 1 5i
3. (4 1 3i)(1 1 2i)
22 1 11i
4. 3i(1 2 2i)
6 1 3i
5. 3i(4 2 i)
3 1 12i
6. 3 2 (22 1 3i) 1 (25 1 i)
22i
7. 4i(6 2 2i)
8 1 24i
8. (5 1 6i) 1 (22 1 4i)
3 1 10i
9. 9(11 1 5i)
99 1 45i
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Reteaching (continued)
4-8
Complex Numbers
• The complex conjugate of a complex number a 1 bi is the complex number a 2 bi.
• (a 1 bi)(a 2 bi) 5 a2 1 b2
• To divide complex numbers, use complex conjugates to simplify the denominator.
Problem
4 1 5i
What is the quotient 2 2 i ?
4 1 5i
22i 5
4 1 5i
21i
5 22i ?21i
5
5
5
5
8 1 4i 1 10i 1 5i2
(2 2 i)(2 1 i)
8 1 4i 1 10i 1 5i2
22 1 12
8 1 14i 1 5(21)
4 1 1
3 1 14i
5
3
5 5 1 14
5i
The complex conjugate of 2 2 i is 2 1 i.
Multiply both numerator and denominator by 2 1 i.
Use FOIL to multiply the numerators.
Simplify the denominator. (a 1 bi)(a 2 bi) 5 a2 1 b2
Substitute 21 for i2.
Simplify.
Write as a complex number a 1 bi.
Exercises
Find the complex conjugate of each complex number.
10. 1 2 2i 1 1 2i
11. 3 1 5i 3 2 5i
12. i 2i
13. 3 2 i 3 1 i
14. 2 1 3i 2 2 3i
15. 25 2 2i 25 1 2i
Write each quotient as a complex number.
3i
16. 1 2 2i 265 1 35i
6
9
17. 3 1 5i 17
2 15
17 i
18.
2 1 2i
2 2 2i
i
2 1 5i 1
19. 3 2 i 10
1 17
10 i
24 2 i
10
20. 2 1 3i 211
13 1 13 i
61i
7
21. 25 2 2i 232
29 1 29 i
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Additional Vocabulary Support
Quadratic Systems
Use the list below to complete the Venn diagram.
solve using substitution
one solution is possible
where two lines intersect
solve using elimination
two solutions are possible
where two parabolas intersect
Quadratic System
of Equations
solve using
substitution
Linear System of
Equations
where two parabolas
intersect
where two lines
intersect
solve using
elimination
two solutions are
possible
one solution is
possible
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Think About a Plan
Quadratic Systems
Business A company’s weekly revenue R is given by the formula R 5 2p2 1 30p,
where p is the price of the company’s product. The company is considering hiring a
distributor, which will cost the company 4p 1 25 per week.
a. Use a system of equations to find the values of the price p for which the
company will still remain profitable if they hire this distributor.
b. Which value of p will maximize the profit after including the distributor cost?
1. What does it mean for the company to be profitable?
It means that the weekly revenue is greater than the distributor cost.
z
z
R 5 2p2 1 30p .
2. The weekly revenue is represented by the function z
z
D 5 4p 1 25
3. The distributor cost D is represented by the function .
R/D
4. Solve this system of equations by graphing.
200
150
5. For what values of p will the company remain profitable?
1 R p R 25
6. How can you find the new weekly revenue of the company
100
50
0
if they hire the distributor?
p
0
10
20
30
40
Subtract the distributor cost function from the weekly revenue function.
7. Write the new weekly revenue function and graph it.
5 2p2 1 26p
2 25z
z R R
200
150
8. How can you find the value of p that will maximize the profit?
100
Find the maximum of the graph of the new weekly revenue
50
function.
0
p
0
9. What value of p will maximize the profit? 13
10. What is the maximum profit? 144
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10
20
30
40
Name
Class
4-9
Date
Practice
Form G
Quadratic Systems
Solve each system by graphing. Check your answers.
1. e
y 5 2x2 1 3x 1 2
(0, 2)
y 5 3x 1 2
2. e
y 5 x2 1 2x 2 3
(22, 23), (2, 5)
y 5 2x 1 1
3. e
y 5 22x2 1 4x 1 3
(21, 23), (2, 3)
y 5 2x 2 1
4. e
y 5 2x2 2 5x
(21, 7), (2, 22)
y 5 23x 1 4
Solve each system by substitution. Check your answers.
5. e
y 5 x2 1 5x 2 2
(22, 28), (0, 22)
y 5 3x 2 2
6. e
y 5 2x2 1 x 1 12
(25, 218), (4, 0)
y 5 2x 2 8
7. e
y 5 x2 2 2x 2 3
(0, 23), (4, 5)
y 5 2x 2 3
8. e
y 5 2x2 2 5x 1 6
(2, 4)
y 5 3x 2 2
9. e
y 5 2x2 1 2x 1 18
(27, 245), (4, 10)
y 5 5x 2 10
10. e
y 5 x2 2 2x 2 2
(23, 13), (2, 22)
y 5 23x 1 4
x1y55
(22, 7), (1, 4)
y 1 1 5 3x2 1 2x
12. e
x 1 y 5 x2 2 6
(22, 0), (2, 24)
x1y1250
14. e
y 1 4 5 x2 2 3x
(1, 26), (5, 6)
y 1 9 5 3x
11. e
x5y25
13. e 2
(22, 3), (1, 6)
x 1 2x 5 y 2 3
15. e
x2 1 y 2 10 5 0
(23, 1), (4, 26)
x1y1250
16. e 2
(27, 14), (1, 6)
x 2 y 5 25x
17. e
y 1 5x 5 x2 2 3
(2, 29), (6, 3)
y 2 3x 5 215
18. e
x1y57
y 2 2x 5 2x2 2 4
(1, 23), (3, 27)
y 1 2x 5 21
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Name
Class
4-9
Date
Practice (continued)
Form G
Quadratic Systems
Solve each system.
19. e
y 5 2x2 1 2x 2 3
(21, 26), (0, 23)
y 5 x2 1 4x 2 3
20. e
y 5 x2 1 2x 2 3
(23, 0), (1, 0)
y 5 2x2 2 2x 1 3
21. e
y 5 2x2 1 x 2 5
(21, 24), (0, 25)
y 5 2x2 2 2x 2 5
22. e
y 5 2x2 1 x 1 2
(21, 0), (3, 24)
y 5 x2 2 3x 2 4
23. e
y 5 x2 1 1
(22, 5), (2, 5)
y 5 2x2 2 3
24. e
y 5 2x2 2 4
(25, 46), (1, 22)
y 5 x2 2 4x 1 1
26. e
y
y . x2 2 4x
6
y , 2x2 1 6 (⫺1,5)
Solve each system by graphing.
25. e
y
y , x2 1 5 (23, 14)
(3, 14)
12
y . 2x2 2 4
8
4
⫺6 ⫺2
27. e
y . x2 2 x
y , x2 1 3 (⫺3, 12)
⫺6
x
2
y
28. e
8
O
x
6
(3,⫺3)
⫺6
6
y
y # 4x2 1 4x
y $ x2 1 4x
8
4
x
⫺6
2
O
⫺6
6
(0, 0)
6 x
29. In business, a break-even point is the point (x, y) at which the graphs of the revenue
and cost functions intersect. For one manufacturing company, the revenue from
producing x items is given by the function y 5 2x 1 12 and the cost of producing x
items is given by y 5 2x2 1 10x 1 5. Find all break-even points. (1, 14), (7, 26)
30. Two skaters are practicing at the same time on the same rink. One skater
follows the path y 5 22x 1 32, while the other skater follows the curve
y 5 22x2 1 18x. Find all points where they might collide if they are not
careful. (2, 28), (8, 16)
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Name
Class
4-9
Date
Practice
Form K
Quadratic Systems
Solve each system by graphing. Check your answers.
y 5 3x2 1 2x 2 5
1. e
y 5 2x 2 2
y
4
⫺8 ⫺4
(⫺1, ⫺4)
y 5 212 x2 2 x 1 3
2. e
y 5 3x 1 14
y
(21, 24),
(1, 0)
⫺8 ⫺4 O
⫺4
y 5 2x2 1 3x 2 1
y 5 2x 2 1
8 y
no solution
4
(1, 0)
x
4
3. e
4
x
4
8
⫺8 ⫺4
⫺4
(0, ⫺1)
(0, 21),
(4, 25)
x
4
8
(4, ⫺5)
⫺8
Solve each system by substitution. Check your answers.
4. e
y 5 22x2 1 4x 2 1
y 5 2x 2 5
5. e
2x 2 5 5 22x2 1 4x 2 1
y 5 x2 1 2
y 5 2x 1 4
(22, 6) and (1, 3)
6. e
y 5 6x2 1 5x 1 3
y 5 2x 1 3
(0, 3) and (21, 4)
(2, 21) and (21, 27)
Solve each system using your graphing calculator.
7. e
y 5 x2 2 2x 1 4
y 5 2x2 1 2x 1 4
8. e
(0, 4) and (2, 4)
y 5 x2 1 3x 2 2
y 5 x2 1 5x 1 4
9. e
y 5 2x2 1 7x 2 2
y 5 2x2 1 3x 1 2
(1, 4)
(23, 22)
10. Reasoning What is the least number of solutions a quadratic system can
have? Explain what that means.
A quadratic system can have no solutions. This means the graphs of the equations
do not intersect.
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Name
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Practice (continued)
Form K
Quadratic Systems
11. You work at a restaurant whose weekly profit is given by the formula
P 5 2c2 1 14c 1 800, where c is the average price of the food, in dollars.
The manager wants to add delivery service, which will cost the restaurant
D 5 5c 1 300 per week.
a. Use a graphing calculator to find the highest average price c the restaurant
can sell its food at and still make a profit if they add delivery. $27.31
b. What will the weekly profit P be if the restaurant sells its food at this average
price and doesn’t offer delivery? $436.50
c. Reasoning Even though these equations have two solutions, why is only
one solution useful? (Hint: Remember this is a real situation.)
The other solution is negative. You cannot have negative price.
Solve each system. When necessary, round solutions to the nearest hundredth.
12. e
y 5 2x2 1 5x 2 2
y 5 3x 2 1
(1, 2)
15. e
y 5 x2 2 4x 1 5
y 5 23x 1 5
14. e
(0, 5) and (1, 2)
y 5 x2 1 7x 2 2
y 5 4x 2 5
no solutions
13. e
16. e
y 5 25x2 2 x 1 3
y 5 2x 2 2
y 5 12x2 1 5x 2 32
y5x29
(23, 212) and (25, 214)
17. e
(21, 21) and (1, 23)
y 5 2x2 2 x 2 4
y5x12
(2.3, 4.3) and (21.3, 0.7)
y 5 2x2 2 x 1 3
has
y 5 23x 1 3
no solutions. Her work is below. What mistake did she make? What is the solution
of this system?
18. Error Analysis A classmate said that the quadratic system e
2x2 2 x 1 3 5 23x 1 3
2x2 1 2x 1 6 5 0
x2 1 x 1 3 5 0
21 4 "12 2 (4)(1)(3)
2(1)
21 4 !211
2
In the second line of her solution, she added 3 to the left side of the equation and
subtracted 3 from the right side instead of adding 3 to both sides; (0, 3) and (21, 6).
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Standardized Test Prep
4-9
Quadratic Systems
Multiple Choice
For Exercises 1−4, choose the correct letter.
y 5 x2 2 4x 1 5
1. What is the solution of the system? e
D
y 5 22x 1 8
(0, 5), (2, 1)
(22, 17), (1, 6)
(21, 10), (4, 5)
(21, 10), (3, 2)
2. What is the solution of the system? e
y 5 2x2 2 2x 1 4
G
y 5 2x 1 2
(22, 4), (2, 0)
(23, 5), (1, 1)
(22, 4), (1, 1)
(23, 1), (2, 0)
3. What is the solution of the system? e
y 5 x2 2 4x 1 3
A
y 5 22x 1 6
(21, 8), (3, 0)
(21, 8), (4, 3)
(22, 10), (3, 0)
(0, 3), (4, 22)
4. What is the solution of the system? e
y 5 2x2 1 4x 1 5
G
y 5 x2 2 2x 2 3
(22, 27), (2, 9)
(21, 0), (3, 8)
(21, 0), (4, 5)
(1, 24), (4, 5)
Short Response
5. What is the solution of the system? Solve by graphing. e
y , 2x2 1 3x
y . x2 2 x 2 6
8 y
[2]
4
⫺6
x
6
⫺6
[1] an error in graphing one of the functions OR the wrong region is shaded
[0] no answer given
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Name
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4-9
Date
Enrichment
Quadratic Systems
y
Write a system of inequalities for the graph at the right.
8
B
1. Write the equation for Graph A in the
6
form y 5 a(x 2 h)2 1 k.
4
y 5 (x 2 3)2 1 1
A
2
2. Rewrite the equation for Graph A as an inequality.
x
⫺8 ⫺6 ⫺4 ⫺2 O
⫺2
2
4
6
8
C
Explain how you decided which inequality to use.
3)2
y K (x 2
1 1; answers may vary. Sample:
The region below the graph is shaded and the
boundary is included.
3. Write an inequality for Graph B. Explain how you decided which inequality to use.
y L 2»x 1 1… 2 4; answers may vary. Sample: The region above the graph is shaded and
the boundary is included.
4. Write an inequality for Graph C. Explain how you decided which inequality to use.
y S 212 (x 1 4)2 1 4.5; answers may vary. Sample: The region above the graph is
shaded and the boundary line is not included.
All three inequalities together describe the system of inequalities for this graph.
Write the system of inequalities for each graph.
y
5.
6.
8
8
y
6
4
2
2
⫺8 ⫺6 ⫺4 ⫺2 O
⫺2
᎐9
x
2
4
6
8
y R (x 1 1)2 2 4, y L (x 2 5)2 1 5
᎐6 ᎐4 ᎐2
᎐2
᎐4
᎐6
᎐8
x
2
4
6
9
y K (x 1 4)2 2 8, y S x2 2 7, y K 2»x… 1 7
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Reteaching
Quadratic Systems
You used graphing and substitution to solve systems of linear equations. You can
use these same methods to solve systems involving quadratic equations.
Problem
What is the solution of the system of equations? e
y 5 2x 2 3
x2 2 2x 2 8 5 2x 2 3
x2 2 4x 2 5 5 0
(x 1 1)(x 2 5) 5 0
x 1 1 5 0 or x 2 5 5 0
x 5 21 or x 5 5
y 5 x2 2 2x 2 8
y 5 2x 2 3
Write one equation.
Substitute x2 2 2x 2 8 for y in the linear equation.
Write in standard form.
Factor the quadratic expression.
Use the Zero-Product Property.
Solve for x.
Because the solutions to the system of equations are ordered pairs of the form
(x, y), solve for y by substituting each value of x into the linear equation. You can
use either equation, but the linear equation is easier.
x 5 21:
x 5 5:
y 5 2x 2 3 5 2(21) 2 3 5 25
y 5 2x 2 3 5 2(5) 2 3 5 7
S
S
(21, 25)
(5, 7)
The solutions are (21, 25) and (5, 7). Check these by graphing the system and
identifying the points of intersection.
Exercises
Solve each system.
1. e
y 5 x2 1 3x 2 5
(21, 27), (1, 21)
y 5 3x 2 4
2. e
y 5 2x2 1 5x 2 1
(1, 3), (5, 21)
y 5 2x 1 4
3. e
y 5 2x2 2 x 2 5
(21, 22), (3, 10)
y 5 3x 1 1
4. e
y 5 x2 1 3x 2 7
(25, 3), (1, 23)
y 5 2x 2 2
5. e
y 5 2x2 2 5x 1 1
(1, 22), (4, 13)
y 5 5x 2 7
6. e
y 5 2x2 2 2x 1 3
(24, 25), (1, 0)
y5x21
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Reteaching (continued)
Quadratic Systems
To solve a system of linear inequalities, you graph each inequality and find the
region where the graphs overlap. You can also use this technique to solve a system
of quadratic inequalities.
Problem
What is the solution of this system of inequalities? e
Step 1
y , 2x2 1 4x
y . x2 2 2x 2 8
Graph the equation y 5 2x2 1 4x. Use a dashed boundary line
because the points on the curve are not part of the solution. Choose a
point on one side of the curve and check if it satisfies the inequality.
y , 2x2 1 4x
?
0 , 2(2)2 1 4(2)
0,4
Check the point (2, 0).
The inequality is true.
᎐8
᎐6
᎐4
᎐2
᎐8᎐6᎐4᎐2
᎐4
y
x
2 4 6 8
Points below the curve satisfy the inequality, so shade that region.
Step 2
y
Graph the equation y 5 x2 2 2x 2 8. Use a dashed boundary line
᎐8
᎐6
because the points on the curve are not part of the solution. Choose a
᎐4
point on one side of the curve and check if it satisfies the inequality.
᎐2
x
2
y . x 2 2x 2 8
᎐8᎐6᎐4᎐2
2 4 6 8
?
0 . (2)2 2 2(2) 2 8
0 . 28
᎐4
᎐6
᎐8
Check the point (2, 0).
The inequality is true.
Points above the curve satisfy the inequality, so shade that region.
Step 3
᎐2
᎐4
᎐6
᎐8
The solution to the system of both inequalities is the set of points
satisfying both inequalities. In other words, the solution is the region
where the graphs overlap. The region contains no boundary points.
᎐8᎐6᎐4᎐2
᎐4
᎐6
Exercises
Solve each system by graphing.
7. e
y , 2x2 1 6
y . x2 2 2
8 y
6
2
᎐8 ᎐4
9. e
y , 2x2 2 2x 1 8
y . 2x2 1 4
2
᎐8 ᎐4
x
᎐4
᎐6
᎐8
y . x2 2 x 2 2
y , 2x2 2 x 1 6
8 y
2
᎐8 ᎐4
468
᎐4
᎐6
᎐8
6
8. e
y
10. e
x
y . x2 2 6x
y , x2 2 6x 1 7
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᎐4
᎐6
᎐8
6
y
2
᎐8 ᎐4
468
x
468
᎐4
᎐6
᎐8
x
2
68
y
x
2 4 6 8
Name
Class
Date
Chapter 4 Quiz 1
Form G
Lessons 4-1 through 4-4
Do you know HOW?
Graph each function.
y
1. y 5 2x2 2 8x 1 3
4
⫺6
O
8 y
2. y 5 2( x 1 3)2 1 4
4
x
⫺4 O
⫺4
6
⫺6
4
x
8
⫺8
3. What are the vertex, axis of symmetry, maximum or minimum value, domain,
and range of the function y 5 2x2 2 4x 1 3?
vertex (22, 7); axis of symmetry x 5 22; maximum value 7; domain: all real numbers;
range: all real numbers K 7
4. Rewrite the equation y 5 4x2 2 4x 2 5 in vertex form. Name the vertex and
the axis of symmetry.
y 5 4 Q x 2 12 R 2 2 6; vertex Q 12, 26 R ; axis of symmetry x 5 12
Write each expression in factored form.
5. 20 2 t 2 t2
(5 1 t)(4 2 t) or 2(t 1 5)(t 2 4)
6. 25w2 1 30w 1 9
(5w 1 3)2
Find an equation in standard form of the parabola passing through the following points.
7. (0, 24), (1, 1), (2, 8)
8. (21, 1), (0, 5), (2, 7)
y 5 x2 1 4x 2 4
y 5 2x2 1 3x 1 5
Write the equation in vertex form for each parabola.
y
9.
⫺4
O
10.
2
x y 5 (x 1 3) 2 9
⫺4
4
2
⫺2 O
⫺8
y
y 5 22(x 2 2)2 1 4
x
2 4 6
⫺4
Do you UNDERSTAND?
11. Compare and Contrast The x-coordinate of the vertices of the parabolas
y1 5 x2 2 3x 1 1 and y2 5 22x2 1 6x 2 2 are the same. Explain why this is so and
explain how the y-coordinates of the vertices are related. The values of a and b in the two
6
23
standard forms are proportional, x 5 2 2(22)
5 2 2(1)
. Because y2 5 22y1, the
y-coordinate of the second parabola is 22 times the y-coordinate of the first parabola.
12. Error Analysis A student says that the graph of the parabola y 5 x2 1 1001 is “one
thousand times larger” than the parabola y 5 x2 1 1. Explain why this is not correct.
“One thousand times larger” would be y 5 1000x2 1 1000. The graph is only shifted
1000 units upward.
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Name
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Date
Chapter 4 Quiz 2
Form G
Lessons 4-5 through 4-9
Do you know HOW?
Complete the square.
1. x2 1 9x 1 81
4
2. 4x2 2 12x 1 9
Evaluate the discriminant of each equation. Determine how many real
solutions each equation has.
3. 2x2 2 4x 1 5 5 0 224; 0
4. 6x2 2 11x 2 10 5 0 361; 2
Simplify each expression.
5. (22 1 4i) 1 (7 2 3i) 5 1 i
6. (3 2 5i) 2 (21 1 7i) 4 2 12i 7. (21 1 2i)(3 1 10i) 223 2 4i
Solve each equation using the Quadratic Formula.
9. 9x2 1 6x 1 1 5 25 22, 43
8. x2 1 4x 5 6 22 6 !10
10. 2x2 2 x 2 5 5 0
1 6 !41
4
!23
11. x2 1 3x 1 8 5 0 2 32 6 2 i
Solve each system of equations.
12.
e
y 5 2x2 2 2x 1 8
(25, 27), (1, 5)
y 5 2x 1 3
13. e
y 5 2x2 2 3x 2 1
(21, 4), (6, 53)
y 5 x2 1 2x 1 5
Do you UNDERSTAND?
14. Compare and Contrast Compare multiplying two binomials in x with
multiplying two complex numbers. They are similar but a product of imaginary numbers
may be simplified using i2 5 21, while x2-terms cannot be further simplified.
15. a. Open Ended Write a system of two quadratic equations that has exactly
one ordered pair as the solution. Answer may vary. Sample: y 5 x2 and y 5 2x2
b. Open Ended Write a system of two quadratic equations that has exactly
two ordered pairs as the solution. Answer may vary. Sample: y 5 x2 and y 5 2x2 1 1
c. Reasoning Is it possible for a system of two quadratic equations to have
exactly three ordered pairs for the solution? If so, give an example. no
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Chapter 4 Test
Form G
Do you know HOW?
1. Write the equation of the parabola in standard form. Find the coordinates of
the points on the other side of the axis of symmetry corresponding to P and Q.
Label these points P9 and Q9, respectively. y 5 2x2 2 6x 2 5; Pr(22, 3), Qr(21, 0)
4
P
y
2
Q
x
⫺4
⫺2
O
Sketch a graph of the quadratic function with the given vertex and through the
given point.
2. vertex (3, 4); point (5, 8)
y
8
4
8 y
4
3. vertex (23, 22); point (1, 2)
⫺6
4 8x
O
Graph each quadratic function. Name the axis of symmetry and the coordinates
of the vertex.
4. y 5 x2 1 5
5. y 5 x2 2 4x 2 3
y
8
4 (0, 5)
x⫽0
⫺4 O 4 x
6. y 5 2x2 1 7x 2 2
y
8
4
O
y
⫺4
⫺4
⫺8
1
7. y 5 2x2 2 6
1
(3 1 , 10 4)
2
1
x ⫽3 2
y
4
O
⫺4⫺4
4 8 x
x
68
(2,⫺7)
x⫽2
x⫽0
4 x
(0, ⫺6)
Simplify each expression.
8. (3 1 i) 2 (7 1 6i) 24 2 5i
9. (3 2 4i)(5 1 2i) 23 2 14i
11. 3 !225 1 4 4 1 15i
10. (24 2 9i) 1 (5 2 7i) 1 2 16i
Solve each quadratic equation.
12. x2 2 16 5 0 4, 24
13. 2x2 2 3x 2 11 5 0
14. x2 1 3x 2 10 5 0 2, 25
15. 3x2 1 48 5 0 w 4i
3 6 !97
4
16. Anthony has 10 ft of framing and wants to use it to make the largest
rectangular picture frame possible. Find the maximum area that can be
enclosed by his frame. 614 ft 2
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⫺4
x
4
Name
Class
Date
Chapter 4 Test (continued)
Form G
Write each function in vertex form. Sketch the graph of the function and label
its vertex.
17. y 5 x2 1 4x 2 7 y 5 (x 1 2)2 2 11
18. y 5 2x2 1 4x 2 1 y 5 2(x 2 2)2 1 3
y
⫺8 ⫺4 O
⫺4
6
4
2
2x
y
⫺6⫺4⫺2O 2
⫺4
⫺8
(2, 3)
x
6
(ⴚ2, ⴚ11)
1
20. y 5 2x2 2 5x 1 12 y 5 12 (x 2 5)2 2 12
19. y 5 3x2 1 18x y 5 3(x 1 3)2 2 27
y
⫺8 O
⫺8
8
y
10
8
6
4
2 O
x
⫺2
2 4 5, ⴚ 1
2
x
⫺16
(ⴚ3, ⴚ27)
Evaluate the discriminant of each equation. Determine how many real
solutions each equation has.
21. x2 1 5x 1 6 5 0 1; 2 real solutions
22. 3x2 2 4x 1 3 5 0 220; no real solutions
23. 22x2 2 5x 1 4 5 0 57; 2 real solutions
24. 16x2 2 8x 1 1 5 0 0; 1 real solution
Solve each system.
25. e
y 5 2x2 1 5x 1 1
(21, 25), (4, 5)
y 5 2x 2 3
26. e
y 5 x2 2 x 1 2
(24, 22), (2, 4)
y 5 2x2 1 x 2 6
Solve the following systems of inequalities by graphing.
27. e
y , x2 1 2x 2 3
y . x2 2 9
6
y
28. e
y . x2 1 3x 2 4
y , 2x2 2 x 1 2
8 y
4
x
O
⫺6
6
⫺6
⫺6
O
⫺4
x
6
⫺8
Do you UNDERSTAND?
29. Reasoning Suppose a parabola has a vertex in Quadrant IV and a , 0 in its equation
y 5 ax2 1 bx 1 c. How many real solutions will the equation ax2 1 bx 1 c 5 0 have?
none
30. Open-Ended Write a complex number with an absolute value between 3 and 8.
Answers may vary. Sample: 3 2 4i
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Date
Chapter 4 Quiz 1
Form K
Lessons 4–1 through 4–4
Do you know HOW?
Graph each function. Identify the axis of symmetry.
1. y 5 (x 2 3)2 1 1 x 5 3
2. y 5 (x 1 5)2 2 5 x 5 25
y
y
6
4
4
O
x
O
⫺10 ⫺4
⫺4
2
2
4
6 x
⫺8
Identify the vertex, the axis of symmetry, the maximum or minimum value, and
the range of each parabola.
3. y 5 2x2 2 8x 2 19
vertex 5 (24, 23); axis of symmetry x 5 24; maximum 5 23;
range 5 all real numbers K 23
4. y 5 x2 2 2x 2 3 vertex 5 (1, 2 4); axis of symmetry x 5 1; minimum 5 2 4;
range 5 all real numbers L 4
Find an equation in standard form of the parabola passing through the points.
5. (24, 3), (21, 26), (3, 10)
y 5 x2 1 2x 2 5
6. (21, 9), (0, 4), (2, 6)
y 5 2x2 2 3x 1 4
Factor each expression.
7. 4x2 1 2x 2x(2x 1 1)
8. t2 1 2t 2 15 (t 1 5)(t 2 3)
9. r 2 1 14r 1 48 (r 1 6)(r 1 8)
10. 8y 2 2 2y 2 1 (4y 1 1)(2y 2 1)
Do you UNDERSTAND?
11. Writing Explain how the sign of a in the equation y 5 a(x 2 h)2 1 k tells
you whether the parabola has a minimum or a maximum value. If a S 0, the parabola
has a minimum value. If a R 0, the parabola has a maximum value.
12. The area of a rectangular garden in square feet is x2 2 5x 2 300. The width is
(x – 20) feet. What is the length of the garden? (x 1 15) ft
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Class
Date
Chapter 4 Quiz 2
Form K
Lessons 4-5 through 4-9
Do you know HOW?
Solve each equation by factoring. Check your answers.
1. x2 2 5x 5 0
x 5 0 and x 5 5
2. x2 1 3x 2 4 5 0
x 5 1 and x 5 24
3. x2 1 10x 5 221
x 5 23 and x 5 27
Solve each quadratic equation by completing the square.
4. x2 1 10x 1 24 5 0
26, 24
5. x2 2 2x 2 2 5 0
1 1 "3, 1 2 "3
6. x2 1 8x 1 11 5 0
24 1 "5, 24 2 "5
8. (2 1 5i)(6 2 i)
17 1 28i
9.
Simplify each expression.
7. (8 1 2i) 1 (3 2 4i)
11 2 2i
4 1 3i
9i
1
4
2
3
9i
Solve each system by substitution. Check your answers.
10. e
y 5 x2 2 2x 2 1
y5x21
(0, 21) and (3, 2)
11. e
y 5 23x2 1 x 1 4
y 5 22x 2 2
12. e
(21, 0) and (2, 26)
y 5 x2 1 4x 2 6
y5x22
(1, 21) and (24, 26)
Do you UNDERSTAND?
13. Open-Ended Write a quadratic equation with a discriminant less than 0.
What does this tell you about the solution of the equation?
Any quadratic equation where b2 2 4ac R 0. When the discriminant is less than zero,
the solution is two complex numbers.
14. Writing Explain how you would use the Quadratic Formula to solve
3x2 1 5 5 x 1 9.
First, rewrite the equation in ax2 1 bx 1 c 5 0 form. Then determine the values
of a, b, and c. Use these in the Quadratic Formula x 5
2b 6 "b2 2 4ac
.
2a
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Chapter 4 Test
Form K
Do you know HOW?
Identify the vertex, the axis of symmetry, the maximum or minimum value, and
the domain and the range of each function.
1. y 5 3(x 2 2)2 1 6
vertex 5 (2, 6); axis of symmetry x 5 2; minimum 5 6; domain 5 all real numbers;
range 5 all real numbers L 6
2. y 5 2(x 1 4)2 23
vertex 5 (24, 23); axis of symmetry x 5 2 4; maximum 5 23; domain 5 all real
numbers; range = all real numbers K 23]
y
3. y 5 2 x2 2 2x 1 3
4. y 5 3x2 2 4x 1 1
y
6
4
4
O
⫺10 ⫺4
⫺4
2
Factor each expression.
⫺4 ⫺2 O
⫺8
2 x
5. 4c2 1 4c 1 1
6. g2 2 49
(2c 1 1)2
(g 1 7)(g 27)
Use a graphing calculator to solve each equation. Give each answer to at most
two decimal places.
7. 5x2 1 9x 1 4 5 0
x 5 2 1 and x 5 2 0.8
8. 23x2 2 2x 1 7 5 0
x 5 1.2 and x 5 21.9
Complete the square.
9. x2 1 14x 1 j
49
10. x2 2 18x 1 j
81
Evaluate the discriminant for each equation. Determine the number of real
solutions.
11. 5x2 1 x 1 6 5 0
2119; 0 real solutions
12. 23x2 2 4x 1 1 5 0
28; 2 real solutions
Plot each complex number and find its absolute value.
13. 7 2 2i !53
8i imaginary axis
4i
14. 8i 8
8i imaginary axis
4i
real axis
⫺8 ⫺4 O
⫺4i
4
⫺8 ⫺4 O
⫺4i
7 ⫺ 2i
⫺8i
real axis
4
⫺8i
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x
Name
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Date
Chapter 4 Test (continued)
Form K
Find all solutions to each quadratic equation.
15. y 5 2x2 2 2x 1 5
3 1
3
1
1 i, 2 i
2
2 2
2
16. y 5 3x2 1 2x 1 4
2
Solve each system by graphing.
17. y . x2
d
6
y
18. y . x2 1 3x
4
y , 2 x2 1 4
d
2
4
2O
2
"11
"11
1
1
1
i, 2 2
i
3
3
3
3
x
2
6
4
y . x2 2 2
2
4
4
y
2O
2
x
2
Do you UNDERSTAND?
19. The parabolic path of a hit tennis ball can be modeled by the table at
the right. The top of the net is at (4, 10).
a. Find a quadratic model for the data. y 5 20.5x2 1 3x 1 4.5
b. Will the ball go over the net? If not, will it hit the net on the way up
or the way down? No; it will hit the net on the way down.
4
x
y
1
2
7
7
8.5
1
20. Writing Explain the relationship between the x-intercepts of quadratic
function and the zeros of a quadratic function.
They are the same thing because the x-intercepts are the x-coordinates where the
quadratic function equals zero.
21. The period of a pendulum is the time the pendulum takes to swing back and
forth. The function l 5 0.81t2 relates the length l in feet of a pendulum to the
period t.
a. If a pendulum is 30 ft long, what is the period of the pendulum in seconds? t 5 6.1 s
b. Reasoning Why does only one of the solutions work for this problem?
The other solution is negative and you cannot have negative time.
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Chapter 4 Performance Tasks
Task 1
Write your own quadratic function in the standard form y 5 ax2 1 bx 1 c, such
that a, b, and c do not equal zero.
a. Describe at least two methods that can be used to determine the graph of your
function.
b. Write your quadratic function in vertex form.
c. Find the maximum or minimum value. Explain how you can determine this
value.
d. Determine the zeros of your function. Give an algebraic reason for the
existence or nonexistence of real-valued zeros of your quadratic function.
[4] Student describes at least two possible methods that clearly demonstrate an in-depth
understanding of the mathematical principles involved. Student writes the function in
vertex form correctly. Student finds the maximum or minimum value and zeros.
[3] Student describes at least one possible graphing method that demonstrates
understanding of the mathematical principles involved. Further detail or more clarity
is needed. Student writes the function in vertex form correctly. Student finds the
maximum or minimum value and zeros.
[2] Student does not describe possible graphing methods in sufficient detail to demonstrate
understanding of the concepts. Student does not write the function in vertex form
correctly. Student finds the maximum or minimum value and zeros.
[1] Student does not write on equation in the correct form. Student does not find all values.
[0] Response is missing or inappropriate.
Task 2
Give complete answers.
a. Indicate whether the graph of the equation y 5 x2 2 4x 1 1 opens up, down,
left, or right. Explain how you know. opens up; The coefficient of x2 is positive.
b. Give the coordinates of the vertex and the equation of the axis of symmetry. (2, 23); x 5 2
c. Sketch the graph of the equation.
2
O
⫺2
y
x
2
4
[4] Student correctly identifies the direction in which the parabola
opens. Explanation indicates an understanding of the direction.
Student identifies the coordinates of the vertex correctly. Equation
of the axis of symmetry is written with no errors. Graphs are drawn
neatly and labeled accurately.
[3] Student correctly identifies the direction in which the parabola opens. Explanation could
be more detailed. Student identifies the coordinates of the vertex. Equation of the axis
of symmetry has minor errors. Graphs could be more neatly drawn and labeled.
[2] Student does not correctly identify the direction in which the parabola opens.
Explanation has minor errors. Student identifies the vertex and axis of symmetry with
minor errors. Graphs have missing information or minor errors.
[1] Student does not correctly identify the direction in which the parabola opens.
Explanation is not sufficient or contains major errors. Student does not identify the
coordinates of the vertex correctly. Equation of the axis of symmetry is written with
significant errors. Graphs are not neatly drawn and labeled.
[0] Response is missing or inappropriate.
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Chapter 4 Performance Tasks (continued)
Task 3
Give complete answers.
a. Explain when to use each method for solving quadratic equations. Give an
example equation for each.
• factoring
• completing the square
• the Quadratic Formula
• graphing
b. Solve the equation 4x2 2 4x 1 7x 2 7 5 2(x 2 1) by whichever method
seems easiest. Explain why you decided to use the method you did. 1, 2 54
c. Give an example of when you would use a quadratic model to model data.
Give an example of when you would use a linear model to model data.
[4] Student explanation of each method is sufficiently detailed to indicate a thorough
understanding of the method. Example equations are appropriate. Equation is
solved correctly and explanation of method chosen is detailed. Examples of linear
and quadratic models are clear and indicate that student understands when each is
appropriate.
[3] Student explanation of each method is detailed enough to indicate an understanding
of each method. Example equations are appropriate. Equation is solved with only
minor errors and explanation of method chosen is provided but could have more detail.
Examples of linear and quadratic models are clear and indicate that student understands
when each is appropriate.
[2] Student explanation of each method is not enough to indicate an understanding of each
method. Example equations are appropriate. Equation is not solved correctly because of
major errors. Explanation of method chosen is provided but does not provide sufficient
detail. Examples of linear and quadratic models are not clear and no explanation is
included.
[1] Student explanation lacks major information. Example equations are not appropriate.
Equation is not solved correctly. Explanation of method chosen is missing or has major
errors. Examples of linear and quadratic models are missing or no explanation is
included.
[0] Response is missing or inappropriate.
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Date
Chapter 4 Cumulative Review
Multiple Choice
For Exercises 1−13, choose the correct letter.
3
2
1. Which line is perpendicular to the graph of y 5 22x 1 3 ? C
3
3
y 5 2x 1 6
y 5 3x 1 23
y 5 23x 1 2
6
y 5 3x 1 23
2. What system describes this graph? I
3
2
1
⫺4 ⫺3
y
x
O
⫺1
⫺1
⫺2
⫺3
1 2
y$x
• x $ 22
x#0
y.x
• x . 22
x,0
y5x
•x $ 2
x#0
y.x
• x $ 22
x#0
3. Which of these is the standard form of y 5 8x 1 12? D
y 2 8x 2 12 5 0
28x 1 y 5 12
y 2 8x 5 12
8x 2 y 5 212
4. What is the solution of the system? e
y 5 2x2 1 3x 2 3
H
y 5 2x 2 5
(0, 23), (2, 21)
(21, 27), (2, 21)
(21, 27), (3, 23)
(21, 210), (3, 2)
5. Which number is irrational? C
27
!144
9
!2
20.5
6. Which point lies on the graph of 2x 2 y 1 z 5 0? I
(0, 4, 28)
(0, 2, 4)
(12, 26, 6)
(0, 25, 25)
7. Which of these is the solution of 27x . 4x 1 33? D
x.3
x,3
x . 23
x , 23
7 1 3i
3 1 3i
8. Simplify (5 1 6i) 1 (2 2 3i). H
4
7 2 3i
9. At which vertex is the objective function C 5 3x 2 4y maximized? B
(9, 0)
(22, 220)
(25, 22)
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(0, 29)
Name
Class
Date
Chapter 4 Cumulative Review (continued)
3x 1 y 1 z 5 7
10. Which ordered triple is a solution to • x 1 3y 2 z 5 13 ? F
y 5 2x 2 1
(2, 3, 22)
(2, 23, 220)
(22, 25, 18)
(22, 3, 10)
5 or 25
4 or 21
11. Which is the solution to |x 2 3| 2 1 5 4? A
8 or 22
28 or 2
12. Which number completes the square of the expression 2x2 2 3x 1 ? G
9
8
9
9
16
9
22
13. Which expression is (3 2 2i)(24 1 i) simplified? D
21 2 i
7 2 3i
212 1 2i
210 1 11i
Short Response
8 y
14. Find the discriminant of each quadratic equation.
a. x2
b. 2x2
1 4x 1 9 5 0 220
y , 2x2 1 x 1 6
15. Solve by graphing. e
y . x2 1 x 2 2
1 3x 5 0 9
4
⫺6
⫺4
x
6
⫺8
16. Open-Ended Write a quadratic function in vertex form. Write the
equation for the axis of symmetry of the parabola. Answers may vary. Sample:
y 5 3(x 2 3)2 1 4, axis of symmetry x 5 3
17. Solve for x: ax 1 by 5 c. x 5 c 2a by
x1y2z52
18. Solve the system: • 2x 1 y 1 z 5 3 Q 2, 212, 212 R
3x 1 y 1 z 5 5
Extended Response
19. Jack’s Bowling Alley charges $1.50 to rent shoes and $4.50 for each game bowled. Jill’s
Bowling Alley charges $2.50 to rent shoes and $4 for each game bowled. How many
games must be bowled in order to make the cost of bowling at Jack’s the same as the cost
of bowling at Jill’s?
[4] Jack’s Alley: c 5 1.50 1 4.5g; Jill’s Alley: c 5 2.50 1 4g; solve 1.50 1 4.5g 5 2.50 1 4g
(OR equivalent equation); 2 games
[3] appropriate methods, but with one computational error
[2] incorrect equation solved correctly or correct equation solved incorrectly
[1] correct number of games, without work shown
20. The length of a rectangle is 5 inches more than its width. The area of the rectangle
is 14 square inches. What are the dimensions of the rectangle?
[4] l 5 5 1 w , A 5 l ? w 5 14, solve (5 1 w) ? w 5 14 (OR equivalent equation); width:
2 in., length: 7 in.
[3] appropriate methods, but with one computational error
[2] incorrect equation solved correctly or correct equation solved incorrectly
[1] correct width and length, without work shown
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TEACHER INSTRUCTIONS
Chapter 4 Project Teacher Notes: On Target
About the Project
The Chapter Project gives students an opportunity to use quadratic equations in real-life
situations. Students use tables and graphs to study the paths of arrows. Students also
make a three-dimensional target or a moving video target. Students display the target
and their other findings.
Introducing the Project
• Encourage students to keep all project-related materials in a separate folder.
• Ask students to show the path of an arrow if it is aimed horizontally. Ask them how the
path changes if the arrow is aimed upward and to name the shape of this path.
• Have students look at Activity 4. Encourage them to start this part of the project now so
they will have time to create good targets.
Activity 1: Graphing
Students graph possible parabolic paths for arrows shot while standing or while seated in a
wheelchair. Then, they identify the similarities of and differences between their graphs.
Activity 2: Analyzing
Students find a parabolic model for the path of an arrow.
Activity 3: Modeling
Students graph data relating the weight of an arrow to its spine, the distance the center
of the arrow bends when a constant weight is attached. Then, they decide whether a linear
model or a quadratic model is a better fit for the data.
Activity 4: Researching
Students research new archery styles using three-dimensional targets or moving video
targets, and then create their own targets.
Finishing the Project
You may wish to plan a project day on which students share their completed projects.
Encourage students to explain their processes as well as their results. Ask students to review
their project work and update their folders.
• Have students review their methods for making their graphs, for writing equations to
model the graphs, and for creating the targets for the project.
• Ask groups to share their insights that resulted from completing the project, such as
shortcuts they found for graphing, for modeling, or for making their targets.
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Chapter 4 Project: On Target
Beginning the Chapter Project
Archery has its roots in prehistoric times. Cave drawings in Spain, France, and
North Africa show hunters using bows and arrows. In the Far East, people made
bows by gluing wood, bone, and animal tendons together. Early Native Americans
also used bows and arrows. Archery has also become a recreational sport. In 1988,
at the Olympic Games in Barcelona, it was an archer who lit the Olympic flame.
In this project, you will research topics such as how archers choose their arrows
and how technology has changed the sport. You may want to finish your project by
making a display or other presentation.
List of Materials
• Calculator
• Graph paper
• Measuring tape
Activities
Activity 1: Graphing
An archer releases an arrow at shoulder height.
• Measure the distance from the floor to your shoulder when you are standing.
Suppose you release an arrow and it hits the target at a point 5 ft above the
ground. Sketch a possible parabolic path of your arrow’s flight using this
information.
Archery is one of only a few sports in which athletes using wheelchairs can
compete with other athletes.
• Measure the distance from the floor to your shoulder while you are sitting
in a chair. Sketch the possible path of an arrow released by someone using a
wheelchair.
• Describe the similarities and differences between your two sketches.
4 cm
4 cm
4 cm
4 cm
Check students’ work.
Activity 2: Analyzing
An archer releases an arrow from a shoulder height of 1.39 m. When
the arrow hits the target 18 m away, it hits point A. When the target is
removed, the arrow lands 45 m away. Find the maximum height of the
arrow along its parabolic path. 1.41 m
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8 cm
A
Name
Class
Date
Chapter 4 Project: On Target (continued)
Activity 3: Modeling
Archers need to use arrows that do not bend easily. The table shows how the weight of
an arrow affects its spine, or the distance the center of the arrow bends when a certain
constant weight is attached. Graph the data in the table to find a linear and a quadratic
model for the data. Use the regression feature on your calculator to find each model.
Which model is a better fit? Explain.
Weight (in grams)
140
150
170
175
205
Weight (in inches)
1.4
1.25
0.93
0.78
0.43
linear: y 5 20.0152x 1 3.513
quadratic: y 5 0.00006x2 2 0.0348x 1 5.172
Answers may vary. Sample: The quadratic model is a better fit. The graph of the quadratic
function that models the data is closer to more of the points than is the graph of the linear
function that models the data.
Activity 4: Researching
Research the new styles of archery that use three-dimensional targets or moving
video targets. Create one of these targets using readily-available materials or a
computer program. Check students’ work.
Finishing the Project
The activities should help you to complete your project. Present your project for
this chapter as a visual display, a demonstration or, if equipment is available, as a
videotape.
Reflect and Revise
Present your information to a small group of classmates. Decide if your work
is complete, clear, and convincing. If needed, make changes to improve your
presentation.
Extending the Project
Interview an archer. Find techniques archers use to increase the range and
accuracy of a shot.
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Date
Chapter 4 Project Manager: On Target
Getting Started
Read the project. As you work on the project, you will need a calculator, a
measuring tape, materials on which you can record your calculations, and
materials to make accurate and attractive graphs. Keep all of your work for the
project in a folder.
Checklist
☐ Activity 1: sketching parabolic paths
☐ Activity 2: finding equations
☐ Activity 3: modeling using regression
☐ Activity 4: researching archery
☐ presentation
Suggestions
☐ Measure standing and sitting height to
your shoulder.
☐ Use given information to write the
equation of a parabola. Find the vertex of
the parabola.
☐ Determine whether there is an archery
club in your area.
☐ How might a videotaped presentation be
more useful in studying parabolic paths
of arrows than a real-life demonstration?
What other objects in sports or real-life
situations follow parabolic paths?
Scoring Rubric
4
Calculations are correct. Graphs are neat, accurate, and labeled correctly,
and they clearly show the differences between the situations. Explanations
are thorough and well thought out. The target is designed well and neatly
made. The display is well organized.
3
Calculations and explanations are mostly correct, with some minor errors.
Graphs are neat and mostly accurate with minor errors in scale. The target
is designed adequately, but is not neatly made. The display presents clear
information, but is not well organized.
2
Calculations contain both minor and major errors. Graphs are not accurate.
Explanations lack detail. The target is poorly designed.
1
Major concepts are misunderstood. Project satisfies few of the requirements
and shows poor organization and effort.
0
Major elements of the project are incomplete or missing.
Your Evaluation of Project Evaluate your work, based on the Scoring Rubric.
Teacher’s Evaluation of Project
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