SHARP ALGEBRA COVER
02.2.19 10:29 AM
Page 3
APPLYING
PRE-ALGEBRA/ALGEBRA
SHARP
U S I N G
T H E
EGraphing
L - 9Calculator
600
D A V I D P. L A W R E N C E
Applying
PRE-ALGEBRA/ALGEBRA
using the
SHARP EL-9600
GRAPHING CALCULATOR
David P. Lawrence
Southwestern Oklahoma State University
This Teaching Resource has been developed specifically for use with the
Sharp EL-9600 graphing calculator. The goal for preparing this book was
to provide mathematics educators with quality teaching materials that
utilize the unique features of the Sharp graphing calculator.
This book, along with the Sharp graphing calculator, offers you and your
students 10 classroom-tested, topic-specific lessons that build skills.
Each lesson includes Introducing the Topic, Calculator Operations, Method
of Teaching, explanations for Using Blackline Masters, For Discussion,
and Additional Problems to solve. Conveniently located in the back
of the book are 38 reproducible Blackline Masters. You’ll find them ideal
for creating handouts, overhead transparencies, or to use as student
activity worksheets for extra practice. Solutions to the Activities are
also included.
We hope you enjoy using this resource book and the Sharp EL-9600
graphing calculator in your classroom.
Other books are also available:
Applying STATISTICS using the SHARP EL-9600 Graphing Calculator
Applying TRIGONOMETRY using the SHARP EL-9600 Graphing Calculator
Applying PRE-CALCULUS and CALCULUS using the SHARP EL-9600 Graphing Calculator
Graphing Calculators: Quick & Easy! The SHARP EL-9600
ALGEBRA USING THE SHARP EL-9600
i
CONTENTS
CHAPTER TOPIC
1
2
3
4
5
6
7
8
9
10
Linear Equations
Quadratic Equations
Formulas or Literal Equations
Graphing Polynomials and Finding the Roots
Solving a System of Equations
Matrix Solutions to Systems of Linear Equations
Inequalities and Systems of Inequalities
Absolute Value Functions, Equations, and Inequalities
Rational Functions
Conic Sections
Blackline Masters
Solutions to the Activities
Dedicated to my grandparents, Harold and Ruby Sharp
Special thanks to Ms. Marina Ramirez and Ms. Melanie Drozdowski for
their comments and suggestions.
Developed and prepared by Pencil Point Studio.
Copyright © 1998 by Sharp Electronics Corporation.
All rights reserved. This publication may not be reproduced, stored in
a retrieval system, or transmitted in any form or by any means, electronic,
mechanical, photocopying, recording, or otherwise without written permission.
The blackline masters in this publication are designed to be used with
appropriate duplicating equipment to reproduce for classroom use.
First printed in the United States of America in 1998.
ii
ALGEBRA USING THE SHARP EL-9600
PAGE
1
6
12
17
22
26
32
38
44
50
57
95
Chapter one
LINEAR EQUATIONS
Introducing the Topic
In this chapter, you and your students will learn how to graph linear equations
and study their characteristics. These characteristics include slope, y-intercept,
perpendicular, and parallel.
A linear equation of y in terms of x can be expressed by the slope-intercept form
y = m • x + b, where m is the slope and b is the y-intercept. We call this equation
a linear equation, since its graph is a straight line. Equations where the
exponents on the x and y are 1 (implied) are considered linear equations.
In graphing linear equations on the calculator, we will let the x-variable be
represented by the horizontal axis and let y be represented by the vertical axis.
Calculator Operations
Turn the calculator on and press Y= . A set of Y prompts will appear on the
viewing screen. Press CL to remove an old Y expression. Move up or down the
set of Y prompts using the arrow keys, or simply touch the screen twice at the Y
prompt desired.
Linear Equations/ALGEBRA USING THE SHARP EL-9600
1
The calculator should be set up with rectangular coordinates and in equation
editor mode. To complete this setup, press 2ndF
SET UP
E
(COORD)
(Note that the alphabet letters are written above and to the right of the access
key and can be used directly within the menu) 1 (Rect) G (EDITOR) and
1 (Equation). Press CL to exit the menu and return to the Y prompts.
The same sequence of keystrokes can be performed by touching the screen.
For example, to set up the calculator in rectangular coordinates, press 2ndF
SET UP , and then touch E COORD once and double touch 1 Rect.
Linear equations given in a form other than the slope intercept form must be
manipulated to isolate the y variable. The calculator will only accept equations
of the form y = ___ (where y is defined explicitly in terms of x). To enter the
linear equation ( y = 3x + 1 ) for Y1, press 3
X/θ/T/n
+
1 . Enter the viewing
window range by pressing ZOOM , touching A ZOOM, and double touching 5
Default. This establishes the default viewing ranges for graphing equations. These
ranges are -10 < x < 10 and -10 < y < 10. The scale for the x and y axes are 1.
To graph another equation, press Y=
CL to access and clear the Y1= prompt.
Then, enter the next equation, and press GRAPH to view the graph. If the line
does not appear in the default viewing window, press ZOOM and double touch
4 OUT to enlarge the viewing window. For future use, please set the zoom
factors to 5 by pressing ZOOM , touching B FACTOR, pressing ENTER
ENTER
5
ENTER and GRAPH to exit the zoom menu and return to the
viewing screen.
2
5
Linear Equations/ALGEBRA USING THE SHARP EL-9600
Method of Teaching
Use Blackline Masters 1.1 and 1.2 to create overheads for graphing linear
equations and investigating slope. Go over in detail how to graph linear
equations, set the default viewing window, zoom out, and how to set the
viewing window manually. Next, use the Blackline Masters 1.3 and 1.4 to create
worksheets for the students. Have the students graph the linear equations to
explore the characteristics of the y intercept, parallel, and perpendicular.
Use the topics For Discussion to supplement the worksheets.
Using Blackline Master 1.2
Graphing a linear equation, discussed above under Calculator Operations, is
presented on Blackline Master 1.1. The following study of slope appears on
Blackline Master 1.2. Graph the equation y = 1x by pressing Y=
CL to remove
an old Y1 expression. To enter the equation press X/θ/T/n . Press ZOOM ,
touch A ZOOM, and double touch 5 Default to view the graph.
Changing the slope to 2 will result in the equation y = 2x. Enter this equation for
Y2 by pressing Y=
▼
2
X/θ/T/n . To view both the graphs on the same
coordinate axes, press GRAPH . The equation Y1=X is displayed first, followed
by the equation Y2=2X. Notice how the "pitch" of the line has changed, with Y2
becoming more steep or climbing faster. Increasing the size of the slope (m > 1)
makes the line more steep.
Linear Equations/ALGEBRA USING THE SHARP EL-9600
3
equation for Y2 by pressing Y=
▼
CL
1
a/b
2
▼
Changing the slope to 1/2 will result in the equation y = 1/2 x. Enter this
X/θ/T/n . To view
both the graphs on the same coordinate axes, press GRAPH .
The equation Y1=X is displayed first, followed by the equation Y2= 1/2 X. Notice
how the "pitch" of the line has changed, with Y2 becoming less steep or climbing
slower. Decreasing the size of the slope (0 < m < 1) makes the line less steep.
Changing the slope to -1 will result in the equation y = -x. Enter this equation for
Y2 by pressing Y= , double touch the expression 1/2 x, press CL
(–)
X/θ/T/n . To view both the graphs on the same coordinate axes, press
GRAPH .
The equation Y1=X is displayed first, followed by the equation Y2= -X. Notice
how the "pitch" of the line has changed, with Y2 decreasing (going down from
left to right) instead of increasing (going up from left to right). Negative slopes
(m < 0) make the line decrease or go down from left to right.
4
Linear Equations/ALGEBRA USING THE SHARP EL-9600
Increasing or decreasing the descent of a line will follow the pattern set by the
discussion of the positive slopes. Slopes between -1 < m < 0 will lessen the
descent, whereas slopes m < -1 increase the descent.
For Discussion
You and your students can discuss how to recognize that lines are the same
(coincident) when expressed in different forms.
Linear Equations/ALGEBRA USING THE SHARP EL-9600
5
Chapter two
QUADRATIC EQUATIONS
Introducing the Topic
In this chapter, you and your students will learn how to graph quadratic
equations and study their characteristics. These characteristics include the
rigid translations (up/down, left/right, reflection), the vertex, and non-rigid
translations (closing/opening, stretching/shrinking, or pinching/spreading).
A quadratic equation of y in terms of x can be expressed by the standard form
y = a(x – h)2 + k, where a is the coefficient of the second degree term (y = ax2 +
bx + c) and (h, k) is the vertex of the parabola formed by the quadratic equation.
An equation where the largest exponent on the independent variable x is 2 is
considered a quadratic equation. In graphing quadratic equations on the
calculator, we will let the x-variable be represented by the horizontal axis and let
y be represented by the vertical axis.
Calculator Operations
Turn the calculator on and press Y= . Y prompts will appear in the viewing
window. Press CL to remove an old Y1 expression. Press ENTER
CL to
remove an old Y2 expression. Repeat to clear additional Y prompts. The
calculator should be setup with rectangular coordinates and in equation editor
mode. To complete this setup, press 2ndF
SET UP , touch E COORD, double
touch 1 Rect, touch G EDITOR, and double touch 1 Equation. Press CL to exit
the set-up menu and return to the Y prompts.
6
Quadratic Equations/ALGEBRA USING THE SHARP EL-9600
Quadratic equations given in a form other than the standard form can be
manipulated (completing the square) into standard form. This form is used for
graph translations and reflections. However, the calculator will accept all
quadratic equations of the form y = ___ (where y is defined explicitly in terms
of x). To enter the quadratic equation ( y = x2 ) for Y1, press Y=
2
2
EZ , touch 1
2
X , touch A Y=AX , and double touch 2 Y=X . Enter the viewing window range
on the screen, and touch 7 Dec. This establishes
the decimal viewing window for graphing equations. These ranges are -6.3 < x <
➧
by pressing ZOOM , touch
6.3 and -3.1 < y < 3.1. The scale for the x- and y-axes are 1.
When 2 is added to the x2, the resulting equation is y = x 2 + 2. Enter this function
using the SUB feature by pressing Y=
▼
EZ , touch 1 X2, double touch
B Y=AX2 + B, touch 1 Y=AX2 + B, press 2ndF
SUB
1
ENTER
2
ENTER
GRAPH . The display shows the graph of this equation and the basic y = x 2.
Notice that the addition of 2 moves the basic x 2 graph up two units on the y-axis.
This is equivalent to a k of 2 within the standard form y = a(x – h)2 + k. This
demonstrates the fact that adding a k > 0 will move the basic graph up k units
on the y-axis.
Quadratic Equations/ALGEBRA USING THE SHARP EL-9600
7
When -2 is added to the x2, the resulting equation is y = x2 – 2. To change Y2 for
this expression, press Y= ▼ 2ndF SUB ENTER (–) 2 ENTER
GRAPH . The display shows the graph this equation and the basic y = x 2.
Notice that the addition of -2 moves the basic x2 graph down two units on the
y-axis. This is equivalent to a k of -2 within the standard form y = a(x – h)2 + k.
This demonstrates the fact that adding a k < 0 will move the basic graph down
k units on the y-axis.
Method of Teaching
Use Blackline Masters 2.1 and 2.2 to create overheads for graphing quadratic
equations and investigating translations and reflections. Go over in detail how
to graph quadratic equations, use the rapid graph feature, the SUB feature, set
the decimal viewing window, and use the SHIFT/CHANGE feature. Next, use
Blackline Master 2.3 to create a work sheet for the students. Have the students
predict the translations and then graph the quadratic equations to verify or
correct their prediction. Show the students how to trace to find the vertex of a
quadratic equation's graph. Use the topics For Discussion to supplement the
worksheets.
Using Blackline Master 2.2
Vertical translations were discussed above under Calculator Operations and
they are presented on Blackline Master 2.1. The following study of horizontal
translations and reflections appears on Blackline Master 2.2. When -2 is added
to the x within the squaring operation (x2), the resulting equation is y = (x – 2)2.
To view this graph, press 2ndF
SHIFT/CHANGE , touch A SHIFT, double touch
2
1 Y=X , double touch at two on the x-axis. The graph, the equation, and the
basic y = x2 graph is shown.
8
Quadratic Equations/ALGEBRA USING THE SHARP EL-9600
Notice that the addition of -2 within the quadratic operation moves the basic x 2
graph right two units on the x-axis. This is equivalent to an h of 2 within the
standard form y = a(x – h)2 + k. This demonstrates the fact that placing an h > 0
in the standard form, will move the basic graph right h units on the x-axis.
When 2 is added to the x within the squaring operation (x 2), the resulting
equation is y = (x + 2)2. To view this graph, press 2ndF
SHIFT/CHANGE ,
2
touch A SHIFT, double touch 1 Y=X , double touch at negative two on the
x-axis. The graph, the equation, and the basic y = x2 graph is shown.
Notice that the addition of 2 within the quadratic operation moves the basic x 2
graph left two units on the x-axis. This is equivalent to an h of -2 within the
standard form y = a(x – h)2 + k. This demonstrates the fact that placing an h < 0
within the standard form will move the basic graph left h units on the x-axis.
When 2 is multiplied to the x 2, the resulting equation is y = 2x 2. To view this
graph, press 2ndF
press ▲
SHIFT/CHANGE , touch B CHANGE, double touch 1 Y=X2,
ENTER . The graph, the equation, and the basic y = x 2 graph is shown.
Quadratic Equations/ALGEBRA USING THE SHARP EL-9600
9
Notice that the multiplication of 2 pinches or closes the basic x 2 graph.
This is equivalent to an a of 2 within the standard form y = a(x – h)2 + k.
This demonstrates the fact that multiplying an a > 1 will pinch or close
the basic graph.
When -2 is multiplied to the x 2, the resultant equation is y = -2x 2. To view this
graph, press 2ndF
SHIFT/CHANGE , touch B CHANGE, double touch 1 Y=X2,
press ▼
ENTER . The graph, the equation, and the basic y = x 2
▼
▼
graph is shown.
Notice that the multiplication of -2 pinches or closes the basic x 2 graph and
flips it (reflects it) across the x-axis. This is equivalent to an a of -2 within the
standard form y = a(x – h)2 + k. This demonstrates the fact that multiplying
an a < -1 will pinch or close the basic graph and flip it (reflect it) across the
x-axis.
10
Quadratic Equations/ALGEBRA USING THE SHARP EL-9600
When 2 divides the x 2, the resulting equation is y = 1/2 x 2. Division by 2 spreads
or opens the basic x 2 graph. This is equivalent to an “a” of 1/2 within the
standard form y = a(x – h)2 + k. This demonstrates the fact that multiplying an
“a” between 0 and 1 will spread or open the basic graph. Also, multiplying an
“a” between -1 and 0 will spread or open the basic graph and flip it across the
x-axis.
For Discussion
You and your students can use the trace feature to locate the vertex of the
parabola and discover that the point is (h, k).
Quadratic Equations/ALGEBRA USING THE SHARP EL-9600
11
Chapter three
FORMULAS OR
LITERAL EQUATIONS
Introducing the Topic
In this chapter, you and your students will learn how to use the graphing
calculator's solver feature to evaluate formulas or literal equations. The
process begins by accessing the solver feature, then entering the equation,
assigning values to the known variables, and then solving for the unknown
variable. Three solving methods are available within the SOLVER mode.
These are the Equation method, Newton's method, and the Graphic method.
The default method is the Equation method, which is used when an exact
solution can be found by simple substitution.
Newton's method can be used for more complicated solutions. It implements
an iterative approach to find the solution once its given a starting point.
When a starting point is not readily available or multiple solutions are expected,
you can use the graphic method. The graphic method plots the left and right
sides of the equation and then locates the intersection(s).
One very useful feature of the EL-9600 graphing calculator is its ability to store
and recall equations.
12
Formulas or Literal Equations/ALGEBRA USING THE SHARP EL-9600
Calculator Operations
Turn the calculator on and press 2ndF
SOLVER to access the solver feature.
A blank screen should appear. If the screen is not blank, then press CL to clear
the screen. Select the Equation method for solving by pressing 2ndF
SOLVER ,
and touching A METHOD, and double touching 1 Equation.
Enter the formula P = L [ 1 – ( 1 + I/12)–N ] – 1
I/12
I
÷
1
2
ALPHA
I
a/b 1 2
)
b
) a (–) 1 .
=
ALPHA L ( a/b
a (–) ALPHA N
b
1
▼
▼
ALPHA
▼
+
P
▼
by pressing ALPHA
–
(
1
ALPHA
This equation is referred to as the amortization formula, where a loan ( L) with a
fixed rate of interest ( I ) is said to be amortized if both the loan and the interest
are paid by a sequence of equal monthly payments ( P). The number of equal
monthly payments to be made is designated by “N”.
Press ENTER to view the variable list. The variables are listed in order of
appearance (left to right) with payment ( P) first followed by loan amount ( L),
interest rate ( I ), and number of months ( N).
Formulas or Literal Equations/ALGEBRA USING THE SHARP EL-9600
13
To find the monthly payment on a $15,000 car loan, made at 9% interest over
four years (48 months), enter the values by pressing ▼
ENTER
•
0
9
ENTER
4
8
ENTER . Press ▲
payment (P) is now highlighted by the cursor, press 2ndF
1
5
▲
0
0
0
▲ and notice the
EXE to solve for
the payment.
Pressing CL will return you to the variable screen. You can now change or
solve for any of the values. Save this formula by pressing 2ndF
SOLVER ,
touching C SAVE, pressing ENTER , and entering the formula name. Give the
formula the name AMORT by pressing A (notice the calculator is locked in
alpha mode) M
O
R
T
ENTER .
Method of Teaching
Use Blackline Masters 3.1 and 3.2 to create overheads for solving formulas or literal equations. Go over in detail how to enter a formula, enter known quantities,
solve for an unknown quantity, and save a formula. Demonstrate how to change
to and use the Graphic method for solving. Next, use the Blackline Master 3.3
to create a worksheet for the students. Have the students use the worksheet
to enter a formula, select Newton's method, enter the known quantities, and
solve for the unknown with an initial guess. Use the topics For Discussion to
supplement the worksheets.
Using Blackline Master 3.2
Using the Equation method for solving a literal equation was discussed
above under Calculator Operations and is presented on Blackline Master 3.1.
The following discussion of the Graphic method appears on Blackline Master 3.2.
14
Formulas or Literal Equations/ALGEBRA USING THE SHARP EL-9600
While viewing the variable list within the solver feature, press CL
CL to return
to and clear the formula entry screen. Select the Graphic method for solving by
pressing 2ndF
SOLVER , touch A METHOD, and double touch 3 Graphic.
ALPHA R ab 2
▼
Enter the formula V = π r 2h by pressing ALPHA
ALPHA
V
ALPHA
=
2ndF π
H .
This equation is the formula for calculating the volume of a cylinder (V) in terms
of the cylinder's height ( H ) and radius ( R ). Press ENTER to view the variable
list. The variables are listed in order of appearance (left to right) with volume
( V ) first, followed by radius ( R ) and height ( H ). To find the radius of a cylinder with a volume of 30 cubic inches, and a height of 10 inches, enter the values
by pressing 3
0
ENTER ▼
1
0
ENTER . Press ▲ and notice the
radius ( R ) is now highlighted by the cursor.
Press 2ndF
EXE to solve for the radius. The graphic solver will prompt you
for a variable range to solve within. This range should be set to hopefully
include the radius you are solving for in the problem. A quick examination of
the formula shows that the radius squared will be no larger than 3 (dividing the
height that we know). Therefore, one could be too small (12 = 1 < 3) and two is
too large (22 = 4 > 3). Use the larger of the values to be safe. So, set the variable
range to 0 and 2 by pressing 0
ENTER
2
ENTER .
Formulas or Literal Equations/ALGEBRA USING THE SHARP EL-9600
15
Press 2ndF
EXE to solve. First, the solver feature will graph the left side of
the equation (volume, y = 30). Second, the solver feature will graph the right
side of the equation (y = 10 r 2), and then will compute the intersection of the
two graphs to find the solution.
Pressing CL will return you to the variable screen. You can now change or
solve for any of the values. Save this formula by pressing 2ndF
SOLVER ,
touching C SAVE, pressing ENTER , and entering the formula name. Give the
formula the name VCYL by pressing V
C
Y
L
ENTER
.
For Discussion
You and your students can discuss how to retrieve an equation or formula from
the calculator's memory and use it to solve a problem. Also, you can discuss
how to rename or delete formulas no longer desired.
Additional Problems
1. Payments on a $10,000 loan at 8% over 36 months.
2. Loan amount possible at 7% over 60 months with a $300 payment.
3. Radius of a cylinder with a volume of 200 in3 and a height of 15 in.
16
Formulas or Literal Equations/ALGEBRA USING THE SHARP EL-9600
Chapter four
GRAPHING POLYNOMIALS
AND FINDING THE ROOTS
Introducing the Topic
In this chapter, you and your students will learn how to graph polynomials and
find their roots through zooming in and tracing, and through the calculate feature.
Calculator Operations
Turn the calculator on and press Y= . Y prompts will appear on the viewing
window. Press CL to remove old Y expressions. The calculator should be
setup with rectangular coordinates, and in equation editor mode. To complete
this setup, press 2ndF
SET UP , touch E COORD, double touch 1 Rect, touch
G EDITOR, and double touch 1 Equation. Press CL to exit the menu and return
to the Y prompts.
To enter the polynomial y = x 3 – 3x 2 + x + 1, press EZ , double touch 5 X3, touch
A Y=AX3+BX2+CX+D, touch 1 Y=AX3+BX2+CX+D, press 2ndF SUB , and enter
the coefficients by pressing 1 ENTER (–) 3 ENTER 1 ENTER 1
➧
ENTER . Press ZOOM , touch
on the screen, and touch 7 Dec to establish
the decimal viewing window and view the graph.
Graphing Polynomials and Finding the Roots/ALGEBRA USING THE SHARP EL-9600
17
▼
Press TRACE to engage the trace feature. Press
to move the cursor near
the left-hand root. Make sure your zoom factors are set to 5 as suggested earlier
in the book by pressing ZOOM , touch B FACTOR, press ENTER
5
5
ENTER
ENTER . Press ZOOM , touch A ZOOM, double touch 3 IN to zoom in on
the left-hand root. Press TRACE and move the tracer to approximate the root.
Press ZOOM , touch H RCL, double touch 2 PreWin to return to the decimal
viewing window. Press TRACE and move the tracer to approximate the middle
root. The root is exactly x = 1, therefore no zooming is needed to find a better
approximate.
18
Graphing Polynomials and Finding the Roots/ALGEBRA USING THE SHARP EL-9600
▼
Press
repeatedly to move the tracer near the right-hand root.
Press ZOOM , touch A ZOOM, double touch 3 In to zoom in. Press TRACE
and move the tracer to approximate the root.
Method of Teaching
Use Blackline Masters 4.1 and 4.2 to create overheads for graphing polynomials
and finding their roots through zooming and the calculate feature. Go over
in detail how to enter polynomials (both manually and rapid graph), set the
viewing window, trace, set zoom factors, zoom in, and how to use the calculate
feature. Next, use the Blackline Master 4.3 to create a worksheet for the
students. Have the students graph the polynomials and approximate their
roots with zooming and the calculate features. Use the topics For Discussion to
supplement the worksheets.
Using Blackline Master 4.2
Finding roots by zooming was discussed above under Calculator Operations and
presented on Blackline Master 4.1. The following discussion of the jump feature
appears on Blackline Master 4.2. Press Y=
CL to return to and clear the Y1
4
▼
▼
prompt. To enter the polynomial y = x4 + x3 – 5x 2 – 3x + 1, press X/θ/T/n ab
+ X/θ/T/n ab 3
– 5 X/θ/T/n x2 – 3 X/θ/T/n + 1 .
View the graph in the decimal viewing window by pressing ZOOM , touching
➧
A ZOOM, touching
on the screen, and touching 7 Dec.
Graphing Polynomials and Finding the Roots/ALGEBRA USING THE SHARP EL-9600
19
Press TRACE and move the cursor near the right-hand most root.
Press 2ndF
CALC to view the calculate menu. Double touch 5 X_Incpt and
the calculator will find the left-hand root.
20
Press 2ndF
CALC , double touch 5 X_Incpt to find the next root.
Press 2ndF
CALC , double touch 5 X_Incpt to find the next root.
Graphing Polynomials and Finding the Roots/ALGEBRA USING THE SHARP EL-9600
Press 2ndF
CALC , double touch 5 X_Incpt to find the next root.
For Discussion
Discuss the process of zooming in more than once to achieve a better
approximate for the root. Emphasize that the zooming and calculation values
are normally approximates for the roots, since the roots can be irrational.
Also, demonstrate the polynomial solver within tool feature. This solver will
find the roots for second and third degree polynomials.
Additional Problems
Find the roots of the following polynomials.
1. y = x 3 + x 2 – 2x – 1 (zooming)
2. y = x 4 – 5x 2 + 3 (zooming)
3. y = x 5 – 2x 4 – 4x 3 + 3x 2 + 2x – 1 (calculate)
Graphing Polynomials and Finding the Roots/ALGEBRA USING THE SHARP EL-9600
21
Chapter five
SOLVING A SYSTEM
OF EQUATIONS
Introduction
In this chapter, you and your students will learn how to solve a system of
equations using the calculate and tool features of the EL-9600.
Calculator Operations
Turn the calculator on and press Y= . Y prompts will appear in the viewing
window. Press CL to clear an old Y1 expression. Press ENTER
CL to clear
additional Y prompts. The calculator should be setup with rectangular
coordinates, and in equation editor mode. To complete this setup, press 2ndF
SET UP , touch E COORD, double touch 1 Rect, touch G EDITOR, and double
touch 1 Equation. Press CL to exit the menu and return to the Y prompts.
To enter the system of equations:
y = x2 – 1
y = 2x
press X/θ/T/n
2
x
–
1
ENTER
X/θ/T/n . View the graphs using the
EZ , touch 5, touch
➧
rapid window by pressing WINDOW
2
on the screen,
double touch 7 –5<X<5, and double touch 4 –10<Y<10. This approach
establishes the viewing window of interest rapidly.
22
Solving a System of Equations/ALGEBRA USING THE SHARP EL-9600
Press TRACE and move the tracer right of the right intersection.
Press 2ndF
CALC to access the calculate feature. Double touch 2 Intsct.
The left-hand intersection will appear on the screen.
Press 2ndF
CALC to access the calculate feature again. Double touch 2 Intsct.
The right-hand intersection will appear on the screen.
Method of Teaching
Use Blackline Masters 5.1 and 5.2 to create overheads for graphing systems of
equations and finding their intersections through the calculate and tool features.
Go over in detail how to enter equations, set the viewing window using the rapid
window feature, use the calculate feature to find intersections, and how to use
Solving a System of Equations/ALGEBRA USING THE SHARP EL-9600
23
the tool feature. Next, use the Blackline Master 5.3 to create a worksheet for the
students. Have the students graph the system of equations and approximate the
solutions with the calculate feature. Then, have them use the tool feature to
solve a system of equations. Use the topics For Discussion to supplement the
worksheets.
Using Blackline Master 5.2
Finding solutions to a system of equations using the calculate feature was
discussed above under Calculator Operations and presented on Blackline Master
5.1. The following discussion of the tool feature appears on Blackline Master 5.2.
Press 2ndF
TOOL to access the tool menu.
Touch B SYSTEM and double touch 2 2 to view the entry screen for solving a linearsystem of equations. Systems up to six variables and six equations can be solved.
To enter the system of equations:
5x + y = 1
-3x + y = –5
Press 5 ENTER 1
(–) 5 ENTER .
24
ENTER
1
ENTER
(–)
3
ENTER
Solving a System of Equations/ALGEBRA USING THE SHARP EL-9600
1
ENTER
Press 2ndF
EXE to solve the system.
For Discussion
Discuss what to do if an intersection of the graphs does not appear on the
viewing screen. Discuss what is happening if the tool feature gives you an error
in its calculation (inconsistent systems/parallel lines have no solution and
dependent/coincident lines have many solutions). Emphasize that solutions are
normally considered approximations since some of the roots are irrational.
Discuss how you know that you have found all the intersections.
Additional Problems
Solve the following system of equations.
1. y = -x 2 + 2 (calculate feature)
y=x+1
2. -2x + y = -2 (tool feature)
1/2 x – y = 1
Solving a System of Equations/ALGEBRA USING THE SHARP EL-9600
25
Chapter six
MATRIX SOLUTIONS TO SYSTEMS
OF LINEAR EQUATIONS
Introducing the Topic
In this chapter, you and your students will learn how to solve a system of linear
equations. Particularly, you will focus on a system of three linear equations,
each consisting of three variables. Equations in three variables cannot be
graphed on the graphing calculator. You can only find the solution of a system
of equations in three variables numerically. A graphical solution is not possible.
Calculator Operations
Press MATRIX to access a menu for entering, editing, and operating on matrices.
26
Matrix Solutions To Systems of Linear Equations/ALGEBRA USING THE SHARP EL-9600
You can select from 10 different matrices. Touch B EDIT and double touch
1 mat A to select matrix A. Now, enter the size or dimension of the matrix.
We will enter the 3 × 3 matrix.
1 2 1
[ ]
2 1 –1
1 1 –2
Press 3
ENTER
3
ENTER to set the dimension of the matrix at three rows
by three columns. The calculator will now prompt you for the matrix.
Enter the elements of the first row by pressing 1
ENTER
2
ENTER
1
ENTER
The calculator will move the prompt to the second row. Enter the
elements of the second row by pressing 2 ENTER 1 ENTER (–) 1
ENTER . Enter the elements of the third row by pressing 1 ENTER 1
ENTER (–) 2 ENTER . The arrow keys can be used to move the prompt
around the matrix. Press 2nd
QUIT to exit the display of matrix A.
Repeat the process to enter a 3 × 3 matrix B =
[ ]
1 2 3
4 5 6
7 8 9 .
Matrix multiplication can be performed if the number of columns of the first
matrix is equal to the number of rows of the second matrix. In the matrix
multiplication A × B, the elements in the first row of A ( 1 2 1 ) are multiplied to
the corresponding elements in the first column of B ( 1 4 7 ). The sum of these
Matrix Solutions To Systems of Linear Equations/ALGEBRA USING THE SHARP EL-9600
27
multiplications (1 • 1 + 2 • 4 + 1 • 7) is placed in the 1,1 (first row, first column)
position of the resulting matrix. This process is repeated until each row of A has
been multiplied to each column of B.
To multiply the matrices A and B together, press MATRIX , touch A NAME,
double touch 1 mat A, press ×
MATRIX , double touch 2 mat B, and press
ENTER . The resulting matrix is shown below.
Method of Teaching
Use Blackline Masters 6.1 and 6.2 to create overheads for entering matrices,
multiplying matrices, finding inverse matrices, and solving systems of equations.
Go over in detail the derivation of the matrix solution (presented below), and
identity matrices. Next, use Blackline Master 6.3 to create a worksheet for the
students. Have the students solve systems of linear equations using matrices.
Use the topics For Discussion to supplement the worksheets.
Using Blackline Master 6.2
The identity matrix I is defined to be a square matrix (n × n) where each position
on the diagonal is 1 and all others are 0. For example, the 3 × 3 identity matrix is:
[ ]
1 0 0
=I
0 1 0
0 0 1
The calculator will directly establish an identity matrix of a given size by
pressing MATRIX , touching C OPE, double touching 05 identity, and pressing
3
ENTER . To save the identity matrix in matrix C, press STO
MATRIX ,
touch A NAME, touch 3 mat C, and press ENTER .
28
Matrix Solutions To Systems of Linear Equations/ALGEBRA USING THE SHARP EL-9600
Confirm that the identity matrix is stored in matrix C by pressing MATRIX ,
touching B EDIT, and double touching 3 mat C.
Press 2ndF
QUIT to exit the matrix editor and press CL to clear the screen.
An inverse matrix A–1 is a matrix that when multiplied to A results in the
identity matrix I (A–1 × A = I). The graphing calculator can find A–1 quickly.
Derivation of A–1 by hand is not discussed here. Find the inverse of the square
matrix A by pressing MATRIX , touching A NAME, double touching 1 mat A,
Press
▼
pressing 2ndF
x–1
ENTER . The resulting matrix is shown below.
to see more of the matrix.
Matrix Solutions To Systems of Linear Equations/ALGEBRA USING THE SHARP EL-9600
29
Some square matrices have no inverse and will generate error statements when
calculating the inverse.
To solve the system of equations
x + 2y + z = 8
using matrices:
2x + y – z = 1
x + y – 2z = -3
Use the matrix A entered previously as the coefficient matrix, and enter
the constants on the right side of the equal sign into matrix B (3 × 1), where
B=
[ ]
8
1
-3
Press 2ndF
QUIT to exit the display of the B matrix. The solution matrix X is
found by multiplying A –1 B. This multiplication is derived from the equation:
AX= B
A–1 A X = A–1 B (multiply both sides by A–1 )
I X = A–1 B (A–1A = I, identity matrix)
X = A–1 B (I X = X )
Note, the multiplication is "order sensitive" and you will not get the correct
answer by multiplying B A–1.
Multiply A–1 B by pressing MATRIX , touching A NAME, double touching
1 mat A, pressing 2ndF x–1 × MATRIX , double touching 2 mat B, and
pressing ENTER . The solution matrix will appear.
30
Matrix Solutions To Systems of Linear Equations/ALGEBRA USING THE SHARP EL-9600
The 1 is the X coordinate, 2 the Y coordinate, and 3 the Z coordinate of
the solution point.
For Discussion
You and your students can discuss why the graphing calculator cannot graph
equations of three variables, the derivation of the X = A–1B formula for solving a
system of linear equations, the size of the matrix generated by multiplication of
two matrices, and multiplication by the identity matrix. Introduce the students
to the TOOL feature and its system solver. Blackline Master 6.3 shows the
method for solving systems of equations using the TOOL feature.
Additional Problems
Solve the following systems of equations:
1. x + 2y + 3z = 4
2x – y – 2z = 0
x – 3y – 3z = -2
2. x – y + z = 3
2x – y + z = 4
x + 2y – z = -1
Matrix Solutions To Systems of Linear Equations/ALGEBRA USING THE SHARP EL-9600
31
Chapter seven
INEQUALITIES AND SYSTEMS
OF INEQUALITIES
Introducing the Topic
In this chapter you and your students will explore graphical techniques to solve
inequalities and systems of inequalities. Graphical solution methods not only
offer instructive visualization of the solution process, but they can be applied to
inequalities that are often very difficult to solve algebraically.
Calculator Operations
To solve an inequality means to find all values that make the inequality true.
Two methods of finding these values for one-variable inequalities are illustrated
in this chapter.
The first method involves rewriting the inequality so that the right-hand side of
the inequality is 0 and the left-hand side is a function of x. Then, find the
solution to f(x) < 0 by determining where the graph of f(x) is below the x-axis, or
find the solution to f(x) > 0 by determining where the graph of f(x) is above the
x-axis, or find the solution to f(x) ≤ 0 by determining where the graph of f(x) is
below or on the x-axis, or find the solution to f(x) ≥ 0 by determining where the
graph of f(x) is above or on the x-axis.
32
Inequalities and Systems of Inequalities /ALGEBRA USING THE SHARP EL-9600
For example, to solve 3(4 – 2x) ≥ 5 – x, rewrite it as 3(4 – 2x) – 5 + x ≥ 0 and
determine the values of x where the function y = 3(4 – 2x) – 5 + x is on or above
the x-axis. To do this, press Y=
CL and enter 3(4 – 2X) – 5 + X in the Y1
location. Set the viewing window of the graph by pressing ZOOM , touch
A ZOOM, and double touch 5 Default. You should be able to clearly view the
x-intercept. Locate the x-intercept at the point (1.4, 0) with the keystrokes
TRACE
2ndF
CALC , and double touch 5 X_Incpt.
Since the graph is above the x-axis to the left of the x-intercept, the solution to
the inequality 3(4 – 2x) – 5 + x ≥ 0 is all values of x such that x ≤ 1.4.
The second method involves graphing each side of the inequality as an
individual function. Then, find the solution to f(x) < g(x) by determining where
the graph of f(x) is below the graph of g(x), or find the solution to f(x) > g(x) by
determining where the graph of f(x) is above the graph of g(x), or find the
solution to f(x) ≤ g(x) by determining where the graph of f(x) is below or
intersects the graph of g(x), or find the solution to f(x) ≥ g(x) by determining
where the graph of f(x) is above or intersects the graph of g(x).
For example, to solve the inequality 3(4 – 2x) ≥ 5 – x, press Y=
CL , enter
3(4 – 2X) for Y1 and 5 – X for Y2. Set the viewing window by pressing ZOOM ,
touching A ZOOM, and double touching 5 Default. Next, shade the set of points
that make the inequality true by pressing 2ndF
DRAW , touching G SHADE,
and double touching 1 Set to access the Set Shade screen. Since the inequality you
is "on the bottom." Do this by touching Y2 on the screen, pressing
▼
are solving is Y1 ≥ Y2 the solution is where the graph of Y1 is “on the top” and Y2
, and
touching Y1 on the screen. Press GRAPH to view the shaded region. Press
TRACE 2ndF CALC , and double touch 2 Intsct to find where the graphs intersect.
Inequalities and Systems of Inequalities/ALGEBRA USING THE SHARP EL-9600
33
Since the shaded region is to the left of x = 1.4, the solution to the inequality
3(4 – 2x) ≥ 5 – x is all values of x such that x ≤ 1.4.
Double Inequalities
The solution to a system of two inequalities in one variable consists of all
values of the variable that make each inequality in the system true.
A system such as
2x – 5 ≥ -1
{ 2x – 5 ≤ 7
, where the same expression appears on both
inequalities, is commonly referred to as a “double” inequality and is often written in
the form –1 ≤ 2x – 5 ≤ 7. Be certain that students notice the order of the inequality
signs and that the double inequality is only used to indicate an expression in x
“trapped” in between two values. Unless both inequality signs are pointing in the
same direction, the double inequality expression is meaningless and should not be
used. Also, by the transitive property of inequalities, a must be less than or equal
to b in the inequality a ≤ f(x) ≤ b or b ≥ f(x) ≥ a.
Turn off the shading by pressing 2ndF
DRAW , touching G SHADE, and
double touching 2 INITIAL. To clear any previously entered functions press Y=
CL . Enter Y1 = –1, Y2 = 2X – 5, and Y3 = 7. Press ZOOM , touch A ZOOM, and
double touch 5 Default to view the line y = 2x – 5 between the lines y = -1 and y = 7.
34
Inequalities and Systems of Inequalities/ALGEBRA USING THE SHARP EL-9600
Press TRACE
2ndF CALC , double touch 2 Intsct to find the point of
intersection of the lines y = 2x – 5 and y = –1 at (2, -1). Press ▲ to move the tracer
to the y = 7 line. Press 2ndF CALC and double touch 2 Intsct to find y = 2x – 5
and y = 7 at (6, 7). The solution to the “double” inequality -1 ≤ 2x – 5 ≤ 7 consists
of all values of x in between, and including, 2 and 6 (i.e., x ≥ 2 and x ≤ 6).
The solution is 2 ≤ x ≤ 6.
Systems of Two-Variable Inequalities
The solution region of a system of two-variable inequalities consists of all points
(a, b) such that when x = a and y = b, all inequalities in the system are true.
For example, to graph the solution region for the system
{ 2xx ++ yy ≤≥ 11
2
, first
rewrite each inequality in the system so that the left-hand-side is y:
{ yy ≥≤ 11 –– x2x
2
Press Y=
CL and enter 1 – 2X in Y1 and 1 – X2 in Y2. Next, shade the points
that have a y-value less than Y2 and greater than Y1. Do this by pressing 2ndF
press
▼
DRAW , touching G SHADE, double touching 1 Set, touch Y1 on the screen,
, and touch Y2 on the screen. Graph the system by pressing ZOOM ,
➧
touching A ZOOM, touching
on the screen, and touching 7 Dec.
Method of Teaching
Use Blackline Master 7.1 to create an overhead or handout for using the first
method to graphically solve an inequality. Remind students of the proper
algebraic procedure to write the inequality in the form f(x) < 0. Next, use
Blackline Master 7.2 to create an overhead or handout for using the second
Inequalities and Systems of Inequalities/ALGEBRA USING THE SHARP EL-9600
35
method to graphically solve an inequality. Demonstrate the process of entering
the left- and right-hand sides of the inequality in the Y1 and Y2 locations and the
process for shading the solution.
When using Blackline Master 7.3, discuss the meaning of a “double” inequality
and its definition in terms of a system of two single-variable inequalities. Use
Blackline Master 7.4 to create an overhead or handout for solving a system of
two-variable inequalities. Emphasize that while the solution to a single-variable
inequality is one dimensional, the solution to a system of inequalities in two
variables is a region of points.
Use Blackline Master 7.5 to create a worksheet for the students. Have the
students choose either method to solve the inequalities in Problem 3.
Use the topics For Discussion to supplement the worksheets.
Using Blackline Master 7.5
The first method for solving an inequality is presented on Blackline Master 7.1.
The second method for solving an inequality is presented on Blackline Master
7.2. The method for solving a double inequality is presented on Blackline Master
7.3. The method for solving a system of two-variable inequalities is presented
on Blackline Master 7.4. Use Blackline Master 7.5 to create a worksheet for the
students to practice what they have learned with Blackline Masters 7.1-4.
For Discussion
Discuss the use of the “free-moving” cursor to graphically determine if a specific
point is in the solution region of a system of two-variable inequalities. After
viewing the shaded solution region for the system of inequalities, determine if a
point (x, y) is a solution by pressing the arrow keys and moving the cursor to
the approximate location of the point and see if it is within the shaded region.
Viewing the region using the decimal window or multiples of those settings
allows for exact free-moving cursor location of most points. Students should
also algebraically confirm that the point (x, y) satisfies both inequalities by
substituting those values in the original inequalities.
36
Inequalities and Systems of Inequalities/ALGEBRA USING THE SHARP EL-9600
Additional Problems
1. Solve 6(x – 2) – 2(x – 5) < 3x using the first method.
2. Solve (4x + 3) – 2 ≥ (x + 3) using the second method.
8
4
3. For which values of x are the following inequalities true?
a.
x3 > 6 + 7x
b.
2(x – 1)2 – 3x > -3(1+x)
c.
10 + x > 4 – x
5
d.
3 < 5 – 2x < 6
4. Solve the system
{
(5x) + 2 ≤ 1
8
(5x) + 2 ≥ 1
8
2
Express your answer as a double inequality.
5. a. Graph the solution region for the system of inequalities
{
2x + 5y > 6
3x + 2.5y < 1.5
b. Are the points (.6, .5) and/or (3.5, 0) solutions to this system?
6. a. Graph the solution region for the system of inequalities
{
x2 + y < 4
y > (x – 1)(x + 2)
b. Are the points (-2, 0) and/or (-1, 2.5) solutions to this system?
Inequalities and Systems of Inequalities/ALGEBRA USING THE SHARP EL-9600
37
Chapter eight
ABSOLUTE VALUE FUNCTIONS,
EQUATIONS, AND INEQUALITIES
Introducing the Topic
This chapter provides a brief introduction to the absolute value function and
discusses methods to graphically solve equations and inequalities involving the
absolute value function.
Calculator Operations
The absolute value of a real number x is defined by the following:
|x| =
x if x ≥ 0
{ –x if x ≤ 0.
The symbol for absolute value is found in the MATH menu. When using the
EQUATION editor, the absolute value appears just as we write it on paper, | |.
Press Y=
CL to clear the Y1 prompt. Press ENTER
CL to clear the
remaining prompts. Enter the function f(x) = |x| in Y1 by pressing MATH ,
touching B NUM, double touching 1 abs( , and pressing X/θ/T/n . View the
➧
graph by pressing ZOOM , touching A ZOOM, touching
on the screen, and
touching 7 Dec.
38
Absolute Value Functions, Equations, and Inequalities/ALGEBRA USING THE SHARP EL-9600
Notice, by the definition of this function with reinforcement from the graphical
view, that the domain of f(x) = |x| is the set of all real numbers and the range is
the set of non-negative real numbers.
Absolute Value Equations
If n is a positive number, there are two solutions to the equation |f(x)| = n because
there are exactly two numbers with the absolute value equal to n: n and -n. The
existence of two distinct solutions is clear when the equation is solved graphically.
For example, to solve |5 – 4x| = 6, press Y=
CL to clear the Y1 prompt. Enter
|5 – 4x| in Y1 with the keystrokes MATH , touch B NUM, double touch 1 abs( , and
press 5 – 4 X/θ/T/n . Next, press ENTER CL 6 to enter 6 in Y2. Press
ZOOM , touch A ZOOM, double touch 5 Default to view the two points of
intersection of the absolute value graph and the horizontal line y = 6.
Press TRACE
2ndF
CALC and double touch 2 Intsct to find one point of
intersection of the two graphs. Press 2ndF
CALC and double touch 2 Intsct
again to find the other point of intersection.
Absolute Value Functions, Equations, and Inequalities/ALGEBRA USING THE SHARP EL-9600
39
Therefore, the solution to the equation |5 – 4x| = 6 consists of the two values
-0.25 and 2.75. Note that although it is not as intuitively obvious, the solution
could also be obtained by finding the x-intercepts of the function y = |5x – 4| – 6.
Absolute Value Inequalities
Absolute value inequalities are of the form |f(x)| < k, |f(x)| ≤ k, |f(x)| > k, or
|f(x)| ≥ k. The graphical solution to an absolute value inequality is found using
either method discussed in Chapter 7.
(6x)
To solve 20 –
< 8 using the first method, rewrite the inequality so that the
5
(6x)
right-hand side of the inequality is zero: 20 –
– 8 < 0.
5
Choose a viewing window by pressing WINDOW
▼
▼
▼
Press Y= and clear all the Y prompts. Enter the left-hand side of the inequality
20 – (6x) – 8 in Y1 by pressing MATH , touching B NUM, double touching
5
– 8 .
1 abs( , pressing 2 0 – a/b 6 X/θ/T/n
5
EZ , double touching 6 on
the screen, double touching 4 –5<X<50, and double touching 4 –10<Y<10.
The x-intercepts of the graph are clearly visible.
40
Absolute Value Functions, Equations, and Inequalities/ALGEBRA USING THE SHARP EL-9600
Press TRACE and move right of the right intercept. Press 2ndF
double touch 5 X_Incpt to find the first x-intercept. Press 2ndF
CALC and
CALC and
touch 5 X_Incpt to find the second x-intercept.
Since the graph is below the x-axis for x in between the two x-intercepts, the
solution is 10 < x < 23.333.
To use the second method to solve the inequality |3.5x + 4| > 10, first press
Y=
CL to clear the Y1 prompt. Press ENTER
CL to clear additional
prompts. Enter the function |3.5x + 4| in Y1 by pressing MATH , touching
B NUM, double touching 1 abs( , and pressing 3
ENTER . Enter 10 in Y2 by pressing 1
•
5
X/θ/T/n
+
4
0 .
Access the Set Shade screen by pressing 2ndF
DRAW , touching G SHADE,
on the screen, and since Y1 is the function "on the top," press
▼
and double touching 1 SET. Since Y2 is the function "on the bottom," touch Y2
and touch Y1
on the screen.
Choose a viewing window by pressing WINDOW
EZ , double touching 2 on
the screen, double touching 5 –10<X<10, and double touching 3 –5<Y<50.
Absolute Value Functions, Equations, and Inequalities/ALGEBRA USING THE SHARP EL-9600
41
Press TRACE
2ndF
CALC , double touch 2 Intsct to locate a point of
intersection. Repeat to find the other. The intersections are (-4, 10) and
(1.714, 10). The solution of the inequality |3.5x + 4| ≥ 10 is all values of x
such that x ≤ –4 or x ≥ 1.714.
Method of Teaching
Use Blackline Masters 8.1 and 8.2 to create overheads or handouts for
investigating the absolute value function. Next, use Blackline Masters 8.3, 8.4
and 8.5 to create overheads or handouts for graphically solving absolute value
equations and inequalities.
Blackline Master 8.6 should be used to create a worksheet for students.
Students should choose either method for solving the absolute value
inequalities. Pay careful attention to the written form of the answer.
Use the topics For Discussion to supplement the worksheets.
Using Blackline Master 8.1
Use Blackline Master 8.1 to demonstrate how to evaluate an absolute value on
the home or computation screen. The students will then graph y = |x|, followed
by graphing y = |x – 1| and y = |x|– 1. Discuss with them the effect on the
graph of y = |x| when 1 is subtracted from the x-term (y = |x – 1|) and when 1
is subtracted from the function (y = |x|– 1).
Blackline Masters 8.3, 8.4 and 8.5 are discussed above under
Calculator Operation.
42
Absolute Value Functions, Equations, and Inequalities/ALGEBRA USING THE SHARP EL-9600
For Discussion
You and your students can use the calculator, with its SHIFT/CHANGE and SUB
features, to explore for specific values of A, B, and C in the function
y = A|x + B| + C:
1. the effect of the value of A on the graph of the absolute value function,
2. the effect of the value of B on the graph of the absolute value function, and
3. the effect of the value of C on the graph of the absolute value function.
Include positive, negative, integer, and fractional values of A, B, and C in
your discussion, and have the students look for patterns in the graphs as
one constant is changed and the others remain fixed.
Additional Problems
1. Is |-2+7| = |-2| + |7|? Evaluate each side of the equation to check your answer.
Is |x+y| = |x| + |y| for all real numbers x and y? If not, when will
|x+y| = |x| + |y|?
2. Is
6–9
6–9
=
Evaluate each side of the equation to check your answer.
1+3
1+3
Investigate with more examples, and decide if you think
for all real numbers x and all real numbers y ≠ .0
x x
=
y y
(2x)
3. Solve 3 –
= 4.
5
4. Find the values of x for which |4x+5| < 3.
5. Solve
(2–5x)
≥ 2.
3
Absolute Value Functions, Equations, and Inequalities/ALGEBRA USING THE SHARP EL-9600
43
Chapter nine
RATIONAL FUNCTIONS
Introducing the Topic
This chapter provides a brief introduction to rational functions and
discusses methods to graphically solve equations and inequalities
involving rational functions.
Calculator Operations
p(x)
A rational function f(x) is defined as the quotient q(x) where p(x) and q(x) are
two polynomial functions such that q(x) ≠ 0. The domain of any rational
function consists of all values of x such that the denominator q(x), is not zero.
A rational function consists of branches separated by vertical asymptotes, and
the values of x that make the denominator q(x) = 0 but do not make the
numerator p(x) = 0 are where the vertical asymptotes occur.
▼
(x–1)
Consider the rational function f(x) = (x 2–1). Press Y= CL to clear the Y1
(x–1)
prompt. Press ENTER CL to clear additional Y prompts. Enter Y1 =
(x 2–1)
2
by pressing a/b X/θ/T/n – 1
X/θ/T/n x – 1 . Graph in the
screen, and double touching 7 Dec.
44
Rational Functions/ALGEBRA USING THE SHARP EL-9600
➧
decimal window by pressing ZOOM , touching A ZOOM, touching
on the
(x–1)
Notice that since f(x) can be written as (x+1) (x–1), the domain consists of all
real numbers x such that x ≠ 1 and x ≠ -1. The function consists of two branches
separated by the vertical asymptote (the line x = -1). However, there is no
vertical asymptote where x = 1 since this value of x also makes the numerator
move the tracer to the coordinates X= 0.9, Y= 0.5263. Press
▼
TRACE to place the tracer in the center of the screen. Press
▼
zero. Look closely at the graph and observe the “hole” in it at x = 1. Press
repeatedly to
to see that the
this rational function. Press
▼
calculator does not display a value for Y at X = 1 since 1 is not in the domain of
once more and observe that the displayed
coordinates are X= 1.1, Y= 0.48. Note that the hole will be seen only when the
window is a "friendly" or decimal window. This means you can trace to the exact
x–value. (For instance, if you graphed this function in a window of -5 ≤ x ≤ 5,
x = 1 is not one of the possible trace values for x and the hole will not be seen.)
The x-intercepts of a rational function f(x), if there are any, occur at the x-values
that make the numerator p(x), but not the denominator q(x), zero. The
y-intercept occurs at f(0). Notice that there are no x-intercepts for the graph of
(x–1)
f(x) = 2 . The y-intercept can easily be found at x = 0, y = 1 by pressing 2ndF
(x –1)
CALC and double touching 6 Y_Incpt.
Rational functions also have horizontal asymptotes, lines of the form y = k (k a
constant) such that the function gets arbitrarily close to, but does not cross, the
horizontal asymptote when |x| is large. (Some rational functions composed of
higher-degree polynomials might cross horizontal asymptotes for small values of
x.). Rational functions may also have an oblique asymptote, a line that is neither
horizontal nor vertical, but which the function gets closer and closer to, but
does not touch, as |x| gets very large. Algebraic methods for finding the
equations of horizontal and oblique asymptotes are covered in most algebra
texts and are not given in this chapter.
Rational Functions/ALGEBRA USING THE SHARP EL-9600
45
(x–1)
Students can observe from the graph of f(x) =
(x2–1)
y = 0 is very likely a horizontal asymptote of f(x).
that the line
Rational Function Inequalities
Inequalities involving rational functions are traditionally solved algebraically
using sign charts. However, the graphical process of solving rational function
inequalities is the same as solving other inequalities. Simply use either method
discussed in Chapters 7 and 8.
x
For instance, to solve the inequality 1 – x2 ≤ 2 using the second method
(intersection method with shading), press Y= CL to clear the Y1 prompt.
B NUM, double touching 1 abs( , pressing a/b
x
2
ENTER . Enter 2 in Y2 by pressing 2
Press 2ndF
X/θ/T/n
▼
Enter the left-hand side of the inequality in Y1 by pressing MATH , touching
1
–
X/θ/T/n
ENTER .
DRAW touch G SHADE, double touch 1 SET to access the Set Shade
the screen first, press
▼
screen. Since Y1 is the value “on the bottom” (the smaller of the two), touch Y1 on
, and since Y2 is the function “on the top” (the larger of
the two), touch Y2 on the screen second. View the graph by pressing ZOOM ,
➧
touching A ZOOM, touching
on the screen, and double touching 7 Dec.
Press 2ndF
CALC and double touch 2 Intsct to locate the x-value of the
leftmost point of intersection, x = -1.281. Press 2ndF CALC and touch
2 Intsct, three more times to locate the x-values of the other three points of
intersection of Y1 and Y2: x = -0.781, 0.781, and 1.281. The solution is all values
of x such that x ≤ -1.281 or -0.781 ≤ x ≤ 0.781 or x ≥ 1.281. Turn off the shading
by pressing 2ndF
46
DRAW , touching G SHADE, and double touching 2 INITIAL.
Rational Functions/ALGEBRA USING THE SHARP EL-9600
Method of Teaching
Use Blackline Masters 9.1 and 9.2 to create overheads or handouts for investigating
rational functions and the solution to rational inequalities. Explain horizontal and
vertical asymptotes and how to find the equation of a vertical asymptote.
The number of branches students should expect to see on the graph of a rational
function is one more than the number of vertical asymptotes. Students should
therefore always determine the number of vertical asymptotes to know when they
are viewing a complete graph of a rational function.
Next, use Blackline Master 9.3 to create overheads or handouts for graphing
rational functions and graphically solving inequalities involving rational
functions. Use the topics For Discussion to supplement the worksheets.
Using Blackline Master 9.3
Graphing rational functions and inequalities was discussed under Calculator
Operations and presented on Blackline Masters 9.1 and 9.2.
In Activity 1, the instructions for entering the left- and right-hand sides of the
rational inequality are given. After entering the rational expression in Y1 and 2
screen first, press
▼
in Y2, have the students shade. Since Y2 is "on the bottom," touch Y2 on the
, and since Y1 is the function "on the top," touch Y1 on the
screen second. Before the graph is drawn, have the students discuss what they
expect to see: an x-intercept at x = 0 and two vertical asymptotes at x = -2 and
x = 2 giving three branches for the rational function. Graph in the decimal
window, changing y-max to 6.2 to provide a good view of the graph.
Rational Functions/ALGEBRA USING THE SHARP EL-9600
47
In Activity 2, have students discuss in groups what they should expect to see for
(x–3)
the graph of the rational function y = 2
before drawing the graph. If
(x –5x +6)
necessary, remind students that when a common factor appears in both the
numerator and denominator of a rational function, that factor results in a hole in
the graph, not a vertical asymptote.
(x–3)
(x–3)
Since (x2–5x +6) = (x–2) (x–3) , the graph should have two branches since there
is only one vertical asymptote, the line x = 2. There is a hole in the graph at x = 3.
Also have students verify by pressing TRACE
2ndF
CALC and double
touching 5 X_Incpt, that there are no x-intercepts for this rational function.
Have them view the graph in the decimal viewing window and then zoom out
horizontally by setting Xmin to -25 and Xmax to 25 with Xscl= 5 to estimate from
the graphs that the line y = 0 is very likely a horizontal asymptote.
For Discussion
Use the rapid window and trace features to explore the graph of a rational
function near a vertical asymptote. For instance, after completing Blackline
Master 9.1, have the students redraw the graph by pressing WINDOW
➧
touching 5, touching
on the screen, double touching 7 –5<X<5, and
double touching 4 –10<Y<10.
48
Rational Functions/ALGEBRA USING THE SHARP EL-9600
EZ ,
▼
Next, have them press TRACE and press
repeatedly to move the tracer along
the graph. Observe the y-values as the graph begins to scroll vertically, going up to
the point (-0.95, 21) and then jumping down to the other branch at (-1.03, -31.5).
Additional Problems
1. a.
Graph the rational function f(x) =(x–3) .
x
b.
What is the domain of f(x)?
c.
Give the equation(s) of the vertical asymptote(s) of f(x).
d.
Find the x- and y-intercepts of f(x).
e.
Estimate, using your graph, the horizontal asymptote of f(x).
2. a.
Graph the rational function g(x) =
x(x+1) .
(x –1)(x+2)
2
b.
What is the domain of g(x)?
c.
Give the equation(s) of the vertical asymptote(s) of g(x), and draw each
vertical asymptote with a dashed line on the graph above.
d.
Find the x- and y-intercepts of g(x).
e.
Estimate, using your graph, the horizontal asymptote of g(x). Draw on
the graph above, using a dashed line, the horizontal asymptote.
3. Solve the inequality
1
≤ 1.6.
(x+1.8)2
4. Determine the value of x for which
(x2+x–2)
> 0.
2(x–1)
Rational Functions/ALGEBRA USING THE SHARP EL-9600
49
Chapter ten
CONIC SECTIONS
Introducing the Topic
When a right circular cone is cut by a plane, the intersection is a curve called
a conic section. This chapter illustrates how to graph conic sections called
parabolas, circles, ellipses, and hyperbolas. The parametric mode of the
calculator is discussed in the procedure for graphing parabolas.
Calculator Operations
Parabolas
Quadratic functions of the form y = ax2 + bx + c and their graphs were discussed
in Chapter 2. The graphs of quadratic equations are called parabolas.
Sometimes the quadratic equations take on the form of x = ay2 + by + c.
Consider, for example, drawing the graph of x = y2 – 2. There is a problem
entering this equation in the calculator graphing list for two reasons: a) it is not
a function, and only functions can be entered in the Y= list locations, and b) the
functions entered in the Y= list must be in terms of x, not y. There are, however,
two methods you can use to draw the graph of a parabola whose equation does
not represent a function:
50
Conic Sections/ALGEBRA USING THE SHARP EL-9600
Method 1: Consider the “top” and “bottom” halves of the parabola as two
different parts of the graph because each individually is a function. Solve the
equation of the parabola for y and enter the two parts (that individually are
functions) in two locations of the Y= list.
Method 2: Choose the parametric graphing mode of the calculator and enter the
parametric equations of the parabola.
Let’s use both methods to graph the parabola x = y2 – 2. To use Method 1, first
solve the equation for y:
x = y2 – 2
x + 2 = y2
y = ± √(x+2)
The graph of the equation y = √ (x+2) is the “top half” of the parabola and the
graph of the equation y = – √(x+2) gives the “bottom half.”
Press Y=
CL to clear the Y1 prompt. Press ENTER
CL to clear additional
prompts. Enter √(x+2) in Y1 with the keystrokes 2ndF √
X/θ/T/n + 2 .
Press ENTER and enter -Y1= – √ (x+2) in Y2with the keystrokes (–)
VARS ,
touch A EQVARS, press ENTER , and double touch 1 Y1. View the graph by
➧
pressing ZOOM , touching A ZOOM, touching
on the screen, and double
touching 7 Dec.
When using Method 2 to draw the graph, it is not necessary to algebraically
solve the equation for y. Change to parametric mode by pressing 2ndF
SET UP , touch E COORD, and double touch 2 Param. Press 2ndF
to exit the set–up screen. Press Y=
QUIT
CL to clear the X1T prompt.
Conic Sections/ALGEBRA USING THE SHARP EL-9600
51
Parametric representations are equation pairs (x = F(t), y = F(t)) that have X and
Y each expressed in terms of a third parameter, T. To rewrite x = y2 – 2 in
parametric form, simply let y = T and substitute in x = y2 –2 to obtain x = T2– 2.
Enter X1T = T2– 2 in the calculator by pressing X/θ/T/n
x2
–
2
ENTER .
(Notice that in parametric mode, pressing X/θ/T/n produces T, not X.)
Enter Y1T = T by pressing X/θ/T/n . View the graph by pressing ZOOM ,
➧
touch A ZOOM, touch
on the screen, and double touch 7 Dec.
press TRACE and
▼
Notice that only half of the parabola is drawn. To understand why this occurs,
to move the tracer right. The tracer starts at T=0 and
increases. Substitute T=0 in X1T and Y1T to obtain X1T= -2 and Y1T=0. This is
the first point graphed by the calculator.
To see the rest of the parabola, press WINDOW and adjust the Tmin. Set Tmin
to -6 by pressing (–) 6 ENTER . The Tstp determines how much T changes
before the next point is plotted on the graph. Even though the dimensions of
the viewing screen are still controlled by Xmin, Xmax, Ymin, and Ymax, the T
settings determine the number of points that are plotted. Press GRAPH to view
the complete parabola.
52
Conic Sections/ALGEBRA USING THE SHARP EL-9600
Circles
The standard form of a circle of radius r, centered at the origin, is x 2 + y 2 = r 2.
To graph the circle x 2 + y 2 = 4 using Method 1, first return to the rectangular
graphing mode by pressing 2ndF
SET UP , touching E COORD, double
touching 1 Rect, and pressing 2ndF
QUIT . Press Y=
CL to clear the Y1
4 –
prompt. Enter Y1= √ (4 – X ) (the top half) by pressing 2ndF √
2
ENTER and then enter Y2 = – √(4 – X2) (the bottom half) by
(–) VARS , touch A EQVARS, press ENTER , touch A XY, and
x2
X/θ/T/n
pressing CL
double touch 1 Y1. View the graph by pressing ZOOM , touching A ZOOM,
➧
touching
on the screen, and double touching 7 Dec.
The standard equation of a circle of radius r that is centered at a point (h, k)
is (x – h)2 + (y – k)2 = r 2. In order to put an equation in standard form so that
you can use entry method 1, it is necessary to solve the equation for y. You
therefore need to use the process of completing the square. Notice that it is
only necessary to complete the square on the y-term in order to enter the
equation in the Y= list. For instance, in order to draw the graph of the circle
x 2 – 2x + y 2 + 4y = 2, proceed as follows:
x 2 – 2x + y 2 + 4y = 2
Place all variable terms on the left and the
constant term on the right-hand side of the
equation.
2
2
x – 2x + y + 4y + 4 = 2 + 4
Complete the square on the y-term.
x 2 – 2x + (y+2)2 = 6
Express the terms in y as a perfect square.
2
2
(y+2) = 6 – x + 2x
Leave only the term involving y on the left
hand side.
y + 2 = ± √(6 – x + 2x)
Take the square root of both sides.
y = ± √ (6 – x 2 + 2x) –2
Solve for y.
2
Conic Sections/ALGEBRA USING THE SHARP EL-9600
53
Notice that if you enter Y1 = √(6 – x 2 + 2x) –2 and Y2 = -Y1, you will not get the
graph of a circle because the “±” does not go with the “-2". To avoid a lot of
keystrokes, enter Y1 = √(6 – x 2 + 2x), Y2 = Y1 – 2 and Y3 = -Y1 – 2. You now have
the top portion and the bottom portion of the circle in Y2 and Y3. Since these are
the only two functions you want to graph, “turn off” Y1 so that it will not graph
▼
by pressing ▲ to move the blinking cursor to the Y1 expression. Press
to
position the blinking cursor over the = sign, and press ENTER so that “=” is no
➧
longer darkened. Press ZOOM , touch
on the screen, and double touch 7 Dec.
This is not a very good graph. We must adjust the screen to see the bottom part
of the circle. Do this using the rapid zoom feature by pressing EZ , double
touching ▼ on the bottom of the screen twice, and double touching ▼ on the
top of the screen twice. Press GRAPH to view the graph in the new window.
The circle looks like a circle because it is displayed in the decimal window.
Adjusting this window equivalently will keep the window "friendly" and a
circle will appear as a circle. Also, zooming in or out from the decimal window
will also keep the window "friendly."(The small separation on the sides is
unavoidable in some views.)
54
Conic Sections/ALGEBRA USING THE SHARP EL-9600
Method 2, parametric graphing, is not discussed in this chapter for conics
other than parabolas. Parametric graphing of circles, ellipses, and hyperbolas
is possible, but the method requires a knowledge of trigonometry.
Ellipses
The standard equation for an ellipse whose center is at the point (h, k) with
2
2
major and minor axes of length a and b is (x – h) + (y – k) = 1.
a2
b2
Method 1 for graphing using the function mode is exactly the same as the circle:
you must solve for y, completing the square if necessary, and enter the “top”
half and the “bottom” half in two locations of the Y= list.
Hyperbolas
The standard equation for a hyperbola can take one of two forms:
(x – h)2 (y – k)2
–
= 1 with vertices at (h ± a, k) or
a2
b2
2
2
(x – k)
(x – h)
–
= 1 with vertices at (h, k ± b).
2
b
a2
Method 1 for graphing using the function mode is exactly the same as the circle
and the ellipse: you must solve for y and enter the “top” half and the “bottom”
half in two locations of the Y= list. Solving for y will involve completing the
square if the equation is not in the standard form.
Method of Teaching
Use Blackline Master 10.1 to create an overhead or handout for investigating
graphing of parabolas using function graphing and parametric graphing. Next, use
Blackline Master 10.2 to create an overhead or handout describing the procedure
to graph a circle. Blackline Master 10.3 should be used to create a worksheet for
students. Use the topics For Discussion to supplement the worksheet.
Using Blackline Master 10.3
For problem 3 on Blackline Master 10.3, students may prefer not to complete the
square and graph this parabola in parametric mode by letting Y1T=T and X1T =
4 – 2T+T2. For problem 4 on Blackline Master 10.3, students should complete the
square on the y-term to obtain y 2 – 6y + 9 = x 2 + 2x + 3 + 9 and then (y–3) 2 = x 2 +
2x + 12. Then graph the hyperbola using rectangular coordinates with
Conic Sections/ALGEBRA USING THE SHARP EL-9600
55
Y1= (x 2 +2x + 12), Y2 = Y1 + 3, and Y3 = –Y1 + 3. (They should turn Y1 off before
graphing.) Zoom out several times from a decimal window to view the graph.
For Discussion
Review the procedure for completing the square on the y-term so that the
equation representing a circle, ellipse, or hyperbola can be solved for y and
entered in the Y= list.
Additional Problems
1. Use rectangular mode to graph x – y 2 = 4.
2. Use parametric mode to graph y 2 + x + 1 = 0.
Classify each of the following as a parabola, circle, ellipse, or hyperbola and
construct the graph.
3. 3x 2 + y =9
4. 2x + (y + 1)2 = 6
5. x 2 + y 2 – 6y = 8
6.
56
y 2 (x + 1)2
–
=1
4
16
Conic Sections/ALGEBRA USING THE SHARP EL-9600
CONTENTS OF
REPRODUCIBLE BLACKLINE MASTERS
Use these reproducible Blackline Masters to create handouts, overhead
transparencies, and activity worksheets.
LINEAR EQUATIONS
BLACKLINE MASTERS 1.1 - 1.4
58 - 61
QUADRATIC EQUATIONS
BLACKLINE MASTERS 2.1 - 2.3
62 - 64
FORMULAS OR LITERAL EQUATIONS
BLACKLINE MASTERS 3.1 - 3.3
65 - 67
GRAPHING POLYNOMIALS AND FINDING THE ROOTS
BLACKLINE MASTERS 4.1 - 4.3
68 - 70
SOLVING A SYSTEM OF EQUATIONS
BLACKLINE MASTERS 5.1 - 5.3
71 - 73
MATRIX SOLUTIONS TO SYSTEMS OF LINEAR EQUATIONS
BLACKLINE MASTERS 6.1 - 6.3
74 - 76
INEQUALITIES, DOUBLE INEQUALITIES, AND
SYSTEMS OF INEQUALITIES
BLACKLINE MASTERS 7.1 - 7.5
77 - 81
ABSOLUTE VALUE FUNCTIONS, EQUATIONS,
AND INEQUALITIES
BLACKLINE MASTERS 8.1 - 8.6
82 - 87
RATIONAL FUNCTIONS AND INEQUALITIES
BLACKLINE MASTERS 9.1 - 9.3
88 - 90
CONIC SECTIONS
BLACKLINE MASTERS 10.1 - 10.3
KEYPAD FOR THE SHARP EL-9600
91 - 93
94
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
57
NAME _____________________________________________________ CLASS __________ DATE __________
1.1
LINEAR EQUATIONS
Steps for graphing a linear equation
1. Turn the calculator on and press Y= .
2. Press CL to remove an old Y1 expression.
3. Set the calculator to rectangular coordinates and in equation editor mode,
by pressing 2ndF
SET UP E (COORD) 1 (Rect) G (EDITOR) and 1
(Equation). Press CL to exit the menu and return to the Y prompts.
4. Enter the linear equation ( y = 3x + 1 ) for Y1 by pressing 3
+
X/θ/T/n
1 .
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
5. Enter the viewing window range by pressing ZOOM ,touching A ZOOM,
and double touching 5 Default. This establishes the default viewing ranges
for graphing equations. These ranges are -10 < x < 10 and -10 < y < 10.
The scale for the x and y axes are 1.
6. To graph another equation, press Y= CL to access and clear the Y1=
prompt.
7. Enter the next equation, and press GRAPH to view the graph.
8. If the line does not appear in the default viewing window, press
ZOOM and double touch 4 OUT to enlarge the viewing window.
9. For future use, please set the zoom factors to 5 by pressing ZOOM ,
touching B FACTOR, pressing ENTER
5
ENTER
5
ENTER and
GRAPH to exit the zoom menu and return to the viewing screen.
58
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
NAME _____________________________________________________ CLASS __________ DATE __________
1.2
LINEAR EQUATIONS
Characteristics of Slope
1. Graph the equation y = 1x by pressing Y= CL to remove an old Y1
expression, and press X/θ/T/n to enter the equation.
2. Press ZOOM , touch A ZOOM, and double touch 5 Default to view the
graph.
3. Changing the slope to 2 will result in the equation y = 2x.
Enter this equation for Y2 by pressing Y=
▼
2
X/θ/T/n .
4. To view both the graphs on the same coordinate axes, press GRAPH .
▼
1
5. Changing the slope to 1 will result in the equation y = 2 x. Enter this
2
equation for Y2 by pressing Y= ▼ CL 1 a/b 2
X/θ/T/n .
6. To view both the graphs on the same coordinate axes, press GRAPH .
How has the line been changed?
______________________________________________________________________
7. Changing the slope to -1 will result in the equation y = - x.
Enter this equation for Y2 by pressing Y= , double touch the expression
1/2 x, press CL (-) X/θ/T/n .
8. To view both the graphs on the same coordinate axes, press GRAPH
.
How has the line been changed?
______________________________________________________________________
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
59
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
How has the line been changed?
NAME _____________________________________________________ CLASS __________ DATE __________
1.3
LINEAR EQUATIONS
Activity 1
Characteristics of the y-intercept
1. Graph the equation y = x by pressing Y= CL to remove an old Y1
expression, and press X/θ/T/n to enter the equation.
2. Press ENTER
CL to clear Y2 or additional prompts.
3. Use the default viewing range, and press ZOOM , touch A ZOOM, double
touch 5 Default to view the graph.
4. Adding 2 will result in the equation y = x + 2. Enter this equation for Y2 by
pressing Y= ▼ X/θ/T/n +
2 .
5. To view both the graphs on the same coordinate axes, press GRAPH .
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
How has the line been changed?
______________________________________________________________________
6. Subtracting 2 will result in the equation y = x – 2. Enter this equation for Y2
by pressing
Y=
▼
CL
X/θ/T/n
–
2
.
7. To view both the graphs on the same coordinate axes, press GRAPH
.
How has the line been changed?
______________________________________________________________________
8. Sketch the three linear equations on the coordinate axes provided below:
60
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
NAME _____________________________________________________ CLASS __________ DATE __________
1.4
LINEAR EQUATIONS
Activity 2
Parallel and Perpendicular Lines
1. Enter the equations Y1 = 3X + 1 and Y2 = 3X + 2 by pressing Y=
X/θ/T/n
+
1
ENTER CL
3
X/θ/T/n
+
2
CL
3
.
2. Use the default viewing range, and press ZOOM , touch A ZOOM, and
double touch 5 Default to view the graphs. What do you notice about the
two lines?
______________________________________________________________________
3. These line are called parallel since they have an equal slope but different
▼
4. Enter the equations Y1 = 3X - 1 and Y2 = -1/3 X + 1 by pressing Y= CL
3
X/θ/T/n –
1 ENTER CL (-) 1 a/b 3
X/θ/T/n
+
1 .
5. Press GRAPH to view the graphs. What do you notice about the two lines?
______________________________________________________________________
6. These line are called perpendicular since they have slopes that are negative
reciprocals of each other (m1 = -1/m2). Notice, these intersecting lines form
four equal angles.
7. Graph two lines with unequal slopes (not negative reciprocals). What do
you see? Are the lines parallel, perpendicular or neither? If they intersect,
what angles are formed by the intersection?
______________________________________________________________________
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
61
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
y-intercepts. These lines will not intersect.
NAME _____________________________________________________ CLASS __________ DATE __________
2.1
QUADRATIC EQUATIONS
Graphing and translations of quadratic equations
1. Turn the calculator on and press Y= . Press CL to remove an old Y1
expression. Press ENTER
CL to remove an old Y2 expression.
2. The calculator should be setup with rectangular coordinates, and in
equation editor mode. To complete this setup, press 2ndF
SET UP ,
touch E COORD, double touch 1 Rect, touch G EDITOR, and double touch
1 Equation. Press CL to exit the set-up menu and return to the Y prompts.
3. To enter the quadratic equation ( y = x 2 ) for Y1, press Y= EZ ,
touch 1 X2, touch A Y=AX2, and double touch 2 Y=X2. Enter the viewing
➧
window range by pressing ZOOM , touch
on the screen, and touch
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
7 Dec.
4. When 2 is added to x 2, the resulting equation is y = x 2 + 2. Enter this
function using the SUB feature by pressing Y= ▼
EZ , touch 1 X2,
double touch B Y=AX2 + B, touch 1 Y=AX2 + B, press 2ndF
ENTER
2
SUB
1
ENTER GRAPH . What does the addition of 2 do?
5. When -2 is added to x2, the resulting equation is y = x 2 – 2. To change Y2
for this expression, press Y= ▼
2ndF
SUB
ENTER (–)
ENTER GRAPH . What does the addition of 2 do?
2
6. Summarize the effect of k within the standard equation y = a(x – h)2 + k.
62
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
NAME _____________________________________________________ CLASS __________ DATE __________
2.2
QUADRATIC EQUATIONS
Graphing and translations of quadratic equations
1. When -2 is added to x within the squaring operation (x2), the resulting
equation is y = (x – 2)2. To view this graph, press 2ndF
SHIFT/CHANGE ,
2
touch A SHIFT, double touch 1 Y=X , double touch at two on the x-axis.
What does the addition of -2 within the quadratic operation do to the basic
x 2 graph?
2. When 2 is added to x within the squaring operation (x2), the resulting
equation is y = (x + 2)2. To view this graph, press 2ndF
SHIFT/CHANGE ,
2
touch A SHIFT, double touch 1 Y=X , double touch at negative two on the
x-axis. What does the addition of 2 within the quadratic operation do to the
basic x2 graph?
graph, press 2ndF
2
1 Y=X , press ▲
SHIFT/CHANGE , touch B CHANGE, double touch
ENTER . What does the multiplication of 2 do to
2
the basic x graph?
4. When -2 is multiplied to the x 2, the resulting equation is y = -2x 2. To view
this graph, press 2ndF
SHIFT/CHANGE , touch B CHANGE, double touch
2
1 Y=X , press ▼
▼ ▼ ENTER . What does the
multiplication of 2 do to the basic x 2 graph?
6. Extend the above by proposing what happens when 2 or - 2 divides the x 2,
1
1
and the resultant equation is y = 2 x 2 or y = - 2 x 2.
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
63
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
3. When 2 is multiplied to x 2, the resulting equation is y = 2x 2. To view this
NAME _____________________________________________________ CLASS __________ DATE __________
2.3
QUADRATIC EQUATIONS
Translations and reflections of quadratic equations
In each of the following quadratic equations given in standard form, predict the
vertical and horizontal translations, opening and closing of the parabolas, and
possible reflections. Then verify or correct your prediction using the EL-9600
graphing calculator.
1. y = x 2 + 1
2. y =
1 2
x
3
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
3. y = -x 2
4. y = (x – 3)2
5. y = 2(x + 1)2
1
6. y = - 4 x 2 – 3
7. y = 3(x – 1)2 + 2
8. y = - 1 (x + 2)2 -1
5
64
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
NAME _____________________________________________________ CLASS __________ DATE __________
3.1
FORMULAS OR LITERAL EQUATIONS
Steps for solving an equation using the equation method
1. Turn the calculator on and press 2ndF
SOLVER to access the solver
feature. A blank screen should appear. If the screen is not blank, then press
CL to clear the screen.
2. Select the Equation method for solving by pressing 2ndF
SOLVER ,
and touching A METHOD, and double touching 1 Equation.
3. Enter the formula P = L [ 1 – ( 1 + I/12)–N ] – 1
I/12
I
÷
1
2
2
)
L
a/b
(-) ALPHA
b
(-)
a
)
(
b
a
1
N
(
–
1
▼
a/b 1
ALPHA
1 .
This equation is referred to as the amortization formula, with a loan ( L) with
a fixed rate of interest ( I).
4. Press ENTER to view the variable list. To find the monthly payment on
a $15,000 car loan made at 9% interest over four years (48 months), enter
the values by pressing ▼
ENTER
4
8
1
ENTER . Press
5
▼
0
▼
0
▼
is now highlighted by the cursor, press 2ndF
0
ENTER
•
0
9
and notice the payment ( P)
EXE
to solve for the
payment.
5. Pressing CL will return you to the variable screen. You can now change
or solve for any of the values. Save this formula by pressing
2ndF
SOLVER , touching C SAVE, pressing ENTER , and entering the formula
name. Give the formula the name AMORT by pressing A
R
T
M
O
ENTER .
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
65
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
ALPHA
I
=
▼
ALPHA
ALPHA
▼ ▼
+
P
▼
Press ALPHA
NAME _____________________________________________________ CLASS __________ DATE __________
3.2
FORMULAS OR LITERAL EQUATIONS
Steps for solving an equation using to graphic method
1. While viewing the variable list within the solver feature, press CL
CL
to return to and clear the formula entry screen. Select the Graphic method
for solving by pressing 2ndF
SOLVER , touch A METHOD, and double
touch 3 Graphic.
π
ALPHA
R
b
a
2
▼
2. Enter the formula V = π r 2h by pressing ALPHA
ALPHA
V
ALPHA
=
2ndF
H .
3. This equation is the formula for calculating the volume of a cylinder ( V) in
terms of the cylinder's height ( H) and radius ( R). Press ENTER to view the
variable list.
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
4. To find the radius of a cylinder with a volume of 30 cubic inches, and a
height of 10 inches, enter the values by pressing 3
0
0
ENTER
▼
1
ENTER . Press ▲ and notice the radius ( R) is now highlighted by
the cursor.
5. Press 2ndF
EXE to solve for the radius. The graphic solver will prompt
you for a variable range to solve within. Set the variable range to 0 and 2 by
pressing 0
6. Press 2ndF
ENTER
2
ENTER .
EXE to solve.
7. Pressing CL will return you to the variable screen. You can now change or
solve for any of the values. Save this formula by pressing 2ndF
SOLVER ,
touching C SAVE, pressing ENTER , and entering the formula name. Give
the formula the name VCYL by pressing V
66
C
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
Y
L
ENTER .
NAME _____________________________________________________ CLASS __________ DATE __________
3.3
FORMULAS OR LITERAL EQUATIONS
Solving an equation using Newton's method
1. Press 2ndF
SOLVER to enter the solver feature. Press CL to clear the
formula entry screen.
2. Select the Newton's method for solving by pressing 2ndF
SOLVER ,
▼
touching A METHOD, and double touching 2 Newton. Enter the formula
1
A = 2 H(B + C) by pressing ALPHA A ALPHA = 1 a/b 2
ALPHA H ( ALPHA B + ALPHA C ) .
3. Press ENTER to view the variable list. To find the height of a trapezoid with
an area of 25 in2, and bases of length 5 and 7 inches, enter the values by
pressing 2
5
ENTER
▼
5
ENTER
7
ENTER
▲
▲ .
4. Press 2ndF
EXE to solve for the height. Newton's method will prompt you
for a guess or starting point. Enter a starting point of 1 by pressing 1
ENTER .
5. Press 2ndF
EXE to solve. A height of 4.1667 will appear on the screen.
6. Pressing CL will return you to the variable screen. You can now change or
solve for any of the values. Save this formula by pressing 2ndF
SOLVER ,
touching C SAVE, pressing ENTER , and entering the formula name.
Give the formula the name "ATRAP," by pressing A
T
R
A
P
ENTER .
7. Find the height of a trapezoid with an area of 50 in2 with bases of 8 and 10.
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
67
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
Notice the height (H) is now highlighted by the cursor.
NAME _____________________________________________________ CLASS __________ DATE __________
4.1
GRAPHING POLYNOMIALS AND FINDING THE ROOTS
Graphing a polynomial and zooming to find the roots
1. Turn the calculator on and press Y= . Y prompts will appear on the viewing
window. Press CL to remove old Y expressions. Setup the calculator with
rectangular coordinates and the equation editor mode by pressing 2ndF
SET UP , touch E COORD, double touch 1 Rect, touch G EDITOR, and
double touch 1 Equation. Press CL to exit the menu and return to the
Y prompts.
2. To enter the polynomial y = x 3 – 3x 2 + x + 1, press EZ , double touch 5 X3,
touch A Y=AX3+BX2+CX+D, touch 1Y=AX3 + BX + CX + D, press 2ndF SUB
and enter the coefficients by pressing 1 ENTER (–) 3 ENTER 1
ENTER
1
ENTER .
on the screen, and touch 7 Dec to establish the
decimal viewing window and view the graph. Press TRACE to engage the
trace feature. Press
▼
to move the cursor near the left-hand root.
4. Set the zoom factors to 5 by pressing ZOOM , touch B FACTOR, press
ENTER
5
ENTER
5
ENTER . Press ZOOM , touch A ZOOM, double
touch 3 IN to zoom in on the left- hand root. Press TRACE and move the
tracer to approximate the root.
5. Press ZOOM , touch H RCL, double touch 2 PreWin to return to the
decimal viewing window. Press TRACE and move the tracer to find the
middle root.
6. Press
▼
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
➧
3. Press ZOOM , touch
repeatedly to move the tracer near the right-hand root. Press
ZOOM , touch A ZOOM, double touch 3 In to zoom in. Press TRACE and
move the tracer to approximate the root.
68
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
NAME _____________________________________________________ CLASS __________ DATE __________
4.2
GRAPHING POLYNOMIALS AND FINDING THE ROOTS
Graphing a polynomial and jumping to find the roots
CL to return to and clear the Y1 prompt.
▼
1. Press Y=
▼
2. To enter the polynomial y = x4 + x 3 –5x 2 –3x + 1, press X/θ/T/n ab 4
– 5 X/θ/T/n x2 – 3 X/θ/T/n + 1 .
+ X/θ/T/n ab 3
3. View the graph in the decimal viewing window by pressing ZOOM ,
on the screen, and touching 7 Dec.
4. Press TRACE and move cursor near the right-hand root. Press 2ndF
CALC to view the calculate menu.
5. Double touch 5 X_Incpt to find the left-hand root.
6. Press 2ndF
CALC , double touch 5 X_Incpt to find the next root.
7. Press 2ndF
CALC , double touch 5 X_Incpt to find the next root.
8. Press 2ndF
CALC , double touch 5 X_Incpt to find the next root.
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
69
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
➧
touching A ZOOM, touching
NAME _____________________________________________________ CLASS __________ DATE __________
4.3
GRAPHING POLYNOMIALS AND FINDING THE ROOTS
Graphing polynomials and zooming to find the roots
Graph the following in a decimal viewing window and sketch the curve in the
box provided (-6.3 < x < 6.3, -3.1 < y < 3.1, scales of 1). Use the zoom feature to
approximate the roots.
(1) y = x 3 – 2x 2 – 7x + 11
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
roots = _______________
Graphing polynomials and using the calculate feature
to find the roots
Graph the following in a decimal viewing window and sketch the curve in the
box provided. Use the calculate feature to approximate the roots.
(2) y = x 3 + 5x 2 + 6x + 1
roots = _______________
70
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
NAME _____________________________________________________ CLASS __________ DATE __________
5.1
SOLVING A SYSTEM OF EQUATIONS
Graphing a system of equations and using the calculate feature to
find the solutions
1. Turn the calculator on and press Y= .
2. Press CL to clear an old Y1 expression. Press ENTER
CL to clear
additional Y prompts.
3. Set the calculator to rectangular coordinates and in equation editor mode.
To complete this setup, press 2ndF
SET UP , touch E COORD, double
touch 1 Rect, touch G EDITOR, and double touch 1 Equation.
Press CL to exit the menu and return to the Y prompts.
y = x2 – 1
4. To enter the system of equations:
press X/θ/T/n
x
–
1
ENTER
2
EZ , touch 5, touch
➧
the rapid window by pressing WINDOW
X/θ/T/n . View the graphs using
on the
screen, double touch 7 –5<X<5, and double touch 4 -10<Y<10.
5. Press 2ndF
CALC to access the calculate feature. Double touch 2 Intsct.
The left-hand intersection will appear on the screen.
6. Press 2ndF
CALC to access the calculate feature again. Double touch
2 Intsct. The right-hand intersection will appear on the screen.
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
71
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
y = 2x
2
NAME _____________________________________________________ CLASS __________ DATE __________
5.2
SOLVING A SYSTEM OF EQUATIONS
Solving a system of linear equations using the tool feature
1. Press 2ndF
TOOL to access the tool menu.
2. Touch B SYSTEM and double touch 2 2 to view the entry screen for solving a
linear-system of equations. Systems up to six variables and six equations
can be solved.
3. To enter the system of equations:
Press 5
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
ENTER
ENTER 1 ENTER
(–) 5 ENTER .
4. Press 2ndF
72
1
5x + y = 1
-3x + y = -5
ENTER (–) 3
EXE to solve the system.
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
ENTER
1
NAME _____________________________________________________ CLASS __________ DATE __________
5.3
SOLVING A SYSTEM OF EQUATIONS
Activity 1
Graphing a system of equations and using the
calculate feature to find the solutions
Graph the following system of equations and sketch the graphs in the box
provided (-5 < x < 5, -5 < y < 5, scales of 1). Use the calculate feature to
approximate the solutions.
solutions = _______________
Activity 2
Using the tool feature to find the solution to a system
of linear equations
x+y=1
-2x + y = – 1
solutions = _______________
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
73
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
y = x2 – 4x + 3
y = -x2 + 2x + 2
NAME _____________________________________________________ CLASS __________ DATE __________
6.1
MATRIX SOLUTIONS TO SYSTEMS
OF LINEAR EQUATIONS
1. Press MATRIX to access the matrix menu.
2. Touch B EDIT and double touch 1 mat A to select matrix A.
3. Now, enter the size or dimension of the matrix. We will enter
the 3 × 3 matrix.
[ ]
1 2 1
2 1 -1
1 1 -2
Press 3
ENTER
3
ENTER to set the dimension of the matrix at
three rows by three columns.
4. The calculator will now prompt you for the matrix. Enter the elements of
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
the matrix by pressing 1 ENTER 2 ENTER 1 ENTER 2 ENTER
1 ENTER (–) 1 ENTER 1 ENTER 1 ENTER (–) 2 ENTER .
5. Press 2nd
QUIT to exit the display of matrix A.
6. Repeat the process to enter a 3 × 3 matrix B =
[ ]
1 2 3
4 5 6
7 8 9
.
7. Matrix multiplication can be performed if the number of columns of the first
matrix is equal to the number of rows of the second matrix. In the matrix
multiplication A × B, the elements in the first row of A are multiplied to the
corresponding elements in the first column of B. The sum of these
multiplications is placed in the 1,1(first row, first column) position of the
resulting matrix. This process is repeated until each row of A has been
multiplied to each column of B.
8. To multiply the matrices A and B together, press MATRIX , touch A NAME,
double touch 1 mat A, press ×
MATRIX , double touch 2 mat B, and
press ENTER .
74
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
NAME _____________________________________________________ CLASS __________ DATE __________
6.2
MATRIX SOLUTIONS TO SYSTEMS
OF LINEAR EQUATIONS
1. The calculator will directly establish an identity matrix of a given size by
pressing MATRIX , touching C OPE, double touching 05 identity, and
pressing 3
ENTER . To save the identity matrix in matrix C, press STO
MATRIX , touch A NAME, touch 3 mat C, and press ENTER . Confirm that
the identity matrix is stored in matrix C by pressing MATRIX , touching
B EDIT, and double touching 3 mat C. Press 2ndF
QUIT to exit the matrix
editor and press CL to clear the screen.
to see more of the matrix.
3. To solve the system of equations
x + 2y + z = 8
2x + y – z = 1
x + y – 2z = -3
using matrices, use the matrix A entered previously as the coefficient matrix,
and enter the constants on the right side of the equal sign into matrix B,
where B =
[]
8
1
-3 .
Press 2ndF
QUIT to exit the display of the B matrix. The solution matrix
X is found by multiplying A-1 B.
4. This multiplication is derived from the equation AX= B,
A-1 A X = A-1 B (multiply both sides by A-1 )
I X = A-1 B (A-1A = I, identity matrix)
X = A-1 B (I X = X )
Multiply A-1 B by pressing MATRIX , touching A NAME, double touching
1 mat A, pressing 2ndF x-1 × MATRIX double touching 2 mat B, and
pressing ENTER . The solution matrix will appear.
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
75
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
Press
▼
2. Find the inverse of the square matrix A by pressing MATRIX , touching
A NAME, double touching 1 mat A, pressing 2ndF x-1 ENTER .
NAME _____________________________________________________ CLASS __________ DATE __________
6.3
MATRIX SOLUTIONS TO SYSTEMS
OF LINEAR EQUATIONS
Activity 1
Solve the following system of linear equations.
x – 2y + 3z = 9
-x + 3y
= -4
2x – 5y + 5z = 17
1. Enter the coefficient matrix A.
2. Enter the constant matrix B.
3. Find A-1 B.
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
4. The solution is (x, y, z) = ( _____, _____, _____ )
Activity 2
x
Solve the following system of linear equations using
the tool feature.
– 3z = -2
3x + y – 2z = 5
2x + 2y + z = 4
1. Press 2ndF
TOOL , touch B SYSTEM, and double touch 3 3.
2. Enter the coefficients and constants in the system solver.
3. Press 2ndF
EXE .
4. The solution is (x, y, z) = ( _____, _____, _____ )
76
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
NAME _____________________________________________________ CLASS __________ DATE __________
7.1
INEQUALITIES
1. To solve 3(4 – 2x) ≥ 5 – x, rewrite it as 3(4 – 2x) – 5 + x ≥ 0 and determine the
values of x where the function y = 3(4 – 2x) – 5 + x is on or above the x-axis.
2. To do this, press Y=
CL and enter 3(4 – 2x) – 5 + X in the Y1 location.
3. Set the viewing window of the graph by pressing ZOOM , touch A ZOOM,
and double touch 5 Default. You should be able to clearly view the
4. Locate the x-intercept at the point (1.4, 0) by pressing TRACE
2ndF
CALC , and double touch 5 X_Incpt.
5. Since the graph is above the x-axis to the left of the x-intercept, the solution
to the inequality 3(4 – 2x) – 5 + x ≥ 0 is all values of x such that x ≤ 1.4.
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
77
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
x-intercept.
NAME _____________________________________________________ CLASS __________ DATE __________
7.2
INEQUALITIES
1. To solve the inequality 3(4 – 2x) ≥ 5 – x, press Y=
CL , enter 3(4 – 2X) for
Y1 and 5 – X for Y2.
2. Set the viewing window by pressing ZOOM , touching A ZOOM, and
double touching 5 Default.
3. Next, shade the set of points that make the inequality true by pressing 2ndF
DRAW , touching G SHADE, and double touching 1 Set to access the Set
Shade screen. Since the inequality you are solving is Y1 ≥ Y2 the solution is
touching Y2 on the screen, pressing
▼
where the graph of Y1 is “on the top” and Y2 is "on the bottom." Do this by
, and touching Y1 on the screen.
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
Press GRAPH to view the shaded region.
4. Press TRACE
2ndF
CALC , and double touch 2 Intsct to find where the
graphs intersect.
5. Since the shaded region is to the left of x = 1.4, the solution to the inequality
3(4 – 2x) ≥ 5 – x is all values of x such that x ≤ 1.4.
78
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
NAME _____________________________________________________ CLASS __________ DATE __________
7.3
DOUBLE INEQUALITIES
1. The inequality -1 ≤ 2x – 5 ≤ 7 is commonly referred to as a “double” inequality.
2. Turn off the shading by pressing 2ndF
DRAW , touching G SHADE, and
double touching 2 INITIAL.
3. Clear any previously entered functions by pressing Y=
CL .
4. Enter Y1 = -1, Y2 = 2X – 5, and Y3 = 7.
6. Press TRACE
2ndF
CALC , double touch 2 Intsct to find the point of
intersection of the lines y = 2x – 5 and y = -1 at (2, -1). Press ▲ to move the
tracer to the y = 7 line. Press 2ndF
CALC and double touch 2 Intsct to
find y = 2x – 5 and y = 7 at (6, 7).
7. The solution to the “double” inequality -1 ≤ 2x – 5 ≤ 7 consists of all values of
x in between, and including, 2 and 6 (i.e., x ≥ 2 and x ≤ 6). The solution is
2 ≤ x ≤ 6.
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
79
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
5. Press ZOOM , touch A ZOOM, and double touch 5 Default to view the line
y = 2x – 5 between the lines y = -1 and y = 7.
NAME _____________________________________________________ CLASS __________ DATE __________
7.4
SYSTEMS OF INEQUALITIES
1. The solution region of a system of inequalities consists of all points (a, b)
such that when x = a and y = b, all inequalities in the system are true.
2. For example, to graph the solution region for the system
2x + y ≥ 1
x2 + y ≤ 1,
first rewrite each inequality in the system so that the left-hand-side is y:
y ≥ 1 – 2x
y ≤ 1 – x2
3. Press Y=
CL and enter 1 – 2X in Y1 and 1 – X2 in Y2.
4. Next, shade the points that have a y-value less than Y2 and greater than Y1.
DRAW , touching G SHADE, double touching
, and touch Y2 on the screen.
5. Graph the system by pressing ZOOM , touching A ZOOM, touching
the screen, and touching 7 Dec.
80
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
➧
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
1 Set, touch Y1 on the screen, press
▼
Do this by pressing 2ndF
on
NAME _____________________________________________________ CLASS __________ DATE __________
7.5
INEQUALITIES AND SYSTEMS OF INEQUALITIES
1. Use the first method to solve the following inequalities:
1
a.
x + 4 ≥ 3x
2
b. 5 – x 2 + 2x < x + 3
2. Use the second method to solve the following inequalities:
3 – 2x
a.
≤ 1 x–4
2
5
b. x3 > (x – 2)2
a.
4x – 5 > 1 – 2x2
b.
2(x + 2)2 – 3 < 1 – x2
c.
√ (4 + x) < x
d.
ln x + 3x ≥ ex
5
4. a.
b.
5. a.
Solve the inequality -3 ≤ 2 – x ≤ 4
6x – 2
Solve the inequality -8 <
≤ -2
5
Graph the solution set of the system of inequalities:
x + 2y ≤ 1
x2 + y ≥ 4
b.
Which of the following points are within the solution
region?
(-1.6, 1.8), (-2, -5), (2.8, -1.4), (-8, 4)
6. Graph the solution set of the system of inequalities:
2y + 3x ≥ 8
2y + 3x ≤ 12
x≥0
y≥1
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
81
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
3. Find the values of x for which:
NAME _____________________________________________________ CLASS __________ DATE __________
8.1
ABSOLUTE VALUE FUNCTIONS
+ –
× ÷
1. Evaluate |-2(5 – 1)| by pressing
to access the home or computation
screen. Enter the absolute value symbol by pressing MATH , touching
B NUM, double touching 1 abs(. You should see |
| on the screen.
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
Enter -2(5-1) inside the absolute value symbols and evaluate by pressing
(–) 2 ( 5 – 1 ) ENTER .
2. Graph y = |x| by pressing Y=
CL to clear the Y1 prompt. Press ENTER
CL to clear the remaining prompts if necessary. Enter |x| in Y1 by pressing
MATH touching B NUM, double touching 1 abs( and pressing X/θ/T/n .
Press ZOOM , touch A ZOOM, and double touch 5 Default to draw the graph.
3. Graph y = |x – 1| by pressing Y=
CL to clear the Y1 prompt. Press EZ ,
double touch 8 |x|, touch A Y = A|X – H| + K, touch 1 Y = A|X – H| + K,
and press 2ndF
SUB
1
ENTER
4. Now, graph y = |x| – 1 by pressing Y=
ENTER (–) 1 ENTER GRAPH .
82
1
ENTER
2ndF
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
0
SUB
ENTER
1
ZOOM
ENTER
0
5 .
NAME _____________________________________________________ CLASS __________ DATE __________
8.2
ABSOLUTE VALUE FUNCTIONS
Activity 1
Direct evaluation of absolute values
1. |-5| = _____
Activity 2
2.
|4(2 – 6)| = _____
3.
-4|-2| = _____
Graphing an absolute value function
Sketch the graph of each function in the box below the problem.
List the domain and range of each function in the blanks below the box.
2.
3.
_____________
_____________
_____________
_____________
_____________
_____________
4. y = -2|x – 1|
5.
_______________
_____________
_______________
_____________
y = -2|x| – 1
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
83
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
y = |x| + 2
y = -2|x|
1. y = |x + 2|
NAME _____________________________________________________ CLASS __________ DATE __________
8.3
ABSOLUTE VALUE EQUATIONS
1. Solve |5 – 4x| = 6 by first pressing Y=
CL to clear the Y1 prompt.
2. Enter |5 – 4x| in Y1 with the keystrokes MATH , touch B NUM,
double touch 1 abs( and press 5
3. Next, press ENTER
CL
–
4
X/θ/T/n .
6 to enter 6 in Y2.
4. Press ZOOM , touch A ZOOM, double touch 5 Default to view the two
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
points of intersection of the absolute value graph and the horizontal line y = 6.
5. Press TRACE
2ndF
CALC and double touch 2 Intsct to find one point
of intersection of the two graphs. Press 2ndF
CALC and double touch
2 Intsct again to find the other point of intersection.
6. The solution to the equation |5 – 4x| = 6 consists of the two
values -0.25 and 2.75.
84
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
NAME _____________________________________________________ CLASS __________ DATE __________
8.4
ABSOLUTE VALUE INEQUALITIES
1. To solve 20 – (6x) < 8 using the first method, rewrite the inequality so that
5
the right-hand side of the inequality is zero: 20 – (6x) – 8 < 0.
5
2. Press Y= and clear all the Y prompts.
20 – (6x)
– 8 in Y1 by pressing
5
MATH , touching B NUM, double touching 1 abs( pressing 2 0 – a/b
– 8 .
6 X/θ/T/n
5
▼
▼
▼
3. Enter the left-hand side of the inequality
4. Choose a viewing window by pressing WINDOW EZ , double touching
6 on the screen, double touching 4 -5<X<50, and double touching 4 -10<Y<10.
5. Press TRACE and move right of the right intercept. Press 2ndF
CALC
and double touch 5 X_Incpt to find the first x-intercept. Press 2ndF
CALC
and double touch 5 X_Incpt to find the second x-intercept.
6. Since the graph is below the x-axis for x in between the two x-intercepts,
the solution is 10 < x < 23.333.
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
85
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
The x-intercepts of the graph are clearly visible.
NAME _____________________________________________________ CLASS __________ DATE __________
8.5
ABSOLUTE INEQUALITIES
1. To use the second method to solve the inequality |3.5x + 4| ≥ 10, first
press Y=
CL to clear the Y1 prompt. Press ENTER
CL to clear
additional prompts.
2. Enter the function |3.5x + 4| in Y1 by pressing MATH , touching B NUM,
double touching 1 abs( and pressing 3
Enter 10 in Y2 by pressing 1
•
5
X/θ/T/n
+
4
ENTER .
0 .
3. Access the Set Shade screen by pressing 2ndF
DRAW , touching
G SHADE, and double touching 1 SET. Since Y2 is the function "on the
press
▼
bottom," touch Y2 on the screen, and since Y1 is the function "on the top,"
and touch Y1 on the screen.
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
4. Choose a viewing window by pressing WINDOW EZ , double touching
2 on the screen, double touching 5 -10<X<10, and double touching 3 -5<Y<50.
5. Press TRACE
2ndF
CALC , double touch 2 Intsct to locate a point of
intersection. Repeat to find the other. The intersections are (-4, 10) and
(1.714, 10). The solution of the inequality |3.5x + 4| > 10 is all values of x
such that x ≤ -4 or x ≥ 1.714.
86
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
NAME _____________________________________________________ CLASS __________ DATE __________
8.6
ABSOLUTE VALUE FUNCTIONS, EQUATIONS,
AND INEQUALITIES
1. Find the values of x for which |3 – 8x| = 7.
2. Solve the equation 5 = |3( x
2
2
) – 4|.
3. Solve the inequality |2x + 3| ≤ 9.
4. For which values of x is |5x – 7| + 3 ≥ 1?
(3x) 1
–
5
2
> 12
6. Solve: |x 2 – 3x – 4| < 10
7. Find the values of x for which |x + 2| ≥ |3 x – 1|.
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
87
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
5. Solve:
NAME _____________________________________________________ CLASS __________ DATE __________
9.1
RATIONAL FUNCTIONS
(x – 1)
1. Graph the rational function f(x) = (x 2 – 1) by first pressing Y=
CL to clear additional Y prompts.
(x – 1)
2. Enter Y1 = (x2 – 1) by pressing a/b
– 1 .
X/θ/T/n
–
1
▼
the Y1 prompt. Press ENTER
CL to clear
X/θ/T/n
x2
3. Graph in the decimal window by pressing ZOOM , touching A ZOOM,
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
➧
touching
on the screen, and double touching 7 Dec.
4. The function consists of two branches separated by the vertical asymptote
(the line x = -1). Look closely at the graph and observe the “hole” in it at
x = 1. Press TRACE and investigate the "hole." Note that the hole will be
seen only when the window is a "friendly" or decimal window.
5. Notice that there are no x-intercepts for the graph. The y-intercept can
easily be found at x = 0, y = 1 by pressing TRACE
2ndF
CALC and
double touching 6 Y_Incpt.
6. Observe that the line y = 0 is very likely a horizontal asymptote of f(x).
88
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
NAME _____________________________________________________ CLASS __________ DATE __________
9.2
RATIONAL INEQUALITIES
x
1. Solve the inequality (1 – x2) ≤ 2 using the intersection method with shading,
by first pressing Y= CL to clear the Y1 prompt.
touching B NUM, double touching 1 abs( pressing a/b
X/θ/T/n
3. Press 2ndF
screen, press
x
X/θ/T/n
ENTER . Enter 2 in Y2 by pressing 2
1
ENTER .
DRAW , touch G SHADE, double touch 1 SET, touch Y1 on the
▼
–
2
▼
2. Enter the left-hand side of the inequality in Y1 by pressing MATH ,
, and touch Y2 on the screen.
➧
4. View the graph by pressing ZOOM , touching A ZOOM, touching
on the
5. Press TRACE
2ndF
CALC and double touch 2 Intsct, repeatedly, to
locate the x-values of the points of intersection, x = -1.281, -0.781, 0.781,
and 1.281. The solution is all values of x such that x ≤ -1.281 or -0.781 ≤ x ≤
0.781 or x ≥ 1.281. Turn off the shading by pressing 2ndF
DRAW ,
touching G SHADE, and double touching 2 INITIAL.
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
89
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
screen, and double touching 7 Dec.
NAME _____________________________________________________ CLASS __________ DATE __________
9.3
RATIONAL FUNCTIONS AND INEQUALITIES
Activity 1
1. Enter
x2
Solve the rational inequality (x2 – 4) ≥ 2
x2
in Y1. Enter 2 in Y2.
(x2 – 4)
2. Set up the shading.
3. View the graph in a decimal window.
4. Find the intersections of the rational function and the line y = 2.
5. Give the solution to the inequality by determining the values of x
corresponding to the shaded regions on the graph.
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
6. Turn off the shading.
Activity 2
1. Enter
x–3
Graph the rational function y =(x2 – 5x + 6)
(x – 3)
in Y1.
(x2 – 5x + 6)
(x – 3)
2. Factor the denominator of y = (x2 – 5x + 6) . How many vertical
asymptotes and how many branches do you expect to see when the
graph of this rational function is drawn?
3. View the graph in a decimal window.
4. What is the domain of this rational function? What do you see in the graph
to support your answer?
90
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
NAME _____________________________________________________ CLASS __________ DATE __________
10.1
CONIC SECTIONS
Steps for graphing a parabola in rectangular mode
1. Graph the parabola x = y 2 – 2 by first rewriting as y = ± √ (x + 2).
2. Press Y=
CL to clear the Y1 prompt. Press ENTER
CL to clear
additional prompts. Enter √ (x + 2) in Y1 with the keystrokes 2ndF
X/θ/T/n + 2 .
3. Press ENTER and enter -Y1 = - √ (x + 2) in Y2 with the key strokes (–)
√
VARS ,
touch A EQVARS, press ENTER , and double touch 1 Y1.
➧
4. View the graph by pressing ZOOM , touching A ZOOM, touching
on the
screen, and double touching 7 Dec.
1. Change to parametric mode by pressing 2ndF
and double touch 2 Param. Press 2ndF
2. Press Y=
SET UP , touch E COORD,
QUIT to exit the set-up screen.
CL to clear the X1T prompt. To rewrite x = y 2 – 2 in parametric
form, simply let y = T and substitute in x = y 2 – 2 to obtain x = T2 – 2. Enter
X1T = T2 – 2 in the calculator by pressing X/θ/T/n
x2
–
2
ENTER .
Enter Y1T = T by pressing X/θ/T/n .
➧
3. View the graph by pressing ZOOM , touch A ZOOM, touch
on the screen,
and double touch 7 Dec.
4. Notice that only a half of the parabola is drawn. To see the rest of the
parabola, press WINDOW and adjust the Tmin. Set Tmin to -6 by pressing
(–)
6
ENTER . Press GRAPH to view the complete parabola.
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
91
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
Steps for graphing a parabola in parametric mode
NAME _____________________________________________________ CLASS __________ DATE __________
10.2
CONIC SECTIONS
Steps for graphing a circle in rectangular mode
1. To graph the circle x 2 + y 2 = 4, first return to the rectangular graphing mode
by pressing 2ndF
SET UP , touching E COORD, double touching 1 Rect,
and pressing 2ndF
QUIT .
2. Press Y=
2ndF
√
CL to clear the Y1 prompt. Enter Y1 = √(4 – X2) by pressing
4 – X/θ/T/n x2 ENTER and then enter Y2 = -√ (4 – X2) by
pressing CL
(–)
VARS , touch A EQVARS, press ENTER , touch A XY,
and double touch 1 Y1. View the graph by pressing ZOOM , touching
➧
A ZOOM, touching
on the screen, and double touching 7 Dec.
Steps for graphing a circle not in standard form
solving for y. For example, draw the graph of the circle x 2 – 2x + y 2 + 4y = 2
by rewriting as y = ± √ (6 – x 2 + 2x) – 2.
2. Enter Y1 = √ (6 – x 2 + 2x), Y2 = Y1 – 2 and Y3 = -Y1 – 2. “Turn off” Y1 so that
it will not graph by pressing ▲ to move the blinking cursor to the Y1
expression. Press
▼
to position the blinking cursor over the = sign, and
press ENTER so that “=” is no longer darkened.
3. To view the graph, press ZOOM , touch
➧
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
1. First solve the equation for y by completing the square on the y-term and
on the screen, and double touch
7 Dec.
4. Adjust the screen to see the bottom part of the circle. Do this using the
rapid zoom feature by pressing EZ , double touching ▼ on the bottom of
the screen twice, and double touching ▼ on the top of the screen twice.
Press GRAPH to view the graph in the new window.
92
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
NAME _____________________________________________________ CLASS __________ DATE __________
10.3
CONIC SECTIONS
1. Graph the circle (x + 3)2 + y2 = 4.
3. Graph the parabola x – y2 = 4 – 2y.
4. Graph the hyperbola x2 + 2x – y2 – 6y + 3 = 0.
Blackline Masters/ALGEBRA USING THE SHARP EL-9600
93
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
2. Graph the ellipse 3(x – 3)2 + (y + 2)2 = 3
KEYPAD FOR
THE SHARP EL-9600 GRAPHING CALCULATOR
STAT PLOT
SPLIT
TBL SET
SUB
FORMAT
CALC
Y=
GRAPH
TABLE
WINDOW
ZOOM
TRACE
SOLVER
CLIP
SHIFT/CHANGE
+ –
× ÷
SLIDE
SHOW
EZ
OFF
PRGM
DRAW
ON
MATRIX
STAT
A-LOCK
TOOL =
INS
SET UP
ALPHA
MATH
DEL
BS
-1
sin A
-1
cos B tan C
cos
sin
,
a b
L
v
Q
L4
V
L1
2
1
<
O
0
i
SPACE
L3
▼
CL
ex
log
F
In
π
K OPTION
VARS
X/θ/T/n
{
O
}
(
S
LIST
X
e
6
W
L2
N
L6
5
4
2
9
R
L5
x
QUIT
D 10x E
STO
w
8
-1
RCL J
I
ab
M
7
x
tan
H a
G
u
▼
P
)
T
FINANCE U
Y
EXE
+
-1
Exp
▼
2nd F
94
Equation Editor
▼
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
EL-9600
+
3
ENTRY
(–)
Keypad/ALGEBRA USING THE SHARP EL-9600
:
ANS
ENTER
Z
SOLUTIONS TO
THE ACTIVITIES
1. LINEAR EQUATIONS
BLACKLINE MASTER 1.3
5. moved up two
7. moved down two
BLACKLINE MASTER 1.4
2. parallel lines
5. perpendicular lines
7. neither, opposite angles
are equal
2. QUADRATIC EQUATIONS
BLACKLINE MASTER 2.3
1. moved up one
2. opening
3. reflection
4. moved right three
5. closing, moved left one
6. reflection, opening,
moved down three
7. closing, moved right one
and up two
8. reflection, opening,
moved left two and
down one
3. FORMULAS OR
LITERAL EQUATIONS
ADDITIONAL PROBLEMS
1. $313.36
2. $15,150.60
3. 2.06
BLACKLINE MASTER 3.3
7. 5.56
4. GRAPHING
POLYNOMIALS AND
FINDING THE ROOTS
ADDITIONAL PROBLEMS
1. -1.8, -.44, 1.3
2. -2.08, -.84, .84, 2.08
3. -1.4, -.75, .41, .79, 2.95
BLACKLINE MASTER 4.2
5. -2.47
6. -.82
7. .24
8. 2.05
BLACKLINE MASTER 4.3
1. -2.52, 1.4, 3.12
2. -3.25, -1.55, -.20
5. SOLVING A SYSTEM
OF EQUATIONS
ADDITIONAL PROBLEMS
1. (-1.62, -.62), (.62, 1.62)
2. (-2.45, -2.22), (-.45, -1.23),
(.90, -.55)
2.
BLACKLINE MASTER 5.1
5. (-.42, -.84)
6. (2.42, 4.86)
3.
BLACKLINE MASTER 5.3
Activity 1 (.18, 2.33), (2.82, .31)
Activity 2 (-1.62, -1.62), (.62,
.62)
6. MATRIX SOLUTIONS
TO SYSTEMS OF LINEAR
EQUATIONS
ADDITIONAL PROBLEMS
1. (1, 0, 1)
2. (1, 0, 2)
4.
5.
BLACKLINE MASTER 6.2
4. (1, 2, 3)
BLACKLINE MASTER 6.3
Activity 1 (1, -1, 2)
Activity 2 (4, -3, 2)
7. INEQUALITIES AND
SYSTEMS OF
INEQUALITIES
ADDITIONAL PROBLEMS
1. All values of x such that
x≤2
2. All values of x such that
x ≥ 9.5
3. a. All values of x between
but not including -2
and -1or all values
greater than 3
b. All real values of x
c. All values of x such
that x > 0
d. All values of x such
that -.5<x<1
4. All values of x such that
-2.4 ≤ x ≤ -1.6
5. b. (.6, .5) is a solution;
(3.5, 0) is not a
solution
6. b. (-2, 0) is not a
solution; (-1, 2.5) is a
solution
BLACKLINE MASTER 7.5
1. a. The graph of
Y1=(1/2)X + 4 - 3X is
above the x-axis for x
< 1.6
6.
b. The graph of Y1=2 + X
- X2 is below the x-axis
for x < -1 or x > 2
a. The graph of Y1=(32x)/5 is below the
graph of Y2=(1/2)X-4
for x>5.1
b. The graph of Y1=x3 is
above the graph of
Y2=(x-2)2 for x > 1
a. All values of x such
that either x < -3 or x
>1
b. All x such that x is
between, but does not
include, -2 and -.667
c. All values of x such
that x > 2.562
d. No real values of x
a. All values of x such
that -2 ≤ x ≤ 5
b. All values of x such
that -6.333 ≤ x ≤ -1.333
b. Points in the solution
region are (2.8, -1.4)
and (-8, 4). Points out
side the solution
region are (-1.6, 1.8)
and (-2, -5). (Note:
Range settings of -9.4
< x < 9.4, -6.2 < y < 6.2
allow you to locate
the points with the
free-moving cursor.)
0 ≤ x ≤ 4.7, 1 ≤ y ≤ 6.2
8. ABSOLUTE VALUE
FUNCTIONS, EQUATIONS,
AND INEQUALITIES
ADDITIONAL PROBLEMS
1. |-2 + 7| = 5; |-2| + |7| =
9; No; When x ≥ 0 and y ≥
0 or x ≤ 0 and y ≤ 0.
2. Both sides equal .75. The
statement is true for all
x and y ≠ 0.
3. x = -2.5, x = 17.5
4. All values of x such that
-2 ≤ x ≤ -.5
5. All values of x such that
x ≤ -.8 or x ≥ 1.6
BLACKLINE MASTER 8.2
ACTIVITY 1
1. 5
2. 16
3. -8
ACTIVITY 2
1. Domain: all real numbers
Range: y ≥ 0
2. Domain: all real numbers
Range: y ≥ 2
3. Domain: all real numbers
Range: y ≤ 0
4. Domain: all real numbers
Range: y ≤ 0
5. Domain: all real numbers
Range: y ≤ -1
BLACKLINE MASTER 8.6
1. x = -0.5, x = 1.25
2. x = 1, x = 4.333
3. -6 ≤ x ≤ 3
4. all real numbers of x
5. x < -8.133 or x > 7.867
6. -2.531 < x < 5.531
7. -.25 ≤ x ≤ 1.5
9. RATIONAL FUNCTIONS
ADDITIONAL PROBLEMS
1. b. all real numbers
except x = 0
c. x = 0
d. x-intercept at x = -3,
no y-intercept
e. y = 1
2. b. all real numbers
except x = -2, -1, 1
c. x = -2, x = 1
d. x-intercept at x = 0,
y-intercept at y = 0
e. y = 0
3. x ≤ -2.591 or x ≥ -1.009
4. all values of x such that
x > -2 except x = 1
BLACKLINE MASTER 9.3
ACTIVITY 1
4. x = -2.828, 2.828
5. -2.828 < x < 2 or 2 < x
< 2.828
ACTIVITY 2
2. 1 vertical asymptote and
2 branches
4. domain: all values of x
such that x ≠ 2, x ≠ 3 a
hole in the graph at x = 3
10. CONIC SECTIONS
ADDITIONAL PROBLEMS
1. parabola opening right,
vertex at (0, 4)
2. parabola opening left,
vertex at (0, -1)
3. ellipse centered at the
origin
4. parabola opening left
5. circle
6. hyperbola, opening up
and down
Answer Key/ALGEBRA USING THE SHARP EL-9600
95
SHARP ALGEBRA COVER
02.2.19 10:29 AM
Page 2
* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project
advertisement
© 2021