Sharp EL-9650, EL-9400, EL-9600c User Manual

Graphing Calculator

EL-9650/9600c/9450/9400

Handbook Vol. 1

Algebra

EL-9650 EL-9450

Contents

1. Linear Equations

1-1 1-2

2. Quadratic Equations

2-1 2-2

3. Literal Equations

3-1 3-2 3-3

4. Polynomials

4-1 4-2

5. A System of Equations

5-1

Slope and Intercept of Linear Equations Parallel and Perpendicular Lines

Slope and Intercept of Quadratic Equations Shifting a Graph of Quadratic Equations

Solving a Literal Equation Using the Equation Method (Amortization) Solving a Literal Equation Using the Graphic Method (Volume of a Cylinder) Solving a Literal Equation Using Newton’s Method (Area of a Trapezoid)

Graphing Polynomials and Tracing to Find the Roots Graphing Polynomials and Jumping to Find the Roots

Solving a System of Equations by Graphing or Tool Feature

6. Matrix Solutions

6-1 6-2

Entering and Multiplying Matrices Solving a System of Linear Equations Using Matrices

7. Inequalities

7-1 7-2 7-3 7-4

Solving Inequalities Solving Double Inequalities System of Two-Variable Inequalities Graphing Solution Region of Inequalities

8. Absolute Value Functions, Equations, Inequalities

8-1 8-2 8-3 8-4 8-5

Slope and Intercept of Absolute Value Functions Shifting a graph of Absolute Value Functions Solving Absolute Value Equations Solving Absolute Value Inequalities Evaluating Absolute Value Functions

9. Rational Functions

9-1 9-2

Graphing Rational Functions Solving Rational Function Inequalities

10. Conic Sections

10-1 10-2 10-3 10-4

Graphing Parabolas Graphing Circles Graphing Ellipses Graphing Hyperbolas

Read this first

1. Always read “Before Starting”

The key operations of the set up condition are written in “Before Starting” in each section. It is essential to follow the instructions in order to display the screens as they appear in the handbook.

2. Set Up Condition

As key operations for this handbook are conducted from the initial condition, reset all memories to the initial condition beforehand.

2nd F

OPTION

Note: Since all memories will be deleted, it is advised to use the CE-LK1P PC link kit (sold

separately) to back up any programmes not to be erased, or to return the settings to the initial

condition (cf. 3. Initial Settings below) and to erase the data of the function to be used.

•

To delete a single data, press

• Other keys to delete data: : to erase equations and remove error displays

CL

: to cancel previous function

2nd F

QUIT

2

E

CL

2nd F

OPTION

and select data to be deleted from the menu.

C

3. Initial settings

Initial settings are as follows:

✩

Set up (

✩

Format (

✩

Stat Plot (

✩

Shade (

✩

Zoom (

✩

Period (

✩

Note:

returns to the default setting in the following operation.

(

2nd F

)

OPTION

): Rad, FloatPt, 9, Rect, Decimal(Real), Equation

SET UP

2nd F

): RectCoord, OFF, OFF, Connect, Sequen

2nd F

FORMAT

): 2. PlotOFF

STAT PLOT

2nd F

): 2. INITIAL

2nd F

): 5. Default

ZOOM

): 1. PmtEnd

2nd F

E1

E

DRAW

G

A

FINANCE

C

ENTER

4. Using the keys

Press to use secondary functions (in yellow).

2nd F

To select “sin

Press to use the alphabet keys (in blue).

ALPHA

To select A:

-1

”: ➔ Displayed as follows:

2nd F

sin

➔ Displayed as follows:

sin

ALPHA

2nd F

ALPHA

sin

A

-1

5. Notes

•

Some features are provided only on the EL-9650/9600c and not on the EL-9450/9400. (Substi-

tution, Solver, Matrix, Tool etc.)

•

As this handbook is only an example of how to use the EL-9650/9600c and 9450/9400,

please refer to the manual for further details.

Using this Handbook

This handbook was produced for practical application of the SHARP EL-9650/9600c and EL-9450/9400 Graphing Calculator based on exercise examples received from teachers actively engaged in teaching. It can be used with minimal preparation in a variety of situations such as classroom presentations, and also as a self-study reference book.

Introduction

Explanation of the section

Example

Example of a problem to be solved in the section

Before Starting

Important notes to read before operating the calculator

Step & Key Operation

A clear step-by-step guide to solving the problems

Display

Illustrations of the calculator

Slope and Intercept of Quadratic Equations

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 = ax 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, let the x- variable be represented by the horizontal axis and let y be represented by the vertical axis. The graph can be adjusted by varying the coefficients a, h, and k.

Example

Graph various quadratic equations and check the relation between the graphs and the values of coefficients of the equations.

1.

Graph y = x2 and y = (x-2)2.

2.

Graph y = x2 and y = x2+2.

3.

Graph y = x2 and y = 2x2.

4.

Graph y = x2 and y = -2x2.

There may be differences in the results of calculations and graph plotting depending on the setting.

Before

Return all settings to the default value and delete all data.

Starting

As the Substitution feature is only available on the EL-9650/9600c, this section does not apply to the EL-9450/9400.

Step & Key Operation

*Use either pen touch or cursor to operate.

Enter the equation y = x2 for Y1.

1

-

1

2

X

/θ/T/

n

Y=

x

Enter the equation y = (x-2)2 for Y2

1

-

2

using Sub feature.

ENTER

1

EZ

*

ENTER ENTER

ALPHA

C

1

*

2nd F

1

ENTER

SUB

2

( )

ENTER

0

View both graphs.

1

-

3

GRAPH

EL-9650/9600c Graphing Calculator

*Use either pen touch or cursor to operate.

2

-

1

Change the equation in Y2 to y = x2+2.

Y=

ENTER ENTER

2

-

2

View both graphs.

GRAPH

Display

3

-

1

Change the equation in Y2 to y = 2x2.

3

-

2

*

ENTER

View both graphs.

GRAPH

4

-

1

Change the equation in Y2 to y = -2x

This shows that placing an h (>0) within the standard form y = a (x - h) h units and placing an h (<0) will move it left h units on the x-axis.

ENTER

4

-

2

View both graphs.

GRAPH

screen for each step

The EL-9650/9600c/9450/9400 allows various quadratic equations to be graphed easily. Also the characteristics of quadratic equations can be visually shown through the relationship between the changes of coefficient values and their

Merits of Using the EL-9650/9600c/9450/9400

graphs, using the Substitution feature.

2-1

Highlights the main functions of the calculator relevant to the section

2

+ bx + c) and (h, k) is the

Step & Key Operation

*

2nd F

SUB

2

Notes

*

2

2nd F

SUB

ENTER

0

Notice that the addition of -2 within the quadratic operation moves the basic y =x right two units (adding 2 moves

2

it left two units) on the x-axis.

.

*

(-)

2nd F

SUB

ENTER

Y=

2

+ k will move the basic graph right

Y=

Notes

Explains the process of each step in the key operations

EL-9650/9600c Graphing Calculator

Display

0

fact that adding k (>0) within the standard form y = a (x -

2

+ K will move the basic graph up K units and placing k

h) (<0) willmove the basic graph down K units on the y axis.

2

graph

2

2-1

ing an a (<-1) in the standard form y = a (x - h) will pinch or close the basic graph and flip it (reflect it) across the x-axis.

Notes

Notice that the addition of 2 moves

2

graph up two units

the basic y =x and the addition of -2 moves the basic graph down two units on the

y-axis.

This demonstrates the

Notice that the multiplication of 2 pinches or closes the basic

2

y=x

graph. This demonstrates the fact that multiplying an a (> 1) in the standard form

2

(x - h)

+ k

will pinch or close

the basic graph.

Notice that the multiplication of

-2 pinches or closes the basic y =

x2 graph and flips it (reflects it) across the x-axis. This demonstrates the fact that multiply-

y = a

2

+ k

• When you see the sign

means same series of key strokes can be done with screen touch on the EL-9650/9600c.

*

( * : for the corresponding key; Key operations may also be carried out with the cursor (not shown).

•

Different key appearance for the EL-9450/9400: for example

on the key:

*

: for the corresponding keys underlined.)

*

X/ /T/

n

➔

X/T

We would like to express our deepest gratitude to all the teachers whose cooperation we received in editing this book. We aim to produce a handbook which is more replete and useful to everyone, so any comments or ideas on exercises will be welcomed.

(Use the attached blank sheet to create and contribute your own mathematical problems.)

Thanks to Dr. David P. Lawrence at Southwestern Oklahoma State University for the use of his teaching resource book (Applying Pre-Algebra/Algebra using the SHARP EL-9650/9600c Graphing Calculator).

Other books available: Graphing Calculator EL-9450/9400 TEACHERS’ GUIDE

EL-9650/9600c/9450/9400 Graphing Calculator

Slope and Intercept of Linear Equations

A linear equation of y in terms of x can be expressed by the slope-intercept form y = mx+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.

Example

Draw graphs of two equations by changing the slope or the y- intercept.

1. Graph the equations y = x and y = 2x.

2. Graph the equations y = x and y = x.

3. Graph the equations y = x and y = - x.

4. Graph the equations y = x and y = x + 2.

1 2

Before

Starting

*Use either pen touch or cursor to operate.

1-1

There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value and delete all data.

(When using EL-9650/9600c)

Display

(When using EL-9650/9600c)

NotesStep & Key Operation

Enter the equation y = x for Y1 and y = 2x for Y2.

1-2

Y=

ENTER

X/

/T/n

View both graphs.

*

X/

/T/n

2

The equation Y1 = x is displayed first, followed by the

GRAPH

equation Y2 = 2x. Notice how Y2 becomes steeper or climbs faster. Increase the size of the slope (m>1) to make the line steeper.

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

2-1

Enter the equation y = x for Y2.

1 2

2-2

Y= CL

a

1

View both graphs.

GRAPH

/b

*

2

X/

/T/n

*

Notice how Y2 becomes less steep or climbs slower. Decrease the size of the slope (0<m<1) to make the line less steep.

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

1-1

EL-9650/9600c/9450/9400 Graphing Calculator

Notes

3-1

3-2

Step & Key Operation

(When using EL-9650/9600c)

*Use either pen touch or cursor to operate.

Enter the equation y = - x for Y2.

Y=

*

CL

(-)

X/

/T/n

View both graphs.

GRAPH

Display

(When using EL-9650/9600c)

Notice how Y2 decreases (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.

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

4-1

Enter the equation y = x + 2 for Y2.

Y= CL + 2

*

X/

/T/n

4-2

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

View both graphs.

GRAPH

Adding 2 will shift the y = x graph upwards.

Making a graph is easy, and quick comparison of several graphs will help students understand the characteristics of linear equations.

1-1

EL-9650/9600c/9450/9400 Graphing Calculator

P arallel and Perpendicular Lines

Parallel and perpendicular lines can be drawn by changing the slope of the linear equation and the y intercept. A linear equation of y in terms of x can be expressed by the slopeintercept form y = mx + b, where m is the slope and b is the y-intercept. Parallel lines have an equal slope with different y-intercepts. Perpendicular lines have slopes that are negative reciprocals of each other (m = - ). These characteristics can be verified by graphing these lines.

Example

Graph parallel lines and perpendicular lines.

1. Graph the equations y = 3x + 1 and y = 3x + 2.

2. Graph the equations y = 3x - 1 and y = - x + 1.

1

3

1

m

Before

Starting

*Use either pen touch or cursor to operate.

1-1

There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value and delete all data.

Set the zoom to the decimal window: * (

ZOOM

ENTER

C

Display

(When using EL-9650/9600c)

ALPHA

)

7

*

NotesStep & Key Operation

Enter the equations y = 3x + 1 for Y1 and y = 3x + 2 for Y2.

1-2

Y=

X/

3

View the graphs.

/T/n

X/

/T/n

3

+

1

+

2

ENTER

*

These lines have an equal slope but different y-intercepts.

GRAPH

They are called parallel, and will not intersect.

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

2-1

Enter the equations y = 3x - 1 for Y1 and y = - x + 1 for Y2.

Y= CL

(-)

CL

+

1

3

X/

a

1

/b

/T/n

—

1

ENTER

3

X/

*

/T/n

*

1-2

EL-9650/9600c/9450/9400 Graphing Calculator

Notes

1

m

2-2

Step & Key Operation

(When using EL-9650/9600c)

*Use either pen touch or cursor to operate.

View the graphs.

GRAPH

Display

(When using EL-9650/9600c)

These lines have slopes that are negative reciprocals of

each other (m = - ). They are called perpendicular. Note that

these intersecting lines form four equal angles.

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

The Graphing Calculators can be used to draw parallel or perpendicular lines while learning the slope or y-intercept of linear equations.

1-2

EL-9650/9600c Graphing Calculator

S lope and I nter cept of Quadratic Equations

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, let the x-variable be represented by the horizontal axis and let y be represented by the vertical axis. The graph can be adjusted by varying the coefficients a, h, and k.

Example

Graph various quadratic equations and check the relation between the graphs and the values of coefficients of the equations.

1. Graph y = x

2. Graph y = x

3. Graph y = x

4. Graph y = x

Before

Starting

Step & Key Operation

*Use either pen touch or cursor to operate.

1-1

1-2

Enter the equation y = x2 for Y1.

Y=

Enter the equation y = (x - 2)2 for Y2 using Sub feature.

ALPHA

2nd F

2

and y = (x - 2)2.

2

and y = x2 + 2.

2

and y = 2x2.

2

and y = -2x2.

There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value and delete all data.

Notes

X/

/T/n

EZ

C

SUB

2

x

1

ENTER ENTER

*

1

ENTER

1

Display

*

2

ENTER

( )

ENTER

0

1-3

View both graphs.

GRAPH

Notice that the addition of -2

within the quadratic operation

2

moves the basic y = x

graph right two units (adding 2 moves it left two units) on the x-axis.

This shows that placing an h (>0) within the standard form y = a (x - h)

2

+ k will move the basic graph right h units and placing an h (<0) will move it left h units on the x-axis.

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

2-1

EL-9650/9600c Graphing Calculator

Notes

x2 graph up two units

2-1

2-2

Step & Key Operation

*Use either pen touch or cursor to operate.

Change the equation in Y2 to y =

Y=

ENTER ENTER

2nd F

*

2

SUB

View both graphs.

GRAPH

Display

x2+2.

0

Notice that the addition of 2 moves the basic y = and the addition of -2 moves the basic graph down two units on the y-axis. This demonstrates the

fact that adding k (>0) within the standard form y = a (x -

2

+ k will move the basic graph up k units and placing k

h) (<0) will move the basic graph down k units on the y-axis.

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

3-1

Change the equation in Y2 to y = 2

Y=

2nd F

*

ENTER

0

SUB

2

ENTER

x2.

3-2

View both graphs.

Notice that the multiplication of 2 pinches or closes the basic

2

graph. This demonstrates

GRAPH

y = x

the fact that multiplying an a (> 1) in the standard form (x - h)2 + k

will pinch or close

y = a

the basic graph.

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

4-1

Change the equation in Y2 to y = -2x2.

4-2

Y=

ENTER

View both graphs.

2nd F SUB

*

(-)

2

Notice that the multiplication of

-2 pinches or closes the basic y =

GRAPH

x2 graph and flips it (reflects

it) across the x-axis. This demonstrates the fact that multiply-

2

ing an a (<-1) in the standard form y = a (x - h)

+ k

will pinch or close the basic graph and flip it (reflect it) across the x-axis.

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

2-1

The EL-9650/9600c allows various quadratic equations to be graphed easily. Also the characteristics of quadratic equations can be visually shown through the relationship between the changes of coefficient values and their graphs, using the Substitution feature.

EL-9650/9600c/9450/9400 Graphing Calculator

Shifting a Graph of Quadratic Equations

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 = a 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, let the x-variable be represented by the horizontal axis and let y be represented by the vertical axis. The relation of an equation and its graph can be seen by moving the graph and checking the coefficients of the equation.

Example

Move or pinch a graph of quadratic equation y = x2 to verify the relation between the coefficients of the equation and the graph.

1. Shift the graph y = x

2. Shift the graph y = x

3. Pinch the slope of the graph y = x

2

upward by 2.

2

to the right by 3.

2

.

x2 + bx + c) and (h, k) is the vertex

Before

Starting

*Use either pen touch or cursor to operate.

There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value and delete all data.

(When using EL-9650/9600c)

1-1 Access Shift feature and select the

2

.

ENTER

A

*

1-2

1-3

equation y = x

2nd F

SHIFT/CHANGE

1

*

Move the graph y = x2 upward by 2.

Save the new graph and observe the changes in the graph and the equation.

ENTER

ALPHA

Display

(When using EL-9650/9600c)

Notice that upward movement of the basic y =

units in the direction of the y- axis means addition of 2 to the

y-intercept. This demonstrates that upward movement of the graph by k units means adding a k (>0) in the standard form y = a(x - h)

NotesStep & Key Operation

2

x

graph by 2

2

+ k.

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

2-2

EL-9650/9600c/9450/9400 Graphing Calculator

Notes

2

graph to the right

2-1

2-2

Step & Key Operation

(When using EL-9650/9600c)

*Use either pen touch or cursor to operate.

Move the graph y = x2 to the right by 3.

CL

(three times)

ENTER

*

Save the new graph and observe the changes in the graph and the equation

Display

(When using EL-9650/9600c)

Notice that movement of the basic y = x by 3 units in the direction of the x-axis is equivalent to the

ENTER

ALPHA

addition of 3 to the x -intercept. This demonstrates that movement of the graph to the right means adding an h (>0) in the standard form

2

y = a (x - h)

+ k and movement to the left means

subtracting an h (<0).

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

3-1

Access Change feature and select the equation y = x2.

3-2

3-3

2nd F

SHIFT/CHANGE

1

*

Pinch the slope of the graph.

ENTER

Save the new graph and observe the changes in the graph and the equation.

ENTER

ALPHA

B

*

Notice that pinching or

2

x

closing the basic y =

graph

is equivalent to increasing an a (>1) within the standard

2

form y = a (x - h)

+ k and broadening the graph is equivalent to decreasing an a (<1).

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

The Shift/Change feature of the EL-9650/9600c/9450/9400 allows visual understanding of how graph changes affect the form of quadratic equations.

2-2

EL-9650/9600c Graphing Calculator

Solving a Literal Equation Using the Equation Method

(Amortization)

The Solver mode is used to solve one unknown variable by inputting known variables, by three methods: Equation, Newton’s, and Graphic. The Equation method is used when an exact solution can be found by simple substitution.

Example

Solve an amortization formula. The solution from various values for known variables can be easily found by giving values to the known variables using the Equation method in the Solver mode.

-1

-N

The formula : P = L

1-(1+ )

12

I / 12

I

P= monthly payment L= loan amount

I= interest rate N=number of months

1. Find the monthly payment on a $15,000 car loan, made at 9% interest over four

years (48 months) using the Equation method.

2. Save the formula as “AMORT”.

3. Find amount of loan possible at 7% interest over 60 months with a $300

payment, using the saved formula.

Before

Starting

1-1

1-2

1-3

There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value and delete all data.

*Use either pen touch or cursor to operate.

Access the Solver feature.

Display

This screen will appear a few

NotesStep & Key Operation

seconds after “SOLVER” is dis-

2nd F

SOL VER

played.

Select the Equation method for solving.

SOL VER

2nd F

1

*

A

*

Enter the amortization formula.

SOL VER

2nd F

a

/b+(1

=

PL

—

ALPHA

(

1

ALPHA

b

a

ALPHA

b

a

/b

1

(-)

ALPHA

N

I

(-)

1

÷

1

a

1

2

)

*

)

*

3-1

EL-9650/9600c Graphing Calculator

Notes

1-4

Step & Key Operation

*Use either pen touch or cursor to operate.

Enter the values L=15,000,

Display

I=0.09, N=48.

15000

*

•09 4

*

The monthly payment (P) is

1-5

ENTER

ENTER ENTER

ENTER

8

Solve for the payment(P).

$373.28.

( )

CL

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

2-1

Save this formula.

*

2nd F

SOL VER

2nd F EXE

*

ENTER

C

*

2-2

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

3-1

3-2

Give the formula the name AMORT.

AMOR T

ENTER

Recall the amortization formula.

SOL VER

2nd F

01

B

*

Enter the values: P = 300, I = 0.01, N = 60

3-3

ENTER ENTER ENTER

•01 61

Solve for the loan (L).

300

*

0

ENTERENTER

*

The amount of loan (L) is $17550.28.

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

2nd F EXE

*

With the Equation Editor , the EL-9650/9600c displays equations, even complicated ones, as they appear in the textbook in easy to understand format. Also it is easy to find the solution for unknown variables by recalling a stored equation and giving values to the known variables in the Solver mode when using the EL-9650/9600c.

3-1

EL-9650/9600c Graphing Calculator

Solving a Literal Equation Using the Graphic Method

(Volume of a Cylinder)

The Solver mode is used to solve one unknown variable by inputting known variables. There are three methods: Equation, Newton’s, and Graphic. The Equation method is used when an exact solution can be found by simple substitution. Newton’s method implements an iterative approach to find the solution once a starting point is given. When a starting point is unavailable or multiple solutions are expected, use the Graphic method. This method plots the left and right sides of the equation and then locates the intersection(s).

Example

Use the Graphic method to find the radius of a cylinder giving the range of the unknown variable.

The formula : V = πr2h ( V = volume r = radius h = height)

1. Find the radius of a cylinder with a volume of 30in

3

and a height of 10in, using

the Graphic method.

2. Save the formula as “V CYL”.

3. Find the radius of a cylinder with a volume of 200in

using the saved formula.

3

and a height of 15in,

Before

Starting

1-1

1-2

1-3

There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value and delete all data.

*Use either pen touch or cursor to operate.

Access the Solver feature.

2nd F

SOL VER

Select the Graphic method for solving.

2nd F

SOL VER

A

*

3

*

Enter the formula V = πr2h.

V

2

ALPHA

x

R

H

2nd F

=ALPHA ALPHA ALPHA

π

Display

NotesStep & Key Operation

This screen will appear a few seconds after “SOLVER” is displayed.

1-4

Enter the values: V = 30, H = 10. Solve for the radius (R).

ENTER

0

30

ENTER

*

2nd F EXE

*

1

*

3-2

EL-9650/9600c Graphing Calculator

30

10

Notes

π

3

π

Step & Key Operation

*Use either pen touch or cursor to operate.

1-5

Set the variable range from 0 to 2.

0

ENTER

*

2

ENTER

Display

The graphic solver will prompt with a variable range for solving.

r2 = = <3 r =1 ➞ r2 = 12 = 1 <3

r =2 ➞ r2 = 22 = 4 >3

Use the larger of the values to be safe.

1-6

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

Solve.

2nd F EXE

(

CL

)

The solver feature will graph the left side of the equation (volume, y = 30), then the right side of the equation (y = 10r

2

and finally will calculate the intersection of the two graphs to find the solution. The radius is 0.98 in.

),

2

Save this formula. Give the formula the name “V CYL”.

*

2nd F

SOL VER

SPACE

VY

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

3-1

3-2

Recall the formula. Enter the values: V = 200, H = 15.

*

2nd F

SOL VER

ENTER ENTER ENTER

15

ENTER

Solve the radius setting the variable range from 0 to 4.

ENTER

C

L

B01

*

ENTER

*

0020

200π 14

r2 = = < 14

15

π

r = 3 ➞ r2 = 32 = 9 < 14

2nd F EXE

*

ENTER 2nd F EXE

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

ENTER

04

r = 4 ➞ r2 = 42 = 16 > 14

Use 4, the larger of the values, to be safe. The answer is : r = 2.06

One very useful feature of the calculator is its ability to store and recall equations. The solution from various values for known variables can be easily obtained by recalling an equation which has been stored and giving values to the known variables. The Graphic method gives a visual solution by drawing a graph.

3-2

EL-9650/9600c Graphing Calculator

Solving a Literal Equation Using Newton's Method (Area of a Trapezoid)

The Solver mode is used to solve one unknown variable by inputting known variables. There are three methods: Equation, Newton’s, and Graphic. The Newton’s method can be used for more complicated equations. This method implements an iterative approach to find the solution once a starting point is given.

Example

Find the height of a trapezoid from the formula for calculating the area of a trapezoid using Newton’s method.

The formula : A= h(b+c)

1

2

(A = area h = height b = top face c = bottom face)

1. Find the height of a trapezoid with an area of 25in

2

and bases of length 5in

and 7in using Newton's method. (Set the starting point to 1.)

2. Save the formula as “A TRAP”.

3. Find the height of a trapezoid with an area of 50in

using the saved formula. (Set the starting point to 1.)

Before

Starting

1-1

1-2

There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value and delete all data.

*Use either pen touch or cursor to operate.

Access the Solver feature.

2nd F

SOL VER

Select Newton's method for solving.

Display

2

with bases of 8in and 10in

NotesStep & Key Operation

This screen will appear a few seconds after “SOLVER” is displayed.

1-3

1-4

2nd F

SOL VER

A

*

2

*

Enter the formula A = h(b+c).

ALPHA

A

ALPHA

H

)

C

Enter the values: A = 25, B = 5, C = 7

ENTER

*

2

5

=

( +

ALPHA ALPHA

ENTER

5

ENTER ENTER

*

7

1

2

1a/b

B

*

2

*

3-3

EL-9650/9600c Graphing Calculator

Step & Key Operation

*Use either pen touch or cursor to operate.

1-5 Solve for the height and enter a

starting point of 1.

Display

Newton's method will prompt with a guess or a

Notes

starting point.

1-6

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

2

Solve.

2nd F EXE

Save this formula. Give the formula

2nd F EXE

*

CL

(

)

1

ENTER

The answer is : h = 4.17

the name “A TRAP”.

*

2nd F

SOL VER

SPACE

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

ENTER

C

APA

R

T

ENTER

3-1

Recall the formula for calculating the area of a trapezoid.

3-2

2nd F

01

Enter the values:

SOL VER

B

*

A = 50, B = 8, C = 10.

ENTER

50 10

ENTER

3-3

ENTER

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

2nd F EXE

*

2nd F EXE

ENTER

*

8

*

The answer is : h = 5.56Solve.

1

One very useful feature of the calculator is its ability to store and recall equations. The solution from various values for known variables can be easily obtained by recalling an equation which has been stored and giving values to the known variables in the Solver mode. If a starting point is known, Newton's method is useful for quick solution of a complicated equation.

3-3

EL-9650/9600c Graphing Calculator

G raphing Polynomials and Tracing to F ind the R oots

A polynomial y = f (x) is an expression of the sums of several terms that contain different powers of the same originals. The roots are found at the intersection of the x-axis and the graph, i. e. when y = 0.

Example

Draw a graph of a polynomial and approximate the roots by using the Zoom-in and Trace features.

1. Graph the polynomial y = x

2. Approximate the left-hand root.

3. Approximate the middle root.

4. Approximate the right-hand root.

3

- 3x2 + x + 1.

Before

Starting

1-1

1-2

There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value and delete all data.

Set the zoom to the decimal window:

Setting the zoom factors to 5 : *

*Use either pen touch or cursor to operate.

Enter the polynomial

3

y = x

- 3x2 + x + 1.

Y= EZ

5

*

ENTER ENTER ENTER

*

Enter the coefficients.

2nd F SUB

1

ENTER

* *

1

(-)

3

ENTER

*

ZOOM

*

ZOOM

B

ENTER

*

ENTER ENTER ENTER

Display

ALPHA

55

(

A

)

7

*

2nd F

QUIT

NotesStep & Key Operation

It may take few seconds for the graph to be drawn. Enter each coefficients when the cursor is displayed.

1-3

Return to the equation display screen.

2nd F EXE

1-4

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

View the graph.

GRAPH

4-1

EL-9650/9600c Graphing Calculator

Step & Key Operation

*Use either pen touch or cursor to operate.

Display

Notes

Tracer

2-1

Move the tracer near the left-hand root.

Note that the tracer is flashing on the curve and the x and y coordinates are shown at the bottom of the screen.

2-2

TRACE

Zoom in on the left-hand root.

ZOOM

(repeatedly)

*

A3

* *

Tracer

2-3

Move the tracer to approximate the

The root is : x -0.42

root.

TRACE

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

3-1

Return to the previous decimal

or

*

(repeatedly)

*

viewing window.

ZOOM

H

*

3-2

2

*

Move the tracer to approximate the middle root.

Tracer

The root is exactly x = 1. (Zooming is not needed to find a better approximate.)

TRACE

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

(repeatedly)

*

Tracer

4

Move the tracer near the right-

The root is : x

2.42 hand root. Zoom in and move the tracer to find a better approximate.

(repeatedly)

*

ZOOM

A3

* *

TRACE

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

or

*

(repeatedly)

*

4-1

The calculator allows the roots to be found (or approximated) visually by graphing a polynomial and using the Zoom-in and Trace features.

EL-9650/9600c/9450/9400 Graphing Calculator

G raphing P olynomials and J umping to F ind the Roots

A polynomial y = f (x) is an expression of the sums of several terms that contain different powers of the same originals. The roots are found at the intersection of the x-axis and the graph, i. e. when y = 0.

Example

Draw a graph of a polynomial and find the roots by using the Calculate feature.

1. Graph the polynomial y = x

2. Find the four roots one by one.

4

+ x3 - 5x2 - 3x + 1.

Before

Starting

*Use either pen touch or cursor to operate.

1-1

1-2

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value and delete all data.

Setting the zoom factors to 5 : *

Step & Key Operation

(When using EL-9650/9600c)

ZOOM

A

ENTER ENTER ENTER 2nd F

AA

Display

(When using EL-9650/9600c)

Notes

Enter the polynomial

4

y = x

+ x3 - 5x2 - 3x + 1

X/

Y= 4

b

35

a

—

b

/T/n X/

a

—

*

X/

/T/n

31

+

*

X/

/T/n

2

x

View the graph.

GRAPH

QUIT

2-1

2-2

Find the first root.

2nd F CALC

5

*

Find the next root.

2nd F CALC

5

*

-2.47

x

Y is almost but not exactly zero. Notice that the root found here is an approximate value.

x -0.82

4-2

EL-9650/9600c/9450/9400 Graphing Calculator

Step & Key Operation

(When using EL-9650/9600c)

*Use either pen touch or cursor to operate.

2-3

2-4

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

Find the next root.

2nd F CALC

5

*

Find the next root.

2nd F CALC

5

*

Display

(When using EL-9650/9600c)

x 0.24

x

Notes

2.05

The calculator allows jumping to find the roots by graphing a polynomial and using the Calculate feature, without tracing the graph.

4-2

EL-9650/9600c/9450/9400 Graphing Calculator

Solving a System of Equations by Graphing or Tool Feature

A system of equations is made up of two or more equations. The calculator provides the Calculate feature and Tool feature to solve a system of equations. The Calculate feature finds the solution by calculating the intersections of the graphs of equations and is useful for solving a system when there are two variables, while the T ool feature can solve a linear system with up to six variables and six equations.

Example

Solve a system of equations using the Calculate or Tool feature. First, use the Calculate feature. Enter the equations, draw the graph, and find the intersections. Then, use the Tool feature to solve a system of equations.

1. Solve the system using the Calculate feature.

y = x2 - 1

{

y = 2x

2. Solve the system using the Tool feature.

Before

Starting

1-1

1-2

5x + y = 1

{

-3x + y = -5

There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default and delete all data.

Choose the viewing window “-5 < X < 5”, “-10 < Y < 10” using Rapid window feature

*

WINDOW

EZ

As the Tool feature is only available on the EL-9650/9600c, the example 2 does not apply to the EL-9450/9400.

(When using EL-9650/9600c)

*Use either pen touch or cursor to operate.

Enter the system of equations

2

y = x

- 1 for Y1 and y = 2x for Y2.

/T/n

2

x

X/

Y= 2

X/

/T/n

View the graphs.

GRAPH

ENTER ENTER ENTER

574

—1

( ) *

ALPHA

*

(When using EL-9650/9600c)

ENTER

*

Display

* *

*

NotesStep & Key Operation

1-3

The find the left-hand intersection using the Calculate feature.

1-4

2nd F CALC

Find the right-hand intersection by

2

*

accessing the Calculate feature again.

2nd F CALC

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

2

*

Note that the x and y coordinates are shown at the bottom of the screen. The answer is : x = -0.41 y = -0.83

The answer is : x = 2.41 y = 4.83

5-1

EL-9650/9600c/9450/9400 Graphing Calculator

2-1

2-2

2-3

Step & Key Operation

(When using EL-9650/9600c)

*Use either pen touch or cursor to operate.

Display

(When using EL-9650/9600c)

Access the Tool menu. Select the number of variables.

*

2nd F

TOOL

B

2

*

Enter the system of equations.

511

ENTER ENTER ENTER

(-)(

31 5

ENTER ENTER

ENTER

)

-

Solve the system.

EXE2nd F

Notes

Using the system function, it is possible to solve simultaneous linear equations. Systems up to six variables and six equations can be solved.

x = 0.75 y = -2.75

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

A system of equations can be solved easily by using the Calculate feature or Tool feature.

5-1

EL-9650/9600c Graphing Calculator

Entering and Multiplying Matrices

A matrix is a rectangular array of elements in rows and columns that is treated as a single element. A matrix is often used for expressing multiple linear equations with multiple variables.

Example

Enter two matrices and execute multiplication of the two.

1. Enter a 3x3 matrix A

2. Enter a 3x3 matrix B

3. Multiply the matrices A and B

A 1 2 1 2 1 -1 1 1 -2

B 1 2 3 4 5 6 7 8 9

Before

Starting

1-1

1-2

1-3

There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value and delete all data.

Step & Key Operation

*Use either pen touch or cursor to operate.

Access the matrix menu.

MATRIX

B

*

1

*

Set the dimension of the matrix at three rows by three columns.

33

ENTER ENTER

Enter the elements of the first row, the elements of the second row, and the elements of the third row.

Display

Notes

121

ENTER ENTER ENTER

21 1

ENTER ENTER

11 2

ENTER ENTER ENTER

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

(-)

ENTER

6-1

EL-9650/9600c Graphing Calculator

Step & Key Operation

*Use either pen touch or cursor to operate.

2

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

3-1

Enter a 3x3 matrix B.

* *

MATRIX

B

23

12

ENTER ENTER ENTER

456

ENTER ENTER ENTER

789

ENTER ENTER ENTER

ENTER ENTER

3

Multiply the matrices A and B together at the home screen.

Display

Matrix multiplication can be performed if the number of col-

Notes

umns of the first matrix is equal

* *

MATRIX MATRIX

A1

X

to the number of rows of the second matrix. The sum of these

2

* *

ENTER

multiplications (1

.

1 + 2.4 + 1.7)

A

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 by each column of B.

3-2

Delete the input matrices for future use.

2nd F

OPTION

C

*

ENTER

2

*

2nd F

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

ENTER

QUIT

Matrix multiplication can be performed easily by the calculator.

6-1

EL-9650/9600c Graphing Calculator

Solving a S ystem of Linear Equations U sing Matrices

Each system of three linear equations consists of three variables. Equations in more than three variables cannot be graphed on the graphing calculator . The solution of the system of equations can be found numerically using the Matrix feature or the System solver in the Tool feature. A system of linear equations can be expressed as AX = B (A, X and B are matrices). The solution matrix X is found by multiplying A and the correct answer will be obtained by multiplying BA-1. An inverse matrix A-1 is a matrix that when multiplied by A results in the identity matrix I (A matrix I is defined to be a square matrix (nxn) where each position on the diagonal is 1 and all others are 0.

Example

Use matrix multiplication to solve a system of linear equations.

-1

B. Note that the multiplication is “order sensitive”

-1

x A=I). The identity

1. Enter the 3x3 identity matrix in matrix A.

2. Find the inverse matrix of the matrix B.

3. Solve the equation system.

x + 2y + z = 8 2x + y - z = 1

{

x + y - 2z = -3

Before

Starting

*Use either pen touch or cursor to operate.

1-1

There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value and delete all data.

Set up 3x3 identity matrix at the home screen.

* *

MATRIX

05 3

C

ENTER

Display

B 1 2 1 2 1 -1 1 1 -2

NotesStep & Key Operation

1-2

1-3

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

Save the identity matrix in matrix A.

*

MATRIX

STO

Confirm that the identity matrix is stored in matrix A.

*

MATRIX

A

B1

*

1

ENTER

*

6-2

EL-9650/9600c Graphing Calculator

Notes

2-1

2-2

Step & Key Operation

*Use either pen touch or cursor to operate.

Enter a 3x3 matrix B.

* *

MATRIX

1

2

ENTER

ENTER ENTER

11

2

B

ENTER ENTER

2 1

ENTER

3

ENTER

1

(-)

3

ENTER

1

2

Exit the matrix editor and find the inverse of the square matrix B.

Display

ENTER

Some square matrices have no inverse and will generate error statements when calculating the inverse.

2nd F

* *

MATRIX

(repeatedly)

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

CL

QUIT

A2

2nd F

-1

ENTER

x

-0.17 0.83 -0.5

0.5 -0.5 0.5

B-1=

0.17 0.17 -0.5

3-1

Enter the constants on the right side of the equal sign into matrix C (3x 1).

* *

MATRIX

8

ENTER ENTER ENTER

33 1

B

1

ENTER ENTER

(-)

3

The system of equations can be expressed as

1 2 1 2 1 -1 1 1 -2

x

y z

=

8 1

-3

Let each matrix B, X, C : BX = C

-1

BX = B-1C (multiply both

3-2

Calculate B-1C.

B sides by B I = B X = B

The 1 is the x coordinate, the 2

-1

)

-1 (B-1

B = I, identity matrix)

-1

C

the y coordinate, and the 3 the z

3-3

*

* *

2nd F

Delete the input matrices for future

CL

x

MATRIX

-1

X

A

MATRIX

2

*

A3

ENTER

coordinate of the solution point. (x, y, z)=(1, 2, 3)

use.

2nd F

OPTION

C

*

2

ENTER

*

QUIT2nd F

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

6-2

The calculator can execute calculation of inverse matrix and matrix multiplication. A system of linear equations can be solved easily using the Matrix feature.

EL-9650/9600c Graphing Calculator

Solving Inequalities

To solve an inequality, expressed by the form of f(x) f(x)

≥

g(x), means to find all values that make the inequality true.

≤

0, f(x) ≥ 0, or form of f (x) ≤

g(x),

There are two methods of finding these values for one-variable inequalities, using graphical techniques. 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. For example, to find the solution to f (x) < 0, determine where the graph of f (x) is below the x-axis. The second method involves graphing each side of the inequality as an individual function. For example, to find the solution to f(x) < g(x), determine where the graph of f(x) is below the graph of g(x).

Example

Solve an inequality in two methods.

1.

Solve 3(4 - 2x) ≥ 5 - x, by rewriting the right-hand side of the inequality as 0.

2.

Solve 3(4 - 2x) ≥ 5 - x, by shading the solution region that makes the inequality true.

Before

Starting

*Use either pen touch or cursor to operate.

There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value and delete all data.

Display

NotesStep & Key Operation

1-1

1-2

1-3

≥

Rewrite the equation 3(4 - 2x)

5 - x so that the right-hand side becomes 0, and enter y = 3(4 - 2x) - 5 + x for Y1.

Y=

5

— +

—

X/

/T/n

()

342

X/

/T/n

View the graph.

GRAPH

Find the location of the x-intercept and solve the inequality.

2nd F CALC

5

*

3(4 - 2x) ≥ 5 - x ➞ 3(4 - 2x) - 5 + x ≥ 0

The x-intercept is located at the point (1.4, 0). Since the graph is above the x-axis to the left of the x-intercept, the solution to the in-

≥

equality 3(4 - 2x) - 5 + x

0 is

all values of x such that

≤

1.4.

x

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

7-1

EL-9650/9600c Graphing Calculator

2-1

2-2

2-3

*Use either pen touch or cursor to operate.

Enter y = 3(4 - 2x) for Y1 and y = 5 - x for Y2.

Y=

ENTER

(7 times) (4 times)

*

5

*

—

X/

/T/n

DEL

View the graph.

GRAPH

Access the Set Shade screen.

2nd F

DRAW

G

*

1

*

Display

NotesStep & Key Operation

2-4

Set up the shading.

—

——

*

Since the inequality being solved is Y1

≥

Y2, the solution is where the graph of Y1 is “on the top” and Y2 is “on the bottom.”

2-5

2-6

View the shaded region.

GRAPH

Find where the graphs intersect and solve the inequality.

The point of intersection is (1.4, 3.6). Since the shaded region is to the left of x = 1.4,

2nd F CALC

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

2

*

the solution to the inequality

≥

3(4 - 2x) of x such that x

5 - x is all values

≤

1.4.

Graphical solution methods not only offer instructive visualization of the solution process, but they can be applied to inequalities that are often difficult to solve algebraically. The EL-9650/9600c allows the solution region to be indicated visually using the Shade feature. Also, the points of intersection can be obtained easily.

7-1

EL-9650/9600c/9450/9400 Graphing Calculator

Solving D ouble 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 f (x) ≥ a, f (x) ≤ b, where the same expression appears on both inequalities, is commonly referred to as a “double” inequality and is often written in the form a ≤ f (x) ≤ b. Be certain that both inequality signs are pointing in the same direction and that the double inequality is only used to indicate an expression in x “trapped” in between two values. Also a must be less than or equal to b in the inequality a ≤ f (x) ≤ b or b ≥ f (x) ≥ a.

Example

Solve a double inequality, using graphical techniques.

2x - 5 ≥ -1 2x -5 ≤ 7

Before

Starting

*Use either pen touch or cursor to operate.

1

Enter y = -1 for Y1, y = 2x - 5 for

There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value and delete all data.

(When using EL-9650/9600c)

Y2, and y = 7 for Y3.

(-)

Y=

2

X/

/T/n

View the lines.

2

GRAPH

3

Find the point of intersection.

2nd F CALC

—

2

1

ENTER

5

*

ENTER

Display

(When using EL-9650/9600c)

NotesStep & Key Operation

The “double” inequality given can also be written to

≤

2x - 5 ≤ 7.

-1

7

*

y = 2x - 5 and y = -1 intersect at (2, -1).

7-2

EL-9650/9600c/9450/9400 Graphing Calculator

NotesStep & Key Operation

≤

2x - 5 ≤ 7 con-

(When using EL-9650/9600c)

*Use either pen touch or cursor to operate.

Move the tracer and find another

4

intersection.

2nd F CALC

5

Solve the inequalities.

2

Display

(When using EL-9650/9600c)

y = 2x - 5 and y = 7 intersect at (6,7).

*

The solution to the “double” inequality -1 sists of all values of x in between, and including, 2 and 6

≥

(i.e., x lution is 2

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

2 and x ≤ 6). The so-

≤

x ≤ 6.

Graphical solution methods not only offer instructive visualization of the solution process, but they can be applied to inequalities that are often difficult to solve algebraically. The EL-9650/9600c/9450/9400 allows the solution region to be indicated visually using the Shade feature. Also, the points of intersection can be obtained easily.

7-2

EL-9650/9600c/9450/9400 Graphing Calculator

System 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. To solve two-variable inequalities, the inequalities must be manipulated to isolate the y variable and enter the other side of the inequality as a function. The calculator will only accept functions of the form y = . (where y is defined explicitly in terms of x).

Example

Solve a system of two-variable inequalities by shading the solution region.

2x + y ≥ 1 x2 + y ≤ 1

Before

Starting

There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value and delete all data.

Set the zoom to the decimal window:

Step & Key Operation

(When using EL-9650/9600c)

*Use either pen touch or cursor to operate.

1

Rewrite each inequality in the system so that the left-hand side is y :

2 Enter y = 1 - 2x for Y1 and y = 1 - x

for Y2.

—

DRAW

*

—

X/

/T/n

G

*

ENTER

X/

/T/n

2

x

*

Y= 1 2

1

Access the set shade screen

3

2nd F

1

*

Shade the points of y -value so that

4

≤

y ≤ Y2.

Y1

———

(

ZOOM

ENTER

A

*

Display

(When using EL-9650/9600c)

2

2nd F

)

7

*

Notes

≥

2x + y

2

x

1 ➞ y ≥ 1 - 2x

+ y ≤ 1 ➞ y ≤ 1 - x

2

Graph the system and find the

5

intersections.

GRAPH

*

22

CALC2nd F CALC2nd F

*

6 Solve the system.

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

The intersections are (0, 1) and (2, -3)

The solution is 0

≤

x ≤ 2.

Graphical solution methods not only offer instructive visualization of the solution process, but they can be applied to inequalities that are often difficult to solve algebraically. The EL-9650/9600c/9450/9400 allows the solution region to be indicated visually using the Shade feature. Also, the points of intersection can be obtained easily.

7-3

EL-9650/9600c/9450/9400 Graphing Calculator

G raphing S olution R egion of Inequalities

The solution region of an inequality consists of all points (a, b) such that when x = a, and y = b, all inequalities are true.

Example

Check to see if given points are in the solution region of a system of inequalities.

1. Graph the solution region of a system of inequalities:

x + 2y ≤ 1 x2 + y ≥ 4

2. Which of the following points are within the solution region?

(-1.6, 1.8), (-2, -5), (2.8, -1.4), (-8,4)

Before

Starting

1-1

1-2

1-3

There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value and delete all data.

(When using EL-9650/9600c)

*Use either pen touch or cursor to operate.

Rewrite the inequalities so that the left-hand side is y.

Enter y = for Y1 and y = 4 - x

Y=

1-x

2

for Y2.

a

/b

1

ENTER

24

*

X/

/T/n

—

— x

X/

/T/n

Set the shade and view the solution region.

*

2nd F

DRAW

1

G

Display

(When using EL-9650/9600c)

2

NotesStep & Key Operation

x + 2y

≤

1 ➞ y

1-x

≤

x2+y ≥ 4 ➞ y ≥ 4 - x

Y2 ≤ y ≤ Y1

2

2-1

7-4

—— —

GRAPH

*

Set the display area (window) to :

-9 < x < 3, -6 < y < 5.

WINDOW

(-)

93

(-)

ENTER

65

ENTERENTER

ENTER ENTER

EL-9650/9600c/9450/9400 Graphing Calculator

2-2

2-3

(When using EL-9650/9600c)

*Use either pen touch or cursor to operate.

Use the cursor to check the position of each point. (Zoom in as necessary).

or or or

GRAPH

Substitute points and confirm whether they are in the solution region.

(-)

X

2

(Continuing key operations omitted.)

•

16

•

1

+

...

8

Display

(When using EL-9650/9600c)

NotesStep & Key Operation

Points in the solution region are (2.8, -1.4) and (-8, 4). Points outside the solution region are (-1.6, 1.8) and (-2, -5).

.

(-1.6, 1.8): -1.6 + 2 ✕ 1.8 = 2

➞ This does not materialize.

.

(-2, -5): -2 + 2 ✕ (-5) = -12

➞ This does not materialize.

.

(2.8, -1.4): 2.8 + 2 ✕ (-1.4) = 0

➞ This materializes.

.

(-8, 4): -8 + 2 ✕ 4 = 0

➞ This materializes.

2

+ (-5) = -1

(-2)

2

+ (-1.4) = 6.44

(2.8)

2

+ 4 = 68

(-8)

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

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. The EL-9650/9600c/9450/9400 allows the solution region to be indicated visually using the Shading feature. Also, the free-moving tracer or Zoom-in feature will allow the details to be checked visually.

7-4

EL-9650/9600c Graphing Calculator

S lope and I nter cept of A bsolute Value Functions

The absolute value of a real number x is defined by the following:

|x| = x if x ≥ 0

-x if x ≤ 0 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. An absolute value function can be presented as y = a|x - h| + k. The graph moves as the changes of slope a, x-intercept h, and y-intercept k.

Example

Consider various absolute value functions and check the relation between the graphs and the values of coefficients.

1. Graph y = |x|

2. Graph y = |x -1| and y = |x|-1 using the Rapid Graph feature.

Before

Starting

1-1

1-2

*Use either pen touch or cursor to operate.

Enter the function y =|x| for Y1.

View the graph.

There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value and delete all data.

Set the zoom to the decimal window: * (

Step & Key Operation

* *

MATH

Y=

B

1

X/

/T/n

ZOOM

A

Display

ENTER 2nd F

)

7

*

Notes

Notice that the domain of f(x) = |x| is the set of all real num-

GRAPH

bers and the range is the set of non-negative real numbers. Notice also that the slope of the graph is 1 in the range of X > 0

≤

and -1 in the range of X

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

2-1

Enter the standard form of an abso-

0.

lute value function for Y2 using the Rapid Graph feature.

2-2

8-1

*

Y=

ENTER

*

EZ

8

ENTER ENTER

*

Substitute the coefficients to graph y = |x - 1|.

2nd F SUB

ENTER

0

ENTER ENTER

11

*

EL-9650/9600c Graphing Calculator

2-3

2-4

2-5

*Use either pen touch or cursor to operate.

View the graph.

GRAPH

Change the coefficients to graph y =|x|-1.

Y=

ENTER ENTER

View the graph.

GRAPH

2nd F SUB ENTER

(-)

1

Display

NotesStep & Key Operation

Notice that placing an h(>0) within the standard form y = a|x - h|+ k will move the graph right h units on the x- axis.

Notice that adding a k(>0) within the standard form y=a|x-h|+k will move the graph up k units on the y-axis.

The EL-9650/9600c shows absolute values with | |, just as written on paper, by using the Equation editor. Use of the calculator allows various absolute value functions to be graphed quickly and shows their characteristics in an easy-to-understand manner.

8-1

EL-9650/9600c/9450/9400 Graphing Calculator

Shifting a graph of A bsolute Value Functions

The absolute value of a real number x is defined by the following:

|x| = x if x ≥ 0

-x if x ≤ 0 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. An absolute value function can be presented as y = a|x - h|+ k. The graph moves as the changes of slope a, x-intercept h, and y-intercept k.

Example

Move and change graphs of absolute value function y =|x| to check the relation between the graphs and the values of coefficients.

1. Move the graph y = |x| downward by 2 using the Shift feature.

2. Move the graph y = |x| to the right by 2 using the Shift feature.

3. Pinch the slope of y = |x| to 2 or minus using the Change feature.

Before

Starting

*Use either pen touch or cursor to operate.

There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value and delete all data.

(When using EL-9650/9600c)

Display

(When using EL-9650/9600c)

1-1 Access the Shift feature.

Select y = |x|.

2nd F

1-2

SHIFT/CHANGE

(

ENTER

Move the graph downward by 2.

ALPHA

A

*

)

8

*

y =|x|changes to y = |x|-2

NotesStep & Key Operation

1-3

8-2

Save the new graph and look at the relationship of the function and the graph.

ENTER

ALPHA

*

The graph of the equation that is highlighted is shown by a solid line. Notice that the y- intercept k in the standard form y = a|x - h|+ k takes charge of vertical movement of the graph.

EL-9650/9600c/9450/9400 Graphing Calculator

Display

(When using EL-9650/9600c)

*Use either pen touch or cursor to operate.

2

-

1

Move the original graph to the right by 2.

ALPHA

2

-2

Save the new graph and look at the relationship of the function and the graph.

ENTER

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

3

Access the Change feature.

-

1

2nd F

ALPHA

SHIFT/CHANGE

*

B

*

ENTER

*

(When using EL-9650/9600c)

y = |x| changes to y = |x-2|

*

Notice that the function h in the standard form y = a|x - h|+ of horizontal movement of the graph.

NotesStep & Key Operation

k takes charge

3

-

2

Select y = |x|.

3

*

3

-

3

Make the slope of the graph steeper. Save the new graph.

ENTER

3

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

Make the slope of the graph minus.

-

4

Save the new graph.

ENTER

-

5

Look at the relationship of the function and the graph.

ALPHA

*

ENTER

*

y = |x|➞ y = 2|x|

y = |x|➞ y = - |x|

Notice that the coefficient a in the standard form y = a |x - h| + k takes charge of changing the slope.

EL-9650/9600c/9450/9400 shows absolute values with | |, just as written on paper , by using the Equation editor. Use of the calculator allows various absolute value functions to be graphed quickly and shows their characteristics in an easy-to-understand manner. The Shift/Change feature of the EL-9650/9600c/9450/9400 allows visual understanding of how graph changes affect the form of absolute value functions.

8-2

EL-9650/9600c/9450/9400 Graphing Calculator

Solving A bsolute Value Equations

The absolute value of a real number x is defined by the following:

|x| = x if x ≥ 0

-x if x 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.

Example

Solve an absolute value equation |5 - 4x| = 6

≤

0

Before

Starting

*Use either pen touch or cursor to operate.

1

Enter y = |5 - 4x| for Y1.

There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value and delete all data.

(When using EL-9650/9600c)

Enter y = 6 for Y2.

* *

Y=

MATH

B1

X/

/T/n

2

View the graph.

GRAPH

Find the points of intersection of

3

ENTER

6

*

54

the two graphs and solve.

2nd F CALC

2

*

2

*

—

Display

(When using EL-9650/9600c)

NotesStep & Key Operation

There are two points of intersection of the absolute value graph and the horizontal line y = 6.

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.

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

The EL-9650/9600c/9450/9400 shows absolute values with | |, just as written on paper, by using the Equation editor. The graphing feature of the calculator shows the solution of the absolute value function visually.

8-3

EL-9650/9600c/9450/9400 Graphing Calculator

Solving A bsolute Value Inequalities

To solve an inequality means to find all values that make the inequality true. Absolute value inequalities are of the form | solution to an absolute value inequality is found using the same methods as for normal inequalities. 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. The second method involves graphing each side of the inequality as an individual function.

Example

Solve absolute value inequalities in two methods.

f (x)

|<

k, |f (x)

|≤

k, |f (x)

|>

k,

or |f (x)

|≥ k. The graphical

1. Solve 20 - < 8 by rewriting the inequality so that the right-hand side of

6x

5

the inequality is zero.

2. Solve 3.5x + 4 > 10 by shading the solution region.

Before

Starting

1-1

1-2

*Use either pen touch or cursor to operate.

Rewrite the equation.

Enter y = |20 - | - 8 for Y1.

There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value and delete all data.

Set viewing window to “-5< x <50,” and “-10< y <10” using Rapid Window feature to solve Q1.

WINDOW

EZ333

(When using EL-9650/9600c)

ENTER ENTER ENTER

*

(When using EL-9650/9600c)

*

Display

NotesStep & Key Operation

6x

|20 - |< 8 ➞|20 - | - 8 < 0.

5

6x

5

6x

5

* *

Y=

B520

MATH

—

a

/b

1-3

1-4

6

X/

/T/n

8

—

View the graph, and find the x-intercepts.

GRAPH

2nd F CALC

5

y = 0.00000006 ( Note)

Solve the inequality.

5

*

➞ x = 10, y = 0

*

➞ x = 23.33333334

*

The intersections with the x- axis are (10, 0) and (23.3, 0)

Note: The value of y in the

( x-intercepts may not appear exactly as 0 as shown in the example, due to an error caused by approximate calculation.)

Since the graph is below the x-axis for x in between the two x-intercepts, the solution is 10 < x < 23.3.

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

8-4

EL-9650/9600c/9450/9400 Graphing Calculator

2-1

2-2

2-3

(When using EL-9650/9600c)

*Use either pen touch or cursor to operate.

Enter the function y =|3.5x + 4|for Y1. Enter y = 10 for Y2.

*

Y= CL

3

•

1

0

MATH

X/

/T/n

54

1

B

*

+

ENTER

Set up shading.

*

2nd F

DRAW

G1

—— —

*

Set viewing window to “-10 < x < 10” and “-5 < y < 50” using Rapid Window feature and view the graph.

WINDOW

ENTER

3

EZ

ENTER ENTER

2

*

5

*

Display

(When using EL-9650/9600c)

*

NotesStep & Key Operation

Since the inequality you are solving is Y1 > Y2, the solution is where the graph of Y2 is “on the bottom” and Y1 in “on the top.”

2

Find the points of intersection.

-

4

Solve the inequality.

The intersections are (-4, 10) and (1.7, 10.0). The solution is all values of x such that

2nd F

CALC

➞ x = -4, y = 10

2

*

➞ x = 1.714285714

2

*

y = 9.999999999 (

Note)

x <-4 or x >1.7.

Note: The value of y in the

( intersection of the two graphs may not appear exactly as 10 as shown in the example, due to an error caused by approximate calculation.)

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

The EL-9650/9600c/9450/9400 shows absolute values with | |, just as written on paper, by using the Equation editor. Graphical solution methods not only offer instructive visualization of the solution process, but they can be applied to inequalities that are often difficult to solve algebraically. The Shade feature is useful to solve the inequality visually and the points of intersection can be obtained easily.

8-4

EL-9650/9600c/9450/9400 Graphing Calculator

E valuating A bsolute Value Functions

The absolute value of a real number x is defined by the following:

|x| = x if x ≥ 0

-x if x ≤ 0 Note that the effect of taking the absolute value of a number is to strip away the minus sign if the number is negative and to leave the number unchanged if it is nonnegative. Thus, |x|≥ 0 for all values of x.

Example

Evaluate various absolute value functions.

1. Evaluate |- 2(5-1)|

2. 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| ?

3. Is | | = ?

6-9

1+3

Evaluate each side of the equation to check your answer. Investigate with more examples, and decide if you think |x / y|=|x|/|y|

|6-9| |1+3|

Before

Starting

*Use either pen touch or cursor to operate.

1-1

1-2

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

Access the home or computation screen.

Enter y = |-2(5-1)| and evaluate.

2-1

There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value and delete all data.

Step & Key Operation

(When using EL-9650/9600c)

*

MATH

B

)

1

Evaluate|-2 + 7|. Evaluate|-2|+|7|.

CL

1

*

ENTER

(-)

2

(

—

5

Display

(When using EL-9650/9600c)

The solution is +8.

|-2 + 7| = 5, |-2| + |7| = 9 ➞|-2 + 7| ≠ |-2| + |7|.

Notes

*

MATH

*

MATH MATH

1

*

7

1

(-)

ENTER

7

ENTER

+

2

8-5

EL-9650/9600c/9450/9400 Graphing Calculator

2-2

(When using EL-9650/9600c)

*Use either pen touch or cursor to operate.

Is |x + y| = |x| +|y|? Think about this problem according to the cases when x or y are positive or negative.

≥

0 and y ≥ 0

If x [e.g.; (x, y) = (2,7)]

≤

0 and y ≥ 0

If x [e.g.; (x, y) = (-2, 7)]

≥

0 and y ≤ 0

If x [e.g.; (x, y) = (2, -7)]

≤

0 and y ≤ 0

If x [e.g.; (x, y) = (-2, -7)]

Display

(When using EL-9650/9600c)

NotesStep & Key Operation

|x +y| = |2 + 7| = 9 |x|+|y| = |2| + |7| = 9 ➞|x + y| = |x| + |y|.

|x +y| = |-2 + 7| = 5 |x|+|y| = |-2| + |7| = 9 ➞|x + y| ≠ |x| + |y|. |x +y| = |2-7| = 5 |x|+|y| = |2| + |-7| = 9 ➞|x + y| ≠ |x| + |y|. |x +y| = |-2-7| = 9 |x|+|y| = |-2| + |-7| = 9

➞|x + y| = |x| + |y|. Therefore |x +y|=|x|+|y|when x

≤

and when x

0 and y ≤ 0.

≥

0 and y ≥ 0,

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

3-1

3-2

Evaluate . Evaluate .

CL

MATH

6-9

1+3

MATH

1

*

1

*

1

a

*

3

—

+

/b

ENTER

3

1

+

69

1

Is |x /y| = |x|/|y|?

6

ENTER

6-9

1+3

—

6-9

= 0.75 , = 0 .75

1+3

9

a

/b

*

6-9

➞ =

1+3

6-9

1+3

6-9

1+3

Think about this problem according to the cases when x or y are positive or negative.

≥

0 and y ≥ 0

If x [e.g.; (x, y) = (2,7)]

|x /y| = |2/7| = 2/7 |x|/|y| = |2| /|7| = 2/7 ➞|x /y| = |x| / |y|

≤

0 and y ≥ 0

If x [e.g.; (x, y) = (-2, 7)]

|x /y| = |(-2)/7| = 2/7 |x|/|y| = |-2| /|7| = 2/7 ➞|x /y| = |x| / |y|

≥

0 and y ≤ 0

If x [e.g.; (x, y) = (2, -7)]

|x /y| = |2/(-7)| = 2/7 |x|/|y| = |2| /|-7| = 2/7 ➞|x /y| = |x| / |y|

≤

0 and y ≤ 0

If x [e.g.; (x, y) = (-2, -7)]

|x /y| = |(-2)/-7| = 2/7 |x|/|y| = |-2| /|-7| = 2/7 ➞|x /y| = |x| / |y| The statement is true for all y ≠ 0.

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

8-5

The EL-9650/9600c/9450/9400 shows absolute values with | |, just as written on paper , by using the Equation editor . The nature of arithmetic of the absolute value can be learned through arithmetical operations of absolute value functions.

EL-9650/9600c/9450/9400 Graphing Calculator

G raphing Rational Functions

A rational function f (x) is defined as the quotient where p (x) and q (x) are two

p (x) q (x)

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 . It also has 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 lar ge.

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).

Example

Graph the rational function and check several points as indicated below.

1.

Graph f (x) = .

2.

Find the domain of f (x), and the vertical asymptote of f (x).

x-1

x2-1

3. Find the x- and y-intercepts of f

4. Estimate the horizontal asymptote of f

Before

Starting

*Use either pen touch or cursor to operate.

1-1

1-2

Enter y = for Y1.

View the graph.

GRAPH

There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value and delete all data.

Set the zoom to the decimal window:

(When using EL-9650/9600c)

x - 1

x2 -1

Y=

a

/b

X/

/T/n

—

1

—

1

X/

/T/n

*

(x)

.

(x)

.

(

A

ZOOM

(When using EL-9650/9600c)

2

x

*

Display

ENTER

ALPHA

*

)

7

*

NotesStep & Key Operation

The function consists of two branches separated by the vertical asymptote.

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

9-1

EL-9650/9600c/9450/9400 Graphing Calculator

NotesStep & Key Operation

x - 1

2

(When using EL-9650/9600c)

*Use either pen touch or cursor to operate.

Find the domain and the vertical asymptote of f (x), tracing the

graph to find the hole at x = 1.

Display

(When using EL-9650/9600c)

Since f (x) can be written as , the domain

(x + 1)(x - 1)

consists of all real numbers x such that x ≠ 1 and x ≠ -1. There is no vertical asymptote

(repeatedly)

TRACE

where x = 1 since this value of x also makes the numerator zero. Next to the coordinates x = 0.9, y = 0.52, see that the calculator does not display a value for y at x = 1 since 1 is not in the domain of this rational function.

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

3

Find the x- and y-intercepts of f (x).

The y-intercept is at (0 ,1). No- tice that there are no x-inter-

2nd F CALC

6

*

cepts for the graph of f (x).

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

4

Estimate the horizontal asymptote of f (x).

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

The line y = 0 is very likely a horizontal asymptote of f (x).

The graphing feature of the EL-9650/9600c/9450/9400 can create the branches of a rational function separated by vertical asymptote. The calculator allows the points of intersection to be obtained easily.

9-1

EL-9650/9600c/9450/9400 Graphing Calculator

Solving Rational F unction I nequalities

A rational function f (x) is defined as the quotient where p (x) and q (x) ar e two

p (x) q (x)

polynomial functions such that q (x) ≠ 0. The solutions to a rational function inequality can be obtained graphically using the same method as for normal inequalities. You can find the solutions by graphing each side of the inequalities as an individual function.

Example

Solve a rational inequality.

x

Solve ≤ 2 by graphing each side of the inequality as an individual function.

Before

Starting

Step & Key Operation

*Use either pen touch or cursor to operate.

2

1 - x

There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value and delete all data.

Set the zoom to the decimal window:

(When using EL-9650/9600c)

(

A

ZOOM

*

ENTER

Display

(When using EL-9650/9600c)

ALPHA

)

7

*

Notes

1 Enter y = for Y1. Enter y = 2

for Y2.

Y=

12

2 Set up the shading.

x

1- x

*

MATH

B1

—

X/

/T/n

2

a

X/

/T/n

/b

*

2

ENTER

x

*

Since Y1 is the value “on the bottom” (the smaller of the

*

2nd F

DRAW

G1

———

*

two) and Y2 is the function “on the top” (the larger of the two), Y1 < Y < Y2.

View the graph.

3

GRAPH

4

Find the intersections, and solve the inequality.

The intersections are when x = -1.3, -0.8, 0.8, and 1.3. The solution is all values of

2nd F CALC

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

Do this four times

2

*

x such that x

-0.8 ≤ x ≤ 0.8 or x ≥ 1.3.

≤

-1.3 or

The EL-9650/9600c/9450/9400 allows the solution region of inequalities to be indicated visually using the Shade feature. Also, the points of intersections can be obtained easily.

9-2

EL-9650/9600c/9450/9400 Graphing Calculator

G raphing P arabolas

The graphs of quadratic equations (y = ax2 + bx + c) are called parabolas. Sometimes the quadratic equation takes on the form of x = ay 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, 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.

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. It is not necessary to algebraically solve the equation for y. 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.

2

+ by + c.

Example

Graph a parabola using two methods.

1. Graph the parabola x = y

2. Graph the parabola x = y

Before

Starting

1-1

1-2

*Use either pen touch or cursor to operate.

Solve the equation for y.

Enter y = √x+2 for Y1 and enter y = -Y1 for Y2.

There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value and delete all data.

Set the zoom to the decimal window:

(When using EL-9650/9600c)

2

-2 in rectangular mode.

-2 in parametric mode.

(

ENTER

A

ZOOM

(When using EL-9650/9600c)

*

Display

ALPHA

)

7

*

x = y2 -2 x + 2 = y

√

+

y =

x + 2

–

*

NotesStep & Key Operation

2

θθ

θ

X

/

/T/

n

+

θθ

√

)

VARS

-

2

ENTERENTER

1A

*

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

1-3

2nd F

Y=

*

(

*

View the graph.

GRAPH

the "bottom half."

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

10-1

EL-9650/9600c/9450/9400 Graphing Calculator

2-1

2-2

2-3

(When using EL-9650/9600c)

*Use either pen touch or cursor to operate.

Change to parametric mode.

2nd F

SET UP

E

*

2

*

Rewrite x = y2 -2 in parametric form. Enter X1T = T

X/

/T/n

Y=

X/

/T/n

2

-2 and Y1T = T.

2

—

x

2

ENTER

*

View the graph. Consider why only half of the parabola is drawn. (To understand this, use Trace feature.)

(

GRAPH

TRACE

)

Display

(When using EL-9650/9600c)

NotesStep & Key Operation

Let y = T and substitute in x = y2 - 2, to obtain x = T2- 2.

The graph starts at T =0 and increases. Since the window setting is T ≥ 0, the region T < 0 is not drawn in the graph.

2-4

2-5

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

Set Tmin to -6.

WINDOW

(-)

6

ENTER

*

View the complete parabola.

GRAPH

The calculator provides two methods for graphing parabolas, both of which are easy to perform.

10-1

EL-9650/9600c/9450/9400 Graphing Calculator

G raphing Cir cles

The standard equation of a circle of radius r that is centered at a point (h, k) is (x - h)2 + (y - k)2 = r2. In order to put an equation in standard form so that you can graph in rectangular mode, it is necessary to solve the equation for y. You therefore need to use the process of completing the square.

Example

Graph the circles in rectangular mode. Solve the equation for y to put it in the standard form.

1. Graph x

2. Graph x

2

+ y2 = 4.

2

- 2x + y2 + 4y = 2.

Before

Starting

*Use either pen touch or cursor to operate.

1-1

Solve the equation for y. Enter y = half). Enter y = -

1-2

View the graph.

There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value and delete all data.

Set the zoom to the decimal window:

(When using EL-9650/9600c)

√

4 - x2 for Y1 (the top

√

4 - x2 for Y2.

Y=

2nd F

*

(

*

4

√

)

-

VARS

A

—

X/

/T/n

ENTERENTER

A

ZOOM

(When using EL-9650/9600c)

2

x

1

*

(

ENTER

*

Display

ALPHA

)

7

*

y2 = 4 - x

+

y =

√4 - x

–

*

NotesStep & Key Operation

2

This is a circle of radius r , centered at the origin.

GRAPH

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

2-1

10-2

Solve the equation for y, completing the square.

2

- 2x + y2 + 4y = 2

x

2

-2x+y2+4y+4=2+4

x

2

- 2x + (y+2)2 = 6

x

2

(y+2)

= 6 -x2 + 2x

y+2 =

y =

√

±

6-x2+2x

√

±

6-x2+2x -2

Place all variable terms on the left and the constant term on the right-hand side of the equation.

Complete the square on the y-term.

Express the terms in y as a perfect square.

Leave only the term involving y on the left hand side.

Take the square root of both sides.

Solve for y.

EL-9650/9600c/9450/9400 Graphing Calculator

2-2

2-3

(When using EL-9650/9600c)

*Use either pen touch or cursor to operate.

Enter y = √6 - x2 + 2x for Y1, y = Y1 - 2 for Y2, and y = -Y1 -2 for Y3.

Y= CL 6

2

+

x

*

VARS

A1

ENTER

2

VARS

)

(

-

2nd F

√

X/

/T/n

2

ENTER

*

ENTER

—

X/

ENTER

—

*

12

*

—

CL

*

"Turn off" Y1 so that it will not graph.

*

ENTER

*

Display

(When using EL-9650/9600c)

/T/n

NotesStep & Key Operation

Notice that if you enter

√

6 - x2 + 2x - 2 for Y1

y =

and y = - Y1 for Y2, you will not get the graph of a circle because the “±” does not go with the “-2”.

Notice that “=” for Y1 is no longer darkened. You now have the top portion and the bottom portion of the circle in Y2 and Y3.

2-4

2-5

View the graph.

GRAPH

Adjust the screen to see the bottom part of the circle using the Rapid feature.

Wait until the graph is displayed after each operation. (It takes few seconds to graph)

EZ

ENTER

2-6

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

View the graph in the new window.

GRAPH

*

ENTER

*

ENTER

*

ENTER

*

Graphing circles can be performed easily on the calculator display. Also, the Rapid Zoom feature of the EL-9650/9600c/9450/9400 allows shifting and adjusting display area (window) of a graph easily.

10-2

EL-9650/9600c/9450/9400 Graphing Calculator

G raphing Ellipses

The standard equation for an ellipse whose center is at the point (h, k) with major and minor axes of length a and b is + = 1.

(x - h)

a

2

(y - k)

b

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. b) the functions entered in the Y = list locations must be in terms of x, not y. To draw a graph of an ellipse, consider the “top” and “bottom” halves of the ellipse as two different parts of the graph because each individual is a function. Solve the equation of the ellipse for y and enter the two parts in two locations of the Y = list.

Example

Graph an ellipse in rectangular mode. Solve the equation for y to put it in the standard form.

Graph the ellipse 3(x -3)2 + (y + 2)2 = 3

Before

Starting

There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value and delete all data.

Set the zoom to the decimal window:

ZOOM

(

A

ENTER

*

ALPHA

)

7

*

(When using EL-9650/9600c)

*Use either pen touch or cursor to operate.

1

Solve the equation for y, completing the square.

Enter

Y1 =

√

3 - 3(x - 3)

2

Y2 = Y1 - 2 Y3 = -Y1 -2

2nd F

Y=

—

X/

/T/n

VARS

A

*

2

1—2

Turn off Y1 so that it will not graph.

2

3

√

)

3

ENTER

*

1

(

)

VARS

-

ENTER

*

—

2

x

—

*

ENTERENTER

*

3

ENTER

Display

(When using EL-9650/9600c)

NotesStep & Key Operation

3(x - 3)2 + (y + 2)2 = 3

2

(y + 2)

y + 2 = y =

(

*

= 3 - 3(x - 3)

+

√

3 - 3(x - 3)

+

√

3 - 3(x - 3)2 - 2

2

10-3

EL-9650/9600c/9450/9400 Graphing Calculator

NotesStep & Key Operation

(When using EL-9650/9600c)

*Use either pen touch or cursor to operate.

3

View the graph.

GRAPH

4

Adjust the screen to see the bottom part of the ellipse using the Rapid Zoom feature.

Display

(When using EL-9650/9600c)

Wait until the graph is displayed after each operation. (It takes few seconds to graph)

EZ

5

View the graph in the new window.

GRAPH

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

ENTER

*

Graphing an ellipse can be performed easily on the calculator display. In addition to the Zoom-in/Zoom-out features, the EL-9650/9600c/9450/9400 have the Rapid Zoom feature to adjust the display easily.

10-3

EL-9650/9600c/9450/9400 Graphing Calculator

G raphing H yperbolas

The standard equation for a hyperbola can take one of two forms:

2

( x - h )

2

a

( x - k )

2

b

( y - k )

- = 1 with vertices at ( h ± a, k ) or

- = 1 with vertices at ( h, k ± b ).

2

b

( y - h)

a

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. b) the functions entered in the Y= list locations must be in terms of x, not y. To draw a graph of a hyperbola, consider the “top” and “bottom” halves of the hyperbola as two different parts of the graph because each individual is a function. Solve the equation of the hyperbola for y and enter the two parts in two locations of the Y= list.

Example

Graph a hyperbola in rectangular mode. Solve the equation for y to put it in the standard form.

Graph the hyperbola x2 + 2x - y2 - 6y + 3 = 0

Before

Starting

There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value and delete all data.

Set the zoom to the decimal window:

ZOOM

(

ENTER

A

*

ALPHA

)

7

*

(When using EL-9650/9600c)

*Use either pen touch or cursor to operate.

Solve the equation for y completing

1

the square.

Enter

√

Y1 =

x2 + 2x + 12

Y2 = Y1 -3 Y3 = -Y1 -3

Y=

2nd F

θθ

X

/

θ

/T/

n

θθ

+

* *

VARS

A

*

VARS

Turn off Y1 so that it will not graph.

2

√

12

ENTER

A

X/

/T/n

1

ENTER

*

ENTER

—

Display

(When using EL-9650/9600c)

NotesStep & Key Operation

x2 + 2x - y2 -6y = -3

2

+ 2x - (y2 + 6y + 9) = -3 -9

x

2

+ 2x - (y +3)2 = -12

x

2

(y + 3) y + 3 =

2

+

x

2

*

3

ENTER

*

3(-)

—

1

*

y =

= x2 + 2x + 12

+

√

x2 + 2x + 12

+

√

x2 + 2x + 12 - 3

10-4

EL-9650/9600c/9450/9400 Graphing Calculator

Display

(When using EL-9650/9600c)

*Use either pen touch or cursor to operate.

3

View the graph.

GRAPH

Zoom out the screen.

4

A

ZOOM

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

4

* *

(When using EL-9650/9600c)

NotesStep & Key Operation

Graphing hyperbolas can be performed easily on the calculator display. In addition to the Zoom-in/Zoom-out features, the EL-9650/9600c/9450/9400 have the Rapid Zoom feature to adjust the display easily. (See the section “Graphing Ellipses (No. 10-3)” about how to use the Rapid Zoom feature.)

10-4

Key pad for the SHARP EL-9650/9600c Calculator

Graphing keys Power supply ON/OFF key Alphabet specification key Secondary function specification key Display screen

Cursor movement keys Clear/Quit key Variable enter key Calculation execute key Communication port for peripheral devices

Key pad for the SHARP EL-9450/9400 Calculator

Graphing keys Power supply ON/OFF key Alphabet specification key Secondary function specification key Display screen

Cursor movement keys Clear/Quit key Variable enter key Calculation execute key Communication port for peripheral devices

Use this form to send us your contribution

Dear Sir/Madam

We would like to take this opportunity to invite you to create a mathematical problem which can be solved with the SHARP graphing calculator EL-9650/9600c/9450/9400. For this purpose, we would be grateful if you would complete the form below and return it to us by fax or mail, specifying which calculator you are writing problems for, the EL-9650/9600c or 9450/9400.

If your contribution is chosen, your name will be included in the next edition of The EL-9650/9600c/9450/ 9400 Graphing Calculator Handbook. We regret that we are unable to return contributions.

We thank you for your cooperation in this project.

Name: ( Mr. Ms.

)

School/College/Univ.: Address:

Post Code: Country:

Phone: Fax: E-mail:

* You are making this sheet for the ( EL-9650/9600c, EL-9450/9400).

SUBJECT : Write a title or the subject you are writing about.

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

INTRODUCTION : Write an explanation about the subject.

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

EXAMPLE : Write example problems.

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

SHARP Graphing Calculator

BEFORE STARTING : Write any conditions to be set up before solving the problems.

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

STEP NOTES

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

SHARP CORPORATION Osaka, Japan

Fax:

SHARP Graphing Calculator

SHARP CORPORATION OSAKA, JAPAN

Sharp EL-9650, EL-9400, EL-9600c User Manual

Specifications and Main Features

Frequently Asked Questions

User Manual