Sharp EL-9900 Handbook Vol. 1 Operation Manual

Graphing Calculator

EL-9900

Handbook Vol. 1

Algebra

For Advanced Levels For Basic Levels

1. Fractions

1-1

Fractions and Decimals

2. Pie Charts

2-1

Pie Charts and Proportions

3. Linear Equations

3-1

Slope and Intercept of Linear Equations

3-2

Parallel and Perpendicular Lines

4. Quadratic Equations

4-1

Slope and Intercept of Quadratic Equations

5. Literal Equations

5-1

Solving a Literal Equation Using the Equation Method (Amortization)

5-2

Solving a Literal Equation Using the Graphic Method (Volume of a Cylinder)

5-3

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

6. Polynomials

6-1

Graphing Polynomials and Tracing to Find the Roots

6-2

Graphing Polynomials and Jumping to Find the Roots

7. A System of Equations

7-1

Solving a System of Equations by Graphing or Tool Feature

8. Matrix Solutions

8-1

Entering and Multiplying Matrices

8-2

Solving a System of Linear Equations Using Matrices

9. Inequalities

9-1

Solving Inequalities

9-2

Solving Double Inequalities

9-3

System of Two-Variable Inequalities

9-4

Graphing Solution Region of Inequalities

10. Absolute Value Functions, Equations, Inequalities

10-1

Slope and Intercept of Absolute Value Functions

10-2

Solving Absolute Value Equations

10-3

Solving Absolute Value Inequalities

10-4

Evaluating Absolute Value Functions

11. Rational Functions

11-1

Graphing Rational Functions

11-2

Solving Rational Function Inequalities

12. Conic Sections

12-1

Graphing Parabolas

12-2

Graphing Circles

12-3

Graphing Ellipses

12-4

Graphing Hyperbolas

Contents

1. Always read “Before Starting”

The key operations of the set up conndition 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.

Note: Since all memories will be deleted, it is advised to use the CE-LK2 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

and select data to be deleted from the menu.

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

: to cancel previous function

3. Initial settings

Initial settings are as follows:

✩

Set up ( ):

Advanced keyboard: Rad, FloatPt, 9, Rect, Decimal(Real), Equation, Auto
Basic keyboard: Deg, FloatPt, 9, Rect, Mixed, Equation, Auto

✩

Format ( ):

Advanced keyboard:

OFF, OFF, ON, OFF, RectCoord

Basic keyboard: OFF, OFF, ON, OFF

Stat Plot (

): 2. PlotOFF

Shade (

): 2. INITIAL

Zoom (

): 5. Default

Period (

): 1. PmtEnd (Advanced keyboard only)

Note:

✩

returns to the default setting in the following operation.

( )

4. Using the keys

Press to use secondary functions (in yellow).

To select “

x

-1

”: ➔ Displayed as follows:

Press to use the alphabet keys (in violet).

To select F:

➔ Displayed as follows:

5. Notes

•

Some features are provided only on the Advanced keyboard and not on the Basic keyboard.

(Solver, Matrix, Tool etc.)

•

As this handbook is only an example of how to use the EL-9900, please refer to the manual

for further details.

Read this first

2nd F

OPTION

E

2

C

2nd F

OPTION

2nd F

CL

CL

QUIT

2nd F

2nd F

2nd F

ZOOM

2nd F

SET UP

FORMAT

DRAW

FINANCE

2nd F

OPTION

2nd F

2nd F

2nd F

ENTER

E

G

A

C

E1

F

ALPHA

ALPHA

ALPHA

x

2

x

-1

x

2

STAT
PLOT

Using this Handbook

This handbook was produced for practical application of the SHARP EL-9900 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.

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.)

Notes

Step & Key Operation

*Use either pen touch or cursor to operate.

Display

EL-9900 Graphing Calculator

4-1

2

-

2

View both graphs.

Notice that the addition of 2 moves
the basic y =x

2

graph up two units
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 -

h)

2

+ k will move the basic graph up k units and placing k
k (<0) will move the basic graph down k units on the y-axis.
axis.

3

-

2

View both graphs.

Notice that the multiplication of
2 pinches or closes the basic
y=x

2

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

y = a

(x - h)

2

+ k

will pinch or close

the basic graph.

4

-

1

Change the equation in Y2 to
y = -2x

2

.

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-

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

2

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

*

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

2

-

1

*

3

-

1

Change the equation in Y2 to y = 2x2.

*

The EL-9900 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.

4

-

2

View both graphs.

2nd F

GRAPH

Y=

SUB

2

0

ENTER ENTER

GRAPH

Y=

2nd F

SUB

2

0

ENTER

ENTER

GRAPH

Y=

2nd F

SUB

(-)

2

ENTER

EL-9900 Graphing 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

2

+ 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.

4-1

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.

Before

Starting

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

Example

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.

1

-

1

Enter the equation y = x 2 for Y1.

Notes

Step & Key Operation

Display

1

-

2

(

)

Notice that the addition of -2
within the quadratic operation
moves the basic

y =x

2

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.

View both graphs.

1

-

3

X

/θ/T/

n

—

Y=

x

2

(

+

2

1

0

ENTER

ENTER

GRAPH

ENTER

ALPHA

H

2nd F

SUB

ALPHA

A

X

/θ/T/

n

)

ALPHA

K

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

Merits of Using the EL-9900

Highlights the main functions of the calculator relevant
to the section

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
screen for each step

Notes

Explains the process of each
step in the key operations

x

2

EL-9900 Graphing Calculator

Fractions and Decimals

To convert a decimal into a fraction, form the numerator by multiplying the decimal by 10n,
where n is the number of digits after the decimal point. The denominator is simply 10n. Then,
reduce the fraction to its lowest terms.

Convert 0.75 into a fraction.

Example

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.
We recommend using the Basic keyboard to calculate fractions.

Choose the manual mode for
reducing fractions.

1

NotesStep & Key Operation Display

H2

2nd F

SET UP

Convert 0.75 into a fraction.

2

ENTER

CL

0

➞b

/c

.

75

The fraction can be reduced
by a factor of 5.

Reduce the fraction.

3

ENTER

Simp

Enter 3 to further reduce the
fraction.

The fraction cannot be re­duced by a factor of 3, even
though the numerator can be.
(15 = 3 x 5)

4

ENTER

Simp

3

Enter 5 to reduce the fraction.

0.75 = 3/4

5

ENTER

Simp

5

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The EL-9900 can easily convert a decimal into a fraction. It also helps
students learn the steps involved in reducing fractions.

Before

Starting

EL-9900 Graphing Calculator

Pie Charts and Proportions

Pie charts enable a quick and clear overview of how portions of data relate to the whole.

A questionnaire asking students about their favourite colour elicited the following results:
Red: 20 students
Blue: 12 students
Green: 25 students
Pink: 10 students
Yellow: 6 students

2-1

NotesStep & Key Operation Display

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.

Before

Starting

1. Make a pie chart based on this data.

2. Find the percentage for each colour.

Example

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

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

Pie charts can be made easily with the EL-9900.

Make another pie chart.

Red: 27.39%
Blue: 16.43%
Green: 34.24%
Pink: 13.69%
Yellow: 8.21%

2-2

GRAPH

Choose the setting for displaying
by percentages.

2-1

STAT
PLOT

A

ENTER

STAT
PLOT

F

2

Make a pie chart.

1-3

GRAPH

Choose the setting for making a
pie chart.

1-2

STAT
PLOT

A

ENTER ENTER

STAT

PLOT

F1

Enter the data.

1-1

A

STAT

ENTER

2

0

ENTER

1

2

ENTER

ENTER

2

5

1

0

ENTERENTER

6

EL-9900 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.

The equation Y1 = x is dis­played first, followed by the
equation Y2 = 2x. Notice how
Y2 becomes steeper or climbs
faster. Increase the size of the
slope (m>1) to make the line
steeper.

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

View both graphs.

1-1

1-2

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

Enter the equation y = x for Y2.

View both graphs.

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

2-1

2-2

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.

3-1

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

2

1

2

1

Y=

ENTER

2

Y= CL

1

a

/b

2

GRAPH

GRAPH

X/

/T/n

X/

/T/n

Before

Starting

NotesStep & Key Operation Display

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.

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

Example

X/

/T/n

Step & Key Operation Display

EL-9900 Graphing Calculator

Notes

Enter the equation y = - x for Y2.

View both graphs.

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.

Adding 2 will shift the y = x
graph upwards.

View both graphs.

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

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

Enter the equation y = x + 2 for
Y2.

3-1

3-2

4-1

4-2

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

3-1

CL

(

-

)

GRAPH

GRAPH

Y=

Y= CL + 2

X/

/T/n

X/

/T/n

EL-9900 Graphing Calculator

P arallel and Perpendicular Lines

3-2

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

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

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

View the graphs.

1-1

Graph parallel lines and perpendicular lines.

1-2

Example

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

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

2-1

These lines have an equal
slope but different y-intercepts.
They are called parallel, and
will not intersect.

1

3

1

3

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.

NotesStep & Key Operation Display

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: ( )

1

m

Y=

ZOOM

C

ENTER

ALPHA

7

3

+

1

ENTER

3

2

+

GRAPH

Before

Starting

X/

/T/n

Y=

CL

31

1

1

3

—

(-)

CL

ENTER

+

a

/b

X/

/T/n

X/

/T/n

X/

/T/n

Step & Key Operation Display

EL-9900 Graphing Calculator

Notes

3-2

View the graphs.

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.

2-2

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

1

m

GRAPH

EL-9900 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.

4-1

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.

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

Example

1. Graph y = x

2

and y = (x - 2)2.

2. Graph y = x

2

and y = x2 + 2.

3. Graph y = x

2

and y = 2x2.

4. Graph y = x

2

and y = -2x2.

1-1

Enter the equation y = x2 for Y1.

NotesStep & Key Operation Display

1-2

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

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

Notice that the addition of -2
within the quadratic operation
moves the basic y = x

2

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.

View both graphs.

1-3

Y=

x

2

GRAPH

Before

Starting

( )

ALPHA

K

A

(

ALPHA

)

+

ALPHA

H

—

21

ENTER

ENTER

2nd F SUB

0

ENTER

X/

/T/n

x

2

X/

/T/n

Step & Key Operation Display

EL-9900 Graphing Calculator

Notes

4-1

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

2-2

View both graphs.

Notice that the addition of 2 moves
the basic y =

x2 graph up two units

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 -
h)

2

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

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

3-2

View both graphs.

Notice that the multiplication of
2 pinches or closes the basic

y = x

2

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

y = a

(x - h)2 + k

will pinch or close

the basic graph.

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

4-1

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

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-

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

2

+ k

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

Change the equation in Y2 to y =

x2+2.

2-1

3-1

Change the equation in Y2 to y = 2

x2.

The EL-9900 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.

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

4-2

View both graphs.

2nd F

GRAPH

Y=

SUB

2

0

ENTER ENTER

GRAPH

Y=

2nd F SUB

2

0

ENTER

ENTER

GRAPH

Y=

2nd F

SUB

(-)

2

ENTER

EL-9900 Graphing Calculator

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.
As the Solver feature is only available on the Advanced keyboard, this section does not apply to the
Basic keyboard.

5-1

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.

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.

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.

The formula : P = L

Example

1-3

Enter the amortization formula.

Select the Equation method for
solving.

1-1

1-2

Access the Solver feature.

P= monthly payment
L= loan amount

I= interest rate
N=number of months

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

1-(1+ )

-N

-1

A

2nd F

SOL VER

2nd F

SOL VER

1

—

(-)

a

/b

+

(

1

I

1

1

2

a

b

I

12

1

(

)

)

ALPHA

ALPHA

N

ALPHA

a

/b

a

/b

a

b

(-)

ALPHA

2nd F

PL

=

I

12

I

12

Before

Starting

Notes

Step & Key Operation

Display

ALPHA

Step & Key Operation Display

EL-9900 Graphing Calculator

Notes

5-1

Save this formula.

2-1

Give the formula the name AMORT.

2-2

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The monthly payment (P) is
$373.28.

Solve for the payment(P).

1-5

( )

Recall the amortization formula.

3-1

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

3-2

The amount of loan (L) is
$17550.28.

Solve for the loan (L).

3-3

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With the Equation Editor, the EL-9900 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
Advanced keyboard.

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1-4

Enter the values L=15,000,
I=0.09, N=48.

2nd F EXE

ENTER

ENTER ENTER

ENTER

ENTER

ENTER

ENTER ENTER ENTER

ENTER

ENTER

2nd F

2nd F

2nd F EXE

SOL VER

SOL VER

CL

15000

•09 4

8

C

A

MORT

B

01

300

•01 60

0

EL-9900 Graphing Calculator

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.
As the Solver feature is only available on the Advanced keyboard, this section does not apply to the
Basic keyboard.

5-2

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).

Solving a Literal Equation Using the Graphic Method

(Volume of a Cylinder)

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

3

and a height of 15in,

using the saved formula.

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)

Example

1-1

Access the Solver feature.

Select the Graphic method for
solving.

1-2

1-3

Enter the formula V = πr2h.

1-4

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

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

2nd F

SOLVER

2nd F

SOL VER

2nd F

2nd F EXE

ENTER ENTER

ENTER

A

3

V

RH

30

1

0

=

ALPHA ALPHA ALPHA

ALPHA

π

Before

Starting

Notes

Step & Key Operation

Display

x

2

Step & Key Operation Display

EL-9900 Graphing Calculator

Notes

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.

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

5-2

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Set the variable range from 0 to 2.

1-5

1-6

Solve.

(

)

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

2

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3-1

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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.

Use the larger of the values to
be safe.

r =1 ➞ r2 = 12 = 1 <3
r =2 ➞ r2 = 22 = 4 >3

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

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

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

r = 3 ➞ r2 = 32 = 9 < 14
r = 4 ➞ r2 = 42 = 16 > 14

r2 = = < 14

3-2

ENTER

0

ENTER

2

2nd F EXE

CL

2nd F

SOL VER

ENTER

L

ENTER

C

C

VY

2nd F

SOL VER

B01

0020

ENTER ENTER ENTER

ENTER

15

2nd F EXE

SPACE

04

ENTER

ENTER 2nd F EXE

r2 = = <3

10

π

30

π

3

15

π

200π 14

EL-9900 Graphing Calculator

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.
As the Solver feature is only available on the Advanced keyboard, this section does not apply to the
Basic keyboard.

5-3

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.

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

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

2

with bases of 8in and 10in

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

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)

Example

1-1

Access the Solver feature.

Select Newton's method
for solving.

1-2

1-3

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

1

2

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

1

2

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

1-4

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

2nd F

SOL VER

2nd F

SOL VER

A

ALPHA

A

2

ALPHA

1

a

/b

2

ALPHA

H

ALPHA

ALPHA

ENTER

=

ENTER

ENTER ENTER

(

+

)

C

2

5

5

7

B

Before

Starting

NotesStep & Key Operation Display

Step & Key Operation Display

Notes

EL-9900 Graphing Calculator

5-3

Enter the values:
A = 50, B = 8, C = 10.

3-2

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Newton's method will
prompt with a guess or a
starting point.

1-5 Solve for the height and enter a

starting point of 1.

The answer is : h = 4.17

Solve.

1-6

(

)

Save this formula. Give the formula
the name “A TRAP”.

2

Recall the formula for calculating
the area of a trapezoid.

3-1

The answer is : h = 5.56Solve.

3-3

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.

ENTER

ENTER

ENTER

ENTER

ENTER

ENTER

ENTER

ENTER

2nd F EXE

2nd F EXE

2nd F EXE

2nd F EXE

1

50
10

01

8

B

T

AP

A

C

1

CL

2nd F

SOL VER

SPACE

2nd F

SOL VER

R

EL-9900 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.

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

1. Graph the polynomial y = x

3

- 3x2 + x + 1.

2. Approximate the left-hand root.

3. Approximate the middle root.

4. Approximate the right-hand root.

Enter the polynomial

y = x

3

- 3x2 + x + 1.

1-1

Example

6-1

Notes

Step & Key Operation

Display

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 :

A

ZOOM

ZOOM

B

7

55

2nd F

QUIT

ENTER

ENTER ENTER

ENTER

ALPHA

Before

Starting

1

3

—

X/

/T/n

3

+

X/

/T/n

Y=

X/

/T/n

a

b

+

1-2

View the graph.

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GRAPH

x

2

Step & Key Operation Display

Notes

EL-9900 Graphing Calculator

(repeatedly)

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

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

Move the tracer near the left-hand
root.

2-1

Zoom in on the left-hand root.

2-2

Move the tracer to approximate the
root.

2-3

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Return to the previous decimal
viewing window.

3-1

Move the tracer to approximate
the middle root.

3-2

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Move the tracer near the right­hand root.
Zoom in and move the tracer to
find a better approximate.

4

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

Tracer

Tracer

Tracer

(repeatedly)

or

(repeatedly)

(repeatedly)

or

(repeatedly)

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6-1

The root is : x -0.42

The root is : x

2.42

Tracer

ZOOM

TRACE

ZOOM

ZOOM

TRACE

TRACE

TRACE

A

2

A3

H

3

EL-9900 Graphing Calculator

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 :

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

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

Find the first root.

2-1

2-2

Example

1. Graph the polynomial y = x

4

+ x3 - 5x2 - 3x + 1.

2. Find the four roots one by one.

NotesStep & Key Operation Display

Enter the polynomial

y = x

4

+ x3 - 5x2 - 3x + 1

1-1

1-2

View the graph.

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Find the next root.

x

-2.47

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

x -0.82

6-2

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.

ZOOMAENTER ENTER ENTER 2nd F

QUIT

AA

Y=

4

35

31

5

5

+

—

—

+

a

b

a

b

2nd F CALC

2nd F CALC

GRAPH

Before

Starting

X/

/T/n

X/

/T/n

x

2

X/

/T/n

X/

/T/n

Step & Key Operation Display

EL-9900 Graphing Calculator

Notes

2-4

2-3

Find the next root.

x

2.05

x

0.24

Find the next root.

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

6-2

5

2nd F CALC

5

2nd F CALC

EL-9900 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 Tool feature can solve a linear
system with up to six variables and six equations.

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

1. Solve the system using the Calculate feature.

2. Solve the system using the Tool feature.

Enter the system of equations

y = x

2

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

View the graphs.

1-1

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.

1-2

Find the right-hand intersection by
accessing the Calculate feature again.

1-4

The answer is : x = 2.41
y = 4.83

Example

Find the left-hand intersection using
the Calculate feature.

1-3

7-1

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y = x2 - 1
y = 2x

5x + y = 1

-3x + y = -5

{
{

Y=

—

1

ENTER

2

2

2

2nd F CALC

2nd F CALC

Before

Starting

NotesStep & Key Operation Display

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 < 5”, “-10 < Y < 10”.

As the Tool feature is only available on the Advanced keyboard, example 2 does not apply to the
Basic keyboard.

ENTER ENTER

5

(-)

5

X/

/T/n

X/

/T/n

GRAPH

WINDOW

x

2

Step & Key Operation Display

EL-9900 Graphing Calculator

Notes

Access the Tool menu. Select the
number of variables.

Enter the system of equations.

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

Solve the system.

2-1

2-2

2-3

7-1

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

x = 0.75
y = -2.75

2nd F

TOOL

B

EXE2nd F

2

5

11

31 5

ENTER

ENTER ENTER

ENTER ENTER

ENTER

(-)(

-

)

EL-9900 Graphing Calculator

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.
As the Matrix feature is only available on the Advanced keyboard, this section does not apply to the
Basic keyboard.

Entering and Multiplying Matrices

1. Enter a 3x3 matrix A

2. Enter a 3x3 matrix B

3. Multiply the matrices A and B

Access the matrix menu.

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

1-1

1-2

Example

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

1-3

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.

8-1

Enter two matrices and execute multiplication of the two.

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

B
1 2 3
4 5 6
7 8 9

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MATRIX

B

1

33

121
21 1
11 2

ENTER ENTER

ENTER ENTER

ENTER

ENTER ENTER

ENTER

ENTER

ENTER

ENTER

(-)
(-)

Before

Starting

NotesStep & Key Operation Display

2nd F

Step & Key Operation Display

Notes

EL-9900 Graphing Calculator

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. The sum of these
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 by
each column of B.

Enter a 3x3 matrix B.

2

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

3-1

Delete the input matrices for
future use.

3-2

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8-1

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Matrix multiplication can be performed easily by the calculator.

MATRIX

MATRIX MATRIX

X

B23

3

12

3
456
78

9

A1

A

2

C

2

2nd F

QUIT

2nd F

OPTION

ENTER

ENTER

ENTER ENTER ENTER

ENTER

ENTER

ENTER

ENTER ENTER ENTER

ENTER

ENTER

ENTER

2nd F

2nd F

2nd F

EL-9900 Graphing Calculator

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.
As the Matrix feature is only available on the Advanced keyboard, this section does not apply to the
Basic keyboard.

NotesStep & Key Operation Display

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

Example

Use matrix multiplication to solve a system of linear equations.

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

Solving a S ystem of Linear Equations U sing Matrices

Set up 3x3 identity matrix at the
home screen.

Save the identity matrix in matrix A.

1-1

1-2

Confirm that the identity matrix is
stored in matrix A.

1-3

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

-1

B. Note that the multiplication is “order sensitive”
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

-1

x A=I). The identity

matrix I is defined to be a square matrix (nxn) where each position on the diagonal is 1
and all others are 0.

8-2

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{

MATRIX

MATRIX

MATRIX

05

3

A1

B1

ENTER

ENTER

STO

C

Before

Starting

2nd F

2nd F

2nd F

Step & Key Operation Display

Notes

EL-9900 Graphing Calculator

3-3

3-1

Delete the input matrices for future
use.

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

The system of equations can
be expressed as

Enter the constants on the right side
of the equal sign into matrix C (3x1).

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

2-1

Enter a 3x3 matrix B.

2-2

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

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

8-2

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

(repeatedly)

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

-0.17 0.83 - 0.5

0.5 -0.5 0.5

0.17 0.17 -0.5

B-1=

3-2

x
y
z

1 2 1

2 1 -1
1 1 -2

8
1

-3

=

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

-1

BX = B-1C (multiply both
sides by B-1)
I = B

-1 (B-1

B = I, identity matrix)

X = B

-1

C

Calculate B-1C.

MATRIX

MATRIX

MATRIX

MATRIX

MATRIX

23 3

1

2

2

1

11

1

1

2

A

2

B33

1

81 3

A2

A3

C

2

2nd F

QUIT

2nd F

OPTION

2nd F

ENTER

ENTERx

-1

X

CL

ENTER

ENTER

ENTER

ENTER ENTER

ENTER

ENTER

ENTER

ENTER ENTER

ENTER

ENTER

ENTER ENTER

ENTER

ENTER

ENTER

CL

2nd F

2nd F

x

-1

QUIT

(-)

(-)
(-)

B

2nd F

2nd F

2nd F

2nd F

2nd F

EL-9900 Graphing Calculator

Solving Inequalities

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

≤

0, f(x) ≥ 0, or form of f (x) ≤

g(x),

f(x)

≥

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

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).

Solve an inequality in two methods.

Example

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.

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.

NotesStep & Key Operation Display

1-2

View the graph.

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

1-3

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 in­equality 3(4 - 2x) - 5 + x

≥

0 is

all values of x such that

x

≤

1.4.

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.

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

1-1

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9-1

2nd F

CALC

5

3

4

2

5

Y=

()

—

—

+

GRAPH

Before

Starting

X/

/T/n

X/

/T/n

NotesStep & Key Operation Display

EL-9900 Graphing Calculator

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

2-1

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

(7 times) (4 times)

2-2

View the graph.

2-3

Access the Set Shade screen.

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

View the shaded region.

2-6

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,
the solution to the inequality
3(4 - 2x)

≥

5 - x is all values

of x such that x

≤

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-9900 allows the solution region to be indicated visually using the
Shade feature. Also, the points of intersection can be obtained easily.

9-1

GRAPH

Y=

DEL

—

—

2

G

1

5

ENTER

2nd F CALC

2nd F

DRAW

GRAPH

X/

/T/n

2nd F

2nd F

VARS

VARS

ENTER

ENTER

A

A

2

1

EL-9900 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.

Solve a double inequality, using graphical techniques.

Example

2x - 5 ≥ -1
2x -5 ≤ 7

1

Enter y = -1 for Y1, y = 2x - 5 for
Y2, and y = 7 for Y3.

The “double” inequality
given can also be written to

-1

≤

2x - 5 ≤ 7.

2

View the lines.

3

Find the point of intersection.

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

9-2

Y=

(-)

1

25

7

2

2nd F CALC

—

ENTER

ENTER

GRAPH

NotesStep & Key Operation Display

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.

Before

Starting

X/

/T/n

EL-9900 Graphing Calculator

NotesStep & Key Operation Display

4

Move the tracer and find another
intersection.

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

5

Solve the inequalities.

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 so-

lution is 2

≤

x ≤ 6.

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

9-2

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-9900 allows the solution region to be indicated visually using the
Shade feature. Also, the points of intersection can be obtained easily.

2

2nd F CALC

EL-9900 Graphing Calculator

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: ( )

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

9-3

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5

Graph the system and find the
intersections.

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

6 Solve the system.

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-9900 allows the solution region to be indicated visually using the Shade feature.
Also, the points of intersection can be obtained easily.

ZOOM

2nd F

A

ENTER

7

GRAPH

2

2

CALC

2nd F

CALC2nd F

Before

Starting

NotesStep & Key Operation Display

1

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

2x + y ≥ 1 ➞ y ≥ 1 - 2x
x2 + y ≤ 1 ➞ y ≤ 1 - x

2

2

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

2

for Y2.

Y= 1 2

1

—

—

ENTER

3

Access the set shade screen

G

1

2nd F

DRAW

4

Shade the points of y -value so that
Y1 ≤ y ≤ Y2.

2nd F

2nd F

VARS

VARS

ENTER

ENTER

A

A

1

2

x

2

X/

/T/n

X/

/T/n

EL-9900 Graphing Calculator

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.

9-4

Graphing Solution Region 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.

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

Example

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)

1-1

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

x + 2y

≤

1 ➞ y

≤

x2+y ≥ 4 ➞ y ≥ 4 - x

2

1-x

2

1-2

Enter y = for Y1 and
y = 4 - x

2

for Y2.

1-x

2

Set the shade and view the solution
region.

1-3

Y2 ≤ y ≤ Y1

Set the display area (window) to :

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

2-1

GRAPH

Y=

a

/b

1

2

4

G1

9

3

6

5

ENTER

ENTER

ENTERENTER

ENTER ENTER

WINDOW

(-)

(-)

—

—

2nd F

DRAW

Before

Starting

NotesStep & Key Operation Display

X/

/T/n

X/

/T/n

2nd F

2nd F

VARS

VARS

ENTER

ENTER

A

A2

1

GRAPH

x

2

EL-9900 Graphing Calculator

NotesStep & Key Operation Display

9-4

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

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).

2-2

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

.

(-1.6, 1.8): -1.6 + 2 ✕ 1.8 = 2

➞ This does not materialize.

.

(-2, -5): -2 + 2 ✕ (-5) = -12

(-2)

2

+ (-5) = -1

➞ This does not materialize.

.

(2.8, -1.4): 2.8 + 2 ✕ (-1.4) = 0

(2.8)

2

+ (-1.4) = 6.44

➞ This materializes.

.

(-8, 4): -8 + 2 ✕ 4 = 0

(-8)

2

+ 4 = 68

➞ This materializes.

...

(Continuing key operations omitted.)

2-3

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

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-9900 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.

or or or

GRAPH

(-)

+

X

•

•

1

6

2

1

8

EL-9900 Graphing Calculator

10-1

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.

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

Example

1. Graph y = |x|

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

1-1

Enter the function y =|x| for Y1.

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

≤

0.

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

1-2

View the graph.

2-1

Enter the standard form of an abso­lute value function for Y2 using the
Rapid Graph feature.

2-2

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

Y=

Y=

ENTER

ENTER ENTER2nd F SUB

MATH

B

1

11

0

GRAPH

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

2nd F

A

7

Notes

Step & Key Operation

Display

ALPHA

H

A

ALPHA—MATH

1B

ALPHA

+ K

X/

/T/n

X/

/T/n

NotesStep & Key Operation Display

EL-9900 Graphing Calculator

2-3

View the graph.

2-5

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

2-4

View the graph.

The EL-9900 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.

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.

10-1

Y=

2nd F SUB ENTER

ENTER ENTER

1

1

(-)

GRAPH

GRAPH

ENTER

0

EL-9900 Graphing Calculator

10-2

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.

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 ≤ 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.

Example

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

1

Enter y = |5 - 4x| for Y1.
Enter y = 6 for Y2.

2

View the graph.

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

3

Find the points of intersection of
the two graphs and solve.

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-9900 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.

GRAPH

Y=

MATH

B

15

4

6

2
2

2nd F

CALC

2nd F

CALC

ENTER

—

Before

Starting

NotesStep & Key Operation Display

X/

/T/n

EL-9900 Graphing Calculator

10-3

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.

Set viewing window to “-5< x <50,” and “-10< y <10”.

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 |

f (x)

|<

k, |f (x)

|≤

k, |f (x)

|>

k,

or |f (x)

|≥ k. The graphical
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.

Solve absolute value inequalities in two methods.

Example

1-2

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

1-1

Rewrite the equation.

|20 - |< 8
➞|20 - | - 8 < 0.

1-3

View the graph, and find the
x-intercepts.

➞ x = 10, y = 0
➞ x = 23.33333334

y = 0.00000006 ( Note)

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.)

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

1-4

Solve the inequality.

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

6x

5

6x

5

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

the inequality is zero.

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

6x

5

WINDOW

5

50

ENTER ENTER

GRAPH

2nd F CALC

2nd F CALC

B12

0

6

8

5

5
5

Y=

MATH

—

—

a

/b

6x

5

Before

Starting

(-)

NotesStep & Key Operation Display

X/

/T/n

EL-9900 Graphing Calculator

NotesStep & Key Operation Display

10-3

2-1

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

2-2

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.”

Set up shading.

2-3

Set viewing window to “-10 < x < 10”
and “-5 < y < 50”, and view the graph.

Y= CL

MATH

+

2nd F

DRAW

WINDOW

ENTERENTER

ENTER

ENTER

10

B

1

35 4

G1

5

0

•

2nd F

2nd F

VARS

VARS

ENTER

ENTER

AA

2

1

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

2

-

4

➞ x = -4, y = 10
➞ x = 1.714285714

y = 9.999999999 ( Note)

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

Find the points of intersection.
Solve the inequality.

The EL-9900 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.

2nd F

2nd F

CALC

CALC

2

2

1

1

(-)

ENTER

0

(-)

ENTER

0

5

ENTER

5

X/

/T/n

EL-9900 Graphing Calculator

10-4

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.

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.

Evaluate various absolute value functions.

Example

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 | | = ?

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

1-1

Access the home or computation
screen.

1-2

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

2-1

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

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

|6-9|
|1+3|

The solution is +8.

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

ENTER

MATH

MATH

MATH

MATH

ENTER

ENTER

(-)

(-)

(-)

+

+

CL

(

)

—

B1

1

7

1

1

2

2

1

7

25

Before

Starting

NotesStep & Key Operation Display

EL-9900 Graphing Calculator

NotesStep & Key Operation Display

= 0.75 , = 0 .75

10-4

2-2

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

If x

≥

0 and y ≥ 0

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

≤

0 and y ≥ 0

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

≥

0 and y ≤ 0

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

≤

0 and y ≤ 0

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

|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

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

3-1

Evaluate . Evaluate .

3-2

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

If x

≥

0 and y ≥ 0

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

≤

0 and y ≥ 0

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

≥

0 and y ≤ 0

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

≤

0 and y ≤ 0

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

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

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

The EL-9900 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.

6-9

1+3

6-9

1+3

6-9

1+3

6-9

1+3

6-9
1+3

6-9

1+3

➞ =

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

≥

0 and y ≥ 0,

and when x ≤ 0 and y ≤ 0.

CL

ENTER

a

/b

MATH

ENTER

a

/b

MATH

+

—

—

+

6

9

1

3

69

1

3

1

1

1

MATH

EL-9900 Graphing Calculator

11-1

G raphing Rational Functions

A rational function f (x) is defined as the quotient 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. 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).

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

Example

1. Graph f (x) = .

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

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

4. Estimate the horizontal asymptote of f (x).

1-1

Enter y = for Y1.

x - 1

x2 -1

1-2

View the graph.

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

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

q (x)

p (x)

x-1

x2-1

GRAPH

Y=

a

/b

—

1

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: (

)

A

ENTER

ALPHA

ZOOM

7

Notes

Step & Key Operation

Display

x

2

X/

/T/n

X/

/T/n

EL-9900 Graphing Calculator

NotesStep & Key Operation Display

The y-intercept is at (0 ,1). No-
tice that there are no x-inter­cepts for the graph of f (x).

11-1

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

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

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

2

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

graph to find the hole at x = 1.

Since f (x) can be written as
, the domain
consists of all real numbers x

such that x ≠ 1 and x ≠ -1.
There is no vertical asymptote
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.

(repeatedly)

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

3

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-9900 can create the branches of a rational
function separated by a vertical asymptote. The calculator allows the points of
intersection to be obtained easily.

x - 1

(x + 1)(x - 1)

TRACE

2nd F

CALC

6

EL-9900 Graphing Calculator

11-2

Solving Rational F unction I nequalities

A rational function f (x) is defined as the quotient where p (x) and q (x) are two
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. Y ou can find the
solutions by graphing each side of the inequalities as an individual function.

Solve a rational inequality.

Example

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

1 Enter y = for Y1. Enter y = 2

for Y2.

2 Set up the shading.

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

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
x such that x

≤

-1.3 or

-0.8 ≤ x ≤ 0.8 or x ≥ 1.3.

Do this four times

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

The EL-9900 allows the solution region of inequalities to be indicated visually
using the Shade feature. Also, the points of intersections can be obtained
easily.

x

1 - x

2

x

1- x

2

Y=

a

/b

MATH

B

1

1

2

G1

2

2nd F CALC

2nd F

DRAW

—

ENTER

q (x)

p (x)

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: (

)

ENTER

A

ZOOM

7

ALPHA

Notes

Step & Key Operation

Display

X/

/T/n

A1

2nd F

2nd F

VARS

VARS

ENTER

ENTER

A

2

3

View the graph.

GRAPH

x

2

X/

/T/n

EL-9900 Graphing Calculator

12-1

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."

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

2

+ by + c.
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.

Graph a parabola using two methods.

Example

1. Graph the parabola x = y

2

-2 in rectangular mode.

2. Graph the parabola x = y

2

-2 in parametric mode.

1-1

Solve the equation for y.

1-2

x = y2 -2
x + 2 = y

2

y =

√

x + 2

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

1-3

View the graph.

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

+

–

GRAPH

ENTER

ENTER

2

1

A(-)

Y= +

VARS

2nd F

√

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: (

)

ENTER

A

ZOOM

7

ALPHA

Notes

Step & Key Operation

Display

2nd F

X/

/T/n

EL-9900 Graphing Calculator

NotesStep & Key Operation Display

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

2-1

Change to parametric mode.

2-2

Rewrite x = y2 -2 in parametric form.
Enter X1T = T2 -2 and Y1T = T.

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

2-3

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

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

Set Tmin to -6.

2-5

View the complete parabola.

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

12-1

GRAPH

GRAPH

Y=

E

(-)

ENTER

WINDOW

TRACE

—

2

2

6

ENTER

2nd F

SET UP

X/

/T/n

x

2

X/

/T/n

EL-9900 Graphing Calculator

12-2

Solve the equation for y.
Enter y =

√

4 - x2 for Y1 (the top

half). Enter y = -

√

4 - x2 for Y2.

x

2

- 2x + y2 + 4y = 2

x

2

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

x

2

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

(y+2)

2

= 6 -x2 + 2x

y+2 =

±

√

6-x2+2x

y =

±

√

6-x2+2x -2

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.

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

Example

1. Graph x

2

+ y2 = 4.

2. Graph x

2

- 2x + y2 + 4y = 2.

1-1

2-1

Solve the equation for y,
completing the square.

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.

y2 = 4 - x

2

y =

√

4 - x

2

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

1-2

View the graph.

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

ENTERENTER

1

4

A(-)

Y=

VARS

2nd F

√

—

GRAPH

+

–

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: (

)

ENTER

A

ZOOM

7

ALPHA

Notes

Step & Key Operation

Display

2nd F

X/

/T/n

x

2

EL-9900 Graphing Calculator

NotesStep & Key Operation Display

2-2

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

Y3.

Notice that if you enter
y =

√

6 - x2 + 2x - 2 for Y1
and y = - Y1 for Y2, you will
not get the graph of a circle
because the “±” does not go
with the “-2”.

2-3

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

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

Adjust the screen so that the whole
graph is shown. Shift 2 units down­wards.

12-2

Y= CL

6

(-)

+

—

VARS

ENTER

CL

ENTER

ENTER

ENTER

VARS

—

—

2

A

1

2

12

2nd F

√

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

Graphing circles can be performed easily on the calculator display.

X/

/T/n

X/

/T/n

2nd F

2nd F

(3 times)

WINDOW

—

2

ENTER

—

2

ENTER

ENTER

x

2

-1.3 < Y < 3.1

-5.1 < Y < 1.1

➝

GRAPH

EL-9900 Graphing Calculator

12-3

Enter

Y1 =

√

3 - 3(x - 3)

2

Y2 = Y1 - 2
Y3 = -Y1 -2

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

Example

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

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.

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.

(x - h)

2

a

2

(y - k)

2

b

2

1

Solve the equation for y, completing
the square.

2

Turn off Y1 so that it will not graph.

Y= 3

(-)

—

VARS

ENTER

ENTER

ENTER

ENTER

VARS

—

—

3

A

3

1

2

2nd F

√

ENTER

(

)

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

2

= 3 - 3(x - 3)

2

y + 2 =

√

3 - 3(x - 3)

2

y =

√

3 - 3(x - 3)2 - 2

+

+

1

—

2

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: (

)

ENTER

A

ZOOM

7

ALPHA

NotesStep & Key Operation Display

2nd F

2nd F

X/

/T/n

x

2

EL-9900 Graphing Calculator

NotesStep & Key Operation Display

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

4

Adjust the screen so that the whole
graph is shown. Shift 2 units down­wards.

3

View the graph.

Graphing an ellipse can be performed easily on the calculator display.

12-3

GRAPH

(3 times)

WINDOW

—

2

ENTER

—

2

ENTER

-3.1 < Y < 3.1

-5.1 < Y < 1.1

➝

GRAPH

EL-9900 Graphing Calculator

12-4

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

Example

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

G raphing H yperbolas

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

- = 1 with vertices at ( h

±

a, k ) or

- = 1 with vertices at ( h, k

±

b ).

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.

2

Turn off Y1 so that it will not graph.

1

Solve the equation for y completing
the square.

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

2

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

x

2

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

(y + 3)

2

= x2 + 2x + 12

y + 3 =

√

x2 + 2x + 12

y =

√

x2 + 2x + 12 - 3

( y - k )

2

b

2

( y - h)

2

a

2

( x - h )

2

a

2

( x - k )

2

b

2

+

+

ENTER

ENTER

ENTER

ENTER

ENTER

Y=

A

2

2nd F

A

1

2

1

3

1

3

(-)

+

—

VARS

+

VARS

—

Enter

Y1 =

√

x2 + 2x + 12

Y2 = Y1 -3
Y3 = -Y1 -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: (

)

ENTER

A

ZOOM

7

ALPHA

NotesStep & Key Operation Display

X/

/T/n

X/

/T/n

2nd F

2nd F

x

2

EL-9900 Graphing Calculator

NotesStep & Key Operation Display

3

View the graph.

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

4

Zoom out the screen.

Graphing hyperbolas can be performed easily on the calculator display.

12-4

GRAPH

A

ZOOM

4

Graphing keys
Power supply ON/OFF key
Secondary function specification key
Alphabet 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-9900 Calculator

Advanced Keyboard

Graphing keys
Power supply ON/OFF key
Secondary function specification key
Alphabet 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-9900 Calculator

Basic Keyboard

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-9900. For this purpose, we would be grateful if you would com­plete the form below and return it to us by fax or mail.

If your contribution is chosen, your name will be included in the next edition of The EL-9900 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:

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

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INTRODUCTION : Write an explanation about the subject.

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EXAMPLE : Write example problems.

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Use this form to send us your contribution

SHARP Graphing Calculator

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

SHARP CORPORATION Osaka, Japan

Fax:

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

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STEP NOTES

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SHARP Graphing Calculator

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

SHARP CORPORATION OSAKA, JAPAN