Graphing Calculator
EL-9900
Handbook Vol. 1
Algebra
For Advanced Levels
For Basic Levels
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Read this first
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.
2nd F
2
OPTION
E
CL
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
Other keys to delete data:
and select data to be deleted from the menu.
2nd F
C
OPTION
:
to erase equations and remove error displays
to cancel previous function
CL
:
2nd F
QUIT
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
): Advanced keyboard: OFF, OFF, ON, OFF, RectCoord
Basic keyboard: OFF, OFF, ON, OFF
2nd F
SET UP
✩
Format (
Stat Plot (
2nd F FORMAT
STAT
):
2. PlotOFF
E
PLOT
Shade
Zoom
Period
(
(
(
): 2. INITIAL
5. Default
): 1. PmtEnd (Advanced keyboard only)
2nd F DRAW
G
):
ZOOM
2nd F
A
FINANCE
C
✩
✩
Note:
returns to the default setting in the following operation.
(
)
2nd F OPTION
ENTER
E
1
4. Using the keys
Press
To select “x ”:
Press
to use secondary functions (in yellow).
2nd F
-1
2
-1
Displayed as follows:
2nd F
x
2nd F
ALPHA
x
✩
to use the alphabet keys (in violet).
ALPHA
2
To select F:
Displayed as follows:
ALPHA
x
F
✩
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.
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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.
Notes
EL-9900 Graphing Calculator
Introduction
Explains the process of each
Slope and Intercept of Quadratic Equations
Explanation of the section
A quadratic equation of y in terms of x can be expressed by the standard form y = a (x -h)2
+
step in the key operations
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
Ex a m p le
L-9900 Graphing Calculator
Graph various quadratic equations and check the relation between the graphs and
the values of coefficients of the equations.
Example of a problem to be
solved in the section
y
N o te s
1.
y = x2 and y = (x-2)2.
2. Graph y = x2 and y = x2+2.
3. Graph y = x2 and y = 2x2
.
4. GGrraapphh y = x2 and y = -2x2
.
There may be differences in the results of
Return all settings to the default value and dele
and graph plotting depending on the setting.
Sta r tin g
calculations
te
a
l
l
data
.
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
Be fo re
Before Starting
Ste p
&
Ke y O p e ra tio n
Disp la
y
N o te s
ng
k
(>0) within the standard form y =
a
(x
-
Important notes to read
before operating the calculator
ove the basic graph up
k
k
units and placing k
units on the y-axis.
2
1
1
-
1
Enter the equation y =
x2
x
for Y1.
the basic graph down
X
/
θ
/
T/n
Y=
2
-2
Enter the equation y = (x-2) for
Y2 using Sub feature.
ALPHA
H
A
(
X
/
θ
/
T
/
n
—
Notice that the multiplication of
pinches or closes the basic
y=x2 graph. This demonstrates
x2
K
ALPHA
)
+
ALPHA
2
2
Step & Key Operation
A clear step-by-step guide
to solving the problems
2nd
F
SUB
1
ENTER
ENTER
the fact that multiplying an
a
a
(> 1) in the standard form y =
(
)
0
ENTER
2
(x
-
h)
+
k
will pinch or close
the basic graph.
Notice that the addition of -2
within the quadratic operation
=x graph
right two units (adding 2 moves
View both graphs.
1
-3
2
moves the basic
y
GRAPH
it left two units) on the x-axis.
This shows that placing an
h
(>0) within the standard
form (x - h)2 k will move thebasic graph right
y
=
a
+
units and placing an
(<0)will move it left
h
units
h
h
on the x-axis.
Notice that the multiplication of
-2 pinches or closes the basic
y =x2 graph and flips it (reflects
it) across the x-axis. This dem-
onstrates the fact that multiply-
Display
4-1
Illustrations of the calculator
screen for each step
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.
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.
Merits of Using the EL-9900
4-1
Highlights the main functions of the calculator relevant
to the section
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.)
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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.
Ex a m p le
Convert 0.75 into a fraction.
Be fo re
Sta r tin g
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.
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
1
2
3
4
5
Choose the manual mode for
reducing fractions.
2nd F
H
2
SET UP
Convert 0.75 into a fraction.
.
➞b
CL
ENTER
0
7
5
/c
Reduce the fraction.
The fraction can be reduced
by a factor of 5.
Simp
ENTER
Enter 3 to further reduce the
fraction.
The fraction cannot be re-
duced by a factor of 3, even
though the numerator can be.
(15 = 3 x 5)
Simp
ENTER
3
Enter 5 to reduce the fraction.
0.75 = 3/4
Simp
ENTER
5
The EL-9900 can easily convert a decimal into a fraction. It also helps
students learn the steps involved in reducing fractions.
1-1
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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.
Ex a m p le
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
1. Make a pie chart based on this data.
2. Find the percentage for each colour.
Be fo re
Sta r tin g
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.
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
1-1
1-2
Enter the data.
A
ENTER
2
ENTER
STAT
0
1
2
ENTER
2
ENTER
1
5
0
ENTER
ENTER
6
Choose the setting for making a
pie chart.
STAT
PLOT
ENTER
ENTER
A
STAT
PLOT
F
1
Make a pie chart.
1-3
2-1
GRAPH
Choose the setting for displaying
by percentages.
STAT
A
F
ENTER
PLOT
STAT
PLOT
2
Make another pie chart.
2-2
Red:
27.39%
Blue: 16.43%
Green: 34.24%
Pink: 13.69%
Yellow: 8.21%
GRAPH
Pie charts can be made easily with the EL-9900.
2-1
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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.
Ex a m p le
Draw graphs of two equations by changing the slope or the y- intercept.
1. Graph the equations y = x and y = 2x.
1
2
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.
Be fo re
Sta r tin g
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.
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
1-1 Enter the equation y = x for Y1
and y = 2x for Y2.
X/ /T/n
Y=
X/ /T/n
ENTER
2
View both graphs.
The equation Y1 = x is dis-
played 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.
1-2
GRAPH
1
2
2-1
2-2
Enter the equation y = x for Y2.
Y=
1
CL
a
2
X/ /T/n
/b
View both graphs.
Notice how Y2 becomes less
steep or climbs slower. De-
crease the size of the slope
(0<m<1) to make the line less
steep.
GRAPH
3-1
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EL-9900 Graphing Calculator
N o te s
Ste p & Ke y O p e ra tio n
Disp la y
Enter the equation y = - x for Y2.
3-1
3-2
( )
-
Y=
CL
X/ /T/n
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
GRAPH
down from left to right.
4-1
4-2
Enter the equation y = x + 2 for
Y2.
Y=
CL
+
2
X/ /T/n
View both graphs.
Adding 2 will shift the y = x
graph upwards.
GRAPH
Making a graph is easy, and quick comparison of several graphs will help
students understand the characteristics of linear equations.
3-1
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EL-9900 Graphing Calculator
Parallel 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 slope-
intercept 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
1
slopes that are negative reciprocals of each other (m = - ). These characteristics can be
m
verified by graphing these lines.
Ex a m p le
Graph parallel lines and perpendicular lines.
1. Graph the equations y = 3x + 1 and y = 3x + 2.
1
3
2. Graph the equations y = 3x - 1 and y = - x + 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.
Be fo re
Sta r tin g
Set the zoom to the decimal window:
(
)
ZOOM
ENTER ALPHA
C
7
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
Enter the equations y = 3x + 1 for
Y1 and y = 3x + 2 for Y2.
1-1
1-2
X/ /T/n
1
ENTER
Y=
+
3
+
3
2
X/ /T/n
View the graphs.
These lines have an equal
slope but different y-intercepts.
They are called parallel, and
will not intersect.
GRAPH
2-1
Enter the equations y = 3x - 1 for
1
3
Y1 and y = - x + 1 for Y2.
—
Y=
CL
+
ENTER
3
X/ /T/n
1
CL
a
( )
-
1
3
X/ /T/n
/b
1
3-2
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EL-9900 Graphing Calculator
N o te s
Ste p & Ke y O p e ra tio n
Disp la y
View the graphs.
2-2
These lines have slopes that
are negative reciprocals of
GRAPH
1
each other (m = - ). They are
m
called perpendicular. Note that
these intersecting lines form
four equal angles.
The Graphing Calculator can be used to draw parallel or perpendicular
lines while learning the slope or y-intercept of linear equations.
3-2
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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.
Ex a m p le
Graph various quadratic equations and check the relation between the graphs and
the values of coefficients of the equations.
1. Graph y = x 2 and y = (x - 2) 2.
2. Graph y = x 2 and y = x 2 + 2.
3. Graph y = x 2 and y = 2x 2.
4. Graph y = x 2 and y = -2x 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.
Be fo re
Sta r tin g
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
Enter the equation y = x 2 for Y1.
1-1
1-2
2
X/ /T/n
x
Y=
Enter the equation y = (x - 2) 2 for
Y2 using Sub feature.
—
ALPHA
H
(
A
X/ /T/n
2
)
ALPHA
x
+
ALPHA
2nd F
K
SUB
ENTER
1
ENTER
2
(
)
ENTER
0
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.
View both graphs.
1-3
GRAPH
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.
4-1
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EL-9900 Graphing Calculator
N o te s
Ste p & Ke y O p e ra tio n
Disp la y
2+2.
2-1
2-2
Change the equation in Y2 to y =
x
2nd F
SUB
Y=
0
ENTER
ENTER
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
GRAPH
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.
Change the equation in Y2 to y = 2x 2
.
3-1
3-2
2nd F
SUB
ENTER
Y=
2
ENTER
0
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.
View both graphs.
GRAPH
4-1
4-2
Change the equation in Y2 to
y = - 2x 2.
( )
-
2nd F
SUB
Y=
2
ENTER
Notice that the multiplication of
-2 pinches or closes the basic
View both graphs.
y =x 2 graph and flips it (reflects
GRAPH
it) across the x-axis. This dem-
onstrates 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.
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.
4-1
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EL-9900 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.
Ex a m p le
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.
-N
)
-1
I
12
1-(1+
I
P= monthly payment
L= loan amount
I= interest rate
N=number of months
The formula : P = L
12
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.
Be fo re
Sta r tin g
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.
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
Access the Solver feature.
This screen will appear a few
seconds after “SOLVER” is dis-
played.
1-1
SOLVER
2nd F
1-2 Select the Equation method for
solving.
SOLVER
2nd F
A
1
Enter the amortization formula.
1-3
ALPHA
ALPHA
2nd F
=
P
1
L
a
(
(
—
1
+
/b
a
)
ALPHA
2
2
I
1
/b
ab
N
1
1
( )
ALPHA
-
a
ALPHA
I
/b
ab
)
( )
-
5-1
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EL-9900 Graphing Calculator
N o te s
Ste p & Ke y O p e ra tio n
Disp la y
1-4
1-5
Enter the values L=15,000,
I=0.09, N=48.
ENTER
ENTER
1
5
0
0
0
•
0
9
ENTER
4
ENTER
8
The monthly payment (P) is
$373.28.
Solve for the payment(P).
2nd F
EXE
(
)
CL
2-1
2-2
Save this formula.
ENTER
2nd F SOLVER
C
Give the formula the name AMORT.
A
M
O
R
T
ENTER
Recall the amortization formula.
3-1
3-2
SOLVER
B
2nd F
0
1
Enter the values: P = 300,
I = 0.01, N = 60
ENTER
ENTER
ENTER
ENTER
3
0
0
0
•
0
1
ENTER
6
0
The amount of loan (L) is
$17550.28.
Solve for the loan (L).
3-3
2nd F
EXE
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.
5-1
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EL-9900 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).
Ex a m p le
Use the Graphic method to find the radius of a cylinder giving the range of the unknown
variable.
The formula : V = πr 2h ( V = volume r = radius h = height)
1. Find the radius of a cylinder with a volume of 30in3 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.
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.
Be fo re
Sta r tin g
As the Solver feature is only available on the Advanced keyboard, this section does not apply to the
Basic keyboard.
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
Access the Solver feature.
This screen will appear a few
seconds after “SOLVER” is dis-
played.
1-1
2nd F SOLVER
1-2 Select the Graphic method for
solving.
2nd F SOLVER
A
3
Enter the formula V = πr 2h.
1-3
1-4
ALPHA
ALPHA
2nd F
ALPHA
V
=
π
2
ALPHA
x
R
H
Enter the values: V = 30, H = 10.
Solve for the radius (R).
ENTER
ENTER
2nd F
1
3
0
0
EXE
ENTER
5-2
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EL-9900 Graphing Calculator
N o te s
Ste p & Ke y O p e ra tio n
Disp la y
Set the variable range from 0 to 2.
1-5
The graphic solver will prompt
with a variable range for solv-
ing.
ENTER
2
ENTER
0
30
10π
3
π
r 2 =
=
<3
r =1 ➞ r 2 = 12 = 1 <3
r =2 ➞ r 2 = 22 = 4 >3
Use the larger of the values to
be safe.
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.
Solve.
1-6
2nd F
EXE
CL
(
)
The radius is 0.98 in.
2
Save this formula.
Give the formula the name “V CYL”.
2nd F SOLVER
C
ENTER
SPACE
V
C
Y
L
ENTER
Recall the formula.
Enter the values: V = 200, H = 15.
3
3
-1
-2
2nd F SOLVER
B
0
1
ENTER
2
0
0
ENTER
0
ENTER
1
5
ENTER
200
15π
14
π
r 2 =
=
< 14
Solve the radius setting the variable
range from 0 to 4.
r = 3 ➞ r 2 = 32 = 9 < 14
r = 4 ➞ r 2 = 42 = 16 > 14
2nd F
2nd F
EXE
EXE
ENTER
0
4
ENTER
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.
5-2
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EL-9900 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.
Ex a m p le
Find the height of a trapezoid from the formula for calculating the area of a trapezoid
using Newton’s method.
1
2
The formula : A= h(b+c)
(A = area h = height b = top face c = bottom face)
1. Find the height of a trapezoid with an area of 25in2 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 50in2 with bases of 8in and 10in
using the saved formula. (Set the starting point to 1.)
There may be differences in the results of calculations and graph plotting depending on the setting.
Be fo re
Sta r tin g
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.
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
Access the Solver feature.
1-1
This screen will appear a few
seconds after “SOLVER” is dis-
played.
2nd F SOLVER
1-2 Select Newton's method
for solving.
2nd F
SOLVER
A
2
1
Enter the formula A = h(b+c).
1-3
1-4
2
a
A
H
ALPHA
ALPHA
C
ALPHA
=
1
2
/b
(
+
ALPHA
ALPHA
B
)
Enter the values: A = 25, B = 5, C = 7
ENTER
2
ENTER
5
5
ENTER
ENTER
7
5-3
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EL-9900 Graphing Calculator
N o te s
Ste p & Ke y O p e ra tio n
Disp la y
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.
2nd F
EXE
ENTER
1
The answer is : h = 4.17
Solve.
1-6
CL
2nd F
EXE
(
)
2
Save this formula. Give the formula
the name “A TRAP”.
2nd F SOLVER
ENTER
C
ENTER
A
SPACE
R
A
P
T
Recall the formula for calculating
the area of a trapezoid.
3-1
3-2
2nd F SOLVER
B
0
1
Enter the values:
A = 50, B = 8, C = 10.
ENTER
ENTER
ENTER
ENTER
8
5
1
0
0
Solve.
The answer is : h = 5.56
3-3
2nd F
EXE
1
ENTER
2nd F
EXE
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.
5-3
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EL-9900 Graphing Calculator
Graphing Polynomials and Tracing to Find 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.
Ex a m p le
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 - 3x 2 + x + 1.
2. Approximate the left-hand root.
3. Approximate the middle root.
4. Approximate the right-hand root.
There may be differences in the results of calculations and graph plotting depending on the setting.
Sta r tin g Return all settings to the default value and delete all data.
Be fo re
Set the zoom to the decimal window:
(
)
A
ALPHA
ZOOM
ENTER
7
Setting the zoom factors to 5 :
ENTER
ENTER
B
5
5
ENTER
QUIT
2nd F
ZOOM
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
Enter the polynomial
1-1
1-2
y = x 3 - 3x 2 + x + 1.
ab
—
Y=
X/ /T/n
3
3
2
X/ /T/n
+
+
x
X/ /T/n
1
View the graph.
GRAPH
6-1
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EL-9900 Graphing Calculator
N o te s
Ste p & Ke y O p e ra tio n
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Tracer
Note that the tracer is flashing
on the curve and the x and y
coordinates are shown at the
bottom of the screen.
2-1
2-2
2-3
Move the tracer near the left-hand
root.
TRACE
(repeatedly)
Zoom in on the left-hand root.
A
3
ZOOM
Tracer
Move the tracer to approximate the
root.
The root is : x -0.42
or
TRACE
(repeatedly)
3-1 Return to the previous decimal
viewing window.
ZOOM
H
2
Tracer
Move the tracer to approximate
the middle root.
The root is exactly x = 1.
(Zooming is not needed to
find a better approximate.)
3-2
TRACE
(repeatedly)
Tracer
Move the tracer near the right-
hand root.
4
The root is : x 2.42
Zoom in and move the tracer to
find a better approximate.
(repeatedly)
3
ZOOM
TRACE
A
or
(repeatedly)
The calculator allows the roots to be found (or approximated) visually by
graphing a polynomial and using the Zoom-in and Trace features.
6-1
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EL-9900 Graphing Calculator
Graphing Polynomials and Jumping to Find 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.
Ex a m p le
Draw a graph of a polynomial and find the roots by using the Calculate feature.
1. Graph the polynomial y = x 4 + x 3 - 5x 2 - 3x + 1.
2. Find the four roots one by one.
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.
Be fo re
Sta r tin g
Setting the zoom factors to 5 :
ZOOM
A
ENTER
A
ENTER
A
ENTER 2nd F QUIT
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
1-1
Enter the polynomial
y = x 4 + x 3 - 5x 2 - 3x + 1
ab
X/ /T/n
4
X/ /T/n
Y=
ab
+
2
—
x
3
5
X/ /T/n
—
X/ /T/n
+
3
1
1-2
2-1
View the graph.
GRAPH
Find the first root.
2nd F CALC
5
x
-2.47
Y is almost but not exactly zero.
Notice that the root found here
is an approximate value.
Find the next root.
2-2
x
-0.82
2nd F CALC
5
6-2
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EL-9900 Graphing Calculator
N o te s
Ste p & Ke y O p e ra tio n
Disp la y
Find the next root.
2-3
x
x
0.24
2.05
2nd F CALC
5
2-4 Find the next root.
2nd F CALC
5
The calculator allows jumping to find the roots by graphing a polynomial
and using the Calculate feature, without tracing the graph.
6-2
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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.
Ex a m p le
Solve a system of equations using the Calculate or Tool feature. First, use the Calcu-
late 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 = x 2 - 1
{
y = 2x
2. Solve the system using the Tool feature.
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 value and delete all data.
Be fo re
Sta r tin g
Set viewing window to “-5 < X < 5”, “-10 < Y < 10”.
(-)
WINDOW
5
ENTER
ENTER
5
As the Tool feature is only available on the Advanced keyboard, example 2 does not apply to the
Basic keyboard.
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
1-1 Enter the system of equations
y = x 2 - 1 for Y1 and y = 2x for Y2.
2
X/ /T/n
—
x
1
ENTER
Y=
2
X/ /T/n
1-2
1-3
1-4
View the graphs.
GRAPH
Note that the x and y coordi-
nates are shown at the bot-
tom of the screen. The answer
is : x = - 0.41 y = -0.83
Find the left-hand intersection using
the Calculate feature.
2nd F CALC
2
Find the right-hand intersection by
accessing the Calculate feature again.
The answer is : x = 2.41
y = 4.83
2nd F CALC
2
7-1
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EL-9900 Graphing Calculator
N o te s
Ste p & Ke y O p e ra tio n
Disp la y
Access the Tool menu. Select the
number of variables.
2-1
2-2
Using the system function, it
is possible to solve simulta-
neous linear equations. Sys-
tems up to six variables and
six equations can be solved.
2nd F TOOL
2
B
Enter the system of equations.
5
ENTER
ENTER
ENTER
1
1
( )
-
( )
-
ENTER
ENTER
3
1
5
ENTER
2-3
Solve the system.
x = 0.75
y = - 2.75
2nd F
EXE
A system of equations can be solved easily by using the Calculate feature
or Tool feature.
7-1
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EL-9900 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.
Ex a m p le
Enter two matrices and execute multiplication of the two.
A
B
1. Enter a 3x3 matrix A
1 2 1
2 1 -1
1 1 -2
1 2 3
4 5 6
7 8 9
2. Enter a 3x3 matrix B
3. Multiply the matrices A and B
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.
Be fo re
Sta r tin g
As the Matrix feature is only available on the Advanced keyboard, this section does not apply to the
Basic keyboard.
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
1-1
1-2
1-3
Access the matrix menu.
MATRIX
2nd F
B
1
Set the dimension of the matrix at
three rows by three columns.
3
ENTER
3
ENTER
Enter the elements of the first row,
the elements of the second row, and
the elements of the third row.
ENTER
1
2
1
ENTER
ENTER
ENTER
2
1
1
ENTER
ENTER
ENTER
1
( )
-
1
2
ENTER
ENTER
( )
-
8-1
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EL-9900 Graphing Calculator
N o te s
Ste p & Ke y O p e ra tio n
Disp la y
2
Enter a 3x3 matrix B.
MATRIX
ENTER
2nd F
ENTER
3
B
2
5
8
2
3
ENTER
ENTER
ENTER
ENTER
ENTER
ENTER
ENTER
ENTER
ENTER
1
4
7
3
6
9
Multiply the matrices A and B
together at the home screen.
Matrix multiplication can
be performed if the num-
ber of columns of the first
matrix is equal to the num-
ber of rows of the second
matrix. The sum of these
3-1
2nd F
MATRIX
2nd F
MATRIX
A
1
X
2
ENTER
A
. .
multiplications (1 1 + 2 4
.
+ 1 7) is placed in the 1,1
(first row, first column) po-
sition of the resulting ma-
trix. This process is re-
peated until each row of A
has been multiplied by
each column of B.
Delete the input matrices for
future use.
3-2
2nd F
2
OPTION
ENTER
QUIT
C
ENTER
2nd F
Matrix multiplication can be performed easily by the calculator.
8-1
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EL-9900 Graphing Calculator
Solving a System of Linear Equations Using 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
-1
solution matrix X is found by multiplying A B. Note that the multiplication is “order sensitive”
-1
-1
and the correct answer will be obtained by multiplying BA . An inverse matrix A is a
-1
matrix that when multiplied by A results in the identity matrix I (A x A=I). The identity
matrix I is defined to be a square matrix (n xn) where each position on the diagonal is 1
and all others are 0.
Ex a m p le
Use matrix multiplication to solve a system of linear equations.
B
1. Enter the 3x3 identity matrix in matrix A.
1 2 1
2. Find the inverse matrix of the matrix B.
2 1 -1
3. Solve the equation system.
1 1 -2
x + 2y + z = 8
2x + y - z = 1
{
x + y - 2z = -3
Be fo re
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.
Sta r tin g
As the Matrix feature is only available on the Advanced keyboard, this section does not apply to the
Basic keyboard.
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
Set up 3 x3 identity matrix at the
home screen.
1-1
MATRIX
2nd F
3
ENTER
C
0
5
1-2
1-3
Save the identity matrix in matrix A.
MATRIX
2nd F
A
1
ENTER
STO
Confirm that the identity matrix is
stored in matrix A.
MATRIX
2nd F
B
1
8-2
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EL-9900 Graphing Calculator
N o te s
Ste p & Ke y O p e ra tio n
Disp la y
Enter a 3 x3 matrix B.
2-1
2-2
2nd F
MATRIX
ENTER
ENTER
ENTER
ENTER
2
3
3
ENTER
B
2
1
1
ENTER
1
2
1
ENTER
ENTER
ENTER
1
( )
-
1
2
ENTER
ENTER
( )
-
Some square matrices have
no inverse and will generate
error statements when calcu-
lating the inverse.
Exit the matrix editor and find the
inverse of the square matrix B.
2nd F
2nd F
QUIT
CL
A
-0.17 0.83 -0.5
-1
MATRIX
2
2nd F
ENTER
x
B-1=
0.5 -0.5
0.5
0.17 0.17 -0.5
(repeatedly)
The system of equations can
be expressed as
Enter the constants on the right side
of the equal sign into matrix C (3 x1).
3-1
x
y
z
8
1
-3
1 2 1
2 1 -1
1 1 -2
MATRIX
ENTER
ENTER
2nd F
B
1
3
3
ENTER
1
=
( )
-
ENTER
ENTER
8
3
Let each matrix B, X, C :
BX = C
B-1BX = B-1C (multiply both
sides by B-1)
I = B-1 (B-1B = I, identity matrix)
X = B-1 C
-1
Calculate B C.
3-2
The 1 is the x coordinate, the 2
the y coordinate, and the 3 the
z coordinate of the solution
point.
MATRIX
2nd F
A
2
CL
-1
MATRIX
x
ENTER
2nd F
X
2nd F
A
3
(x, y, z)=(1, 2, 3)
3-3 Delete the input matrices for future
use.
2nd F OPTION
C
2
ENTER
QUIT
2nd F
The calculator can execute calculation of inverse matrix and matrix
multiplication. A system of linear equations can be solved easily using the
Matrix feature.
8-2
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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).
Ex a m p le
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.
Be fo re
Sta r tin g
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.
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
1-1 Rewrite the equation 3(4 - 2x)
≥
5 - x
3(4 - 2x) ≥ 5 - x
so that the right-hand side becomes 0,
© 3(4 - 2x) - 5 + x ≥ 0
and enter y = 3(4 - 2x) - 5 + x for Y1.
—
(
)
3
5
2
X/ /T/n
Y=
—
4
X/ /T/n
+
View the graph.
1-2
1-3
GRAPH
Find the location of the x-intercept
and solve the inequality.
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-in-
tercept, the solution to the in-
equality 3(4 - 2x) - 5 + x ≥ 0 is
all values of x such that
x ≤ 1.4.
CALC
2nd F
5
9-1
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EL-9900 Graphing Calculator
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
Enter y = 3(4 - 2x) for Y1 and
y = 5 - x for Y2.
2-1
(7 times)
(4 times)
Y=
DEL
—
X/ /T/n
ENTER
5
2-2
2-3
View the graph.
GRAPH
Access the Set Shade screen.
2nd F DRAW
G
1
Since the inequality being
solved is Y1 ≥ Y2, the solu-
tion is where the graph of Y1
is “on the top” and Y2 is “on
the bottom.”
Set up the shading.
2-4
2-5
2-6
—
VARS
VARS
A
2nd F
A
ENTER
2
2nd F
ENTER
1
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,
the solution to the inequality
3(4 - 2x) ≥ 5 - x is all values
of x such that x ≤ 1.4.
2nd F CALC
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.
9-1
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EL-9900 Graphing Calculator
Solving Double Inequalities
The solution to a system of two inequalities in one variable consists of all values of the variable
that make each inequality in the system true. A system 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.
Ex a m p le
Solve a double inequality, using graphical techniques.
2x - 5 ≥ -1
2x -5 ≤ 7
Be fo re There may be differences in the results of calculations and graph plotting depending on the setting.
Sta r tin g
Return all settings to the default value and delete all data.
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
The “double” inequality
given can also be written to
-1 ≤ 2x - 5 ≤ 7.
1
Enter y = -1 for Y1, y = 2x - 5 for
Y2, and y = 7 for Y3.
( )
-
ENTER
ENTER
Y=
1
—
2
X/ /T/n
5
7
View the lines.
2
3
GRAPH
y = 2x - 5 and
y = -1 intersect at (2, -1).
Find the point of intersection.
2nd F CALC
2
9-2
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EL-9900 Graphing Calculator
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
y = 2x - 5 and y = 7
intersect at (6,7).
Move the tracer and find another
intersection.
4
5
2nd F CALC
2
Solve the inequalities.
The solution to the “double”
inequality -1 ≤ 2x - 5 ≤ 7 con-
sists of all values of x in be-
tween, and including, 2 and 6
(i.e., x ≥ 2 and x ≤ 6). The so-
lution is 2 ≤ 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-9900 allows the solution region to be indicated visually using the
Shade feature. Also, the points of intersection can be obtained easily.
9-2
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EL-9900 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).
Ex a m p le
Solve a system of two-variable inequalities by shading the solution region.
2x + y ≥ 1
x2 + y ≤ 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.
Be fo re
Sta r tin g
Set the zoom to the decimal window:
(
)
ENTER
ZOOM
A
2nd F
7
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
1
2
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 - x2
Enter y = 1 - 2x for Y1 and y = 1 - x 2
for Y2.
—
Y=
1
2
ENTER
X/ /T/n
2
—
x
X/ /T/n
1
Access the set shade screen
3
4
2nd F
G
DRAW
1
Shade the points of y -value so that
Y1 ≤ y ≤ Y2.
2nd F
A
VARS
A
ENTER
1
2nd F
VARS
ENTER
2
The intersections are (0, 1)
and (2, -3)
Graph the system and find the
intersections.
5
6
GRAPH
2nd F
2
CALC
2
2nd F CALC
The solution is 0 ≤ x ≤ 2.
Solve the system.
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.
9-3
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EL-9900 Graphing Calculator
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.
Ex a m p le
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
x 2 + 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)
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.
Be fo re
Sta r tin g
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
1-x
x + 2y ≤ 1 © y ≤
Rewrite the inequalities so that the
left-hand side is y.
1-1
1-2
2
x2+y ≥ 4 © y ≥ 4 - x 2
1-x
Enter y =
for Y1 and
2
y = 4 - x 2 for Y2.
a
2
—
4
Y=
1
X/ /T/n
/b
2
—
x
ENTER
X/ /T/n
Set the shade and view the solution
region.
Y2 ≤ y ≤ Y1
1-3
2nd F DRAW
2nd F VARS
G
A
1
ENTER
A
2
2nd F
GRAPH VARS
ENTER
1
GRAPH
Set the display area (window) to :
-9 < x < 3, -6 < y < 5.
2-1
WINDOW
( )
ENTER
ENTER
ENTER
ENTER
-
9
3
( )
-
ENTER
5
6
9-4
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EL-9900 Graphing Calculator
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
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
2-3
or
or
or
GRAPH
.
(-1.6, 1.8): -1.6 + 2 © 1.8 = 2
Substitute p oints and confirm
whether they are in the solution
region.
©©© This does not materialize.
.
(-2, -5): -2 + 2 © (-5) = -12
(-2)2 + (-5) = -1
©©© This does not materialize.
( )
1
•
-
6
8 ...+
.
(2.8, -1.4): 2.8 + 2 © (-1.4) = 0
(2.8)2 + (-1.4) = 6.44
X
2
•
1
©©© This materializes.
(Continuing key operations omitted.)
.
(-8, 4): -8 + 2 © 4 = 0
(-8)2 + 4 = 68
©©© This materializes.
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.
9-4
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EL-9900 Graphing Calculator
Slope and Intercept of Absolute 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.
Ex a m p le
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 Rapid Graph feature.
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.
Be fo re
Sta r tin g
Set the zoom to the decimal window:
(
)
ENTER
ZOOM
A
2nd F
7
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
1-1
1-2
Enter the function y =| x| for Y1.
1
Y=
MATH
X/ /T/n
B
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.
View the graph.
GRAPH
2-1
Enter the standard form of an abso-
lute value function for Y2 using the
Rapid Graph feature.
Y=
ALPHA
ALPHA
A
H
MATH
B
1
—
ALPHA
+
K
X/ /T/n
Substitute the coefficients to graph
y = | x - 1| .
2-2
2nd F
SUB
ENTER
ENTER
1
1
0
ENTER
10-1
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EL-9900 Graphing Calculator
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
2-3
2-4
View the graph.
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.
GRAPH
Change the coefficients to graph
y =| x| -1.
Y=
2nd F
SUB
ENTER
1
( )
-
0
ENTER
ENTER
1
ENTER
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.
View the graph.
2-5
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.
10-1
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EL-9900 Graphing Calculator
Solving Absolute 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.
Ex a m p le
Solve an absolute value equation | 5 - 4x| = 6
Be fo re
Sta r tin g
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.
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
1
Enter y = | 5 - 4x| for Y1.
Enter y = 6 for Y2.
—
MATH
1
5
Y=
4
B
ENTER
X/ /T/n
6
There are two points of in-
tersection of the absolute
value graph and the hori-
zontal line y = 6.
2
3
View the graph.
GRAPH
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 intu-
itively obvious, the solution
could also be obtained by
finding the x-intercepts of the
function y = | 5x - 4| - 6.
Find the points of intersection of
the two graphs and solve.
2nd F
2nd F
CALC
CALC
2
2
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.
10-2
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EL-9900 Graphing Calculator
Solving Absolute 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.
Ex a m p le
Solve absolute value inequalities in two methods.
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.
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.
Be fo re
Sta r tin g
Set viewing window to “-5< x <50,” and “-10< y <10”.
WINDOW
( )
-
5
ENTER
ENTER
5
0
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
1-1
1-2
6x
Rewrite the equation.
| 20 -
| < 8
5
6x
©| 20 -
| - 8 < 0.
5
6x
Enter y = | 20 -
| - 8 for Y1.
5
a
—
Y=
0
MATH
B
1
2
/b
5
6
X/ /T/n
—
8
1-3
1-4
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 calcu-
lation.)
View the graph, and find the
x-intercepts.
GRAPH
© x = 10, y = 0
2nd F CALC
2nd F CALC
5
5
© x = 23.33333334
y = 0.00000006 ( Note)
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.
10-3
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EL-9900 Graphing Calculator
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
Enter the function
y =| 3.5x + 4| for Y1.
Enter y = 10 for Y2.
2-1
Y=
CL
MATH
B
1
+
ENTER
3
•
5
X/ /T/n
4
1
0
Since the inequality you are
solving is Y1 > Y2, the solu-
tion is where the graph of Y2
is “on the bottom” and Y1 in
“on the top.”
2-2
2-3
Set up shading.
2nd F DRAW
2nd F VARS
G
A
1
ENTER
2
A
2nd F
VARS
ENTER
1
Set viewing window to “-10 < x < 10”
and “-5 < y < 50”, and view the graph.
WINDOW
( )
-
ENTER
0
1
1
0
( )
ENTER
ENTER
ENTER
ENTER
5
0
-
5
ENTER
5
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 approxi-
mate calculation.)
Find the points of intersection.
Solve the inequality.
2
-4
© x = -4, y = 10
2nd F CALC
2nd F CALC
2
2
© x = 1.714285714
y = 9.999999999 ( Note)
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.
10-3
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EL-9900 Graphing Calculator
Evaluating Absolute 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.
Ex a m p le
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| ?
6-9
1+3
| 6-9|
| 1+3|
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|
Be fo re
Sta r tin g
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.
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
Access the home or computation
screen.
1-1
1-2
+
The solution is 8.
Enter y = | -2(5-1)| and evaluate.
( )
-
(
—
MATH
2
5
B
1
)
1
ENTER
2-1
| -2 + 7| = 5, | -2| + | 7| = 9
Evaluate| -2 + 7| . Evaluate| -2| +| 7| .
©| -2 + 7| ≠ | -2| + | 7| .
CL
( )
-
2
2
ENTER
MATH
MATH
MATH
1
+
1
1
7
( )
-
+
7
ENTER
10-4
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EL-9900 Graphing Calculator
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
Is | x + y| = | x| +| y| ? Think about
this problem according to the cases
when x or y are positive or negative.
2-2
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| .
If x ≤ 0 and y ≥ 0
[e.g.; (x, y) = (-2, 7)]
| x +y| = | -2 + 7| = 5
| x| +| y| = | -2| + | 7| = 9
©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©| x + y| ≠ | x| + | y| .
If x ≥ 0 and y ≤ 0
[e.g.; (x, y) = (2, -7)]
| x +y| = | 2-7| = 5
| x| +| y| = | 2| + | -7| = 9
©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©| x + y| ≠ | x| + | y| .
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| .
Therefore | x +y| =| x| +| y| when x ≥ 0 and y ≥ 0,
and when x ≤ 0 and y ≤ 0.
6-9
1+3
6-9
1+3
6-9
1+3
6-9
1+3
Evaluate
. Evaluate
.
3-1
3-2
= 0.75 ,
= 0 .75
6-9
1+3
6-9
1+3
a
—
CL
MATH
1
+
9
/b
6
ENTER
9
©
=
1
1
1
3
a
—
MATH
MATH
6
/b
3
ENTER
1
+
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)]
| x /y| = | 2/7| = 2/7
| x| /| y| = | 2| /| 7| = 2/7
©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©| x /y| = | x| / | y|
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|
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|
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|
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.
10-4
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EL-9900 Graphing Calculator
Graphing Rational Functions
p (x)
q (x)
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 large.
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).
Ex a m p le
Graph the rational function and check several points as indicated below.
x-1
1. Graph f (x) =
.
x2-1
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).
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.
Be fo re
Sta r tin g
Set the zoom to the decimal window:
(
)
ZOOM
ALPHA
A
ENTER
7
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
x - 1
Enter y =
for Y1.
1-1
x2 -1
Y=
2
a
—
x
X/ /T/n
/b
X/ /T/n
1
—
1
View the graph.
The function consists of two
branches separated by the verti-
cal asymptote.
1-2
GRAPH
11-1
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EL-9900 Graphing Calculator
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
Find the domain and the vertical
asymptote of f (x), tracing the
graph to find the hole at x = 1.
2
Since f (x) can be written as
x - 1
, the domain
(x + 1)(x - 1)
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 numera-
tor zero. Next to the coordi-
nates 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)
TRACE
3
The y-intercept is at (0 ,1). No-
tice that there are no x-inter-
cepts for the graph of f (x).
Find the x- and y-intercepts of f (x).
2nd F
CALC
6
4
The line y = 0 is very likely a
horizontal asymptote of f (x).
Estimate the 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.
11-1
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EL-9900 Graphing Calculator
Solving Rational Function Inequalities
p (x)
q (x)
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. You can find the
solutions by graphing each side of the inequalities as an individual function.
Ex a m p le
Solve a rational inequality.
x
Solve
≤ 2 by graphing each side of the inequality as an individual function.
1 - x2
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.
Be fo re
Sta r tin g
Set the zoom to the decimal window:
(
)
ENTER
ZOOM
A
ALPHA
7
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
x
1
2
Enter y =
for Y2.
for Y1. Enter y = 2
1- x2
a
MATH
B
X/ /T/n
/b
Y=
1
2
—
1
x
ENTER
X/ /T/n
2
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.
Set up the shading.
2nd F
2nd F
2nd F
DRAW
VARS
VARS
G
1
ENTER
A
1
A
2
ENTER
View the graph.
3
4
GRAPH
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
Find the intersections, and solve the
inequality.
Do this four times
2nd F CALC
2
-0.8 ≤ x ≤ 0.8 or x ≥ 1.3.
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.
11-2
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EL-9900 Graphing Calculator
Graphing Parabolas
The graphs of quadratic equations (y = ax2 + bx + c) are called parabolas. Sometimes the quadratic
equation takes on the form of x = ay2 + 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.
Ex a m p le
Graph a parabola using two methods.
1. Graph the parabola x = y2 -2 in rectangular mode.
2. Graph the parabola x = y2 -2 in parametric mode.
Be fo re There may be differences in the results of calculations and graph plotting depending on the setting.
Sta r tin g
Return all settings to the default value and delete all data.
Set the zoom to the decimal window:
(
)
ALPHA
ZOOM
ENTER
A
7
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
x = y2 -2
Solve the equation for y.
1-1
1-2
x + 2 = y2
+
y =
x + 2
√
–
Enter y =
√x+2 for Y1 and enter
y = -Y1 for Y2.
2
X/ /T/n
Y=
2nd F
+
√
( )
-
A
ENTER
2nd F
VARS
ENTER
1
The graph of the equation y =
1-3
View the graph.
x+ 2 is the "top half" of the
√
GRAPH
parabola and the graph of the
equation y = - x + 2 gives
√
the "bottom half."
12-1
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EL-9900 Graphing Calculator
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
Change to parametric mode.
2-1
2-2
2nd F SET UP
E
2
Rewrite x = y2 -2 in parametric form.
Let y = T and substitute in x
Enter X1T = T 2 -2 and Y1T = T.
= y2 - 2, to obtain x = T2- 2.
2
x
—
X/ /T/n
2
ENTER
Y=
X/ /T/n
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.
View the graph. Consider why only
half of the parabola is drawn.
(To understand this, use Trace fea-
ture.)
2-3
GRAPH
TRACE
(
)
Set Tmin to -6.
2-4
2-5
( )
-
WINDOW
ENTER
6
View the complete parabola.
GRAPH
The calculator provides two methods for graphing parabolas, both of which
are easy to perform.
12-1
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EL-9900 Graphing Calculator
Graphing Circles
The standard equation of a circle of radius r that is centered at a point (h, k) is (x - h) 2 +
(y - k) 2 = r 2. In order to put an equation in standard form so that you can graph in rectangular
mode, it is necessary to solve the equation for y. You therefore need to use the process of
completing the square.
Ex a m p le
Graph the circles in rectangular mode. Solve the equation for y to put it in the
standard form.
1. Graph x 2 + y2 = 4.
2. Graph x 2 - 2x + y2 + 4y = 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.
Be fo re
Sta r tin g
Set the zoom to the decimal window:
(
)
ZOOM
A
ENTER
ALPHA
7
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
y2 = 4 - x2
Solve the equation for y.
1-1
Enter y = 4 - x2 for Y1 (the top
4 - x2
+
y =
–
√
√
half). Enter y = - 4 - x2 for Y2.
√
2
x
2nd F
—
X/ /T/n
Y=
4
√
( )
-
A
2nd F
VARS
ENTER
ENTER
1
This is a circle of radius r ,
centered at the origin.
1-2
2-1
View the graph.
GRAPH
x2 - 2x + y2 + 4y = 2
Place all variable terms on the
left and the constant term on
the right-hand side of the
equation.
Solve the equation for y,
completing the square.
x2-2x+y2+4y+4=2+4 Complete the square on the
y-term.
x2 - 2x + (y+2)2 = 6
Express the terms in y as a
perfect square.
(y+2)2 = 6 -x2 + 2x
Leave only the term involving
y on the left hand side.
y+2 = ± 6-x2+2x
Take the square root of both
sides.
√
y = ± 6-x2+2x -2
√
Solve for y.
12-2
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EL-9900 Graphing Calculator
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
Enter y = 6 - x2 + 2x for Y1,
y = Y1 - 2 for Y2, and y = -Y1 -2 for
Y3.
Notice that if you enter
2-2
√
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”.
2nd F
A
—
CL
—
6
X/ /T/n
Y=
CL
√
2
x
2
X/ /T/n
ENTER
+
2nd F
ENTER
VARS
ENTER
2nd F
1
2
( )
-
—
VARS
ENTER
1
2
"Turn off" Y1 so that it will not
graph.
2-3
2-4
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.
ENTER
-1.3 < Y < 3.1
-5.1 < Y < 1.1
Adjust the screen so that the whole
graph is shown. Shift 2 units down-
wards.
(3 times)
—
WINDOW
2
ENTER
—
GRAPH
ENTER
2
Graphing circles can be performed easily on the calculator display.
12-2
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EL-9900 Graphing Calculator
Graphing Ellipses
The standard equation for an ellipse whose center is at the point (h, k) with major and
(x - h) 2
(y - k) 2
= 1.
minor axes of length a and b is
+
a 2
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.
Ex a m p le
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
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.
Be fo re
Sta r tin g
Set the zoom to the decimal window:
(
)
ZOOM
A
ENTER
ALPHA
7
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
3(x - 3)2 + (y + 2)2 = 3
(y + 2)2 = 3 - 3(x - 3)2
1
Solve the equation for y, completing
the square.
Enter
3 - 3(x - 3)2
+
y + 2 =
√
Y1 = 3 - 3(x - 3)2
Y2 = Y1 - 2
Y3 = -Y1 -2
y =
3 - 3(x - 3)2 - 2
√
+
√
(
—
2nd F
Y=
X/ /T/n
2nd F
2
3
)
3
√
2
—
3
x
ENTER
—
ENTER
2nd F
A
1
VARS
ENTER
—
( )
-
VARS
ENTER
1
2
Turn off Y1 so that it will not graph.
2
ENTER
12-3
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EL-9900 Graphing Calculator
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
3
4
View the graph.
GRAPH
Adjust the screen so that the whole
graph is shown. Shift 2 units down-
wards.
-3.1 < Y < 3.1
-5.1 < Y < 1.1
(3 times)
—
WINDOW
ENTER
2
—
2
ENTER
GRAPH
Graphing an ellipse can be performed easily on the calculator display.
12-3
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EL-9900 Graphing Calculator
Graphing Hyperbolas
The standard equation for a hyperbola can take one of two forms:
( y - k )2
( x - h )2
-
-
= 1 with vertices at ( h ± a, k ) or
a2
b2
( y - h)2
( x - k )2
b2
= 1 with vertices at ( h, k ± b ).
a2
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.
Ex a m p le
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
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.
Be fo re
Sta r tin g
Set the zoom to the decimal window:
(
)
ZOOM
A
ENTER
ALPHA
7
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
x2 + 2x - y2 -6y = -3
Solve the equation for y completing
the square.
1
x2 + 2x - (y2 + 6y + 9) = -3 -9
x2 + 2x - (y +3)2 = -12
(y + 3)2 = x2 + 2x + 12
Enter
Y1 = x2 + 2x + 12
Y2 = Y1 -3
Y3 = -Y1 -3
√
+
y + 3 =
x2 + 2x + 12
√
x2 + 2x + 12 - 3
+
y =
√
2
2nd F
+
X/ /T/n
x
Y=
+
2
3
√
X/ /T/n
2nd F
2
ENTER
1
—
VARS
2nd F
A
ENTER
ENTER
1
( )
-
—
3
VARS
A
ENTER
1
Turn off Y1 so that it will not graph.
2
ENTER
12-4
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EL-9900 Graphing Calculator
Ste p & Ke y O p e ra tio n
Disp la y
N o te s
3
View the graph.
GRAPH
Zoom out the screen.
4
ZOOM
A
4
Graphing hyperbolas can be performed easily on the calculator display.
12-4
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Key pad for the SHARP EL-9900 Calculator
Advanced Keyboard
Cursor movement keys
Clear/Quit key
Graphing keys
Power supply ON/OFF key
Secondary function specification key
Alphabet specification key
Display screen
Variable enter key
Calculation execute key
Communication port for peripheral devices
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Key pad for the SHARP EL-9900 Calculator
Basic Keyboard
Cursor movement keys
Clear/Quit key
Graphing keys
Power supply ON/OFF key
Secondary function specification key
Alphabet specification key
Display screen
Variable enter key
Calculation execute key
Communication port for peripheral devices
Download from Www.Somanuals.com. All Manuals Search And Download.
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-9900. For this purpose, we would be grateful if you would com-
plete 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:
E-mail:
Fax:
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
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BEFORE STARTING : Write any conditions to be set up before solving the problems.
STEP
NOTES
SHARP CORPORATION Osaka, Japan
Fax:
SHARP Graphing Calculator
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SH ARP CORPORATION OSAKA, JAPAN
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