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EL9600C - Graphing Calculator
Handbook
60 pgs1.02 Mb0
Teachers Manual
35 pgs1.52 Mb0

Sharp EL9600C - Graphing Calculator, EL-9600, EL-9400 Handbook

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Graphing Calculator
EL-9600/9400
Handbook Vol. 1
Algebra
EL-9600
EL-9400
Read this first
1. Always read “Before Start”
2. Set Up Condition
As key operations for this handbook are conducted from the initial condition, reset all memories to the initial condition beforehand.
2nd F
OPTION
Note: Since all memories will be deleted, it is advised to use the CE-LK1 PC link kit (sold
separately) to back up any programmes not to be erased, or to return the settings to the initial condition (cf. 3. Initial Settings below) and to erase the data of the function to be used.
•
To delete a single data, press
• Other keys to delete data: : to erase equations and remove error displays
CL
: to cancel previous function
2nd F
QUIT
2
E
CL
2nd F
OPTION
and select data to be deleted from the menu.
C
3. Initial settings
Initial settings are as follows:
✩
Set up (
✩
Format (
✩
Stat Plot (
✩
Shade (
✩
Zoom (
✩
Period (
✩
Note:
returns to the default setting in the following operation.
(
2nd F
)
OPTION
): Rad, FloatPt, 9, Rect, Decimal(Real), Equation
2nd F
SET UP
): RectCoord, OFF, OFF, Connect, Sequen
FORMAT
2nd F
): 2. PlotOFF
STAT PLOT
2nd F
): 2. INITIAL
2nd F
ZOOM
): 1. PmtEnd
2nd F
2nd F
E1
E
DRAW
G
): 5. Default
A
FINANCE
C
ENTER
4. Using the keys
Press to use secondary functions (in yellow).
2nd F
-1
To select “sin
Press to use the alphabet keys (in blue).
ALPHA
To select A:
”: ➔ Displayed as follows:
2nd F
sin
➔ Displayed as follows:
sin
ALPHA
2nd F
ALPHA
sin
A
-1
5. Notes
•
Some features are provided only on the EL-9600 and not on the EL-9400. (Substitution,
Solver, Matrix, Tool etc.)
•
As this handbook is only an example of how to use the EL-9600 and 9400, please refer to the
manual for further details.
Using this Handbook
This handbook was produced for practical application of the SHARP EL-9600 and 9400 Graphing Calculator based on exercise examples received from teachers actively engaged in teaching. It can be used with minimal preparation in a variety of situations such as classroom presentations, and also as a self-study reference book.
Introduction
Explanation of the section
Example
Example of a problem to be solved in the section
Before Start
Important notes to read before operating the calculator
Step & Key Operation
A clear step-by-step guide to solving the problems
Display
Slope and Intercept of Quadratic Equations
A quadratic equation of y in terms of x can be expressed by the standard form y = a (x - h)2+ k, where a is the coefficient of the second degree term ( y = ax vertex of the parabola formed by the quadratic equation. An equation where the largest exponent on the independent variable x is 2 is considered a quadratic equation. In graphing quadratic equations on the calculator, let the x- variable be represented by the horizontal axis and let y be represented by the vertical axis. The graph can be adjusted by varying the coefficients a, h, and k.
Example
Graph various quadratic equations and check the relation between the graphs and the values of coefficients of the equations.
2
1. Graph y = x
and y = (x-2)2.
2
2. Graph y = x
and y = x2+2.
2
3. Graph y = x
and y = 2x2.
2
4. Graph y = x
and y = -2x2.
There may be differences in the results of calculations and graph plotting depending on the setting.
Before
Return all settings to the default value or to delete all data.
Start
As Substitution feature is only available on the EL-9600, this section does not apply to the EL-9400.
Step & Key Operation
*Use either pen touch or cursor to operate.
Enter the equation y = x2 for Y1.
1-1
2
X
/θ/T/
n
Y=
x
Enter the equation y = (x-2)2 for Y2
1-2
using Sub feature.
ENTER
1
EZ
ENTER ENTER
ALPHA
C
1
*
2nd F
1
ENTER
SUB
( )
ENTER
0
View both graphs.
1-3
GRAPH
2-1
2-2
Display
3-1
*
3-2
*
2
ENTER
4-1
This shows that placing an h (>0) within the standard form y = a (x - h) h units and placing an h (<0) will move it left h units on the x-axis.
4-2
Illustrations of the calculator screen for each step
Merits of Using the EL-9600/9400
2-1
Highlights the main functions of the calculator relevant to the section
EL-9600 Graphing Calculator
Notes
Explains the process of each
2
+ bx + c) and (h, k) is the
Step & Key Operation
*Use either pen touch or cursor to operate.
Change the equation in Y2 to y = x2+2.
*
2nd F
SUB
2
Notes
*
2
2nd F
SUB
ENTER
0
Notice that the addition of -2 within the quadratic operation moves the basic y =x right two units (adding 2 moves
2
it left two units) on the x-axis.
.
*
(-)
2nd F
SUB
0
ENTER
2
graph
2
Y=
ENTER ENTER
View both graphs.
GRAPH
Change the equation in Y2 to y = 2x2.
Y=
View both graphs.
GRAPH
Change the equation in Y2 to y = -2x
2
+ k will move the basic graph right
Y=
ENTER
View both graphs.
GRAPH
The EL-9600/9400 allows various quadratic equations to be graphed easily. Also the characteristics of quadratic equations can be visually shown through the relationship between the changes of coefficient values and their graphs, using Substitution feature.
step in the key operations
EL-9600 Graphing Calculator
Display
fact that adding k (>0) within the standard form y = a (x -
2
+ k will move the basic graph up k units and placing an
h) k (<0) will move the basic graph down k units on the y­axis.
2-1
ing an a (<-1) in the standard form y = a (x - h) will pinch or close the basic graph and flip it (reflect it) across the x-axis.
Notes
Notice that the addition of 2 moves
2
graph up two units
the basic y =x and the addition of -2 moves the basic graph down two units on the y-axis. This demonstrates the
Notice that the multiplication of 2 pinches or closes the basic
2
y=x
graph. This demonstrates the fact that multiplying an a (> 1) in the standard form
2
+ k
will pinch or close
(x - h) the basic graph.
Notice that the multiplication of
-2 pinches or closes the basic y =
x2 graph and flips it (reflects it) across the x-axis. This dem­onstrates the fact that multiply-
y = a
2
+ k
• When you see the sign
means same series of key strokes can be done with screen touch on the EL-9600.
*
( * : for the corresponding key; Key operations may also be carried out with the cursor (not shown).
•
Different key appearance for the EL-9400: for example
on the key:
*
: for the corresponding keys underlined.)
*
X/ /T/
n
➔
X/T
We would like to express our deepest gratitude to all the teachers whose cooperation we received in editing this book. We aim to produce a handbook which is more replete and useful to everyone, so any comments or ideas on exercise will be welcomed.
(Use the attached blank sheet to create and contribute your own mathematical problems.)
Thanks to Dr. David P. Lawrence at Southwestern Oklahoma State University for the use of his teaching resource book (Applying Pre-Algebra/Algebra using the SHARP EL-9600 Graphing Calculator).
Other books available: Graphing Calculator EL-9600 TEACHERS’ GUIDE Graphing Calculator EL-9400 TEACHERS’ GUIDE
Contents
1. Linear Equations
1-1
Slope and Intercept of Linear Equations
1-2
Parallel and Perpendicular Lines
2. Quadratic Equations
2-1
Slope and Intercept of Quadratic Equations
2-2
Shifting a Graph of Quadratic Equations
3. Literal Equations
3-1
Solving a Literal Equation Using the Equation Method (Amortization)
3-2
Solving a Literal Equation Using the Graphic Method (Volume of a Cylinder)
3-3
Solving a Literal Equation Using Newton’s Method (Area of a Trapezoid)
4. Polynomials
4-1
Graphing Polynomials and Tracing to Find the Roots
4-2
Graphing Polynomials and Jumping to Find the Roots
5. A System of Equations
5-1
Solving a System of Equations by Graphing or Tool Feature
6. Matrix Solutions
6-1
Entering and Multiplying Matrices
6-2
Solving a System of Linear Equations Using Matrices
7. Inequalities
7-1
Solving Inequalities
7-2
Solving Double Inequalities
7-3
System of Two-Variable Inequalities
7-4
Graphing Solution Region of Inequalities
8. Absolute Value Functions, Equations, Inequalities
8-1
Slope and Intercept of Absolute Value Functions
8-2
Shifting a graph of Absolute Value Functions
8-3
Solving Absolute Value Equations
8-4
Solving Absolute Value Inequalities
8-5
Evaluating Absolute Value Functions
9. Rational Functions
9-1
Graphing Rational Functions
9-2
Solving Rational Function Inequalities
10. Conic Sections
10-1 10-2 10-3 10-4
Graphing Parabolas Graphing Circles Graphing Ellipses Graphing Hyperbolas
EL-9600/9400 Graphing Calculator
Slope and Intercept of Linear Equations
A linear equation of y in terms of x can be expressed by the slope-intercept form y = mx+b, where m is the slope and b is the y- intercept. W e call this equation a linear equation since its graph is a straight line. Equations where the exponents on the x and y are 1 (implied) are considered linear equations. In graphing linear equations on the calculator, we will let the x variable be represented by the horizontal axis and let y be represented by the vertical axis.
Example
Draw graphs of two equations by changing the slope or the y- intercept.
1. Graph the equations y = x and y = 2x.
2. Graph the equations y = x and y = x.
3. Graph the equations y = x and y = - x.
4. Graph the equations y = x and y = x + 2.
1 2
Before
1-1
1-2
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
2-1
There may be differences in the results of calculations and graph plotting depending on the setting.
Start
Return all settings to the default value or to delete all data.
Step & Key Operation
*Use either pen touch or cursor to operate.
(When using EL-9600)
Enter the equation y = x for Y1 and y = 2x for Y2.
*
X/
Y=
View both graphs.
GRAPH
Enter the equation y = x for Y2.
Y= CL
/T/n X/
ENTER
*
/T/n
2
1 2
Display
(When using EL-9600)
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.
Notes
*
a
/b
2
1
2-2
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
View both graphs.
GRAPH
X/
/T/n
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.
1-1
EL-9600/9400 Graphing Calculator
Notes
3-1
3-2
Step & Key Operation
(When using EL-9600)
*Use either pen touch or cursor to operate.
Enter the equation y = - x for Y2.
(
Y=
CL
*
)
-
X/
/T/n
View both graphs.
GRAPH
Display
(When using EL-9600)
Notice how Y2 decreases (going down from left to right) instead of increasing (going up from left to right). Negative slopes (m<0) make the line decrease or go down from left to right.
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
4-1
Enter the equation y = x + 2 for Y2.
4-2
Y= CL
*
View both graphs.
GRAPH
+ 2
X/
/T/n
Adding 2 will shift the y = x graph upwards.
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
Making a graph is easy, and quick comparison of several graphs will help students understand the characteristics of linear equations.
1-1
EL-9600/9400 Graphing Calculator
P arallel and Perpendicular Lines
Parallel and perpendicular lines can be drawn by changing the slope of the linear equation and the y intercept. A linear equation of y in terms of x can be expressed by the slope­intercept form y = mx + b, where m is the slope and b is the y-intercept. Parallel lines have an equal slope with different y-intercepts. Perpendicular lines have slopes that are negative reciprocals of each other (m = - ). These characteristics can be verified by graphing these lines.
Example
Graph parallel lines and perpendicular lines.
1. Graph the equations y = 3x + 1 and y = 3x + 2.
2. Graph the equations y = 3x - 1 and y = - x + 1.
1
3
1
m
Before
1-1
There may be differences in the results of calculations and graph plotting depending on the setting.
Start
Return all settings to the default value or to delete all data. Set the zoom to the decimal window:
Step & Key Operation
*Use either pen touch or cursor to operate.
(When using EL-9600)
* (
ZOOM
C
(When using EL-9600)
ENTER
ALPHA
Display
)
7
*
*
Notes
Enter the equations y = 3x + 1 for Y1 and y = 3x + 2 for Y2.
1-2
Y=
3
View the graphs.
X/
/T/n
3
+
X/
/T/n
1
+
2
ENTER
*
These lines have an equal
slope but different y- inter-
GRAPH
cepts. They are called paral-
lel, and will not intersect.
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
2-1
Enter the equations y = 3x - 1 for Y1 and y = - x + 1 for Y2.
Y= CL
(
CL
+
1
1 3
3
)
1
-
—
X/
/T/n
a
/b
1
ENTER
3
X/
*
*
/T/n
1-2
EL-9600/9400 Graphing Calculator
Notes
1
m
2-2
Step & Key Operation
(When using EL-9600)
*Use either pen touch or cursor to operate.
View the graphs.
GRAPH
Display
(When using EL-9600)
These lines have slopes that are negative reciprocals of
each other (m = - ). They are called perpendicular. Note that
these intersecting lines form four equal angles.
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
The Graphing Calculators can be used to draw parallel or perpendicular lines while learning the slope or y-intercept of linear equations.
1-2
EL-9600 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 = ax2 + bx + c) and (h, k) is the vertex of the parabola formed by the quadratic equation. An equation where the largest exponent on the independent variable x is 2 is considered a quadratic equation. In graphing quadratic equations on the calculator, let the x-variable be represented by the horizontal axis and let y be represented by the vertical axis. The graph can be adjusted by varying the coefficients a, h, and k.
Example
Graph various quadratic equations and check the relation between the graphs and the values of coefficients of the equations.
/T/n
2
and y = (x - 2)2.
2
and y = x2 + 2.
2
and y = 2x2.
2
and y = -2x2.
2
x
1. Graph y = x
2. Graph y = x
3. Graph y = x
4. Graph y = x
Before
1-1
1-2
There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value or to delete all data.
Start
As Substitution feature is only available on the EL-9600, this section does not apply to the EL-9400.
Step & Key Operation
*Use either pen touch or cursor to operate.
Enter the equation y = x2 for Y1.
X/
Y=
Enter the equation y = (x - 2)2 for Y2 using Sub feature.
EZ
ALPHA
C
2nd F
SUB
ENTER
1
ENTER ENTER
1
1
*
ENTER
Display
*
2
*
ENTER
Notes
( )
ENTER
0
View both graphs.
1-3
GRAPH
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
Notice that the addition of -2 within the quadratic operation
2
moves the basic y = x
graph right two units (adding 2 moves it left two units) on the x-axis.
This shows that placing an h (>0) within the standard form y = a (x - h)
2
+ k will move the basic graph right h units and placing an h (<0) will move it left h units on the x-axis.
2-1
EL-9600 Graphing Calculator
Notes
x2 graph up two units
2-1
2-2
Step & Key Operation
(When using EL-9600)
*Use either pen touch or cursor to operate.
Change the equation in Y2 to y =
2
*
2nd F
SUB
Y=
ENTER ENTER
View both graphs.
GRAPH
Display
(When using EL-9600)
x2+2.
0
Notice that the addition of 2 moves the basic y = and the addition of -2 moves the basic graph down two units on the y-axis. This demonstrates the
fact that adding k (>0) within the standard form y = a (x -
2
+ k will move the basic graph up k units and placing an
h) k (<0) will move the basic graph down k units on the y­axis.
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
3-1
Change the equation in Y2 to y = 2
Y=
*
ENTER
0
2nd F
SUB
2
ENTER
x2.
3-2
View both graphs.
GRAPH
Notice that the multiplication of 2 pinches or closes the basic
2
y = x
graph. This demonstrates the fact that multiplying an a (> 1) in the standard form
2
+ k
(x - h)
will pinch or close
y = a
the basic graph.
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
4-1
4-2
Change the equation in Y2 to y = -2x
ENTER
Y=
2
.
2nd F
*
SUB
(-)
View both graphs.
2
Notice that the multiplication of
-2 pinches or closes the basic y =
GRAPH
x2 graph and flips it (reflects
it) across the x-axis. This dem­onstrates the fact that multiply-
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-9600/9400 allows various quadratic equations to be graphed easily. Also the characteristics of quadratic equations can be visually shown through the relationship between the changes of coefficient values and their graphs, using Substitution feature.
2-1
EL-9600/9400 Graphing Calculator
Shifting a Graph of Quadratic Equations
A quadratic equation of y in terms of x can be expressed by the standard form y = a (x - h)2 + k, where a is the coefficient of the second degree term (y = a of the parabola formed by the quadratic equation. An equation where the largest exponent on the independent variable x is 2 is considered a quadratic equation. In graphing quadratic equations on the calculator, let the x-variable be represented by the horizontal axis and let y be represented by the vertical axis. The relation of an equation and its graph can be seen by moving the graph and checking the coefficients of the equation.
Example
Move or pinch a graph of quadratic equation y = x2 to verify the relation between the coefficients of the equation and the graph.
1. Shift the graph y = x
2. Shift the graph y = x
3. Pinch the slope of the graph y = x
2
upward by 2.
2
to the right by 3.
2
.
x2 + bx + c) and (h, k) is the vertex
Before
1-1 Access Shift feature and select the
1-2
1-3
There may be differences in the results of calculations and graph plotting depending on the setting.
Start
Return all settings to the default value or to delete all data.
Step & Key Operation
*Use either pen touch or cursor to operate.
(When using EL-9600)
2
equation y = x
2nd F
SHIFT/CHANGE
1
*
Move the graph y = x2 upward by 2.
Save the new graph and observe the changes in the graph and the equation.
ENTER
ALPHA
.
ENTER
A
*
*
Display
(When using EL-9600)
Notice that upward movement of the basic y =
units in the direction of the y- axis means addition of 2 to the
y-intercept. This demonstrates that upward movement of the graph by k units means adding a k (>0) in the standard form y = a(x - h)2 + k.
Notes
2
x
graph by 2
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
2-2
EL-9600/9400 Graphing Calculator
Notes
2
graph to the right
2-1
2-2
Step & Key Operation
(When using EL-9600)
*Use either pen touch or cursor to operate.
Move the graph y = x2 to the right by 3.
CL
(three times)
ENTER
*
Save the new graph and observe the changes in the graph and the equation
Display
(When using EL-9600)
Notice that movement of the basic y = x by 3 units in the direction of the x-axis is equivalent to the
ENTER
ALPHA
addition of 3 to the x -intercept. This demonstrates that movement of the graph to the right means adding an h (>0) in the standard form y = a (x - h)
2
+ k and movement to the left means
subtracting an h (<0).
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
3-1
Access Change feature and select the equation y = x
2
.
2nd F
3-2
3-3
SHIFT/CHANGE
1
*
Pinch the slope of the graph.
ENTER
Save the new graph and observe the changes in the graph and the equation.
ENTER
ALPHA
B
*
Notice that pinching or closing the basic y =
2
x
graph
is equivalent to increasing an a (>1) within the standard form y = a (x - h)
2
+ k and broadening the graph is equivalent to decreasing an a (<1).
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
The Shift/Change feature of the EL-9600/9400 allows visual understanding of how graph changes affect the form of quadratic equations.
2-2
EL-9600 Graphing Calculator
Solving a Literal Equation Using the Equation Method
(Amortization)
Solver mode is used to solve one unknown variable by inputting known variables, by three methods: Equation, Newton’s, and Graphic. The Equation method is used when an exact solution can be found by simple substitution.
Example
Solve an amortization formula. The solution from various values for known variables can be easily found by giving values to the known variables using Equation method in Solver mode.
-1
-N
The formula : P = L
1-(1+ )
12
I / 12
I
P= monthly payment L= loan amount
I= interest rate N=number of months
1. Find the monthly payment on a $15,000 car loan, made at 9% interest over four
years (48 months) using the Equation method.
2. Save the formula as “AMORT”.
3. Find amount of loan possible at 7% interest over 60 months with a $300
payment, using the saved formula.
1-1
1-2
1-3
Before
There may be differences in the results of calculations and graph plotting depending on the setting.
Start
Return all settings to the default value or to delete all data. As Solver feature is only available on the EL-9600, this section does not apply to the EL-9400.
Step & Key Operation
*Use either pen touch or cursor to operate.
Access the Solver feature.
SOL VER
2nd F
Select the Equation method for solving.
SOL VER
2nd F
1
*
A
*
Enter the amortization formula.
SOL VER
2nd F
a
/b
=
P
1
L
—
(
ALPHA
1
Display
Notes
This screen will appear a few seconds after “SOLVER” is dis­played.
+(
a
1
ALPHA
b
(-)
a
ALPHA
I
b
(-)
a
/b
ALPHA
÷
1
1
N
1
2
2
)
*
*
)
*
3-1
EL-9600 Graphing Calculator
Notes
1-4
Step & Key Operation
(When using EL-9600)
*Use either pen touch or cursor to operate.
Enter the values L=15,000,
Display
(When using EL-9600)
I=0.09, N=48.
15000
*
•09 4
*
The monthly pay ment (P) is
1-5
ENTER
ENTER ENTER
*
8
ENTER
Solve for the payment(P).
$373.28.
( )
CL
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
2-1
2-2
Save this formula.
*
2nd F
SOL VER
Give the formula the name AMORT.
C
2nd F
*
ENTER
EXE
*
AMORT
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
3-1
3-2
Recall the amortization formula.
SOL VER
2nd F
01
*
Enter the values: P = 300,
B
ENTER
*
I = 0.01, N = 60
3-3
ENTER ENTER ENTER
•01 61
300
*
Solve for the loan (L).
0
ENTERENTER
*
The amount of loan (L) is $17550.28.
2nd F
*
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
EXE
With the Equation Editor, the EL-9600/9400 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 Solver mode when using the EL-9600.
3-1
EL-9600 Graphing Calculator
Solving a Literal Equation Using the Graphic Method
(Volume of a Cylinder)
Solver mode is used to solve one unknown variable by inputting known variables. There are three methods: Equation, Newton’s, and Graphic. The Equation method is used when an exact solution can be found by simple substitution. Newton’s method implements an iterative approach to find the solution once a starting point is given. When a starting point is unavailable or multiple solutions are expected, use the Graphic method. This method plots the left and right sides of the equation and then locates the intersection(s).
Example
Use the Graphic method to find the radius of a cylinder giving the range of the unknown variable.
The formula : V = πr2h ( V = volume r = radius h = heiqht)
1. Find the radius of a cylinder with a volume of 30in
3
and a height of 10in, using
the Graphic method.
2. Save the formula as “V CYL”.
3. Find the radius of a cylinder with a volume of 200in
using the saved formula.
3
and a height of 15 in,
1-1
1-2
1-3
Before
There may be differences in the results of calculations and graph plotting depending on the setting.
Start
Return all settings to the default value or to delete all data. As Solver feature is only available on the EL-9600, this section does not apply to the EL-9400.
*Use either pen touch or cursor to operate.
Access the Solver feature.
2nd F
SOL VER
Select the Graphic method for solving.
2nd F
SOL VER
A
*
3
*
Enter the formula V = πr2h.
V
2
R
ALPHA
x
2nd F
=ALPHA ALPHA ALPHA
H
π
Display
NotesStep & Key Operation
This screen will appear a few seconds after “SOLVER” is dis­played.
1-4
Enter the values: V = 30, H = 10. Solve for the radius (R).
ENTER ENTER
0
30 1
ENTER
*
*
2nd F EXE
*
3-2
EL-9600 Graphing Calculator
10
30
Notes
π
3
π
1-5
Step & Key Operation
(When using EL-9600)
*Use either pen touch or cursor to operate.
Set the variable range from 0 to 2.
0
ENTER
*
2
ENTER
Display
(When using EL-9600)
The graphic solver will prompt with a variable range for solv­ing.
r2 = = <3 r =1 ➞ r2 = 12 = 1 <3
r =2 ➞ r2 = 22 = 4 >3
Use the larger of the values to be safe.
1-6
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Solve.
2nd F EXE
CL
(
)
The solver feature will graph the left side of the equation (volume, y = 30), then the right side of the equation (y = 10r
2
and finally will calculate the intersection of the two graphs to find the solution. The radius is 0.98 in.
),
2
Save this formula. Give the formula the name “V CYL”.
*
2nd F
SOL VER
SPACE
VY
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
3-1
3-2
Recall the formula. Enter the values: V = 200, H = 15.
*
2nd F
SOL VER
ENTER ENTER ENTER
15
ENTER
Solve the radius setting the variable range from 0 to 4.
ENTER
C
C
L
B01
*
ENTER
*
0020
200π 14
r2 = = < 14
15
π
r = 3 ➞ r2 = 32 = 9 < 14
2nd F EXE
*
ENTER 2nd F EXE
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
00
ENTER
r = 4 ➞ r2 = 42 = 16 > 14
Use 4, the larger of the values, to be safe. The answer is : r = 2.06
One very useful feature of the calculator is its ability to store and recall equations. The solution from various values for known variables can be easily obtained by recalling an equation which has been stored and giving values to the known variables. The Graphic method gives a visual solution by drawing a graph.
3-2
EL-9600 Graphing Calculator
Solving a Literal Equation Using Newton's Method (Area of a Trapezoid)
Solver mode is used to solve one unknown variable by inputting known variables. There are three methods: Equation, Newton’s, and Graphic. The Newton’s method can be used for more complicated equations. This method implements an iterative approach to find the solution once a starting point is given.
Example
Find the height of a trapezoid from the formula for calculating the area of a trapezoid useing Newton’s method.
The formula : A= h(b+c)
1
2
(A = area h = height b = top face c = bottom face)
1. Find the height of a trapezoid with an area of 25in
2
and bases of length 5 in
and 7 in using Newton's method. (Set the starting point to 1.)
2. Save the formula as “A TRAP”.
3. Find the height of a trapezoid with an area of 50in
using the saved formula. (Set the starting point to 1.)
Before
1-1
1-2
1-3
There may be differences in the results of calculations and graph plotting depending on the setting.
Start
Return all settings to the default value or to delete all data. As Solver feature is only available on the EL-9600, this section does not apply to the EL-9400.
*Use either pen touch or cursor to operate.
Access the Solver feature.
2nd F
SOL VER
Select Newton's method for solving.
2nd F
SOL VER
A
*
2
*
Enter the formula A = h(b+c).
1
2
Display
2
with bases of 8 and 10
NotesStep & Key Operation
This screen will appear a few seconds after “SOLVER” is dis­played.
A
ALPHA
ALPHA
C
1-4
Enter the values: A = 25, B = 5, C = 7
ENTER
ALPHA
H
)
25
5
*
=
(
ALPHA ALPHA
ENTER
ENTER ENTER
*
7
1
B
*
a
2
/b
*
+
3-3
EL-9600 Graphing Calculator
Step & Key Operation
(When using EL-9600)
*Use either pen touch or cursor to operate.
1-5 Solve for the height and enter a
starting point of 1.
Display
(When using EL-9600)
Newton's method will prompt with a guess or a
Notes
starting point.
2nd F EXE
*
1-6
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
2
Solve.
2nd F EXE
CL
(
)
Save this formula. Give the formula
ENTER
1
The answer is : h = 4.17
the name “A TRAP”.
*
2nd F
SOL VER
SPACE
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
3-1
Recall the formula for calculating
C
R
TAPA
ENTER
ENTER
the area of a trapezoid.
2nd F
SOL VER
B
*
01
Enter the values:
3-2
A = 50, B = 8, C = 10.
ENTER
ENTER
50
10
3-3
2nd F EXE
*
ENTER
2nd F EXE
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
ENTER
ENTER
*
8
*
The answer is : h = 5.56Solve.
1
One very useful feature of the calculator is its ability to store and recall equations. The solution from various values for known variables can be easily obtained by recalling an equation which has been stored and giving values to the known variables in the Solver mode. If a starting point is known, Newton's method is useful for quick solution of a complicated equation.
3-3
EL-9600 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.
Example
Draw a graph of a polynomial and approximate the roots by using zoom-in and Trace features.
1. Graph the polynomial y = x
2. Approximate the left-hand root.
3. Approximate the middle root.
4. Approximate the right-hand root.
There may be differences in the results of calculations and graph plotting depending on the setting.
Before
Start
Return all settings to the default value or to delete all data. Set the zoom to the decimal window:
Setting the zoom factors to 5 : As Substitution feature is only available on the EL-9600, this section does not apply to the EL-9400.
3
- 3x2 + x + 1.
ZOOM
ZOOM
B
(
A
*
ENTER ENTER ENTER
*
ENTER
ALPHA
55
)
7
*
*
2nd F
QUIT
*Use either pen touch or cursor to operate.
1-1
1-2
1-3
Enter the polynomial
3
- 3x2 + x + 1.
y = x
Y= EZ
5
*
ENTER ENTER ENTER
*
Enter the coefficients.
2nd F SUB
1 1
ENTER
*
ENTER ENTER
1
*
Return to the equation display screen.
2nd F EXE
1-4
View the graph.
(-)
3
*
ENTER
Display
*
NotesStep & Key Operation
It may take few seconds for the graph to be drawn. Enter each coefficients when the cursor is displayed.
*
GRAPH
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
4-1
EL-9600 Graphing Calculator
Step & Key Operation
(When using EL-9600)
*Use either pen touch or cursor to operate.
Display
(When using EL-9600)
Notes
Tracer
2-1
Move the tracer near the left-hand root.
Note that the tracer is flashing on the curve and the x and y coordinates are shown at the bottom of the screen.
2-2
TRACE
(repeatedly)
*
Zoom in on the left-hand root.
ZOOM
A3
* *
Tracer
2-3
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
3-1
Move the tracer to approximate the root.
TRACE
or
*
(repeatedly)
*
Return to the previous decimal
The root is : x -0.42
viewing window.
ZOOM
H
*
2
*
Tracer
3-2
Move the tracer to approximate the middle root.
The root is exactly x = 1. (Zooming is not needed to find a better approximate.)
TRACE
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
(repeatedly)
*
Tracer
4
Move the tracer near the right-
The root is : x
2.42 hand root. Zoom in and move the tracer to find a better approximate.
(repeatedly)
*
ZOOM
A3
* *
TRACE
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
or
*
(repeatedly)
*
The calculator allows the roots to be found (or approximated) visually by graphing a polynomial and using the Zoom-in and Trace features.
4-1
EL-9600/9400 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.
Example
Draw a graph of a polynomial and find the roots by using the Calculate feature.
1. Graph the polynomial y = x
2. Find the four roots one by one.
4
+ x3 - 5x2 - 3x + 1.
Before
Start
1-1
1-2
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value or to delete all data. Setting the zoom factors to 5 :
(When using EL-9600)
*Use either pen touch or cursor to operate.
Enter the polynomial
4
+ x3 - 5x2 - 3x + 1
y = x
Y=
b
a
35
—
b
X/
/T/n X/
a
4
—
*
X/
/T/n
31
+
+
*
X/
/T/n
ZOOMAENTER ENTER ENTER 2nd F
x
*
/T/n
2
AA
Display
(When using EL-9600)
QUIT
NotesStep & Key Operation
View the graph.
GRAPH
2-1
2-2
Find the first root.
2nd F CALC
5
*
Find the next root.
2nd F CALC
5
*
-2.47
x
Y is almost but not exactly zero. Notice that the root found here is an approximate value.
x -0.82
4-2
EL-9600/9400 Graphing Calculator
Step & Key Operation
(When using EL-9600)
*Use either pen touch or cursor to operate.
2-3
2-4
Find the next root.
2nd F CALC
5
*
Find the next root.
2nd F CALC
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
5
*
Display
(When using EL-9600)
x 0.24
x
Notes
2.05
The calculator allows jumping to find the roots by graphing a polynomial and using the Calculate feature, without tracing the graph.
4-2
EL-9600/9400 Graphing Calculator
Solving a System of Equations by Graphing or Tool Feature
A system of equations is made up of two or more equations. The calculator provides the Calculate feature and Tool feature to solve a system of equations. The Calculate feature finds the solution by calculating the intersections of the graphs of equations and is useful for solving a system when there are two variables, while the Tool feature can solve a linear system up to six variables and six equations.
Example
Solve a system of equations using the Calculate or Tool feature. First, use the Calcu­late feature. Enter the equations, draw the graph, and find the intersections. Then, use the Tool feature to solve a system of equations.
1. Solve the system using the Calculate feature.
y = x2 - 1
{
y = 2x
2. Solve the system using the Tool feature.
5x + y = 1
{
-3x + y = -5
Before
1-1
1-2
1-3
There may be differences in the results of calculations and graph plotting depending on the setting.
Start
Return all settings to the default value or to delete all data. Choose the viewing window “-5 < X < 5”, “-10 < Y < 10” using Rapid window feature
WINDOW
EZ
As Tool feature is only available on the EL-9600, the example 2 does not apply to the EL-9400.
Step & Key Operation
*Use either pen touch or cursor to operate.
(When using EL-9600)
Enter the system of equations
2
- 1 for Y1 and y = 2x for Y2.
y = x
Y=
X/
2
X/
/T/n
/T/n
x
View the graphs.
GRAPH
Find the left-hand intersection using Calculate feature.
2nd F CALC
2
574
*
2
—1
(
ENTER ENTER ENTER
*
ALPHA
ENTER
)
*
*
Display
(When using EL-9600)
* *
*
Notes
Note that the x and y coordi­nates are shown at the bot­tom of the screen. The answer
*
is : x = -0.41 y = -0.83
1-4
Find the right-hand intersection by accessing the Calculate feature again.
2nd F CALC
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
2
*
The answer is : x = 2.41 y = 4.83
5-1
EL-9600/9400 Graphing Calculator
Notes
2-1
Step & Key Operation
(When using EL-9600)
*Use either pen touch or cursor to operate.
Access the Tool menu. Select the number of variables.
Display
(When using EL-9600)
Using system function, it is possible to solve simultaneous linear equations. Systems up
*
2nd F
TOOL
B
2-2
2-3
2
*
Enter the system of equations.
ENTER ENTER ENTER
511
(-)(
ENTER
ENTER ENTER
31 5
)
-
Solve the system.
to six variables and six equa­tions can be solved.
x = 0.75 y = -2.75
EXE2nd F
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
A system of equations can be solved easily by using the Calculate feature or Tool feature.
5-1
EL-9600 Graphing Calculator
Entering and Multiplying Matrices
A matrix is a rectangular array of elements in rows and columns that is treated as a single element. A matrix is often used for expressing multiple linear equations with multiple variables.
Example
Enter two matrices and execute multiplication of the two.
1. Enter a 3x3 matrix A
2. Enter a 3x3 matrix B
3. Multiply the matrices A and B
A 1 2 1 2 1 -1 1 1 -2
B 1 2 3 4 5 6 7 8 9
Before
1-1
1-2
1-3
There may be differences in the results of calculations and graph plotting depending on the setting.
Start
Return all settings to the default value or to delete all data. As Matrix feature is only available on the EL-9600, this section does not apply to the EL-9400.
Step & Key Operation
*Use either pen touch or cursor to operate.
Access the matrix menu.
MATRIX
B
*
1
*
Set the dimension of the matrix at three rows by three columns.
33
ENTER ENTER
Enter the elements of the first row, the elements of the second row, and the elements of the third row.
Display
Notes
121
ENTER ENTER ENTER
21 1
ENTER ENTER
11 2
ENTER ENTER ENTER
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
(-) (
ENTER
)
-
6-1
EL-9600 Graphing Calculator
Step & Key Operation
(When using EL-9600)
*Use either pen touch or cursor to operate.
2
Enter a 3x3 matrix B.
* *
MATRIX
123
456 789
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
3-1
Multiply the matrices A and B
23 3
B
ENTER ENTER ENTER
ENTER ENTER ENTER
ENTER ENTER ENTER
ENTER ENTER
together at the home screen.
Display
(When using EL-9600)
Matrix multiplication can be performed if the number of col-
Notes
umns of the first matrix is equal
MATRIX
* *
A
* *
A
2
ENTER
1
MATRIX
X
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.
3-2
Delete the input matrices for future use.
2nd F
OPTION
C
*
2
ENTER ENTER
*
2nd F
QUIT
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
Matrix multiplication can be performed easily by the calculator.
6-1
EL-9600 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 solution matrix X is found by multiplying A and the correct answer will be obtained by multiplying BA-1. An inverse matrix A-1 is a matrix that when multiplied by A results in the identity matrix I (A matrix I is defined to be a square matrix (nxn) where each position on the diagonal is 1 and all others are 0.
Example
Use matrix multiplication to solve a system of linear equations.
1. 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
-1
B. Note that the multiplication is “order sensitive”
-1
x A=I). The identity
B 1 2 1 2 1 -1 1 1 -2
Before
1-1
1-2
1-3
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
There may be differences in the results of calculations and graph plotting depending on the setting.
Start
Return all settings to the default value or to delete all data. As Matrix feature is only available on the EL-9600, this section does not apply to the EL-9400.
Step & Key Operation
*Use either pen touch or cursor to operate.
Set up 3x3 identity matrix at the home screen.
* *
MATRIX
C053
Save the identity matrix in matrix A.
*
MATRIX
STO
Confirm that the identity matrix is stored in matrix A.
*
MATRIX
A
B1
1
ENTER
*
*
ENTER
Display
Notes
6-2
EL-9600 Graphing Calculator
Notes
2-1
2-2
Step & Key Operation
(When using EL-9600)
*Use either pen touch or cursor to operate.
Enter a 3x3 matrix B.
* *
MATRIX
12 2 11
23 3
B
ENTER ENTER
ENTER
ENTER ENTER
1
ENTER ENTER
ENTER
ENTER
1
(-)
1
)
(
2
-
ENTER
ENTER
Exit the matrix editor and find the inverse of the square matrix B.
Display
(When using EL-9600)
Some square matrices have no inverse and will generate error statements when calculating the inverse.
2nd F
MATRIX
(repeatedly)
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
CL
QUIT
* *
A2
2nd F
-1
ENTER
x
-0.17 0.83 -0.5
0.5 - 0.5 0.5
B-1=
0.17 0.17 -0.5
3-1
Enter the constants on the right side of the equal sign into matrix C (3x1).
* *
B33 1
MATRIX
81 3
ENTER ENTER ENTER
ENTER ENTER
(-)
The system of equations can be expressed as
1 2 1 2 1 -1 1 1 -2
x y z
=
8 1
-3
Let each matrix B, X, C : BX = C
-1
BX = B-1C (multiply both
3-2
Calculate B-1C.
B sides by B I = B X = B
The 1 is the x coordinate, 2 the y
-1
)
-1 (B-1
B = I, identity matrix)
-1
C
coordinate, and the 3 the z coor-
*
MATRIX
X
A2
MATRIX
A3
*
ENTER
dinate of the solution point. (x, y, z)=(1, 2, 3)
3-3
CL
-1
* *
2nd F
x
Delete the input matrices for future use.
2nd F
OPTION
C
*
ENTER
2
*
QUIT2nd 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.
6-2
EL-9600/9400 Graphing Calculator
Solving Inequalities
To solve an inequality, expressed by the form of f(x) f(x)
≥
g(x), means to find all values that make the inequality true.
≤
0, f (x) ≥ 0, or form of f (x) ≤
g(x),
There are two methods of finding these values for one-variable inequalities, using graphical techniques. The first method involves rewriting the inequality so that the right-hand side of the inequality is 0 and the left-hand side is a function of x. For example, to find the solution to f(x) < 0, determine where the graph of f (x) is below the x-axis. The second method involves graphing each side of the inequality as an individual function. For example, to find the solution to f(x) < g(x), determine where the graph of f(x) is below the graph of g( x).
Example
Solve an inequality in two methods.
1.
Solve 3(4 - 2x) ≥ 5 - x, by rewriting the right-hand side of the inequality as 0.
2.
Solve 3(4 - 2x) ≥ 5 - x, by shading the solution region that makes the inequality true.
Before
There may be differences in the results of calculations and graph plotting depending on the setting.
Start
Return all settings to the default value or to delete all data.
*Use either pen touch or cursor to operate.
(When using EL-9600)
Display
(When using EL-9600)
NotesStep & Key Operation
Rewrite the equation 3(4 - 2x)
1-1
so that the right-hand side becomes 0, and enter y = 3(4 - 2x) - 5 + x for Y1.
()
342
Y=
5
—+
View the graph.
1-2
GRAPH
1-3
Find the location of the x-intercept and solve the inequality.
2nd F CALC
5
*
—
X/
/T/n
≥
5 - x
X/
/T/n
3(4 - 2x) ≥ 5 - x ➞ 3(4 - 2x) - 5 + x ≥ 0
The x-intercept is located at the point (1.4, 0). Since the graph is above the x-axis to the left of the x-in- tercept, the solution to the in-
≥
equality 3(4 - 2x) - 5 + x all values of x such that
≤
1.4.
x
0 is
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
7-1
EL-9600/9400 Graphing Calculator
2-1
2-2
2-3
2-4
Step & Key Operation
(When using EL-9600)
*Use either pen touch or cursor to operate.
Enter y = 3(4 - 2x) for Y1 and y = 5 - x for Y2.
Y=
ENTER
(7 times) (4 times)
*
5
*
—
X/
DEL
/T/n
View the graph.
GRAPH
Access the Set Shade screen.
2nd F
DRAW
G
*
1
*
Set up the shading.
—
——
*
*
*
Display
(When using EL-9600)
Notes
Since the inequality being solved is Y1
≥
Y2, the solu­tion is where the graph of Y1 is “on the top” and Y2 is “on the bottom.”
2-5
2-6
View the shaded region.
GRAPH
Find where the graphs intersect and solve the inequality.
The point of intersection is (1.4, 3.6). Since the shaded region is to the left of x = 1.4,
2nd F CALC
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
2
*
the solution to the inequality
≥
3(4 - 2x) of x such that x
5 - x is all values
≤
1.4.
Graphical solution methods not only offer instructive visualization of the solution process, but they can be applied to inequalities that are often difficult to solve algebraically. The EL-9600/9400 allows the solution region to be indicated visually using the Shade feature. Also, the points of intersection can be obtained easily.
7-1
EL-9600/9400 Graphing Calculator
Solving D ouble Inequalities
The solution to a system of two inequalities in one variable consists of all values of the variable that make each inequality in the system true. A system f (x) ≥ a, f (x) ≤ b, where the same expression appears on both inequalities, is commonly referred to as a “double” inequality and is often written in the form a ≤ f (x) ≤ b. Be certain that both inequality signs are pointing in the same direction and that the double inequality is only used to indicate an expression in x “trapped” in between two values. Also a must be less than or equal to b in the inequality a ≤ f (x) ≤ b or b ≥ f (x) ≥ a.
Example
Solve a double inequality, using graphical techniques.
2x - 5 ≥ -1 2x -5 ≥ 7
Before
1
There may be differences in the results of calculations and graph plotting depending on the setting.
Start
Return all settings to the default value or to delete all data.
*Use either pen touch or cursor to operate.
(When using EL-9600)
Enter y = -1 for Y1, y = 2x - 5 for Y2, and y = 7 for Y3.
(-)
Y=
2
X/
/T/n
View the lines.
2
GRAPH
3
Find the point of intersection.
2nd F CALC
—
1
3
ENTER
5
*
*
ENTER
Display
(When using EL-9600)
NotesStep & Key Operation
The “double” inequality given can also be written to
≤
2x - 5 ≤ 7.
-1
7
*
y = 2x - 5 and y = -1 intersect at (2, -1).
7-2
EL-9600/9400 Graphing Calculator
Step & Key Operation
(When using EL-9600)
*Use either pen touch or cursor to operate.
Move the tracer and find another
4
intersection.
2nd F CALC
5
Solve the inequalities.
2
Display
(When using EL-9600)
y = 2x - 5 and y = 7 intersect at (6,7).
*
The solution to the “double” inequality -1
Notes
≤
2x - 5 ≤ 7 con­sists of all values of x in be­tween, and including, 2 and 6
≥
(i.e., x lution is 2
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
2 and x ≤ 6). The so-
≤
x ≤ 6.
Graphical solution methods not only offer instructive visualization of the solution process, but they can be applied to inequalities that are often difficult to solve algebraically. The EL-9600/9400 allows the solution region to be indicated visually using the Shade feature. Also, the points of intersection can be obtained easily.
7-2
EL-9600/9400 Graphing Calculator
System of Two-Variable Inequalities
The solution region of a system of two-variable inequalities consists of all points (a, b) such that when x = a and y = b, all inequalities in the system are true. To solve two-variable inequalities, the inequalities must be manipulated to isolate the y variable and enter the other side of the inequality as a function. The calculator will only accept functions of the form y = . (where y is defined explicitly in terms of x).
Example
Solve a system of two-variable inequalities by shading the solution region.
2x + y ≥ 1 x2 + y ≤ 1
Before
1
There may be differences in the results of calculations and graph plotting depending on the setting.
Start
Return all settings to the default value or to delete all data. Set the zoom to the decimal window:
Step & Key Operation
*Use either pen touch or cursor to operate.
(When using EL-9600)
Rewrite each inequality in the system so that the left-hand-side is y :
2
Enter y = 1 - 2x for Y1 and y = 1 - x for Y2.
Y= 1 2
1
Access the set shade screen
3
2nd F
1
*
Shade the points of y -value so that
4
≤
Y1
———
—
—
X/
DRAW
2
/T/n
x
G
*
y ≤ Y2.
*
*
ENTER
X/
/T/n
*
ZOOM
*
(
ENTER
A
*
Display
(When using EL-9600)
2
2nd F
)
7
*
2x + y
≥
2
+ y ≤ 1 ➞ y ≤ 1 - x
x
Notes
1 ➞ y ≥ 1 - 2x
2
Graph the system and find the
5
intersections.
GRAPH
*
Solve the system.
6
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
22
CALC2nd F CALC2nd F
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 EL­9600/9400 allows the solution region to be indicated visually using the Shade feature. Also, the points of intersection can be obtained easily.
The intersections are (0, 1) and (2, -3)
*
The solution is 0
≤
x ≤ 2.
7-3
EL-9600/9400 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.
Example
Check to see if given points are in the solution region of a system of inequalities.
1. Graph the solution region of a system of inequalities:
x + 2y ≤ 1 x2 + y ≥ 4
2. Which of the following points are within the solution region?
(-1.6, 1.8), (-2, -5), (2.8, -1.4), (-8,4)
Before
1-1
There may be differences in the results of calculations and graph plotting depending on the setting.
Start
Return all settings to the default value or to delete all data.
Step & Key Operation
*Use either pen touch or cursor to operate.
(When using EL-9600)
Rewrite the inequalities so that the left-hand-side is y.
1-2
Enter y = for Y1 and y = 4 - x
Y=
Set the shade and view the solution
1-3
1-x
2
2
for Y2.
a
/b
1
ENTER
24
X/
—
*
region.
*
2nd F
DRAW
—— —
GRAPH
1
G
*
*
/T/n
—
*
Display
(When using EL-9600)
x + 2y x2+y ≥ 4 ➞ y ≥ 4 - x
/T/n
2
x
X/
Notes
≤
1 ➞ y
1-x
≤
2
2
Y2 ≤ y ≤ Y1
2-1
Set the display area (window) to :
-9 < x < 3, -6 < y < 5.
WINDOW
(-)
93
(
)
ENTER
7-4
65
-
ENTER ENTER
ENTERENTER
EL-9600/9400 Graphing Calculator
Step & Key Operation
(When using EL-9600)
*Use either pen touch or cursor to operate.
2-2
2-3
Use the cursor to check the position of each point. (Zoom in as necessary).
or or or
GRAPH
Substitute points and confirm whether they are in the solution region.
(-)
X
2
(Continuing key operations omitted.)
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
•
16
•
1
...
8
+
Display
(When using EL-9600)
Points in the solution region are (2.8, -1.4) and (-8, 4). Points outside the solution region are (-1.6, 1.8) and (-2, -5).
.
(-1.6, 1.8): -1.6 + 2 ✕ 1.8 = 2
➞ This does not materialize.
.
(-2, -5): -2 + 2 ✕ (-5) = -12
➞ This does not materialize.
.
(2.8, -1.4): 2.8 + 2 ✕ (-1.4) = 0
➞ This materializes.
.
(-8, 4): -8 + 2 ✕ 4 = 0
➞ This materializes.
Notes
(-2)
(2.8)
(-8)
2
+ (-5) = -1
2
+ (-1.4) = 6.44
2
+ 4 = 68
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-9600/9400 allows the solution region to be indicated visually using the Shading feature. Also, the free-moving tracer or Zoom-in feature will allow the details to be checked visually.
7-4
EL-9600 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.
Example
Consider various absolute value functions and check the relation between the graphs and the values of coefficients.
1. Graph y = |x|
2. Graph y = |x -1| and y = |x|-1 using Rapid Graph feature.
Before
1-1
1-2
There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value or to delete all data.
Start
Set the zoom to the decimai window: As Substitution feature is only available on the EL-9600, this section does not apply to the EL-9400.
*Use either pen touch or cursor to operate.
ZOOM
A
(
ENTER 2nd F
*
Display
)
7
*
Enter the function y =|x| for Y1.
* *
MA TH
Y=
View the graph.
1
X/
B
/T/n
Notice that the domain of f(x)
NotesStep & Key Operation
= |x| is the set of all real num-
GRAPH
bers and the range is the set of non-negative real numbers. Notice also that the slope of the graph is 1in the range of X > 0
≤
and -1in the range of X
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
2-1
Enter the standard form of an abso-
0.
lute value function for Y2 using Rapid Graph feature.
*
2-2
Y=
ENTER
Substitute the coefficients to graph
EZ
*
y = |x - 1|.
2nd F SUB
ENTER
0
8-1
ENTER ENTER
11
8
ENTER ENTER
*
*
EL-9600 Graphing Calculator
2-3
Step & Key Operation
(When using EL-9600)
*Use either pen touch or cursor to operate.
View the graph.
Display
(When using EL-9600)
Notice that placing an h(>0)
Notes
within the standard form
GRAPH
y = a|x - h|+ k will move the graph right h units on the x­axis.
2-4
Change the coefficients to graph y =|x|-1.
2-5
Y=
ENTER ENTER
View the graph.
GRAPH
2nd F SUB ENTER
(
)
1
-
1
Notice that adding a k(>0) within the standard form y=a|x-h|+k will move the graph up k units on the y-axis.
The EL-9600/9400 shows absolute values with | | just as written on paper by using the Equation editor. Use of the calculator allows various absolute value functions to be graphed quickly and shows their characteristics in an easy-to-understand manner.
8-1
EL-9600/9400 Graphing Calculator
Shifting a graph 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.
Example
Move and change graphs of absolute value function y =|x| to check the relation between the graphs and the values of coefficients.
1. Move the graph y = |x| downward by 2 using the Shift feature.
2. Move the graph y = |x| to the right by 2 using the Shift feature.
3. Pinch the slope of y = |x| to 2 or minus using the Change feature.
Before
1-1
1-2
There may be differences in the results of calculations and graph plotting depending on the setting.
Start
Return all settings to the default value or to delete all data.
*Use either pen touch or cursor to operate.
(When using EL-9600)
Access the Shift feature. Select y = |x|.
2nd F
SHIFT/CHANGE
(
ENTER
ALPHA
Move the graph downward by 2.
ENTER
Display
(When using EL-9600)
A
*
)
8
*
*
y =|x|changes to y = |x|-2
*
NotesStep & Key Operation
1-3
Save the new graph and look at the relationship of the function and the graph.
ENTER
ALPHA
8-2
The graph of the equation that is highlighted is shown by a solid line. Notice that the y­intercept k in the standard form y = a|x - h|+ k takes charge of vertical movement of the graph.
EL-9600/9400 Graphing Calculator
Step & Key Operation
(When using EL-9600)
*Use either pen touch or cursor to operate.
2
-
1
Move the original graph to the right by 2.
ALPHA
2
-2
Save the new graph and look at the relationship of the function and the graph.
ENTER
ALPHA
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
Access the Change feature.
3
-
1
2nd F
SHIFT/CHANGE
3
-
2
Select y = |x|.
*
*
B
*
ENTER
*
*
Display
(When using EL-9600)
y = |x| changes to y = |x-2|
Notice that the function h in the standard form y = a|x - h|+ of horizontal movement of the graph.
Notes
k takes charge
3
*
3
-
3
Make the slope of the graph steeper. Save the new graph.
ENTER
ENTER
Make the slope of the graph minus.
3
-
4
Save the new graph.
ENTER
3
-
5
Look at the relationship of the function and the graph.
ALPHA
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
ENTER
*
*
*
y = |x|➞ y = 2|x|
y = |x|➞ y = - |x|
Notice that the coefficient a in the standard form y = a |x - h| + k takes charge of changing the slope.
EL-9600/9400 shows absolute values with | | just as written on paper by using the Equation editor. Use of the calculator allows various absolute value functions to be graphed quickly and shows their characteristics in an easy-to-understand manner . The Shift/Change feature of the EL-9600/9400 allows visual understanding of how graph changes affect the form of absolute value functions.
8-2
EL-9600/9400 Graphing Calculator
Solving A bsolute Value Equations
The absolute value of a real number x is defined by the following:
|x| = x if x ≥ 0
-x if x ≤ 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
Before
1
There may be differences in the results of calculations and graph plotting depending on the setting.
Start
Return all settings to the default value or to delete all data.
Step & Key Operation
*Use either pen touch or cursor to operate.
(When using EL-9600)
Enter y = |5 - 4x| for Y1. Enter y = 6 for Y2.
* *
Y=
MATH
B1
X/
/T/n
2
View the graph.
GRAPH
Find the points of intersection of
3
ENTER
*
54
6
the two graphs and solve.
2nd F CALC
2nd F CALC
2
*
2
*
—
Display
(When using EL-9600)
Notes
There are two points of in­tersection of the absolute value graph and the hori­zontal line y = 6.
The solution to the equation |5 - 4x|= 6 consists of the two values -0.25 and 2.75. Note that although it is not as intu­itively obvious, the solution could also be obtained by finding the x-intercepts of the function y = |5x - 4| - 6.
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
The EL-9600/9400 shows absolute values with | | just as written on paper by using the Equation editor. The graphing feature of the calculator shows the solution of the absolute value function visually.
8-3
EL-9600/9400 Graphing Calculator
Solving A bsolute Value Inequalities
To solve an inequality means to find all values that make the inequality true. Absolute value inequalities are of the form | solution to an absolute value inequality is found using the same methods as for normal inequalities. The first method involves rewriting the inequality so that the right-hand-side of the inequality is 0 and the left-hand-side is a function of x. The second method involves graphing each side of the inequality as an individual function.
Example
Solve absolute value inequalities in two methods.
f (x)
|<
k, |f (x)
|≤
k, |f (x)
|>
k,
or |f (x)
|≥ k. The graphical
1. Solve 20 - < 8 by rewriting the inequality so that the right-hand side of
6x
5
the inequality is zero.
2. Solve 3.5x + 4 > 10 by shading the solution region.
Before
1-1
1-2
There may be differences in the results of calculations and graph plotting depending on the setting.
Start
Return all settings to the default value or to delete all data. Choose viewing windows “-5< x <50,” and “-10< y <10” using Rapid Window feature to solve Q1.
WINDOW
EZ 3 3 3
Step & Key Operation
*Use either pen touch or cursor to operate.
(When using EL-9600)
ENTER ENTER ENTER
*
Rewrite the equation.
*
(When using EL-9600)
*
Display
|20 - |< 8 ➞|20 - | - 8 < 0.
Enter y = |20 - | - 8 for Y1.
* *
Y=
MA TH
6x
5
B120
—
a
/b
6x
5
Notes
6x
5
6
X/
/T/n
8
—
1-3
View the graph, and find the
*
1
x-intercepts.
GRAPH
2nd F CALC
2nd F CALC
➞ x = 10, y = 0
5
*
➞ x = 23.33333334
5
*
y = 0.00000006 ( Note)
1-4
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
Solve the inequality.
*
The intersections with the x- axis are (10, 0) and (23.3, 0)
Note: The value of y in the
( x-intercepts may not appear exactly as 0 as shown in the example, due to an error caused by approximate calcu­lation.)
Since the graph is below the x-axis for x in between the two x-intercepts, the solution is 10 < x < 23.3.
8-4
EL-9600/9400 Graphing Calculator
2-1
2-2
2-3
Step & Key Operation
(When using EL-9600)
*Use either pen touch or cursor to operate.
Enter the function y =|3.5x + 4|for Y1. Enter y = 10 for Y2.
*
Y= CL
35 4
•
1
0
MATH
X/
/T/n
B
+
1
*
ENTER
*
Set up shading.
*
2nd F
DRAW
—— —
G
*
1
*
*
*
Choose viewing windows “-10 < x < 10,” and “-5 < y < 50” using Rapid Window feature and view the graph.
WINDOW
ENTER
3
EZ
ENTER ENTER
2
*
5
*
*
Display
(When using EL-9600)
Notes
Since the inequality you are solving is Y1 > Y2, the solu­tion is where the graph of Y2 is “on the bottom” and Y1 in “on the top.”
2
Find the points of intersection.
-
4
Solve the inequality.
The intersections are (-4, 10) and (1.7, 10.0). The solution is all values of x such that
2nd F
2nd F
CALC
CALC
➞ x = -4, y = 10
2
*
➞ x = 1.714285714
2
*
y = 9.999999999 ( Note)
x <-4 or x >1.7.
Note: The value of y in the
( intersection of the two graphs may not appear exactly as 10 as shown in the example, due to an error caused by approxi­mate calculation.)
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
The EL-9600/9400 shows absolute values with | | just as written on paper by using the Equation editor. Graphical solution methods not only offer instructive visualization of the solution process, but they can be applied to inequalities that are often difficult to solve algebraically. The Shade feature is useful to solve the inequality visually and the points of intersection can be obtained easily.
8-4
EL-9600/9400 Graphing Calculator
E v aluating A bsolute Value Functions
The absolute value of a real number x is defined by the following:
|x| = x if x ≥ 0
-x if x ≤ 0 Note that the effect of taking the absolute value of a number is to strip away the minus sign if the number is negative and to leave the number unchanged if it is nonnegative. Thus, |x|≥ 0 for all values of x.
Example
Evaluate various absolute value functions.
1. Evaluate |- 2(5-1)|
2. Is |-2+7| = |-2| + |7|?
Evaluate each side of the equation to check your answer. Is |x + y| =|x|+ |y| for all real numbers x and y ? If not, when will |x + y| = |x|+|y| ?
3. Is | | = ?
6-9
1+3
Evaluate each side of the equation to check your answer. Investigate with more examples, and decide if you think |x / y|=|x|/|y|
|6-9| |1+3|
Before
1-1
1-2
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
2-1
There may be differences in the results of calculations and graph plotting depending on the setting.
Start
Return all settings to the default value or to delete all data.
Step & Key Operation
*Use either pen touch or cursor to operate.
(When using EL-9600)
Access the home or computation screen.
Enter y = |-2(5-1)| and evaluate.
*
MATH
B
)
1
Evaluate|-2 + 7|. Evaluate|-2|+|7|.
CL
1
*
ENTER
(-)
2
(
5
Display
(When using EL-9600)
The solution is +8.
—
|-2 + 7| = 5, |-2| + |7| = 9 ➞|-2 + 7| ≠ |-2| + |7|.
Notes
*
(
1
1
)
-
(-)
ENTER
2
2
MATH
*
MATH MATH
1
7
*
7
ENTER
+
+
8-5
EL-9600/9400 Graphing Calculator
2-2
Step & Key Operation
(When using EL-9600)
*Use either pen touch or cursor to operate.
Is |x + y| = |x| +|y|? Think about
Display
(When using EL-9600)
Notes
this problem according to the cases when x or y are positive or negative.
≥
0 and y ≥ 0
If x [e.g.; (x, y) = (2,7)]
|x +y| = |2 + 7| = 9 |x|+|y| = |2| + |7| = 9 ➞|x + y| = |x| + |y|.
≤
0 and y ≥ 0
If x [e.g.; (x, y) = (-2, 7)]
|x +y| = |-2 + 7| = 5 |x|+|y| = |-2| + |7| = 9
➞|x + y| ≠ |x| + |y|.
≥
0 and y ≤ 0
If x [e.g.; (x, y) = (2, -7)]
|x +y| = |2-7| = 5 |x|+|y| = |2| + |-7| = 9 ➞|x + y| ≠ |x| + |y|.
≤
0 and y ≤ 0
If x [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
≤
and when x
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
0 and y ≤ 0.
0 and y ≥ 0,
3-1
3-2
Evaluate . Evaluate .
CL
MATH
MATH
6-9
1+3
MATH
13
*
*
1
*
1
a
/b
1
*
+
69
13
ENTER
—
+
Is |x /y| = |x|/|y|?
6-9
1+3
—
69
a
/b
*
ENTER
6-9
= 0.75 , = 0 .75
1+3
6-9
➞ =
1+3
6-9
1+3
6-9
1+3
Think about this problem according to the cases when x or y are positive or negative.
≥
0 and y ≥ 0
If x [e.g.; (x, y) = (2,7)]
|x /y| = |2/7| = 2/7 |x|/|y| = |2| /|7| = 2/7 ➞|x /y| = |x| / |y|
≤
0 and y ≥ 0
If x [e.g.; (x, y) = (-2, 7)]
|x /y| = |(-2)/7| = 2/7 |x|/|y| = |-2| /|7| = 2/7 ➞|x /y| = |x| / |y|
≥
0 and y ≤ 0
If x [e.g.; (x, y) = (2, -7)]
|x /y| = |2/(-7)| = 2/7 |x|/|y| = |2| /|-7| = 2/7 ➞|x /y| = |x| / |y|
≤
0 and y ≤ 0
If x [e.g.; (x, y) = (-2, -7)]
|x /y| = |(-2)/-7| = 2/7 |x|/|y| = |-2| /|-7| = 2/7 ➞|x /y| = |x| / |y| The statement is true for all y ≠ 0.
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
The EL-9600/9400 shows absolute values with | | just as written on paper by using the Equation editor. The nature of arithmetic of the absolute value can be learned through arithmetical operations of absolute value functions.
8-5
EL-9600/9400 Graphing Calculator
G raphing Rational F unctions
A rational function f (x) is defined as the quotient where p (x) and q (x) are two
p (x) q (x)
polynomial functions such that q (x) ≠ 0. The domain of any rational function consists of all values of x such that the denominator q (x) is not zero. A rational function consists of branches separated by vertical asymptotes, and the values of x that make the denominator q (x) = 0 but do not make the numerator p (x) = 0 are where the vertical asymptotes occur. It also has horizontal asymptotes, lines of the form y = k (k, a constant) such that the function gets arbitrarily close to, but does not cross, the horizontal asymptote when |x| is 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).
Example
Graph the rational function and check several points as indicated below.
1.
Graph f (x) = .
2.
Find the domain of f (x), and the vertical asymptote of f (x).
3. Find the x- and y-intercepts of f
4. Estimate the horizontal asymptote of f
x-1
x2-1
(x)
.
(x)
.
Before
1-1
1-2
There may be differences in the results of calculations and graph plotting depending on the setting.
Start
Return all settings to the default value or to delete all data. Set the zoom to the decimal window:
*Use either pen touch or cursor to operate.
(When using EL-9600)
Enter y = for Y1.
Y=
a
/b
x - 1
x2 -1
—
X/
/T/n
1—
1
*
View the graph.
(
ENTER
ZOOM
A
*
(When using EL-9600)
2
x
X/
/T/n
ALPHA
Display
)
*
The function consists of two
7
*
NotesStep & Key Operation
branches separated by the verti-
GRAPH
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
cal asymptote.
9-1
EL-9600/9400 Graphing Calculator
2
Step & Key Operation
(When using EL-9600)
*Use either pen touch or cursor to operate.
Find the domain and the vertical asymptote of f (x), tracing the
graph to find the hole at x = 1.
Display
(When using EL-9600)
Since f (x) can be written as , the domain
(x + 1)(x - 1)
Notes
x - 1
consists of all real numbers x such that x ≠ 1 and x ≠ -1. There is no vertical asymptote
(repeatedly)
TRACE
where x = 1 since this value of x also makes the numera­tor zero. Next to the coordi­nates x = 0.9, y = 0.52, see that the calculator does not display a value for y at x = 1 since 1 is not in the domain of this rational function.
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
3
Find the x- and y-intercepts of f (x).
The y-intercept is at (0 ,1). No­tice that there are no x-inter-
2nd F CALC
6
*
cepts for the graph of f (x).
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
4
Estimate the horizontal asymptote of f (x).
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
The line y = 0 is very likely a horizontal asymptote of f (x).
The graphing feature of the EL-9600/9400 can create the branches of rational function separated by vertical asymptote. The calculator allows the points of intersection to be obtained easily.
9-1
EL-9600/9400 Graphing Calculator
Solving Rational Function Inequalities
A rational function f (x) is defined as the quotient where p (x) and q (x) are two
p (x) q (x)
polynomial function 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.
Example
Solve a rational inequality.
x
Solve ≤ 2 by graphing each side of the inequality as an individual function.
Before
Start
Step & Key Operation
*Use either pen touch or cursor to operate.
1 Enter y = for Y1. Enter y = 2
for Y2.
2
1 - x
There may be differences in the results of calculations and graph plotting depending on the setting. Return all settings to the default value or to delete all data.
Set the zoom to the decimal window:
(When using EL-9600)
ZOOM
(
ENTER
A
*
Display
(When using EL-9600)
ALPHA
)
7
*
*
Notes
x
2
1- x
*
MATH
Y=
12
—
B1
X/
/T/n
2 Set up the shading.
a
X/
/T/n
/b
*
2
x
ENTER
*
*
Since Y1 is the value “on the bottom” (the smaller of the
*
2nd F
DRAW
———
3
View the graph.
GRAPH
4
Find the intersections, and solve the
1
G
*
*
*
inequality.
two) and Y2 is the function “on the top” (the larger of the two), Y1 < Y < Y2.
The intersections are when x = -1.3, -0.8, 0.8, and 1.3. The solution is all values of
2nd F CALC
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
Do this four times
2
*
x such that x
≤
x ≤ 0.8 or x ≥ 1.3.
-0.8
≤
-1.3 or
The EL-9600/9400 allows the solution region of inequalities to be indicated visually using the Shade feature. Also, the points of intersections can be obtained easily.
9-2
G raphing Parabolas
EL-9600/9400 Graphing Calculator
The graphs of quadratic equations (y = ax2 + bx + c) are called parabolas. Sometimes the quadrafic equation takes on the form of x = ay There is a problem entering this equation in the calculator graphing list for two reasons: a) it is not a function, and only functions can be entered in the Y= list locations, b) the functions entered in the Y= list must be in terms of x, not y. There are, however, two methods you can use to draw the graph of a parabola.
Method 1: Consider the "top" and "bottom" halves of the parabola as two different parts of the graph because each individually is a function. Solve the equation of the parabola for y and enter the two parts (that individually are functions) in two locations of the Y= list.
Method 2: Choose the parametric graphing mode of the calculator and enter the parametric equations of the parabola. It is not necessary to algebraically solve the equation for y. Parametric representations are equation pairs x = F(t), y = F(t) that have x and y each expressed in terms of a third parameter, t.
2
+ by + c.
Example
Graph a parabola using two methods.
1. Graph the parabola x = y
2. Graph the parabola x = y
2
-2 in rectangular mode.
2
-2 in parametric mode.
Before
1-1
1-2
There may be differences in the results of calculations and graph plotting depending on the setting.
Start
Return all settings to the default value or to delete all data. Set the zoom to the decimal window:
Step & Key Operation
*Use either pen touch or cursor to operate.
(When using EL-9600)
Solve the equation for y.
ZOOM
(
ENTER
A
*
Display
(When using EL-9600)
ALPHA
)
7
*
x = y2 -2 x + 2 = y y =
*
√
+
–
Notes
2
x + 2
Enter y = √x+2 for Y1 and enter y = -Y1 for Y2.
(-)
X
/θ/T/
n
+
√
VARS
2
ENTERENTER
1A
*
The graph of the equation y =
√
x+ 2 is the "top half" of the
parabola and the graph of the
√
equation y = -
x + 2 gives
1-3
Y=
2nd F
*
*
View the graph.
GRAPH
the "bottom half."
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
10-1
EL-9600/9400 Graphing Calculator
2-1
2-2
2-3
2-4
Step & Key Operation
(When using EL-9600)
*Use either pen touch or cursor to operate.
Change to parametric mode.
2nd F
SET UP
E
*
2
*
Rewrite x = y2 -2 in parametric form. Enter X1T = T
X/
Y=
X/
/T/n
/T/n
2
-2 and Y1T = T.
2
—
x
2
ENTER
*
View the graph. Consider why only half of the parabola is drawn. (To understand this, use Trace fea­ture.)
(
GRAPH
TRACE
)
Set Tmin to -6.
Display
(When using EL-9600)
Notes
Let y = T and substitute in x
2
= y
- 2, to obtain x = T2- 2.
The graph starts at T =0 and increases. Since the window setting is T ≥ 0, the region T < 0 is not drawn in the graph.
(-)
WINDOW
2-5
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
View the complete parabola.
GRAPH
6
ENTER
*
The calculator provides two methods for graphing parabolas both of which are easy to perform.
10-1
EL-9600/9400 Graphing Calculator
G raphing C ir 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. Y ou therefore need to use the process of completing the square.
Example
Graph the circles in rectangular mode. Solve the equation for y to put it in the standard form.
1. Graph x
2. Graph x
2
+ y2 = 4.
2
- 2x + y2 + 4y = 2.
Before
1-1
1-2
There may be differences in the results of calculations and graph plotting depending on the setting.
Start
Return all settings to the default value or to delete all data. Set the zoom to the decimal window:
*Use either pen touch or cursor to operate.
(When using EL-9600)
Solve the equation for y. Enter y = half). Enter y = -
Y=
4 - x2 for Y1 (the top
√
2nd F
*
(
-
*
√
4 - x2 for Y2.
—
4
√
)
VARS
A
X/
ENTERENTER
View the graph.
/T/n
ZOOM
2
x
1
*
(
ENTER
A
*
Display
(When using EL-9600)
ALPHA
)
7
*
*
NotesStep & Key Operation
y2 = 4 - x
+
y =
√4 - x
–
This is a circle of radius r ,
2
2
centered at the origin.
GRAPH
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
2
- 2x + y2 + 4y = 2
2-1
Solve the equation for y, completing the square.
x
Place all variable terms on the left and the constant term on the right-hand side of the equation.
2
x
-2x+y2+4y+4=2+4
Complete the square on the y-term.
2
x
- 2x + (y+2)2 = 6
Express the terms in y as a perfect square.
2
(y+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.
10-2
EL-9600/9400 Graphing Calculator
2-2
2-3
2-4
Step & Key Operation
(When using EL-9600)
*Use either pen touch or cursor to operate.
Enter y = √6 - x2 + 2x for Y1, y = Y1 - 2 for Y2, and y = -Y1 -2 for Y3.
Y= CL 6
2
x
+
*
VARS
A1
ENTER
2
VARS
)
(
-
2nd F √
X/
/T/n
2
ENTER
*
ENTER
—
X/
/T/n
ENTER
*
12
*
—
—
CL
*
"Turn off" Y1 so that it will not graph.
*
ENTER
*
View the graph.
Display
(When using EL-9600)
Notes
Notice that if you enter
√
6 - x2 + 2x - 2 for Y1
y =
and y = - Y1 for Y2, you will not get the graph of a circle because the “±” does not go with the “-2”.
Notice that “=” for Y1 is no longer darkened. You now have the top portion and the bottom portion of the circle in Y2 and Y3.
GRAPH
2-5
Adjust the screen to see the bottom part of the circle using the Rapid feature.
Wait until the graph is dis­played after each operation. (It takes few seconds to graph)
EZ
ENTER
2-6
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View the graph in the new window.
GRAPH
*
*
ENTER
*
ENTER
*
*
ENTER
*
Graphing circles can be performed easily on the calculator display. Also, the Rapid Zoom feature of the EL-9600/9400 allows shifting and adjusting display area (window) of a graph easily.
10-2
G raphing E llipses
EL-9600/9400 Graphing Calculator
The standard equation for an ellipse whose center is at the point (h, k) with major and minor axes of length a and b is + = 1.
(x - h)
a
2
2
(y - k)
b
2
2
There is a problem entering this equation in the calculator graphing list for two reasons: a) it is not a function, and only functions can be entered in the Y = list locations. b) the functions entered in the Y = list locations must be in terms of x, not y. To draw a graph of an ellipse, consider the “top” and “bottom” halves of the ellipse as two different parts of the graph because each individual is a function. Solve the equation of the ellipse for y and enter the two parts in two locations of the Y = list.
Example
Graph an ellipse in rectangular mode. Solve the equation for y to put it in the standard form.
Graph the ellipse 3(x -3)2 + (y + 2)2 = 3
Before
There may be differences in the results of calculations and graph plotting depending on the setting.
Start
Return all settings to the default value or to delete all data. Set the zoom to the decimal window:
ZOOM
(
A
ENTER
*
ALPHA
)
7
*
*
Step & Key Operation
*Use either pen touch or cursor to operate.
1
(When using EL-9600)
Solve the equation for y, completing the square.
Enter
Y1 =
√
3 - 3(x - 3)
2
Y2 = Y1 - 2 Y3 = -Y1 -2
Y= 3
2nd F √
—
X/
/T/n
A
VARS
2
1—2
Turn off Y1 so that it will not graph.
2
*
)
3
(-)
1
*
ENTER
*
—
2
x
—
*
VARS
ENTERENTER
ENTER
*
3
ENTER
Display
(When using EL-9600)
Notes
3(x - 3)2 + (y + 2)2 = 3
2
(y + 2)
y2 + 2 = y =
(
*
= 3 - 3(x - 3)
+
√
3 - 3(x - 3)
+
√
3 - 3(x - 3)2 - 2
2
2
10-3
EL-9600/9400 Graphing Calculator
Step & Key Operation
(When using EL-9600)
*Use either pen touch or cursor to operate.
3
View the graph.
GRAPH
4
Adjust the screen to see the bottom part of the ellipse using the Rapid Zoom feature.
Display
(When using EL-9600)
Wait until the graph is dis­played after each operation. (It takes few seconds to
Notes
graph)
EZ
5
View the graph in the new window.
GRAPH
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*
ENTER
*
Graphing ellipse can be performed easily on the calculator display. In addition to the Zoom-in/Zoom-out features, the EL-9600/9400 have the Rapid Zoom feature to adjust the display easily.
10-3
EL-9600/9400 Graphing Calculator
G raphing Hyperbolas
The standard equation for a hyperbola can take one of two forms:
2
( x - h )
2
a
( x - k )
2
b
( y - k )
- = 1 with vertices at ( h ± a, k ) or
- = 1 with vertices at ( h, k ± b ).
2
b
( y - h)
a
There is a problem entering this equation in the calculator graphing list for two reasons: a) it is not a function, and only functions can be entered in the Y= list locations. b) the functions entered in the Y= list locations must be in terms of x, not y. To draw a graph of a hyperbola, consider the “top” and “bottom” halves of the hyperbola as two different parts of the graph because each individual is a function. Solve the equation of the hyperbola for y and enter the two parts in two locations of the Y= list.
Example
Graph a hyperbola in rectangular mode. Solve the equation for y to put it in the standard form.
2
2
2
2
Graph the hyperbola x2 + 2x - y2 - 6y + 3 = 0
Before
1
There may be differences in the results of calculations and graph plotting depending on the setting.
Start
Return all settings to the default value or to delete all data. Set the zoom to the decimal window:
*Use either pen touch or cursor to operate.
(When using EL-9600)
ZOOM
A
Solve the equation for y completing the square.
Enter
√
Y1 =
x2 + 2x + 12
Y2 = Y1 -3 Y3 = -Y1 -3
2
/T/n
x
ENTER
—
1
1
+
2
*
3
ENTER
*
—
*
3
12
ENTER
A
X/
ENTER
Y=
2nd F √
X
/θ/T/
n
+
* *
VARS
A
*
(-)
VARS
(
ENTER
*
(When using EL-9600)
ALPHA
Display
)
7
*
*
NotesStep & Key Operation
x2 + 2x - y2 -6y = -3
2
+ 2x - (y2 + 6y + 9) = -3 -9
x
2
+ 2x - (y +3)2 = -12
x
2
(y + 3)
y + 3 = y =
= x2 + 2x + 12
+
√
x2 + 2x + 12
+
√
x2 + 2x + 12 - 3
Turn off Y1 so that it will not graph.
2
10-4
*
ENTER
*
EL-9600/9400 Graphing Calculator
Step & Key Operation
(When using EL-9600)
*Use either pen touch or cursor to operate.
3
View the graph.
GRAPH
Zoom out the screen.
4
A
ZOOM
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4
* *
Display
(When using EL-9600)
Notes
Graphing hyperbolas can be performed easily on the calculator display. In addition to the Zoom-in/Zoom-out features, the EL-9600/9400 have the Rapid Zoom feature to adjust the display easily. (See the section “Graphing Ellipses (No. 10-3)” about how to use the Rapid Zoom feature.)
10-4
Key pad for the SHARP EL-9600 Calculator
Graphing keys Power supply ON/OFF key Alphabet specification key Secondary function specification key Display screen
Cursor movement keys Clear/Quit key Variable enter key Calculation execute key Communication port for peripheral devices
Key pad for the SHARP EL-9400 Calculator
Graphing keys Power supply ON/OFF key Alphabet specification key Secondary function specification key Display screen
Cursor movement keys Clear/Quit key Variable enter key Calculation execute key Communication port for peripheral devices
Use this form to send us your contribution
Dear Sir/Madam
W e would like to take this opportunity to invite you to create a mathematical problem which can be solved with the SHARP graphing calculator EL-9600/9400. For this purpose, we would be grateful if you would complete the form below and return it to us by fax or mail, specifying which calculator you are writing problems for, the EL-9600 or 9400.
If your contribution is chosen, your name will be included in the next edition of The EL-9600/9400 Graphing Calculator Handbook. We regret that we are unable to return contributions.
We thank you for your cooperation in this project.
Name: ( Mr. Ms.
)
School/College/Univ.: Address:
Post Code: Country:
Phone: Fax: E-mail:
* You are making this sheet for the ( EL-9600, EL-9400).
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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SHARP Graphing Calculator
BEFORE START : Write any conditions to be set up before solving the problems.
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STEP NOTES
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SHARP CORPORATION Osaka, Japan
Fax:
SHARP Graphing Calculator
SHARP CORPORATION OSAKA,JAPAN
FAX:06-628-1653
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