Graphing Technology Lab Casio fx-CG10
Family of Exponential Functions
An exponential function is a function of the form y = abx, where a ≠ 0, b > 0, and b ≠ 1. You
have studied the effects of changing parameters in linear functions. You can use a graphing
calculator to analyze how changing the parameters a and b affects the graphs in the family of
exponential functions.
Activity 1 b in y = bx, b > 1
Graph the set of equations on the same screen. Describe any similarities and differences among the
graphs.
y = 2x, y = 3x, y = 6x
y = 3x
Clear the calculator memory first.
KEYSTROKES:
.&/6 System ' ' '
y = 6x
Use the Graph Func menu to enter the equations.
KEYSTROKES:
.&/6 Graph 2
9θ5 &9& 3
9θ5
y = 2x
&9& 6
9θ5 &9& 4)*'5 [V-Window] –
10 &9& 10
&9& 1 &9&
–
10 &9& 100 &9& 1 &9& '
[-10, 10] scl: 1 by [-10, 100] scl: 10
There are many similarities in the graphs. The domain for each
function is all real numbers, and the range is all positive real
numbers. The functions are increasing over the entire domain.
The graphs do not display any line symmetry.
Use the [Zoom] menu to investigate the key features of the graphs.
Zooming in twice on a point near the origin allows closer
inspection of the graphs.
Use the Y-ICEPT feature from the [G-Solv] menu to find
the y-intercepts. The y-intercept is 1 for all three graphs.
Tracing along the graphs reveals that there are no x-intercepts,
no maxima and no minima.
The graphs are different in that the graphs for the equations in
which b is greater are steeper.
The effect of b on the graph is different when 0 < b < 1.
[-2.5, 2.5] scl: 1 by [-9.55..., 17.94...] scl: 10
Activity 2 b in y = bx, 0 < b < 1
Graph the set of equations on the same screen.
Describe any similarities and differences among the graphs.
(_2 )
(_3 )
x
x
1
y= 1 ,y= 1 ,y= _
(6)
x
Delete the equations from Activity 1.
KEYSTROKES:
' ' '
' '
' '
Use the Graph Func menu to enter the equations.
KEYSTROKES:
1 ÷ 2 9θ5
&9&
1 ÷ 6 9θ5
&9& 4)*'5 [V-Window] –
10
&9& 10 &9& 1 &9&
1 ÷ 3 9θ5 &9&
–
10 &9& 100 &9& 1 &9& '
Use the [Zoom] menu to investigate the key features of the graphs.
Use the Y-ICEPT feature from the [G-Solv] menu to find
the y-intercepts.
The domain for each function is all real numbers, and the range is all
positive real numbers. The function values are all positive and the
functions are decreasing over the entire domain. The graphs display
no line symmetry. There are no x-intercepts, and the y-intercept is
1 for all three graphs. There are no maxima or minima.
However, the graphs in which b is lesser are steeper.
⎛ 1 ⎞x
y = ⎝3⎭
⎛ 1 ⎞x
y = ⎝6⎭
⎛ 1 ⎞x
y = ⎝2⎭
[-10, 10] scl: 1 by [-10, 100] scl: 10
Activity 3 a in y = abx, a > 0
Graph each set of equations on the same screen. Describe
any similarities and differences among the graphs.
_
y = 2x, y = 3(2x), y = 1 (2x)
6
y = 2x
Delete the equations from Activity 2.
KEYSTROKES:
' ' '
' '
' '
y = 3(2)x
Use the Graph Func menu to enter the equations.
KEYSTROKES:
2
9θ5
1 ÷ 6 &9& 3 2 2 9θ5
9θ5
y = 1 (2)x
6
&9&
&9&
4)*'5 [V-Window] –
10 &9& 10 &9& 1
&9&
[-10, 10] scl: 1 by [-10, 100] scl: 10
–
10 &9& 100 &9& 1 &9& '
The domain for each function is all real numbers, and the
range is all positive real numbers. The functions are increasing
over the entire domain. The graphs do not display any line
symmetry.
Use the [Zoom] menu to investigate the key features
of the graphs.
Zooming in twice on a point near the origin allows closer
inspection of the graphs.
Use the Y-ICEPT feature from the [G-Solv] menu to find the
y-intercepts.
Tracing along the graphs reveals that there are no x-intercepts,
no maxima and no minima.
However, the graphs in which a is greater are steeper.
The y-intercept
1
1 x
in y = _
(2 ).
is 1 in the graph of y = 2 x, 3 in y = 3(2 x), and _
6
6
[-2.5, 2.5] scl: 1 by [-9.55..., 17.94...] scl: 10
Activity 4 a in y = abx, a < 0
Graph each set of equations on the same screen. Describe any
similarities and differences among the graphs.
_
y = -2x, y = -3(2x), y = - 1 (2x)
6
Do not clear the calculator memory first. Add negatives before
each equation in Graph Func.
KEYSTROKES:
.&/6 Graph –
&9& –
&9& –
&9& 4)*'5
[V-Window] –
10 &9& 10 &9& 1 &9&
–
100 &9&
10 &9& 1 &9& '
Use the [Zoom] menu to investigate the key features of the graphs.
The domain for each function is all real numbers, and the
range is all negative real numbers. The functions are
decreasing over the entire domain. The graphs do not
display any line symmetry.
Use the Y-ICEPT feature from the [G-Solv] menu to find
the y-intercepts.
y = -2x
y = -3(2)x
There are no x-intercepts, no maxima and no minima.
However, the graphs in which the absolute value of a is
greater are steeper. The y-intercept is -1 in the graph of
y = -2x, -3 in
[-10, 10] scl: 1 by [-100, 10] scl: 10
1
1 x
in y = -_
(2 ).
y = -3(2x), and -_
6
6
Model and Analyze
1. How does b affect the graph of y = abx when b > 1 and when 0 < b < 1? Give examples.
2. How does a affect the graph of y = abx when a > 0 and when a < 0? Give examples.
x
1
. Verify your
3. Make a conjecture about the relationship of the graphs of y = 3x and y = _
3
conjecture by graphing both functions.
()
y = - 1 (2)x
6