Pdf 2nd Pass

Graphing Technology Lab Graphing Technology Lab Casio fx-CG10Casio fx-CG10

Family of Exponential FunctionsFamily 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

Clear the calculator memory first.

KEYSTROKES: System

Use the Graph Func menu to enter the equations.

KEYSTROKES: Graph 2 θ 3 θ 6

θ [V-Window] – 10 10

1 – 10 100 1

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.

y = 2x

y = 3x

y = 6x

[-10, 10] scl: 1 by [-10, 100] scl: 10

[-2.5, 2.5] scl: 1 by [-9.55..., 17.94...] scl: 10

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Pdf Pass

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.

y = ( 1 _ 2 )

x , y = ( 1 _

3 )

x , y = ( 1 _

6 )

x

Delete the equations from Activity 1.

KEYSTROKES:

Use the Graph Func menu to enter the equations.

KEYSTROKES: 1 ÷ 2 θ

1 ÷

3 θ

1 ÷ 6 θ [V-Window] – 10

10 1 – 10 100 1

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. [-10, 10] scl: 1 by [-10, 100] scl: 10

y = 13x⎛

⎝⎞⎭

y = 12x⎛

⎝⎞⎭

y = 16x⎛

⎝⎞⎭

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Pdf 2nd Pass

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 _ 6 (2x)

Delete the equations from Activity 2.

KEYSTROKES:

Use the Graph Func menu to enter the equations.

KEYSTROKES: 2 θ 3 2 θ

1 ÷ 6

2 θ

[V-Window] – 10 10 1

– 10 100 1

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

is 1 in the graph of y = 2 x , 3 in y = 3( 2 x ), and 1 _ 6 in y = 1 _

6 ( 2 x ).

[-10, 10] scl: 1 by [-10, 100] scl: 10

y = 16

y = 2x

y = 3(2)x

(2)x

[-2.5, 2.5] scl: 1 by [-9.55..., 17.94...] scl: 10

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Pdf Pass

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 _

6 (2x)

Do not clear the calculator memory first. Add negatives before each equation in Graph Func.

KEYSTROKES: Graph – – –

[V-Window] – 10 10 1 – 100

10 1

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.

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

y = -3(2x), and - 1 _

6 in y = -

1 _

6 (2x).

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.

3. Make a conjecture about the relationship of the graphs of y = 3x and y = ( 1 _ 3 )

x . Verify your

conjecture by graphing both functions.

[-10, 10] scl: 1 by [-100, 10] scl: 10

y = 16

y = -2x

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

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