Introduction
In this lesson, different methods will be used to graph exponential functions and analyze the key features of the graph. In an exponential function , the graph is a smooth line with a rounded curve. All exponential functions have an asymptote , or a line that the graph gets closer and closer to, but never crosses or touches.
Every exponential function has a y -intercept, where the graph crosses the y -axis. There is, at most, one x intercept, where the function crosses the x -axis.
1
3.4.2: Graphing Exponential Functions

Key Concepts
• To find the y -intercept of an exponential function, evaluate f (0).
• The y -intercept has the coordinates (0, f (0)).
• To locate the y -intercept of a graphed function, determine the coordinates of the function where the line crosses the y -axis.
• To find the x -intercept in function notation, set f ( x ) = 0 and solve for x .
• The x -intercept has the coordinates ( x , 0).
2
3.4.2: Graphing Exponential Functions

Key Concepts, continued
• To locate the x -intercept of a graphed function, determine the coordinates of the line where the line crosses the x -axis.
• Not all exponential functions cross the x -axis.
• The asymptote of exponential functions of the form f ( x ) = ab x is always the x -axis, or y = 0.
• If the exponential function is of the form f ( x ) = ab x + k , then the function will be shifted vertically by the same number of units as k.
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3.4.2: Graphing Exponential Functions

Key Concepts, continued
• The asymptote is then y = k .
• The end behavior , or the behavior of the graph as x becomes larger or smaller, will always be one of three descriptions: infinity, negative infinity, or the asymptote.
• It is easiest to first graph the function and then observe what happens to the value of y as the value of x increases and decreases.
• Graph complex exponential models using technology as values can become quite large or small very quickly.
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3.4.2: Graphing Exponential Functions

Key Concepts, continued
Graphing Equations Using a TI-83/84:
Step 1: Press [Y=].
Step 2: Key in the equation using [X, T, q
, n ] for x .
Step 3: Press [WINDOW] to change the viewing window, if necessary.
Step 4: Enter in appropriate values for Xmin, Xmax,
Xscl, Ymin, Ymax, and Yscl, using the arrow keys to navigate.
Step 5: Press [GRAPH].
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3.4.2: Graphing Exponential Functions

Key Concepts, continued
Graphing Equations Using a TI-Nspire:
Step 1: Press the home key.
Step 2: Arrow over to the graphing icon (the picture of the parabola or the U-shaped curve) and press [enter].
Step 3: Enter in the equation and press [enter].
Step 4: To change the viewing window: press [menu], arrow down to number 4: Window/Zoom, and click the center button of the navigation pad.
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3.4.2: Graphing Exponential Functions

Key Concepts, continued
Step 5: Choose 1: Window settings by pressing the center button.
Step 6: Enter in the appropriate XMin, XMax, YMin, and YMax fields.
Step 7: Leave the XScale and YScale set to auto.
Step 8: Use [tab] to navigate among the fields.
Step 9: Press [tab] to “OK” when done and press
[enter].
3.4.2: Graphing Exponential Functions
7

Common Errors/Misconceptions
• incorrectly plotting points
• mistaking the y -intercept for the x -intercept and vice versa
• not being able to identify key features of an exponential model
• confusing the value of a function for its corresponding x -coordinate
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3.4.2: Graphing Exponential Functions

Guided Practice
Example 2
Create a table of values for the exponential function f ( x ) = –1(3) x – 2. Identify the asymptote and y -intercept of the function. Plot the points and sketch the graph of the function, and describe the end behavior.
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3.4.2: Graphing Exponential Functions

Guided Practice: Example 2, continued
Choose values of x and solve for the corresponding values of f ( x ).
1
2
3 x
–1
0
f(x)
–2.33
–3
–5
–11
–29
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3.4.2: Graphing Exponential Functions

Guided Practice: Example 2, continued
The asymptote of the function is always the constant, k .
In the function f ( x ) = –1(3) x – 2, the value of k is –2.
The asymptote of the function is y = –2.
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3.4.2: Graphing Exponential Functions

Guided Practice: Example 2, continued
The y -intercept of the function is the value of f ( x ) when x is equal to 0.
It can be seen in the table that when x = 0, f ( x ) = –3.
The y -intercept is (0, –3).
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3.4.2: Graphing Exponential Functions

Guided Practice: Example 2, continued
Use the table of values to create a graph of the function.
3.4.2: Graphing Exponential Functions
13

Guided Practice: Example 2, continued
The end behavior is what happens at the ends of the graph.
As x becomes larger, the value of the function approaches negative infinity.
As x becomes smaller, the value of the function approaches the asymptote, –2.
✔
14
3.4.2: Graphing Exponential Functions

Guided Practice: Example 2, continued
3.4.2: Graphing Exponential Functions

Guided Practice
Example 3
Create a table of values for the exponential function f ( x )
=
4
ç
æ
è
2
3
÷
ö
ø x
-
3
. Identify the asymptote and y -intercept of the function. Plot the points and sketch the graph of the function, and describe the end behavior.
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3.4.2: Graphing Exponential Functions

Guided Practice: Example 3, continued
Choose values of x and solve for the corresponding values of f ( x ). x
–4
–2
0
2
4
f(x)
17.25
6
1
–1.22
–2.2099
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3.4.2: Graphing Exponential Functions

Guided Practice: Example 3, continued
The asymptote of the function is always the constant, k .
In the function f ( x )
=
4
ç
æ
è
2
3
ö
ø
÷ x
-
3 , the value of
The asymptote of the function is y = –3. k is –3.
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3.4.2: Graphing Exponential Functions

Guided Practice: Example 3, continued
The y -intercept of the function is the value of f ( x ) when x is equal to 0.
It can be seen in the table that when x = 0, f ( x ) = 1.
The y -intercept is (0, 1).
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3.4.2: Graphing Exponential Functions

Guided Practice: Example 3, continued
Use the table of values to create a graph of the function.
3.4.2: Graphing Exponential Functions
20

Guided Practice: Example 3, continued
The end behavior is what happens at the ends of the graph.
As x becomes larger, the value of the function approaches the asymptote, –3.
As x becomes smaller, the value of the function approaches infinity.
Since the function approaches infinity
✔ as x becomes smaller, the graph shows exponential decay.
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3.4.2: Graphing Exponential Functions

Guided Practice: Example 3, continued
3.4.2: Graphing Exponential Functions
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