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Graphs of Exponential Equations

Author: Sophia
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what's covered
In this lesson, you will graph exponential functions. Specifically, this lesson will cover:

Table of Contents

1. Graphing f   (x  )  =  a    ⋅  bˣ When b > 1

Given f open parentheses x close parentheses equals a times b to the power of x comma let’s look at f open parentheses 0 close parentheses colon

f open parentheses x close parentheses equals a times b to the power of x Original expression
f open parentheses 0 close parentheses equals a times b to the power of 0 Replace x with 0.
equals a times 1 Any nonzero number raised to the 0 power is 1.
equals a Simplify.

Since f open parentheses 0 close parentheses equals a comma this means that the y-intercept of an exponential function is open parentheses 0 comma space a close parentheses. Thus, by looking at the equation, the coefficient of the power term is the y-intercept, or “starting value”.

Consider three exponential functions: f open parentheses x close parentheses equals 2 to the power of x comma g open parentheses x close parentheses equals 3 times 2 to the power of x comma and h open parentheses x close parentheses equals short dash 5 times 2 to the power of x.

The tables of values for each function are shown here:

bold italic x bold italic f open parentheses bold x close parentheses bold equals bold 2 to the power of bold x bold italic g open parentheses bold x close parentheses bold equals bold 3 bold times bold 2 to the power of bold x bold italic h open parentheses bold x close parentheses bold equals bold short dash bold 5 bold times bold 2 to the power of bold x
-3 0.125 0.375 -0.625
-2 0.25 0.75 -1.25
-1 0.5 1.5 -2.5
0 1 3 -5
1 2 6 -10
2 4 12 -20
3 8 24 -40

The graphs and descriptions of each function are shown below. Note the vertical scales in each.

bold italic f open parentheses bold x close parentheses bold equals bold 2 to the power of bold x
A graph with an x-axis ranging from −5 to 5 and a y-axis ranging from 0 to 9. The graph has a curve that rises to the second quadrant along the negative x-axis; passes through the marked points at (−3, 0.125), (−2, 0.25), (−1, 0.5), (0, 1), (1, 2), (2, 4), and (3, 8); and extends upward into the first quadrant. As x decreases indefinitely, f open parentheses x close parentheses approaches 0 from above and as x increases indefinitely, f open parentheses x close parentheses also increases indefinitely.

Since f open parentheses 0 close parentheses equals 2 to the power of 0 equals 1 comma a equals 1 and the y-intercept is open parentheses 0 comma space 1 close parentheses.

The graph has horizontal asymptote y equals 0.

The domain of f open parentheses x close parentheses is the set of all real numbers, or using interval notation, open parentheses short dash infinity comma space infinity close parentheses.

The range of f open parentheses x close parentheses is the set of all positive numbers, or using interval notation, open parentheses 0 comma space infinity close parentheses.
bold italic g open parentheses bold x close parentheses bold equals bold 3 bold times bold 2 to the power of bold x
A graph with an x-axis ranging from −5 to 5 and a y-axis ranging from 0 to 45 at intervals of 5. The graph has a curve that rises to the second quadrant along the negative x-axis; passes through the marked points at (−3, 0.375), (−2, 0.75), (−1, 1.5), (0, 3), (1, 6), (2, 12), and (3, 24); and extends upward into the first quadrant. As x decreases indefinitely, g open parentheses x close parentheses approaches 0 from above and as x increases indefinitely, g open parentheses x close parentheses also increases indefinitely.

Since g open parentheses 0 close parentheses equals 3 times 2 to the power of 0 equals 3 comma a equals 3 and the y-intercept is open parentheses 0 comma space 3 close parentheses.

The graph has horizontal asymptote y equals 0.

The domain of g open parentheses x close parentheses is the set of all real numbers, or using interval notation, open parentheses short dash infinity comma space infinity close parentheses.

The range of g open parentheses x close parentheses is the set of all positive numbers, or using interval notation, open parentheses 0 comma space infinity close parentheses.
bold italic h open parentheses bold x close parentheses bold equals bold short dash bold 5 bold times bold 2 to the power of bold x
A graph with an x-axis ranging from −5 to 5 and a y-axis ranging from −80 to 10 at intervals of 10. The graph has a curve that descends the along the negative x-axis; passes through the marked points at (−3, −0.625), (−2, −1.25), (−1, −2.5), (0, −5), (1, −10), (2, −20), and (3, −40); and extends downward into the fourth quadrant. As x decreases indefinitely, h open parentheses x close parentheses approaches 0 from below and as x increases indefinitely, h open parentheses x close parentheses also decreases indefinitely.

Since h open parentheses 0 close parentheses equals short dash 5 times 2 to the power of 0 equals short dash 5 open parentheses 1 close parentheses equals short dash 5 comma a equals short dash 5 and the y-intercept is open parentheses 0 comma space short dash 5 close parentheses.

The graph has horizontal asymptote y equals 0.

The domain of h open parentheses x close parentheses is the set of all real numbers, or using interval notation, open parentheses short dash infinity comma space infinity close parentheses.

The range of h open parentheses x close parentheses is the set of all negative numbers, or using interval notation, open parentheses short dash infinity comma space 0 close parentheses.

In summary, the graph of f open parentheses x close parentheses equals a times b to the power of x can be generalized into two shapes: one where a less than 0 and one where a greater than 0.

General Shape and Behavior When bold italic a bold less than bold 0 General Shape and Behavior When bold italic a bold greater than bold 0
A graph with an x-axis and a y-axis intersecting at the origin. The graph has a curve that descends along the negative x-axis, opens downward in the third quadrant, passes through the marked point (0, a) on the negative y-axis near the origin, and extends downward into the fourth quadrant. A graph with an x-axis and a y-axis intersecting at the origin. The graph has a curve that rises along the negative x-axis, opens upward in the second quadrant, passes through the marked point (0, a) on the positive y-axis near the origin, and extends upward into the first quadrant.
The graph decreases over its entire domain.
Domain: open parentheses short dash infinity comma space infinity close parentheses
Range: open parentheses short dash infinity comma space 0 close parentheses
Horizontal asymptote: y equals 0 		The graph increases over its entire domain.
Domain: open parentheses short dash infinity comma space infinity close parentheses
Range: open parentheses 0 comma space infinity close parentheses
Horizontal asymptote: y equals 0

Here is one for you to try:

try it
Consider the function f open parentheses x close parentheses equals 3 open parentheses 1.25 close parentheses to the power of x.
Is the graph of f   (x  ) increasing or decreasing?
It is increasing since a equals 3 and 1.25 greater than 1.
What are the domain and range of f   (x  )?
The domain is open parentheses short dash infinity comma space infinity close parentheses and the range is open parentheses 0 comma space infinity close parentheses.
What is the y-intercept of the graph of f   (x  )?
The y-intercept is open parentheses 0 comma space 3 close parentheses.
What is the horizontal asymptote of f   (x  )?
Its equation is y equals 0.

Let’s do a similar exploration, but this time when 0 less than b less than 1.

2. Graphing f   (x  )  =  a    ⋅  bˣ When 0 < b < 1

Consider three exponential functions: f open parentheses x close parentheses equals open parentheses 1 half close parentheses to the power of x comma g open parentheses x close parentheses equals 4 open parentheses 1 half close parentheses to the power of x comma and h open parentheses x close parentheses equals short dash 3 open parentheses 1 half close parentheses to the power of x.

The tables of values for each function are shown here:

bold italic x bold italic f open parentheses bold x close parentheses bold equals open parentheses bold 1 over bold 2 close parentheses to the power of bold x bold italic g open parentheses bold x close parentheses bold equals bold 4 open parentheses bold 1 over bold 2 close parentheses to the power of bold x bold italic h open parentheses bold x close parentheses bold equals bold short dash bold 3 open parentheses bold 1 over bold 2 close parentheses to the power of bold x
-3 8 32 -24
-2 4 16 -12
-1 2 8 -6
0 1 4 -3
1 0.5 2 -1.5
2 0.25 1 -0.75
3 0.125 0.5 -0.375

The graphs and descriptions of each function are shown below. Note the vertical scales in each.

bold italic f open parentheses bold x close parentheses bold equals open parentheses bold 1 over bold 2 close parentheses to the power of bold x
A graph with an x-axis ranging from −7 to 7 and a y-axis ranging from 0 to 10. The graph has a curve that descends from the second quadrant; passes through the marked points at (−3, 8), (−2, 4), (−1, 2), (0, 1), (1, 0.5), (2, 0.25), and (3, 0.125); and extends into the first quadrant along the positive x-axis. As x decreases indefinitely, f open parentheses x close parentheses increases indefinitely; and x increases indefinitely, f open parentheses x close parentheses approaches 0 from above.

Since f open parentheses 0 close parentheses equals open parentheses 1 half close parentheses to the power of 0 equals 1 comma a equals 1 and the y-intercept is open parentheses 0 comma space 1 close parentheses.

The graph has horizontal asymptote y equals 0.

The domain of f open parentheses x close parentheses is the set of all real numbers, or using interval notation, open parentheses short dash infinity comma space infinity close parentheses.

The range of f open parentheses x close parentheses is the set of all positive numbers, or using interval notation, open parentheses 0 comma space infinity close parentheses.
bold italic g open parentheses bold x close parentheses bold equals bold 4 open parentheses bold 1 over bold 2 close parentheses to the power of bold x
A graph with an x-axis ranging from −5 to 5 and a y-axis ranging from 0 to 40 in increments of 5. The graph has a curve that descends from the second quadrant; passes through the marked points at (−3, 32), (−2, 16), (−1, 8), (0, 4), (1, 2), (2, 1), and (3, 0.5); and extends into the first quadrant along the positive x-axis. As x decreases indefinitely, g open parentheses x close parentheses increases indefinitely; and x increases indefinitely, g open parentheses x close parentheses approaches 0 from above.

Since g open parentheses 0 close parentheses equals 4 open parentheses 1 half close parentheses to the power of 0 equals 4 open parentheses 1 close parentheses equals 4 comma a equals 4 and the y-intercept is open parentheses 0 comma space 4 close parentheses.

The graph has horizontal asymptote y equals 0.

The domain of g open parentheses x close parentheses is the set of all real numbers, or using interval notation, open parentheses short dash infinity comma space infinity close parentheses.

The range of g open parentheses x close parentheses is the set of all positive numbers, or using interval notation, open parentheses 0 comma space infinity close parentheses.
bold italic h open parentheses bold x close parentheses bold equals bold short dash bold 3 open parentheses bold 1 over bold 2 close parentheses to the power of bold x
A graph with an x-axis ranging from −5 to 5 and a y-axis ranging from −35 to 5 in increments of 5. The graph has a curve that rises from the third quadrant; passes through the marked points at (−3, −24), (−2, −12), (−1, −6), (0, −3), (1, −1.5), (2, −0.75), and (3, −0.375); and extends upward into the fourth quadrant along the positive x-axis. As x decreases indefinitely, h open parentheses x close parentheses decreases indefinitely; and x increases indefinitely, h open parentheses x close parentheses approaches 0 from below.

Since h open parentheses 0 close parentheses equals short dash 3 open parentheses 1 half close parentheses to the power of 0 equals short dash 3 open parentheses 1 close parentheses equals short dash 3 comma a equals short dash 3 and the y-intercept is open parentheses 0 comma space short dash 3 close parentheses.

The graph has horizontal asymptote y equals 0.

The domain of h open parentheses x close parentheses is the set of all real numbers, or using interval notation, open parentheses short dash infinity comma space infinity close parentheses.

The range of h open parentheses x close parentheses is the set of all negative numbers, or using interval notation, open parentheses short dash infinity comma space 0 close parentheses.

In summary, the graph of f open parentheses x close parentheses equals a times b to the power of x when 0 less than b less than 1 can be generalized into two shapes: one where a less than 0 and one where a greater than 0.

General Shape and Behavior When bold italic a bold less than bold 0 General Shape and Behavior When bold italic a bold greater than bold 0
A graph with an x-axis and a y-axis intersecting at the origin. The graph has a curve that rises upward from the third quadrant, opens downward, passes through the marked point (0, a) on the negative y-axis near the origin, and extends into the fourth quadrant along the positive x-axis. A graph with an x-axis and a y-axis intersecting at the origin. The graph has a curve that descends from the second quadrant, passes through the marked point (0, a) on the positive y-axis near the origin, and extends into the first quadrant along the positive x-axis.
The graph increases over its entire domain.
Domain: open parentheses short dash infinity comma space infinity close parentheses
Range: open parentheses short dash infinity comma space 0 close parentheses
Horizontal asymptote: y equals 0 		The graph decreases over its entire domain.
Domain: open parentheses short dash infinity comma space infinity close parentheses
Range: open parentheses 0 comma space infinity close parentheses
Horizontal asymptote: y equals 0

watch
This video covers the example of graphing f open parentheses x close parentheses equals 1 half open parentheses 3 over 4 close parentheses to the power of short dash x end exponent and describing the graph’s characteristics.

Here is one for you to try:

try it
Consider the function f open parentheses x close parentheses equals short dash 5 open parentheses 5 over 6 close parentheses to the power of x.
Is the graph of f   (x  ) increasing or decreasing?
It is increasing since a equals short dash 5 comma which is negative, and 5 over 6 less than 1.
What are the domain and range of f   (x  )?
The domain is open parentheses short dash infinity comma space infinity close parentheses and the range is open parentheses short dash infinity comma space 0 close parentheses.
What is the y-intercept of the graph of f   (x  )?
The y-intercept is open parentheses 0 comma space short dash 5 close parentheses.
What is the horizontal asymptote of f   (x  )?
Its equation is y equals 0.

The shape of the graph of an exponential function f open parentheses x close parentheses equals a times b to the power of x is affected by the sign of a and the value of b. There are four possible general shapes of the graph of an exponential function.

summary
In this lesson, you learned that the shape of the graph of an exponential function f open parentheses x close parentheses equals a times b to the power of x is affected by the sign of a and the value of b. The graph of an exponential function when b > 1 and the graph of an exponential function when 0 < b < 1 can each be generalized into two shapes: one where a less than 0 and one where a greater than 0 (a total of four possible general shapes).

Source: THIS TUTORIAL HAS BEEN ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE. ACCESS FOR FREE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License

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