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Chapter 4Exponential Functions

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4.3 Graphing Exponential Functions

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Example: Graphing an Exponential Function with b > 1

Graph f(x) = 2x by hand.

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Solution

First, list input-output pairs of the function f in a table. Note that as the value of x increases by 1, the value of y is multiplied by 2 (the base).

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Solution

Next, plot the solutions from the table and sketch an increasing curve that contains the plotted points.

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Solution

We can set up a window to verify our graph on a graphing calculator.

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Exponential Curve

The graph of an exponential function is called an exponential curve.

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Base Multiplier Property

For an exponential function of the form y = abx, if the value of the independent variable increases by 1, the value of the dependent variable is multiplied by b.

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Increasing or Decreasing Property

Let f(x) = abx, where a > 0. Then,

• If b > 1, then the function f is increasing. We say the function grows exponentially.

• If 0 < b < 1, then the function f is decreasing. We say the function decays exponentially.

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y-Intercept of an Exponential Function

For an exponential function of the form

y = abx,

the y-intercept is (0, a).

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y-Intercept of an Exponential Function

Warning

For an exponential function of the form y = bx (rather than y = abx), the y-intercept is not (0, b). By writing y = bx = 1bx, we see the y-intercept is (0, 1).

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Example: Intercepts and Graph of an Exponential Function

Let

1. Find the y-intercept of the graph of f.2. Find the x-intercept of the graph of f.3. Graph f by hand.

1( ) 6 .

2

x

f x

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Solution

1. Since is of the form f(x) = abx, the1

( )2

6x

f x

y-intercept is (0, a), or (0, 6).

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Solution

2. By the base multiplier property, as the value of x increases by 1, the value of y is multiplied by one half. Values are shown in the table below.

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Solution

2. When we halve a number, it becomes smaller. But no number of halvings will give a result that is zero. So, as x grows large, y will become extremely close to, but never equal, 0. Likewise, the graph of f gets arbitrarily close to, but never reaches, the x-axis. In this case, we call the x-axis a horizontal asymptote. We conclude that the function f has no x-intercepts.

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Solution

3. Plot the points from the table and sketch a decreasing exponential curve that contains the points.

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Reflection Property

The graphs of f(x) = –abx and g(x) = abx are reflections of each other across the x-axis.

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Horizontal Asymptote

For all exponential functions, the x-axis is a horizontal asymptote.

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Domain and Range of an Exponential Function

The domain of any exponential function f(x) = abx is the set of real numbers.

The range of an exponential function f(x) = abx is the set of all positive real numbers if a > 0, and the range is the set of all negative real numbers if a < 0.

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Example: Finding Values of a Function from Its Graph

The graph of an exponential function f is shown below.

1. Find f(2).2. Find x when f(x) = 2.3. Find x when f(x) = 0.

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Solution

1. The blue arrows show that the input x = 2 leads to the output y = 8. We conclude that f(2) = 8.

2. The red arrows show that the output y = 2 originates from the input x = –2. So, x = –2 when f(x) = 2.

3. Recall that the graph of an exponential functions gets close to, but never reaches, the x-axis. So, there is no value of x where f(x) = 0.