1 6.3 Exponential Functions In this section, we will study the following topics: Evaluating...

1

6.3 Exponential Functions

In this section, we will study the following topics:

Evaluating exponential functions with base a

Graphing exponential functions with base a

Evaluating exponential functions with base e

Graphing exponential functions with base e

2

Transcendental Functions

In this chapter, we continue our study of functions with

two very important ones—

exponential and logarithmic functions.

These two functions are types of nonalgebraic

functions, known as transcendental functions.

So, in an exponential function, the variable is in the exponent.

4

Exponential Functions

Which of the following are exponential functions?

3( )f x x

( ) 3xf x

( ) 5f x

( ) 1xf x

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Graphs of Exponential Functions

Just as the graphs of all quadratic functions have the same

basic shape, the graphs of exponential functions have

the same basic characteristics.

They can be broken into two categories—

exponential growth

exponential decay (decline)

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The Graph of an Exponential Growth Function

We will look at the graph of an exponential function that increases as

x increases, known as the exponential growth function.

It has the form

Example: f(x) = 2x

( ) where a > 1. xf x a

Notice the rapid increase in the graph as x increasesThe graph increases

slowly for x < 0.

y-intercept is (0, 1)

Horizontal asymptote is y = 0.

x f(x)

-5

-4

-3

-2

-1

0

1

2

3

f(x)=2x

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The Graph of an Exponential Decay (Decline) Function

We will look at the graph of an exponential function that

decreases as x increases, known as the exponential decay

function.

It has the form

Example: g(x) = 2-x

( ) where a > 1. xf x a

Notice the rapid decline in the graph for x < 0.

The graph decreases more slowly as x increases.

y-intercept is (0, 1)

Horizontal asymptote is y = 0.

x f(x)

-3

-2

-1

0

1

2

3

4

5

g(x)=2-x

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Notice that f(x) = 2x and g(x) = 2-x are reflections of one another about the y-axis.

Both graphs have: y-intercept (___,___) and horizontal asymptote y = .

The domain of f(x) and g(x) is _________; the range is _______.

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Also, note that , using the properties of exponents.

So an exponential function is a DECAY function if

The base a is greater than one and the function is written as f(x) = a-x

-OR-

The base a is between 0 and 1 and the function is written as f(x) = ax

1( ) 2

2

xxg x

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Examples:

( ) 0.25xf x ( ) 5.6 xf x

In this case, a = 0.25 (0 < a < 1). In this case, a = 5.6 (a > 1).

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Transformations of Graphs of Exponential Functions

Look at the following shifts and reflections of the graph of f(x) = 2x.

The new horizontal asymptote is y = 3

( ) 2xf x

( ) 2xf x

( ) 2 3xg x

2( ) 2xg x

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( ) 2xf x

( ) 2xf x

( ) 2xg x

( ) 2 xg x

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Describe the transformation(s) that the graph of must undergo in order to obtain the graph of each of the following functions.

State the domain, range and the horizontal asymptote for each.

1.

( ) 2xf x

( ) 2 5xf x

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2.

( ) 2xf x

1( ) 2xf x

16




3.

( ) 2xf x

( ) 2 4xf x

17




4.

( ) 2xf x

3( ) 2xf x

A) B)

C) D)

Graph using transformations and determine the domain, range and horizontal asymptote.

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It may seem hard to believe, but when working with exponents and logarithms, it is often convenient to use the irrational number e as a base.

The number e is defined as

This value approaches as x approaches infinity.

Check this out using the TABLE on your calculator:

Enter and look at the value of y as x gets larger and larger.

Natural base e

1lim 1

x

xe

x

2.718281828e

1 (1 1/ ) ^y x x

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Evaluating the Natural Exponential Function

To evaluate the function f(x) = ex, we will use our calculators to find an approximation. You should see the ex button on your graphing calculator (Use ).

Example: Evaluate to three decimal places.

e-0.5 ≈ ____________

e ≈ _______________

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Graphing the Natural Exponential Function

( ) xf x e Domain:___________________

Range: ___________________

Asymptote: _______________

x-intercept: _______________

y-intercept: _______________

Increasing/decreasing over ___________

List four points that are on the graph of f(x) = ex.

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Solving Exponential Equations

3 -1Solve: 2 32x

42 13

1Solve: x x

xe e

e

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End of Section 6.3

1 6.3 Exponential Functions In this section, we will study the following topics: Evaluating...

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