Section+6 3

Math 1023

Section 6.3

Exponential Functions

In this section we will look at what happens when you raise a positive number, which is not equal to one, to a power. Ex. 1: Find the approximate value of e3.2. Round to 3 decimal places. An exponential function is a function of the form f(x) = ax, where a is a positive real number (a > 0) and a ≠ 1. If a = 1, then f is simply a constant function f(x) = 1 since 1 to any power = 1. The domain of an exponential function is the set of all real numbers. Ex. 2: Let us consider the exponential function f(x) = 2x. What is the y-intercept? f(0) = 1 Find f(1) = 2

f(-1) = 1/2

f(2) = 4

f(-2) = 1/4

Graph f showing all the points. (2,4)

4

(1,2)

2

(-2,

1 4

)

(-1,

1 2

(0,1)

)

-5

5

-2

Notice that the range of f is all positive numbers and that the graph is always increasing. As x approaches negative infinity, the graph gets closer and closer to the x-axis but never touches it. That means there is a horizontal asymptote at the x-axis or y = 0. -4

The graph of f(x) = 2x is typical of all exponential functions that have a base larger than one.

What do you think the graph of 3x would look like? 5x? Graph them on the same axes as 2x. x

8

1 Ex. 3: Now let’s consider the graph of g(x) =   . 2 6

Find g(0) = 1

g(1) = 1/2

g(-1) = 2

4

Graph g(x). (-1,2)

2

(0,1) (1,1/2) -10

-5

5

10

-2

Notice again that the range of g is all positive numbers, y > 0. The graph of g is always decreasing. As x approaches infinity, the graph gets closer and closer to the x-axis but never touches it. This means that the x-axis is a horizontal asymptote or y = 0. The graph of g is typical of the graph of all exponential functions whose base is between 0 and 1. -4

-6

-8

Characteristics of the Graph of an Exponential Function f(x) = ax. 1. The domain is all real numbers. 2. The range is the set of all positive numbers. 3. There are no x-intercepts; the y-intercept is (0, 1). 4. The x-axis (y = 0) is a horizontal asymptote. 5. The graph is increasing if a > 1, decreasing if 0 < a < 1. 6. f is a one-to-one function. 7. The graph contains the points (0, 1) and (1, a). 8. The graph of f is smooth and continuous.

a>1 0
The number e is approximately 2.7. Your calculator has an ex key on it. Ex. 4: Now let’s graph f(x) = ex.

________________________

Now that we know the basic exponential functions we will look at transformations of exponential functions 6

4

Ex. 5: Given f(x) = ex + 3 a.) determine the graph from the following: 6

6

2

6

4

4

-5

5

4

2

2

-2 2

-5

-5

5

5

-4 -5

5

-2

-2

-2

-4

b.) Determine the domain. Write in interval notation. -4

-4

-6 -6

-6

c.) Determine the range. Write in interval notation.

d.) List the equation of the horizontal asymptote:

-6

Ex. 6: Given graph of the function below, determine the function. 15

(2,16)

10

5

(1,4) (0,1)

-20

6

-10

10

-5 4

Ex. 7: Given the graph of a function below, match the graph to one of the following functions. -10 x A. y = 2 E. y = -2x 2

-5

-15

B. y = 2x – 1

F. y = 2-x

C. y = -2-x

G. y = 2x + 1

D. y = 2x – 1

H. y = 2x + 1

5

-2

-4

Solving exponential equations with the same base. If au = av, then u = v. In other words, if the bases are the same and the sides are equal then the exponents must be equal. This is a one-to-one property. Ex. 8: Solve the exponential equation. Give exact answer. 2 x1

 1    5

 25

20