5.2 Exponential Functions & Graphs

Graph exponential equations and exponential functions.

Solve applied problems involving exponential functions and their graphs.

Exponential Function

The function f(x) = ax, where x is a real number, a > 0 and a 1, is called the exponential function, base a.

The base needs to be positive in order to avoid the complex numbers that would occur by taking even roots of negative numbers.

The following are examples of exponential functions:

f (x) 2x f (x) 1

2

x

f (x) (3.57)x

Graphing Exponential Functions

To graph an exponential function, follow the steps

listed:

1. Compute some function values and list the results in a table.

2. Plot the points and connect them with a smooth curve. Be sure to plot enough points to determine how steeply the curve rises.

Graph

2xy

x y

-3

-2

-1

0

1

2

3

Graph

1

2

x

y

x y

-3

-2

-1

0

1

2

3

2xy 1

2

x

y

Graphs of Exponential Functions

Observe the following graphs of exponential functions and look for patterns in them.

Example

Graph y = 2x – 2.The graph is the graph of y = 2x shifted to right 2 units.

Example

Graph y = 5 – 0.5x .

The graph is a reflection of the graph of y = 2x across the y-axis, followed by a reflection across the x-axis and then a shift up 5 units.

y 5 0.5x 5 1

2

x

5 2 x.

Find the exponential function whose graph is shown.

Example

Application

The amount of money A to which a principal P will grow to after t years at interest rate r (in decimal form), compounded n times per year, is given by the formula

A P 1r

n

nt

.

Example

a) Find a function for the amount to which the investment grows after t years.

b) Find the amount of money in the account at t = 0, 4, 8, and 10 yr.

c) Graph the function.

Suppose that $100,000 is invested at 6.5% interest, compounded semiannually.

A P 1r

n

nt

.

a) Since P = $100,000, r = 6.5%=0.065, and n = 2, we can substitute these values and write the following function

A t 100,000 10.065

2

2t

$100,000 1.0325 2t

b) We can calculate the values directly on a calculator.

c) Graph.

The Number e

e is a very special number in mathematics. Leonard Euler named this number e. The decimal representation of the number e does not terminate or repeat; it is an irrational number that is a constant;

e 2.7182818284…

A P 1r

n

nt

.

Example

Find each value of ex, to four decimal places, using the ex key on a calculator.

a) e3 b) e0.23

c) e0 d) e1

a) e3 ≈ 20.0855 b) e0.23 ≈ 0.7945

c) e0 = 1 d) e1 ≈ 2.7183

Solution:

Graphs of Exponential Functions, Base e Example

Graph f (x) = ex and g(x) = e–x.Use the calculator and enter y1 = ex and y2 = e–x. Enter numbers for x.

Graphs of Exponential Functions, Base e - Example (continued)

The graph of g is a reflection of the graph of f across the y-axis.

ExampleGraph f (x) = ex + 3.Solution: The graph f (x) = ex + 3 is a translation of the graph of y = ex left 3 units.

ExampleGraph f (x) = e–0.5x.Solution: The graph f (x) = e–0.5x is a horizontal stretching of the graph of y = ex followed by a reflection across the y-axis.

ExampleGraph f (x) = 1 e2x.Solution: The graph f (x) = 1 e2x is a horizontalshrinking of the graph of y = ex followed by a reflection across the y-axis and then across the x-axis, followed by a translation up 1 unit.

Find the exponential function given

f(0) = 4 and f(-2) = 16.

Example

a. Determine an exponential function for the population after t years given the table below. The population is given in thousands.

b. What is the population when t = 40?

Example

t P(t)

0 10

5 20

10 40