Exponential Growth

Exponential Decay

ExampleGraph the exponential function given by ( ) 3 .xf x

Solution

x y, or f(x)

01

–12

–23

13

1/39

1/927 4 3 2 1 0 1 2 3 4

2

1

1

2

3

4

5

6

7

8

9

10

Example

Graph the exponential function given by 1

( ) .3

xf x

Solution

x y, or f(x)

01–12–2–3

11/33

1/99

27 4 3 2 1 0 1 2 3 4

2

1

1

2

3

4

5

6

7

8

9

10

4 3 2 1 0 1 2 3 4

2

1

1

2

3

4

5

6

7

8

9

10

( ) 3xy f x 1

( )3

xy f x

From the previous two examples, we can make the following observations.A. For a > 1, the graph of f (x ) = ax increases from left to right. The greater the value of a, the steeper the curve, this is called exponential growth.B. For 0 < a < 1, the graph of f (x ) = ax decreases from left to right. For smaller values of a, the graph becomes steeper, this is called exponential decay.

D. All graphs of f (x) = ax have the x-axis as the asymptote.

E. If f (x ) = ax, with a > 0, a not 1, the domain of f is all real numbers, and the range of f is all positive real numbers.

, 1,

exponential growth

xy a a , 0 1,

exponential decay

xy a a

intercept (0,1)y

0

horizontial asymptote

y

C. All graphs of f (x ) = ax go through the y-intercept (0, 1).

3Consider the exponential function ( ) 4 5.

(a) Graph ( ) in the window [ 7,2] [ 7,7].

(b) Find the zero of ( ).

(c) Find the equation of the horizontal asymptote.

xf x

f x

f x

Example

Horizontal asymptote x = -5

Zero x = -1.839

Equations with x and y Interchanged

It will be helpful in later work to be able to graph an equation in which the x and y in y = ax are interchanged x = ay .

Applications of Exponential Functions

Example Interest compounded annually.

The amount of money A that a principal P will be worth after t years at interest rate i, compounded annually, is given by the formula

(1 ) .tA P i

Example

Solution

Suppose that $60,000 is invested at 5% interest,

compounded annually.

a) Find a function for the amount in the account after t years.

b) Find the amount of money in the account at t = 6.

a) ( ) (1 )tA t P i = $60000(1 + 0.05)t

= $60000(1.05)t

b) A(6) = $60000(1.05)6 $80,405.74.

Linear vs Polynomial vs Exponential