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8.1 Exponential Functions. ©2001 by R. Villar All Rights Reserved. Exponential Functions. Exponential Function: An equation in the form f(x) = Ca x . Recall that if 0 < a < 1 , the graph represents exponential decay

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8 1 exponential functions

8.1 Exponential Functions

©2001 by R. Villar

All Rights Reserved

exponential functions

Exponential Functions

Exponential Function: An equation in the formf(x) = Cax.

Recall that if 0 < a < 1 , the graph represents exponential decay

and that if a > 1, the graph represents exponential growth

Examples: f(x) = (1/2)x f(x) = 2x

Exponential Decay

Exponential Growth

We will take a look at how these graphs “shift” according to changes in their equation...

take a look at how the following graphs compare to the original graph of f x 1 2 x

Take a look at how the following graphs compare to the original graph of f(x) = (1/2)x :

f(x) = (1/2)x f(x) = (1/2)x + 1 f(x) = (1/2)x – 3

Vertical Shift: The graphs of f(x) = Cax + k are shifted vertically by k units.

take a look at how the following graphs compare to the original graph of f x 2 x

Take a look at how the following graphs compare to the original graph of f(x) = (2)x :

(3,1)

(0,1)

(-2,-2)

f(x) = (2)x f(x) = (2)x – 3 f(x) = (2)x + 2 – 3

Notice that f(0) = 1

Notice that this graph

is shifted 3 units to the

right.

Notice that this graph

is shifted 2 units to the

left and 3 units down.

Horizontal Shift: The graphs of f(x) = Cax – h are shifted horizontally by h units.

take a look at how the following graphs compare to the original graph of f x 2 x5

Take a look at how the following graphs compare to the original graph of f(x) = (2)x :

(0,1)

(0,-1)

(-2,-4)

f(x) = (2)x f(x) = –(2)x f(x) = –(2)x + 2 – 3

Notice that f(0) = 1

This graph

is a reflection of

f(x) = (2)x . The graph is

reflected over the x-axis.

Shift the graph of

f(x) = (2)x ,2 units to the left. Reflect the graph over the x-axis. Then, shift the graph 3 units down