8-1: Exponential 8-1: Exponential GrowthGrowth

Objective CA 12: Students know the laws of fractional exponents, understanding exponential functions, and use these functions in problems

involving exponential growth and decay.

An exponential function involves the expression

where the base b is a positive number other than 1

xb

Graph the function ( ) 2xf x

xx--33

22-3 -3 ==

1/81/8--22

22-2 -2 = ¼= ¼

--11

22-1-1 = ½ = ½

00 2200 = 1 = 111 2211 = 2 = 222 2222 = 4 = 4

2xf x

Question:What do you notice about the x-values and the f(x) values?

4

2

f x = 2x

Notice the end behavior as , ( )As , ( ) , which means that the graph has the line y = 0 as an asymptote.

x f xx f x

An asymptote is a line that a graph approaches as you move away from the

origin.

Example 1: Graphing Example 1: Graphing Exponential Functions of the Exponential Functions of the

Form y = abForm y = abxx 2Graph the function: 23

xy

2 13 3Plot the points: (0, ), (1,1 )

4

2

-5

1, 3/2

0, 1/2 f x =

12

3x

23Graph the function: ( )xy

Plot the points: (0, 1), (1, 1.5)

-2

5

1, - 3/2 0, -1

f x = -32

x

To graph a general exponential function:

x hy ab k

Start by graphing . Then translate the graph by h units horizontally and k units vertically.

xy ab

Example 2: Graphing a Example 2: Graphing a General Exponential FunctionGeneral Exponential Function

1Graph: 3 2 4. State the domain and range.xy

Begin by sketching the graph 3 2xy

Plot :(0, 3)) and (1, 6). 10

5

f x = 32x

Translate the graph 1 unit to the right and down 4 units

8

6

4

2

-2

-5 5g x = 32x-1-4

f x = 32x

Homework:Homework:Page 469 #19-33 OddPage 469 #19-33 Odd

Table of Values needed for credit

Investigating Graph of Investigating Graph of Exponential Functions page Exponential Functions page

46546511. Graph 2 and 3 2 Compare the 3

graphs with the graph of 2 .12. Graph 2 and 5 2 . Compare the5

graphs with the graph of 2

3. Describe the effect of on the graph of 2whe

x x

x

x x

x

x

y y

y

y y

y

a y a

n is positive and when is negative.a a

4

2

-5 5

h x = 2x

g x = 32x

f x = 13

2x

1The graph of 2 lies below 2 and 3

has a y-intercept of 1/3, while 3 2 lies aboveand has a y-intercept of 3. All three graphs havethe same end behavior and same general slope.

x x

x

y y

y

2

-2

-4

-5 5

h x = 2x

g x = -52x f x = -15

2x

1The graph of 2 lies closer to the x-axis 5

than that of 2 so that y approaches the x-axis from below as , and

x

x

y

yx y

If 2 where reflected in the x-axis it would1lie between 2 and 5 2 .5

x

x x

y

y y

Describe the effect of a on the graph of 2when a is positive and when a is negative.

xy

If 0 1 then the graph of 2 lies below

2 while if 1 the graph of 2 lies

above the graph 2 . In either case, the graphhas the same end behavior and the same generalshape.

x

x x

x

a y a

y a y a

y

If 1 0 then the graph 2 lies closer

to the x-axis than the graph 2 but below thex-axis instead of above it.

If 1 the graph of 2 lies below the x-axis but grows away from the x-axis mo

x

x

x

a y a

y

a y a

re

quickly than that of 2xy

2xy a Note the following about the graph .

The graph passes through the point (0, a). The y-intercept is a.The x-axis is an asymptote of the graph.The domain is all real numbers.The range is y > 0 if a > 0 and y < 0 if a<0,

The characteristics of the graph 2 listed on

the previous slide are true of the graph of .

If 1 and 1, the function is an Exponential Growth Function.

x

x

x

y a

y ab

a b y ab