State the domain and range of each function. 3.1 Graphs of Exponential Functions.

State the domain and range of each function

21)y x

: ; : 0Domain all reals Range y

2)y x

: 0; : 0Domain x Range y

3.1 Graphs of Exponential Functions

Exponential Function

Notice that now the variable is the exponent.

Where base (b), b > 0, b 1, and x is any real number.

( ) xf x b

Graphs of Exponential Equations

To see the basic shape of an exponential function such as

make a table of values and plot points.

( ) 2xf x

x y-2

-1

0123

1

41

21248

Graph f(x) = 2x

Notice the end behavior of the graph. As x → +∞, f(x) → +∞, which means that the graph moves up to the right.

As x → -∞, f(x) → 0, which means that the graph has the line y=0 as an asymptote.

An asymptote is a line that a graph approaches (gets close to) as you move away from the origin.

1Graph (2 ) and 3(2 ).

3

Compare the graphs with the graph of 2

x x

x

y y

y

1Graph (2 ) and 5(2 ).

5

Compare the graphs with the graph of 2

x x

x

y y

y

Describe the effect of on the graph of (2 )

when is positive and when a is negative.

xa y a

a

1(2 )

3xy

3(2 )xy 2xy

1(2 )

5xy

5(2 )xy 2xy

Exponential Growth Function:

The graph passes through the point (0,a)

[the y-intercept].

Have these characteristics…

xy abWhere a > 0 and b > 1

The x-axis is an asymptote of the graph.

The domain is all real numbers.

The range is y > 0 if a > 0

Identify the y-intercept and asymptoteState the domain and range

11) (2 )

3xy

1int : 0,

3y

: 0Asymptote y

:Domain all reals

: 0Range y

2) 3(2 )xy

int : 0,3y

: 0Asymptote y :Domain all reals

: 0Range y

Identify the y-intercept and asymptoteState the domain and range

3) 2xy

int : 0,1y

: 0 or -Asymptote y x axis:Domain all reals

: 0Range y

What do you think would happen to the shift of the graph?

x hy ab k h determines the horizontal shift

k determines the vertical shift

4) 4xy 15) 3xy Domain: All Real Numbers

Range: All Positive Real Numbers

Domain: All Real Numbers

Range: All Positive Real Numbers

Asymptote: y = 0 Asymptote: y = 0

26) 2xy 7) 5 3xy Domain: All Real Numbers

Range: All Positive Real Numbers

Domain: All Real Numbers

Range: y > -3

Asymptote: y = -3Asymptote: y = 0

18) 5 2xy 39) 2 1xy Domain: All Real Numbers

Range: y > 2

Domain: All Real Numbers

Range: y > 1

Asymptote: y = 2 Asymptote: y = 1

Exponential Decay Function

Where a > 0, base b is 0 < b < 1 (b is a fraction), and x is any real number.

( ) xf x ab

Decide whether f(x) is an exponential growth or exponential decay function.

2( ) 5

3

x

f x

3( ) 8

2

x

f x

Because 0 < b < 1 f is an exponential decay function

Because b > 1 f is an exponential growth function

1)

2)

Graphs of Exponential Equations

To see the basic shape of an exponential function such as

make a table of values and plot points.

1( )

2

x

f x

x y

-2-1

01

2

8421

½¼

Graph f(x) = (½)x

-3

State the domain, range, make a table of values and graph the following

2)3/1()( xxf

1)4/3(3)( xxf

where e is an irrational constant whose value is

2.718281828459045...e

The exponential function with base e is called “The Natural Exponential Function”

( ) xy f x e

The Number e

The Natural Exponential Function

Homework:Page 185 #7-13 odd, 15-18, 29-37 odd