+

Exponential Growth FunctionsHow do we graph exponential

growth functions?M2 Unit 5a: Day 5

+Exponential Growth Function:

A function of the form y abx where a>0 and b>1

+Example:Graph the function

y 5x

1

1

x -1 0 1

y

15

1 5

Draw a smooth curve from left to right just above the x-axis that moves up and to the right.

We will call functions like this “parent functions” because they haven’t been translated

Translations in Exponential Functions….

yabx h k•h moves the function to the right or left•k moves the function up or down

Example: The graph of is translated up 3 units.

What is the equation of the translation?

Example: The graph of is translated left 2 units

and down 5 units. What is the equation of the

translation?

y5x

y4(2x )

y5x 3

y4(2x2 ) 5

The graph of is translated

left 3 units and down 1 unit. What is the

equation of the translation?

34

2

x

yæö÷ç= × ÷ç ÷çè ø

33

4 12

x

y+æö÷ç= -÷ç ÷çè ø

You Try:

+Asymptotes and y-intercepts

y-intercept:

To find the y-intercept, plug in zero for x. In an exponential growth function, your y-intercept is always a when your function is in the form

Asymptote:

a line that a graph approaches more and more closely. Exponential functions have a horizontal asymptote at y=k when your function is in the form .

.xy ab=

x hy ab k-= +

+Example: What are the asymptote and y-intercept for the function on the left?

Since this is a “parent function”, it’s asymptote is y = 0.

The y-intercept is always a in a “parent function”, so the y-intercept is 1.

+Domain and Range

The domain of an exponential function will always be ALL REAL NUMBERS

The range of an exponential function will depend on where the asymptote is

+Example: Graph y = 3x. Analyze the graph.

“Parent functions” of exponential growth function have a horizontal asymptote at y = 0.

The DOMAIN is all real numbers and the RANGE is y > 0

x -1 0 1

y 1/3 1 31

1

Example: Graph . Find the asymptote, domain, and range.

y23x 2 2

Start by sketching the graph of .

This is the “parent function”.

y23x

Then, translate the graph __________ 2 units and __________ 2 units.

rightdown

This graph has an asymptote at the line ______________.

The domain is ____________ and the range is ________________.

y = -2

All Real #s y > -2

x -1 0 1

y 2 6

23

+Graph the function . Find the asymptote, domain, and range.

16

4xy = ×

x -1 0 1

y 1/4 3/21/24

The DOMAIN is all real numbers and the RANGE is y > 0

“Parent functions” have a horizontal asymptote at y = 0.

You Try:

+Reflections of Exponential FunctionsSometimes, you may see an exponential

growth function in which a<0. When this occurs, the graph is reflected over the x-axis.Example:

Graph 4 3xy =- -x -1 0 1

y Dow

n 3

x -1 0 1

y

-1/4 -1 -4

-3 ¼ -4 -7

+End BehaviorAfter you graph your function, decide what it is doing as x goes to -∞(to the left) and ∞(to the right)

Ex: Describe the end behavior

As , ( ) and as x - , ) 3(x f x f x® ¥ ® ® ¥ ®¥As , ( ) __ and as x - , ( ) __x f x f x® ¥ ® ® ¥ ®

+

a. As x - ∞, f(x) 0; as x ∞, f(x) - ∞

b. As x - ∞, f(x) 0; as x ∞, f(x) ∞

c. As x - ∞, f(x) ∞; as x ∞, f(x) 0

d. As x - ∞, f(x) - ∞; as x ∞, f(x) ∞

Describe the end behavior of the following graph.

+average rate of change

Average Rate of Change – the “slope”

The most steep part of the graph has the highest rate of change (ROC)

2 1

2 1

y yx x

--

+ROC

Where would the rate of change be highest for this function:

A. Between -6 and -4

B. Between -4 and -2

C. Between -2 and 0

D. Between 0 and 2

+ROC

Where would the rate of change be highest for this function:

A. Between 6 and 8

B. Between 4 and 6

C. Between 2 and 4

D. Between 0 and 2

+

Asymptote:

Y-intercept:

Domain:

Range:

Describe the translation:

Describe the End Behavior:

-æö÷ç ÷= +ç ÷ç ÷çè ø

132 12xy

y = 1

All Real #s

x -1 0 1

y

x 0 1 2

y1

13

2 31

23

3 4

y > 1

As x - ∞, f(x) 1; as x ∞, f(x) ∞

Graph

Right 1, Up 1

2

+Graph and analyze the function.

Asymptote:

Y-intercept:

Domain:

Range:

Describe the translation:

Describe the End Behavior:

12 3xy +=- -

As x - ∞, f(x) ___; as x ∞, f(x) __

+Graph and analyze the function.

Asymptote

Y-intercept

Domain:

Range:

Describe the translation:

Describe the End Behavior:

53 3xy -= ×

As x - ∞, f(x) ___; as x ∞, f(x) __

+ HW: pg 122 #1-3 On pg. 122#5, 7 and pg 123 #1, 2, 4

1. Graph each function

2. Describe the translation

3. Find the asymptote

4. Find the y-int

5. Find the domain and range

6. Describe the end behavior

As x - ∞, f(x) ___; as x ∞, f(x) __