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Exponential Growth Functions

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Exponential Growth Functions

How do we graph exponential growth functions?

M2 Unit 5a: Day 5

5

1

1

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

1

We will call functions like this “parent functions”

because they haven’t been translated

- 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?

You Try:

The graph of is translated left 3 units and down 1 unit. What is the equation of the translation?

y-intercept:

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

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.

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

1/3

3

1

1

1

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

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

Example: Graph .

Find the asymptote, domain, and range.

Start by sketching the graph of .

This is the “parent function”.

6

2

Then, translate the graph __________ 2 units

and __________ 2 units.

right

down

This graph has an asymptote at the line ______________.

The domain is ____________ and the range is ________________.

y = -2

All Real #s

y > -2

You Try:

1/24

3/2

1/4

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

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

Sometimes, you may see an exponential growth function in which a<0. When this occurs, the graph is reflected over the x-axis.

Example:

Graph

-1/4

-1

-4

Down 3

-3 ¼

-4

-7

End Behavior

- After 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

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) ∞

Average Rate of Change – the “slope”

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

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

- Between -6 and -4
- Between -4 and -2
- Between -2 and 0
- Between 0 and 2

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

- Between 6 and 8
- Between 4 and 6
- Between 2 and 4
- Between 0 and 2

- Asymptote:
- Y-intercept:
- Domain:
- Range:
- Describe the translation:
- Describe the End Behavior:

y = 1

2

All Real #s

y > 1

Right 1, Up 1

As x - ∞, f(x) 1; as x ∞, f(x) ∞

Asymptote:

Y-intercept:

Domain:

Range:

Describe the translation:

Describe the End Behavior:

As x - ∞, f(x) ___; as x ∞, f(x) __

Asymptote

Y-intercept

Domain:

Range:

Describe the translation:

Describe the End Behavior:

As x - ∞, f(x) ___; as x ∞, f(x) __

HW: pg 122 #1-3

On pg. 122#5, 7 and pg 123 #1, 2, 4

Graph each function

Describe the translation

Find the asymptote

Find the y-int

Find the domain and range

Describe the end behavior

As x - ∞, f(x) ___; as x ∞, f(x) __