Exponential Growth Functions

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Exponential Growth Functions

• How do we graph exponential growth functions?

M2 Unit 5a Day 5
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Exponential Growth Function
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ExampleGraph the function
x -1 0 1
y
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 havent been translated
4
Translations in Exponential Functions.

• 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?
5
You Try
The graph of is translated
left 3 units and down 1 unit. What is the
equation of the translation?
6
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 yk when your function is in the
  form .

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Example What are the asymptote and y-intercept
for the function on the left?
Since this is a parent function, its asymptote
is y 0.
The y-intercept is always a in a parent
function, so the y-intercept is 1.
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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

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Example Graph y 3x. Analyze the graph.

x -1 0 1
y
1/3
1
3
1
1
The DOMAIN is all real numbers and the RANGE is y
gt 0
Parent functions of exponential growth function
have a horizontal asymptote at y 0.
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Example Graph .
Find the asymptote, domain, and range.
Start by sketching the graph of
. This is the parent function.
x -1 0 1
y
2
6
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 gt -2
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Graph the function . Find
the asymptote, domain, and range.
You Try
x -1 0 1
y
3/2
1/24
1/4
The DOMAIN is all real numbers and the RANGE is y
gt 0
Parent functions have a horizontal asymptote at
y 0.
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Reflections of Exponential Functions

• Sometimes, you may see an exponential growth
  function in which alt0. When this occurs, the
  graph is reflected over the x-axis.

Example Graph
x -1 0 1
y
-1/4
-1
-4
Down 3
x -1 0 1
y
-3 ¼
-4
-7
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End Behavior

• After you graph your function, decide what it is
  doing as x goes to -8(to the left) and 8(to the
  right)
• Ex Describe the end behavior

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Describe the end behavior of the following graph.

• a. As x ? - 8, f(x) ? 0 as x ? 8, f(x) ? - 8
• b. As x ? - 8, f(x) ? 0 as x ? 8, f(x) ? 8
• c. As x ? - 8, f(x) ? 8 as x ? 8, f(x) ? 0
• d. As x ? - 8, f(x) ? - 8 as x ? 8, f(x) ? 8

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average rate of change

• Average Rate of Change the slope
• The most steep part of the graph has the highest
  rate of change (ROC)

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ROC

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

1. Between -6 and -4
2. Between -4 and -2
3. Between -2 and 0
4. Between 0 and 2

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ROC

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

1. Between 6 and 8
2. Between 4 and 6
3. Between 2 and 4
4. Between 0 and 2

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Graph
x 0 1 2
y
x -1 0 1
y

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

y 1
2
All Real s
y gt 1
Right 1, Up 1
As x ? - 8, f(x) ? 1 as x ? 8, f(x) ? 8
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Graph and analyze the function.

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

As x ? - 8, f(x) ? ___ as x ? 8, f(x) ?__
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Graph and analyze the function.

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

As x ? - 8, f(x) ? ___ as x ? 8, f(x) ?__
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• HW pg 122 1-3
• On pg. 1225, 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 ? - 8, f(x) ? ___ as x ? 8, f(x) ?__