Transforming Exponential

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Transforming Exponential and Logarithmic
Functions
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Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
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Warm Up How does each function compare to its
parent function?
1. f(x) 2(x 3)2 4
vertically stretched by a factor of 2, translated
3 units right, translated 4 units down
2. g(x) (x)3 1
reflected across the y-axis, translated 1 unit up
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Objectives
Transform exponential and logarithmic functions
by changing parameters. Describe the effects of
changes in the coefficients of exponents and
logarithmic functions.
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You can perform the same transformations on
exponential functions that you performed on
polynomials, quadratics, and linear functions.
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Example 1 Translating Exponential Functions
Make a table of values, and graph g(x) 2x 1.
Describe the asymptote. Tell how the graph is
transformed from the graph of the function f(x)
2x.
x 3 2 1 0 1 2
g(x) 9 5 3 2 1.5 1.25
The asymptote is y 1, and the graph approaches
this line as the value of x increases. The
transformation reflects the graph across the
y-axis and moves the graph 1 unit up.
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Check It Out! Example 1
Make a table of values, and graph f(x) 2x 2.
Describe the asymptote. Tell how the graph is
transformed from the graph of the function f(x)
2x.
x 2 1 0 1 2
f(x) 1
The asymptote is y 0, and the graph approaches
this line as the value of x decreases. The
transformation moves the graph 2 units right.
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Example 2 Stretching, Compressing, and
Reflecting Exponential Functions
Graph the function. Find y-intercept and the
asymptote. Describe how the graph is transformed
from the graph of its parent function.
parent function f(x) 1.5x
asymptote y 0
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Example 2 Stretching, Compressing, and
Reflecting Exponential Functions
B. h(x) ex 1
parent function f(x) ex
y-intercept e
asymptote y 0
The graph of h(x) is a reflection of the parent
function f(x) ex across the y-axis and a shift
of 1 unit to the right. The range is yy gt 0.
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Check It Out! Example 2a
Graph the exponential function. Find y-intercept
and the asymptote. Describe how the graph is
transformed from the graph of its parent function.
parent function f(x) 5x
asymptote 0
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Check It Out! Example 2b
g(x) 2(2x)
parent function f(x) 2x
y-intercept 2
asymptote y 0
The graph of g(x) is a reflection of the parent
function f(x) 2x across the y-axis and vertical
stretch by a factor of 2.
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Because a log is an exponent, transformations of
logarithm functions are similar to
transformations of exponential functions. You can
stretch, reflect, and translate the graph of the
parent logarithmic function f(x) logbx.
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Examples are given in the table below for f(x)
logx.
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Example 3A Transforming Logarithmic Functions
Graph each logarithmic function. Find the
asymptote. Describe how the graph is transformed
from the graph of its parent function.
g(x) 5 log x 2
asymptote x 0
The graph of g(x) is a vertical stretch of the
parent function f(x) log x by a factor of 5 and
a translation 2 units down.
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Example 3B Transforming Logarithmic Functions
Graph each logarithmic function. Find the
asymptote. Describe how the graph is transformed
from the graph of its parent function.
h(x) ln(x 2)
asymptote x 2
The graph of h(x) is a reflection of the parent
function f(x) ln x across the y-axis and a
shift of 2 units to the right. Dxx lt 2
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Check It Out! Example 3
Graph the logarithmic p(x) ln(x 1) 2.
Find the asymptote. Then describe how the graph
is transformed from the graph of its parent
function.
asymptote x 1
The graph of p(x) is a reflection of the parent
function f(x) ln x across the x-axis 1 unit
left and a shift of 2 units down.
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Example 4A Writing Transformed Functions
Write each transformed function.
f(x) 4x is reflected across both axes and moved
2 units down.
f(x) 4x
Begin with the parent function.
g(x) 4x
To reflect across the y-axis, replace x with x.
To reflect across the x-axis, multiply the
function by 1.
g(x) 4x
(4x) 2
To translate 2 units down, subtract 2 from the
function.
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Example 4B Writing Transformed Functions
g(x) ln2(x 3)
When you write a transformed function, you may
want to graph it as a check.
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Check It Out! Example 4
Write the transformed function when f(x) log x
is translated 3 units left and stretched
vertically by a factor of 2.
g(x) 2 log(x 3)
When you write a transformed function, you may
want to graph it as a check.
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Example 5 Problem-Solving Application
The temperature in oF that milk must be kept at
to last n days can be modeled by T(n) 75 16
ln n. Describe how the model is transformed from
f(n) ln n. Use the model to predict how long
milk will last if kept at 34oF.
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Example 5 Continued
The answers will be the description of the
transformations in T(n) 75 16ln n and the
number of days the milk will last if kept at 34oF.

• List the important information
• The model is the function T(n) 75 16ln n.
• The function is a transformation of f(n) ln n.
• The problem asks for n when T is 34.

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Example 5 Continued
Rewrite the function in a more familiar form, and
then use what you know about the effect of
changing the parent function to describe the
transformations. Substitute known values into
T(n) 75 16ln n, and solve for the unknown.
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Example 5 Continued
Rewrite the function, and describe the
transformations.
T(n) 75 16 ln n
Commutative Property
T(n) 16 ln n 75
The graph of f(n) ln n is reflected across the
x-axis, vertically stretched by a factor of 16,
and translated 75 units up.
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Example 5 Continued
Find the number of days the milk will last at
34oF.
34 16ln n 75
Substitute 34 for T(n).
41 16ln n
Subtract 75 from both sides.
Divide by 16.
Change to exponential form.
n 13
The model predicts that the milk will last about
13 days.
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Example 5 Continued
It is reasonable that milk would last 13 days if
kept at 34oF.
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Check It Out! Example 5
A group of students retake the written portion of
a drivers test after several months without
reviewing the material. A model used by
psychologists describes retention of the material
by the function a(t) 85 15log(t 1), where a
is the average score at time t (in months).
Describe how the model is transformed from its
parent function. When would the average score
drop below 0. Is your answer reasonable?
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Check It Out! Example 5 Continued
The answers will be the description of the
transformations in a(t) 85 15log(t 1) and
the number of months when the score falls below 0.

• List the important information
• The model is the function a(t) 85 15log (t
  1).
• The function is a transformation of f(t)
  log(t).
• The problem asks for t when t 0.

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Check It Out! Example 5 Continued
Rewrite the function in a more familiar form, and
then use what you know about the effect of
changing the parent function to describe the
transformations. Substitute known values into
a(t) 85 15 log(t 1), and solve for the
unknown.
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Check It Out! Example 5 Continued
Rewrite the function, and describe the
transformations.
a(t) 85 15 log(t 1)
Commutative Property
a(t) 15 log(t 1) 85
The graph of f(t) ln n is reflected across the
x-axis, vertically stretched by a factor of 15,
and translated 85 units up and 1 unit left.
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Check It Out! Example 5 Continued
Find the time when the average score drops to 0.
Substitute 0 for a(t).
0 15 log(t1) 85
85 15 log(t 1)
Subtract 85 from both sides.
5.67 log(t 1)
Divide by 15.
Change to exponential form.
105.6667 t 1
464,194 t
Change from months to years.
38,683 t
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Check It Out! Example 5 Continued
It is unreasonable that scores would drop to zero
38,683 years after the students take the test
without reviewing the material.
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Lesson Quiz Part I
1. Graph g(x) 20.25x 1. Find the asymptote.
Describe how the graph is transformed from the
graph of its parent function.
y 1 the graph of g(x) is a horizontal stretch
of f(x) 2x by a factor of 4 and a shift of 1
unit down.
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Lesson Quiz Part II
2. Write the transformed function f(x) ln x is
stretched by a factor of 3, reflected across the
x-axis, and shifted by 2 units left.
g(x) 3 ln(x 2)