holt algebra 2 7-7 transforming exponential and logarithmic functions 7-7 transforming exponential...

Holt Algebra 2

7-7 Transforming Exponential and Logarithmic Functions7-7 Transforming Exponential and Logarithmic Functions

Holt Algebra 2

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

Holt Algebra 2

7-7 Transforming Exponential and Logarithmic Functions

Warm Up

How does each function compare to its parent function?

vertically stretched by a factor of 2, translated 3 units right, translated 4 units down

1. f(x) = 2(x – 3)2 – 4

reflected across the y-axis, translated 1 unit up

2. g(x) = (–x)3 + 1

Holt Algebra 2


Transform exponential and logarithmic functions by changing parameters.

Describe the effects of changes in the coefficients of exponents and logarithmic functions.

Objectives

Holt Algebra 2


You can perform the same transformations on exponential functions that you performed on polynomials, quadratics, and linear functions.

Holt Algebra 2


Holt Algebra 2


It may help you remember the direction of the shift if you think of “h is for horizontal.”

Helpful Hint

Holt Algebra 2


Make a table of values, and graph g(x) = 2–x + 1. Describe the asymptote. Tell how the graph is transformed from the graph of the function f(x) = 2x.

Example 1: Translating Exponential Functions

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.

Holt Algebra 2


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.

Check It Out! Example 1

x –2 –1 0 1 2

f(x) 1116

1 8

1 4

1 2

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.

Holt Algebra 2


Graph the function. Find y-intercept and the asymptote. Describe how the graph is transformed from the graph of its parent function.

Example 2: Stretching, Compressing, and Reflecting Exponential Functions

A. g(x) = (1.5x)23

parent function: f(x) = 1.5x

asymptote: y = 0

y-intercept: 23

The graph of g(x) is a vertical compression of the parent function f(x) 1.5x by a factor of . 2

3

Holt Algebra 2


Example 2: Stretching, Compressing, and Reflecting Exponential Functions

B. h(x) = e–x + 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 {y|y > 0}.

Holt Algebra 2


Graph the exponential function. Find y-intercept and the asymptote. Describe how the graph is transformed from the graph of its parent function.

h(x) = (5x)13

parent function: f(x) = 5x

asymptote: 0

y-intercept 13

The graph of h(x) is a vertical compression of the parent function f(x) = 5x by a factor of . 13

Check It Out! Example 2a

Holt Algebra 2


g(x) = 2(2–x)

parent function: f(x) = 2x

asymptote: y = 0

y-intercept: 2

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.

Check It Out! Example 2b

Holt Algebra 2


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.

Holt Algebra 2


Examples are given in the table below for f(x) = logx.

Holt Algebra 2


Transformations of ln x work the same way because lnx means logex.

Remember!

Holt Algebra 2


Graph each logarithmic function. Find the asymptote. Describe how the graph is transformed from the graph of its parent function.

Example 3A: Transforming Logarithmic Functions

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.

Holt Algebra 2


Graph each logarithmic function. Find the asymptote. Describe how the graph is transformed from the graph of its parent function.

Example 3B: Transforming Logarithmic Functions

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. D:{x|x < 2}

Holt Algebra 2


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.

Holt Algebra 2


Example 4A: Writing Transformed Functions

Write each transformed function.

Begin with the parent function.f(x) = 4x

f(x) = 4x is reflected across both axes and moved 2 units down.

g(x) = 4–x

g(x) = –4–x

= –(4–x) – 2

To reflect across the y-axis, replace x with –x.

To reflect across the x-axis, multiply the function by –1.

To translate 2 units down, subtract 2 from the function.

Holt Algebra 2


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.

f(x) = ln x is compressed horizontally by a factor of and moved 3 units left.1

2

Holt Algebra 2



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.

Holt Algebra 2


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.

Example 5: Problem-Solving Application

Holt Algebra 2


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.

Example 5 Continued

11 Understand the Problem

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.

Holt Algebra 2


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.

22 Make a Plan

Example 5 Continued

Holt Algebra 2


Rewrite the function, and describe the transformations.

Solve33

T(n) = 75 – 16 ln n

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.

Commutative Property

Example 5 Continued

Holt Algebra 2


Find the number of days the milk will last at 34oF.

Solve33

34 = –16ln n + 75

The model predicts that the milk will last about 13 days.

Substitute 34 for T(n).

–41 = –16ln n

–41–16 = ln n

Subtract 75 from both sides.

Divide by –16.

Change to exponential form.e = n 4116

n ≈ 13

Example 5 Continued

Holt Algebra 2


It is reasonable that milk would last 13 days if kept at 34oF.

Look Back44

Example 5 Continued

Holt Algebra 2


A group of students retake the written portion of a driver’s 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?


Holt Algebra 2


11 Understand the Problem

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.

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.

Check It Out! Example 5 Continued

Holt Algebra 2


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.

22 Make a Plan


Holt Algebra 2


Rewrite the function, and describe the transformations.

Solve33

a(t) = 85 – 15 log(t + 1)

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.

Commutative Propertya(t) = –15 log(t + 1) + 85


Holt Algebra 2


Find the time when the average score drops to 0.

Solve33

Subtract 85 from both sides.

Substitute 0 for a(t).

Divide by –15.

Change to exponential form.

0 = –15 log(t+1) + 85

–85 = –15 log(t + 1)

5.67 = log(t + 1)

105.6667 = t + 1

464,194 ≈ t Change from months to years.38,683 ≈ t


Holt Algebra 2


It is unreasonable that scores would drop to zero 38,683 years after the students take the test without reviewing the material.

Look Back44


Holt Algebra 2


Lesson Quiz: Part I1. 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.

Holt Algebra 2


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)

holt algebra 2 7-7 transforming exponential and logarithmic functions 7-7 transforming exponential...

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