1. 7-7 Warm Up Lesson Presentation Lesson Quiz Transforming Exponential
and Logarithmic Functions
7-7
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2

2. How does each function compare to its parent function?
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

3. Objectives
Transform exponential and logarithmic functions by changing parameters.
Describe the effects of changes in the coefficients of exponents and logarithmic functions.

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

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

7. Example 1: Translating Exponential Functions
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. x –3 –2 –1 1 2
g(x) 9 5 3
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.

8. 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 1 2
f(x) 1 16 8 4 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.

9. 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.
A. g(x) = (1.5x) 2 3
The graph of g(x) is a vertical compression of the parent function f(x) 1.5x by a factor of 2 3
parent function: f(x) = 1.5x
y-intercept: 2 3
asymptote: y = 0

10. 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}.

11. 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.
h(x) = (5x) 1 3
The graph of h(x) is a vertical compression of the parent function f(x) = 5x by a factor of 1 3
parent function: f(x) = 5x
y-intercept 1 3
asymptote: 0

12. Check It Out! Example 2b
g(x) = 2(2–x)
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.

13. 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.

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

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

16. 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.

17. 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. D:{x|x < 2}

18. 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.

19. 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) = 4–x
To reflect across the y-axis, replace x with –x.
To reflect across the x-axis, multiply the function by –1.
g(x) = –4–x
= –(4–x) – 2
To translate 2 units down, subtract 2 from the function.

20. Example 4B: Writing Transformed Functions
f(x) = ln x is compressed horizontally by a factor of and moved 3 units left. 1 2
g(x) = ln2(x + 3)
When you write a transformed function, you may want to graph it as a check.

21. 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.

22. 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.

23. Understand the Problem
Example 5 Continued 1
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.
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.

24. Example 5 Continued 2
Make a Plan
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.

25. Example 5 Continued
Solve 3
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.

26. Find the number of days the milk will last at 34oF.
Example 5 Continued
Solve 3
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.
–41
–16
= ln n
Divide by –16.
e = n 41 16
Change to exponential form.
n ≈ 13
The model predicts that the milk will last about 13 days.

27. Example 5 Continued
Look Back 4
It is reasonable that milk would last 13 days if kept at 34oF.

28. Check It Out! Example 5
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?

29. Understand the Problem
Check It Out! Example 5 Continued 1
Understand the Problem
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.

30. Check It Out! Example 5 Continued2
Make a Plan
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.

31. Check It Out! Example 5 Continued
Solve 3
Rewrite the function, and describe the transformations.
a(t) = 85 – 15 log(t + 1)
a(t) = –15 log(t + 1) + 85
Commutative Property
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.

32. Check It Out! Example 5 Continued
Solve 3
Find the time when the average score drops to 0.
0 = –15 log(t+1) + 85
Substitute 0 for a(t).
–85 = –15 log(t + 1)
Subtract 85 from both sides.
5.67 = log(t + 1)
Divide by –15.
= t + 1
Change to exponential form.
464,194 ≈ t
Change from months to years.
38,683 ≈ t

33. Check It Out! Example 5 Continued
Look Back 4
It is unreasonable that scores would drop to zero 38,683 years after the students take the test without reviewing the material.

34. 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.

35. 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)