7-7
Transformations of Logarithmic
Functions
TEKS FOCUS
VOCABULARY
TEKS (5)(A) Determine the effects on the key attributes
on the graphs of f(x) = bx and f(x) = logb(x) where b is
2, 10, and e when f(x) is replaced by af(x), f(x) + d, and
f(x - c) for specific positive and negative real values of
a, c, and d.
TEKS (1)(D) Communicate mathematical ideas,
reasoning, and their implications using multiple
representations, including symbols, diagrams, graphs,
and language as appropriate.
ĚLogarithmic parent function – The simplest example of
a logarithmic function is the logarithmic parent function,
written f(x) = logb(x), where b is a positive real number, b ≠ 1.
ĚImplication – a conclusion that follows from previously
stated ideas or reasoning without being explicitly stated
ĚRepresentation – a way to display or describe information.
You can use a representation to present mathematical
ideas and data.
ESSENTIAL UNDERSTANDING
You can apply the four types of transformations—stretches, compressions,
reflections, and translations—to logarithmic functions.
Concept Summary Logarithmic Function Family
f (x) = log b x, b 7 0, b ≠ 1
Parent function
Translation
y = log b x + d
y = log b (x - c)
d70
shifts up 0 d 0 units
c70
shifts to the right 0 c 0 units
d60
shifts down 0 d 0 units
c60
shifts to the left 0 c 0 units
Stretch, Compression, and Reflection
y = a log b x
304
0a0 7 1
0 6 0a0 6 1
vertical stretch
a60
reflection across the x-axis
Lesson 7-7
vertical compression (shrink)
Transformations of Logarithmic Functions
Problem 1
P
TEKS Process Standard (1)(D)
Analyzing y = af (x) for f (x) = log2 x
Graph each function on the same set of axes as the parent function y = log 2 x.
Describe the effect of the transformation on the parent function. What is the
effect of the transformation on the domain?
1
A y = 3 log 2 x
Multiplying the parent function by 13 will shrink the graph of the function
vertically because 13 6 1.
x log2 x 13 log2 x
Which point do these
functions have in
common?
Each function has the
same x-intercept, (1, 0).
This point does not
change when the y-value
is multiplied by a.
1
2
-1
- 13
1
0
0
2
1
4
2
1
3
2
3
y
y 5 log2 x
2
y = 13 log2 x
1
x
O
1
2
3
4
5
⫺1
The domain remains x 7 0.
B y = −3
~ log2 x
Multiplying the parent function by -3 will stretch the graph of the function
vertically and reflect it across the x-axis.
x
log2 x
1
2
-1
3
1
0
0
2
1
-3
−3 ∙ log2 x
3
y
2
1
(1, 0)
O
y = log2 x
x
2
3
4
5
-1
-2
y = −3 ∙ log2 x
-3
The domain remains x 7 0.
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Problem 2
P
Analyzing y = af (x) for f (x) = log10 x
Graph each function on the same set of axes as the parent function y = log 10 x.
Describe the effect of the transformation on the parent function. What is the
effect of the transformation on the x-intercept?
A y=2
How does a negative
value for a affect
the graph of the
parent function
f(x) = log10 x?
When a is negative, the
graph of the parent
function is reflected
across the x-axis.
~ log10 x
y
The x-intercept remains (1, 0).
y 5 2log10 x
2
Multiplying the parent function by 2 will
stretch the graph of the function vertically
because 2 7 1.
O
⫺2
2
x y 5 log10 x
4
6
8 10
1
y 5 2 2 log10 x
1
B y = − 2 log 10 x
Multiplying the parent function by - 12 will
shrink the graph of the function vertically
and reflect it across the x-axis.
The x-intercept remains (1, 0).
Problem
bl
3
TEKS Process Standard (1)(D)
Analyzing y = f (x) + d for f (x) = log2 x
Graph each function on the same set of axes as the parent function y = log 2 x.
Describe the effect of the transformation on the parent function. What is the
effect of the transformation on the asymptote?
A y = log 2 x + 6
What is the domain of
each function?
Logarithms are not
defined for negative
values. The domain for
each function in this
problem is x 7 0.
Adding 6 to the parent function will preserve the
shape of the function and shift it up 6 units.
The asymptote remains x = 0.
B y = log 2 x − 2
Subtracting 2, or adding -2, to the parent
function will preserve the shape of the function
and shift it down 2 units.
The asymptote remains x = 0.
y
8
6
4
2
O
⫺2
⫺4
306
Lesson 7-7
Transformations of Logarithmic Functions
y 5 log2 x + 6
y 5 log2 x
x
2
3
4
y 5 log2 x − 2
Problem 4
P
Analyzing y = f (x) + d for f (x) = log10 x
Graph each function on the same set of axes as the parent function y = log 10 x.
Describe the effect of the transformation on the parent function. What is the
effect of the transformation on the range?
A y = log 10 x + 3
Adding 3 to the parent function will preserve the shape of the
function and shift it up 3 units.
The range remains the set of all real numbers.
4
y
y 5 log10 x + 3
2
How is the shape of
the graph affected by
a translation?
The shape of the graph
is not changed by a
vertical translation. The
graph of the parent
function is simply shifted
up or down.
O
2
4
y 5 log10 x
x
6
8 10
⫺2
B y = log 10 x − 1
Subtracting 1, or adding -1, to the parent function will preserve the
shape of the function and shift it down 1 unit.
The range remains the set of all real numbers.
4
2
O
⫺2
y
y 5 log10 x
x
2
4
6
8 10
y 5 log10 x − 1
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Problem 5
P
TEKS Process Standard (1)(D)
Analyzing y = f (x − c) for f (x) = log2 x
A Use the table to sketch the graph of the function y = log 2 (x − 4). Identify
the x-intercept and the vertical asymptote of the graph.
How do you choose
the values of x?
Start by completing
the middle column of
the table with values
that you want to use to
evaluate the logarithm.
Add 4 to each value to
find the value of x. Apply
the function to find the
value of y.
x
(x 2 4)
y 5 log2(x 2 4)
4.5
0.5
21
5
1
6
2
1
8
4
2
4
0
y
y 5 log2 (x − 4)
2
O
2
4
6
x
10
8
⫺2
The x-intercept of the graph is (5, 0). The line x = 4 is a vertical asymptote.
B Sketch the graphs of y = log 2 x, y = log 2 (x − 4), and y = log 2 (x + 5) on
the same set of axes. Describe how to use the parent function to graph the
translations. What is the effect of the transformation on the x-intercept for
each function?
What happens to the
vertical asymptote
under a horizontal
translation?
The vertical asymptote
will shift left or right the
same number of units as
the parent function.
4
⫺4
⫺2
O
y y 5 log2 (x + 5)
2
4
6
y 5 log2 x
y 5 log2 (x − 4)
x
8 10
⫺2
Graph the parent function first. To sketch the graph of y = log 2 (x - 4), translate
each point on the graph of the parent function 4 units to the right. To sketch the
graph of y = log 2 (x + 5), translate each point on the graph of the parent function
5 units to the left.
The x-intercept for y = log 2 (x - 4) is now (5, 0) and the x-intercept for
y = log 2 (x + 5) is now ( -4, 0).
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Lesson 7-7
Transformations of Logarithmic Functions
Problem 6
P
Analyzing y = f (x − c) for f (x) = log10 x
Graph each function on the same set of axes as the parent function y = log 10 x.
Describe the effect of the transformation on the parent function. What is the effect
of the transformation on the x-intercept?
A y = log 10 (x − 1)
How do you know
that subtracting
−2 from the parent
function will shift the
graph to the left?
Since -2 6 0, the graph
of f(x) shifts 0 -2 0 , or
2 units to the left.
1
Subtracting 1 from x before applying the
logarithmic function will preserve the shape of
the parent function and shift it right 1 unit.
The x-intercept is now (2, 0) instead of (1, 0).
y
y 5 log10 (x + 2)
0.5
x
⫺2 ⫺1
B y = log 10 (x + 2)
Subtracting -2, or adding 2, to the parent
function will preserve the shape of the function
and shift it left 2 units.
O
1
2
3
4
⫺0.5
y 5 log10 (x − 1)
⫺1
y 5 log10 x
The x-intercept is now ( -1, 0) instead of (1, 0).
Problem
P
bl
7
Analyzing Change in Logarithmic Functions
A small telecommunications company has just created a new division for
satellite television distribution. Monthly profits P for the division, measured
in thousands of dollars, are given by the function
P(x) = −25 + 30 log 2 x
How does the −25
affect the graph of
the parent function
f(x) = log2 x?
Since - 25 6 0, the
graph of f(x) shifts down
25 units.
where
x represents the number of subscribers in hundreds.
w
A Explain how the graph of the profit function is related to the graph of the
parent function y = log 2 x.
The function will be stretched vertically (by a factor of 30) and shifted down 25 units.
B A change in equipment costs causes the company to formulate a new
profit function:
Q(x) = −45 + 30 log 2 x
Describe how the graph of the new function is related to the graph of the
original profit function.
This function will have the same shape as the original profit function, but it will
be shifted down 20 units below P(x).
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HO
ME
RK
O
NLINE
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PRACTICE and APPLICATION EXERCISES
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go to PearsonTEXAS.com.
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Use Multiple Representations to Communicate Mathematical Ideas (1)(D)
The graph of y = log 2 x is shown. Copy the graph, and then sketch each
function on the same set of axes. Describe the effects on the graph of the parent
function. What is the effect of the transformation on the domain and range?
y
4
y 5 log2 x
2
x
O
2
4
6
8
⫺2
⫺4
1. y = -log 2 x
4. y = log 2 x + 3
2. y = 0.25log 2 x
3. y = 2log 2 x
5. y = log 2 x - 1
6. y = log 2 x + 0.5
7. y = log 2 (x + 3)
8. y = log 2 (x - 1)
9. log 2 (x + 0.5)
The graph of y = log 10 x is shown. Copy the graph, and then sketch each
function on the same set of axes. Describe the effects on the graph of the
parent function. What is the effect of the transformation on the asymptote?
2
y
1.5
1
y 5 log10 x
0.5
⫺10 O
10
20
30
x
40
⫺0.5
⫺1
1
2
10. y = 4 log 10 x
13. y = log 10 x - 1.5
11. y = -2log 10 x
14. y = log 10 x + 1
12. y = - 3 log 10 x
15. y = log 10 x - 0.4
16. y = log 10 (x + 2)
17. y = log 10 (x - 5)
18. y = log 10 (x - 0.5)
For each given function, explain the effects of the transformations on the
graph of the parent function.
310
Lesson 7-7
19. y = log 2 (x - 1) + 3
20. y = 4log 2 (x - 1)
22. y = log 10 (x + 5) + 9
23. y = -log 10 (x - 6)
Transformations of Logarithmic Functions
21. y = -2log 2 x + 3
24. y = 0.5log 10 x - 12
Write an equation that models the function described.
25. Shifts the parent function, y = log2 x, 1 unit left
26. Shifts the parent function, y = log10 x, 6 units down
27. Shifts the parent function, y = log2 x, 3 units up
28. Stretches the parent function, y = log2 x, by a factor of 4 and reflects it across
the x-axis
29. Shifts the parent function, y = log10 x, 2 units right
30. Shrinks the parent function, y = log2 x, by a factor of 0.3
31. Apply Mathematics (1)(A) The Italian economist Vilfredo Pareto
(1848–1923) perceived that wealth is not evenly distributed throughout
a country. Most of the wealth is owned by a relatively few members
of the population. Pareto’s Principle is an equation which relates the
level of wealth (the amount of money a person has) to the number of
people in the country that have that much money. For a certain country,
the equation is shown below.
y = 3.9 - 2.5 log10 W
Explain how the constants in this equation transform the graph of the
parent function y = log10 x.
32. Select Techniques to Solve Problems (1)(C) Write a logarithmic function that
has the same shape as the parent function y = log2 x and passes through point
(6, -3).
TEXAS Test Practice
T
Use the graph at the right to answer the following questions.
3
33. Which graph represents the function y = log2 (x + 2)?
A. Graph A
C. Graph C
B. Graph B
D. Graph D
34. Which graph represents the function y = log2 x + 2?
F. Graph A
H. Graph C
G. Graph B
J. Graph D
35. Which graph represents the function y = log2 (x - 2)?
A. Graph A
C. Graph C
B. Graph B
D. Graph D
y
A
2
1
⫺2 ⫺1
O
B
D
C
1
2
3
x
4
⫺1
⫺2
36. A function g(x) shifts the graph of the function y = log10 x down 3 units and to
the left 8 units. It stretches the function by a factor of 4 and reflects it across the
x-axis. Write an equation for g(x).
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