SLT 25 Identify the effect of
transformations on exponential
& logarithmic graphs.
MCPS Math
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Printed: June 8, 2015
AUTHOR
MCPS Math
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Chapter 1. SLT 25 Identify the effect of transformations on exponential & logarithmic graphs.
C HAPTER
1
SLT 25 Identify the effect of
transformations on exponential &
logarithmic graphs.
Transformations of the graphs of exponential and logarithmic functions will be explored in this SLT.
Watch This
https://learnzillion.com/lessons/3335-shift-exponential-and-logarithmic-functions
https://learnzillion.com/lessons/3342-stretch-and-reflect-exponential-and-logarithmic-functions
Guidance
Graphs of Exponential Functions
Let’s consider the graph of f (x) = 2x . The graph below shows this function, with several points marked in blue.
FIGURE 1.1
Notice that as x approaches ∞ , the function grows without bound. However, if x approaches −∞ , the function values
get closer and closer to 0. Therefore the function is asymptotic to the x- axis. This is the graphical result of the fact
that the range of the function is limited to positive y values.
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Graphing Exponential Functions Using Transformations
From your prior studies of function transformations , you should recognize the graph of g(x) = 2x + 3 as a vertical
shift of the graph of f (x) = 2x . In general, we can produce a graph of an exponential function with base 2 if we
analyze the equation of the function in terms oftransformations . The table below summarizes the different kinds of
transformations of f (x) = 2x .
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Chapter 1. SLT 25 Identify the effect of transformations on exponential & logarithmic graphs.
Example A
Use transformations to graph the function a(x) = 3x + 2
Solution:
a(x) = 3x + 2
This graph represents a shift of y = 3x two units to the left. The graph below shows this relationship between the
graphs of these two functions:
FIGURE 1.2
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Example B
Use transformations to graph the function b(x) = -3x + 4
Solution:
b(x) = -3x + 4
This graph represents a reflection over the y- axis and a vertical shift of 4 units. You can produce a graph of b(x)
using three steps: sketch y = 3x , reflect the graph over the x- axis, and then shift the graph up 4 units. The graph
below shows this process:
FIGURE 1.3
While you can always quickly create a graph using a graphing utility, using transformations will allow you to sketch
a graph relatively quickly on your own. If we start with a parent function such as y = 3x , you can quickly plot several
points: (0, 1), (2, 9), (-1, 1/3), etc. Then you can transform the graph, as we did in the previous example.
Notice that when we sketch a graph, we choose x values, and then use the equation to find yvalues.
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Chapter 1. SLT 25 Identify the effect of transformations on exponential & logarithmic graphs.
Graphs of Logarithmic Functions
In a previous lesson, log functions were identified as the inverses of exponential functions, in this lesson we explore
that fact visually through the graphs of logarithmic functions.
As you can see below, because the function f (x) = log2 x is the inverse of the function g(x) = 2x , the graphs of these
functions are reflections over the line y = x.
We can verify that the functions are inverses by looking at the graph. For example, the graph of g(x) = 2x contains
the point (1, 2), while the graph of f (x) = log2 x contains the point (2, 1).
Also, note that while that the graph of g(x) = 2x is asymptotic to the x-axis, the graph of f (x) = log2 x is asymptotic to
the y-axis. This behavior of the graphs gives us a visual interpretation of the restricted range of g and the restricted
domain of f.
Graphing Logarithmic Functions Using Transformations
Consider again the log function f (x) = log2 x. The table below summarizes how we can use the graph of this function
to graph other related functions.
TABLE 1.1:
Equation
g(x) = log2 (x - a), for a >0
g(x) = log2 (x+a) for a >0
g(x) = log2 (x) + a for a >0
g(x) = log2 (x) - a for a >0
g(x) = alog2 (x) for a >0
Relationship to f (x) = log2 x
Obtain a graph of g by shifting the
graph of f a units to the right.
Obtain a graph of g by shifting the
graph of f a units to the left.
Obtain a graph of g by shifting the
graph of f up a units.
Obtain a graph of g by shifting the
graph of f down a units.
Obtain a graph of g by vertically
stretching the graph of f by a factor
of a.
Domain
x >a
x >-a
x >0
x >0
x >0
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TABLE 1.1: (continued)
Equation
g(x) = -alog2 (x) , for a >0
g(x) = log2 (-x)
Relationship to f (x) = log2 x
Obtain a graph of g by vertically
stretching the graph of f by a factor
of a,
and by reflecting the graph over the
x-axis.
Obtain a graph of g by reflecting the
graph of f over the y-axis.
Domain
x >0
x <0
Example C
Graph the functions:
f (x) = log2 (x)
g(x) = log2 (x) + 3
h(x) = log2 (x + 3)
Solution:
The graph below shows these three functions together:
Notice that the location of the 3 in the equation makes a difference! When the 3 is added to log2 x , the shift is
vertical. When the 3 is added to the x, the shift is horizontal. It is also important to remember that adding 3 to the x
is a horizontal shift to the left. This makes sense if you consider the function value when x = -3:
h(-3) = log2 (-3 + 3) = log2 0 = undefined
This is the vertical asymptote!
Note that in order to graph these functions, we evaluated them by investigating specific values of x. If we want to
know what the x value is for a particular y value, we need to solve a logarithmic equation.
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Chapter 1. SLT 25 Identify the effect of transformations on exponential & logarithmic graphs.
Example D
Graph the function y = log2 (x − 3)
Solution:
Start by making a table:
TABLE 1.2:
x
3
4
5
6
7
8
y
undefined
0
1
1.585
2
2.32
Since 20 = 1 that means x − 3 = 1 and x = 4
That means that when y = 0, x = 4
Since 21 = 2 that means x − 3 = 2 and x = 5
That means that when y = 1, x = 5
The graph looks like:
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Guided Practice
1) Which of the following functions is graphed in the image below?
a) y = log4 x
b) y = −log4 x
c) y = log4 (−x)
2) Graph the function y = log3 x
3) Transform the graph of y = log3 x from problem #2 into the graph of y = log3 (x + 3)
Answers
1) All three functions are varieties of log4 x
The image shows the function reflected across the y axis, therefore:
c) y = log4 (−x) is correct.
2) To graph y = log3 x we will start with a table of values:
TABLE 1.3:
x
3
9
27
81
y
1
2
3
4
Plotting those points and drawing a smooth curve between them gives:
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Chapter 1. SLT 25 Identify the effect of transformations on exponential & logarithmic graphs.
3) The graph of y = log3 x above ran up the positive side of the y-axis to reach the x-axis.
The "+3" inside the parentheses of y = log3 (x + 3) means there is a shift of 3 to the left.
The image of y = log3 x shifted 3 units to the left looks like this:
Explore More
Identify the domain and range, then sketch the graph.
1. y = log2 (x + 1) − 5
2. y = log2 (x − 1) + 3
3. y = −log2 (−x)
Sketch the graph of each function.
4. y = 2 · 3x
5. y = 4 · 21
2
6. y = 2 · 12
x+1
+2
Use graphing transformation rules to make a conjecture about what the graph of each function will look like.
7.
8.
9.
10.
f (x) = 3x−4
f (x) = −4x
f (x) = 3x − 2
f (x) = −5x+2
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11. f (x) = 5x−4 − 3
Describe the transformations applied to the parent graph f (x) = nx to obtain the graph of each function.
12.
13.
14.
15.
g(x) = 13 (2x )
m(x) = −23x
s(x) = 2x−5
t(x) = −3(2)x−5 − 4
Look at the graphs below and identify the function that the graph represents from the functions listed below.
a)
b)
c)
d)
e)
16.
17.
18.
10
f (x) = −log10 x
f (x) = log10 x
f (x) = −log10 (−2x)
f (x) = log10 (−3x)
f (x) = log10 x + 3
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Chapter 1. SLT 25 Identify the effect of transformations on exponential & logarithmic graphs.
19.
20.
Graph the following logarithmic functions.
21. y = log10 (2x)
22. y = log1/2 (x + 2)
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23. y = log3 (2x + 2)
24. The table below shows the x and y values of the points on an exponential curve. Switch them and identify the
corresponding coordinates of the points that would appear on the logarithmic curve. Can you identify the function?
TABLE 1.4:
Point on exponential curve
(-3, 1/8)
(-2, 1/4)
(-1, 1/2)
(0, 1)
(1, 2)
(2, 4)
(3, 8)
Corresponding point on logarithmic curve
(1/8, -3)
14. (__ , __)
15. (__ , __)
16. (__ , __)
17. (__ , __)
18. (__ , __)
19. (__ , __)
Graph logarithmic functions, using the inverse of the related exponential function. Then graph the pair of functions
on the same axes.
25. y = log3 x
26. y = log5 x
12