4-7
Transforming Exponential
and Logarithmic Functions
CC.9-12.F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) … find the value of k given
the graphs. … illustrate … using technology. Also CC.9-12.A.CED.2, CC.9-12.A.CED.3, CC.9-12.F.IF.5
Who uses this?
Psychologists can use transformations of
exponential functions to describe knowledge
retention rates over time. (See Example 5.)
Objectives
Transform exponential
and logarithmic functions
by changing parameters.
Describe the effects
of changes in the
coefficients of
exponential and
logarithmic functions.
You can perform the same transformations
on exponential functions that you performed on
polynomial, quadratic, and linear functions.
The hippocampus, in orange,
directs the storage of memory
in the brain.
Transformations of Exponential Functions
Transformation
f(x) Notation
y=2
2 units right
y=2
x+1
1 unit left
y = 6(2 )
y = 1 (2 x)
2
x
Vertical stretch
or compression
af(x)
Horizontal stretch
or compression
1x
f (_
)
y=2
-f(x)
y = -2 x
1
_
y=2
b
Reflection
EXAMPLE
6 units down
x-2
y=2 -6
f(x - h)
Horizontal translation
3 units up
x
y=2 +3
f(x) + k
Vertical translation
V
It may help you
remember the
direction of the shift
if you think of “h is
for horizontal.”
Examples
x
f(-x)
y=2
(_15 x)
3x
-x
stretch by 6
_
compression by 1
2
stretch by 5
_
compression by 1
3
across x-axis
across y-axis
Translating Exponential Functions
Make a table of values, and graph the function g(x) = 2 x - 4. Describe the
asymptote. Tell how the graph is transformed from the graph of f (x) = 2 x.
x
-2
-1
0
1
2
3
g(x)
-3.75
-3.5
-3
-2
0
4
doc-stock/Alamy
The asymptote is y = -4, and the graph
approaches this line as the value of
x decreases. The transformation moves
the graph of f (x) = 2 x down 4 units. The
⎧
⎫
range changes to ⎨ y ⎪ y > -4 ⎬.
⎩
⎭
4
(2, 4)
2
(0, 1)
-4
-2
y
f
(2, 0) x
0
(0, -3)
2
1. Make a table of values, and graph j(x) = 2 x - 2. Describe the
asymptote. Tell how the graph is transformed from the graph
of f (x) = 2 x.
4- 7 Transforming Exponential and Logarithmic Functions
CC13_A2_MESE647074_C04L07.indd 281
4
g
281
5/4/11 3:31:52 PM