7-7
Transforming Exponential and Logarithmic Functions
LEARNING GOALS – LESSON 7.7
7.7.1: Transform exponential and logarithmic functions by
changing
parameters.
7.7.2: Describe the effects of changes in the coefficients of
exponents and logarithmic functions.
TRANSFORMING EXPONNTIAL FUNCTIONS
Translation:
Vertical Translation:
Rule:
f(x) + k
Example 1:
3 units up: g(x) = 2x+3
6 units down:
Horizontal Translation:
f(x-h)
Vertical Stretch/Comp.
af(x) stretch by 6: g(x) = 6(2x)
comp. by ¼:
Horiz. Stretch/Comp.
f
 
1
b
x
3 units right: g(x) = 2x – 3
1 unit left:
1
5
stretch by 5: g(x)  2
compress by ⅓:

x

Reflection (x-axis) -f(x)
x-axis reflection: g(x) =-2x
(y-axis) f(-x)
y-axis reflection: g(x) = 2x
Example 1: Translating Exponential Functions
g(x) = 2–x + 1. Tell how the graph is transformed from
the graph of the function
f(x) = 2x.
The asymptote is y = _____.
The transformation:
Holt Algebra 2
7-7
Transforming Exponential and Logarithmic Functions
Check It Out! Example 1
f(x) = 2x – 2. Describe the asymptote. Tell how the graph is transformed
from the graph of the function f(x) = 2x.
The asymptote is y = ____.
The transformation moves the graph:
Example 2: Stretching, Compressing, and Reflecting Exponential Functions
Describe how the graph is transformed from
the graph of its parent function.
A. g(x) = ⅔(1.5x)
y-intercept:
parent function: f(x) =
asymptote: y =
B. h(x) = e–x + 1
parent function: f(x) = ex
y-intercept: e
asymptote: y =
CAUTION!
Really is:
h(x) = e(-(x-1))
Holt Algebra 2
DISTRIBUTE!
7-7
Transforming Exponential and Logarithmic Functions
Check It Out! Example 2b
g(x) = 2(2–x)
g
parent function: f(x) =
f
y-intercept:
asymptote: y =
TRANSFORMING LOGARITHMIC FUNCTIONS:
Translation:
Rule:
Vertical Translation:
f(x) + k
Example:
3 units up: g(x) = log(x)+3
6 units down:
Horizontal Translation:
f(x-h)
2 units right: g(x) = log(x-2)
1 unit left:
Vertical Stretch/Comp.
af(x)
stretch by 6: g(x) = 6log(x)
comp. by ¼:
Horiz. Stretch/Comp.

f
 x
1
b
stretch by 5: g(x) = log(⅕x)
compress by ⅓:
Reflection: (x-axis)
-f(x)
x-axis reflection:
(y-axis)
f(-x)
y-axis reflection: g(x) = log (-x)
Holt Algebra 2
g(x) = -log(x)
7-7
Transforming Exponential and Logarithmic Functions
Example 3A: Transforming Logarithmic Functions
Find the asymptote. Describe how the graph is transformed
from the graph of its parent function.
g(x) = 5 log x – 2
asymptote: x =
Example 3B: Transforming Logarithmic Functions
Find the asymptote. Describe how the graph is
transformed from the graph of its parent function.
h(x) = ln(–x + 2)
asymptote: x =
CAUTION!
Really is:
h(x) = ln(-(x-2))
DISTRIBUTE!
Holt Algebra 2
7-7
Transforming Exponential and Logarithmic Functions
Example 4A: Writing Transformed Functions
Write each transformed function.
A. f(x) = 4x is reflected across both axes and move units
down.
B. f(x) = ln x is compressed horizontally by a factor of ½
and moved 3 units left.
CAUTION!
Be sure to apply
horizontal stretches
and compressions last
(enabling you to
distribute!)
C. f(x) = log x is translated 3 units left and stretched
vertically by a factor of 2.
Holt Algebra 2