Download Transforming Exponential and Logarithmic Functions

7-7
Transforming Exponential
and Logarithmic Functions
Holt Algebra 2
7-7
Transforming Exponential
and Logarithmic Functions
What is the exponential parent function?
Any base raised to a power is a parent function.
For example, f(x) = 5𝑥 is the parent function for
𝑔 𝑥 = −5𝑥+1 , and f(x) = 𝑒 𝑥 is the parent function for
1
𝑥
2
g(x) = 3𝑒 .
All exponential parent functions have the properties:
• Horizontal asymptote: y = 0
• y-intercept: (0, 1)
• Domain: (-∞, ∞)
• Range: (0, ∞)
Holt Algebra 2
7-7
Transforming Exponential
and Logarithmic Functions
Example 1: Translating Exponential Functions
Given: g(x) = 2–x + 1. Tell how the graph is
transformed from f(x) = 2x. What is the
asymptote? y-intercept? Domain and range?
Reflect over y-axis.
Vertical shift up 1.
Asymptote: y = 1.
y-intercept: (0, 1).
Domain: (-∞, ∞).
Range: (1, ∞).
Holt Algebra 2
7-7
Transforming Exponential
and Logarithmic Functions
Check It Out! Example 1
Given: g(x) = 2x – 2. Tell how the graph is
transformed from f(x) = 2x. What is the
asymptote? y-intercept? Domain and range?
Horizontal shift right 2.
Asymptote: y = 0.
y-intercept: 20-2 = 2-2 = ¼: (0, ¼)
Domain: (-∞, ∞).
Range: (0, ∞).
Holt Algebra 2
7-7
Transforming Exponential
and Logarithmic Functions
Check It Out! Example 1
𝟐
Given: g(x) = (𝟏. 𝟓)𝒙 . Tell how the graph is
𝟑
transformed from its parent function. What is
the asymptote? y-intercept? Domain and range?
Parent function: (1.5)x
Vertical compression by 2/3.
Asymptote: y = 0.
y-intercept: (0, 2/3)
Domain: (-∞, ∞).
Range: (0, ∞).
Holt Algebra 2
7-7
Transforming Exponential
and Logarithmic Functions
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.
g(x) = –4–x
To reflect across the x-axis, multiply the function
by –1.
= –(4–x) – 2 To translate 2 units down, subtract 2 from the
function.
Holt Algebra 2
7-7
Transforming Exponential
and Logarithmic Functions
Example 4A: Writing Transformed Functions
Write each transformed function.
f(x) = ex is shifted up 1 unit, horizontally
compressed by a factor of ½, and vertically
stretched by a factor of 3.
f(x) = ex
Begin with the parent function.
g(x) = e2x
Horizontal compression: 1 = 2.
1
2
g(x) = 3e2x
Vertical stretch: place a 3 in front of the base.
g(x) = 3e2x + 1
To translate 1 unit up, add 1 to the function.
Holt Algebra 2
7-7
Transforming Exponential
and Logarithmic Functions
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Holt Algebra 2