Introduction
It is important to understand the relationship between a
function and the graph of a function. In this lesson, we
will explore how a function and its graph change when a
constant value is added to the function. When a
constant value is added to a function, the graph
undergoes a vertical shift. A vertical shift is a type of
translation that moves the graph up or down depending
on the value added to the function. A translation of a
graph moves the graph either vertically, horizontally, or
both, without changing its shape. A translation is
sometimes called a slide. A translation is a specific type
of transformation.
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3.7.2: Tranformations of Linear and Exponential Functions
Introduction, continued
A transformation moves a graph. Transformations can
include reflections and rotations in addition to
translations. We will also examine translations of graphs
and determine how they are similar or different.
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3.7.2: Tranformations of Linear and Exponential Functions
Key Concepts
• Vertical translations can be performed on linear and
exponential graphs using f(x) + k, where k is the value
of the vertical shift.
• A vertical shift moves the graph up or down k units.
• If k is positive, the graph is translated up k units.
• If k is negative, the graph is translated down k units.
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3.7.2: Tranformations of Linear and Exponential Functions
Key Concepts, continued
•
Translations are one type of transformation.
•
Given the graphs of two functions that are vertical
translations of each other, the value of the vertical
shift, k, can be found by finding the distance between
the y-intercepts.
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3.7.2: Tranformations of Linear and Exponential Functions
Common Errors/Misconceptions
• mistaking vertical shift for horizontal shift
• mistaking a y-intercept for the value of the vertical
translation
• incorrectly graphing linear or exponential functions
• incorrectly combining like terms when changing a
function rule
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3.7.2: Tranformations of Linear and Exponential Functions
Guided Practice
Example 2
Given f(x) = 2x + 1 and
the graph of f(x) to the
right, graph
g(x) = f(x) – 5.
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3.7.2: Tranformations of Linear and Exponential Functions
Guided Practice: Example 2, continued
1. Graph g(x).
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3.7.2: Tranformations of Linear and Exponential Functions
Guided Practice: Example 2, continued
2. How are f(x) and g(x) related?
g(x) is a vertical shift down 5 units of f(x).
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3.7.2: Tranformations of Linear and Exponential Functions
Guided Practice: Example 2, continued
3. What are the steps you need to follow to
graph g(x)?
For each point on f(x), plot a point 5 units lower on
the graph and connect the points.
✔
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3.7.2: Tranformations of Linear and Exponential Functions
Guided Practice: Example 2, continued
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3.7.2: Tranformations of Linear and Exponential Functions
Guided Practice
Example 3
The graphs of two
functions f(x) and
g(x) are shown to the
right. Write a rule for
g(x) in terms of f(x).
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3.7.2: Tranformations of Linear and Exponential Functions
Guided Practice: Example 3, continued
1. Write a function rule for the graph of f(x).
f(x) = –x – 4
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3.7.2: Tranformations of Linear and Exponential Functions
Guided Practice: Example 3, continued
2. Write a function rule for the graph of g(x).
g(x) = –x + 3
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3.7.2: Tranformations of Linear and Exponential Functions
Guided Practice: Example 3, continued
3. How are f(x) and g(x) related?
g(x) is a vertical shift up 7 units from f(x), since the
vertical distance is the distance between the yintercepts (–4 and 3), and 3 – (–4) = 7. You could
also count the units on the graph.
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3.7.2: Tranformations of Linear and Exponential Functions
Guided Practice: Example 3, continued
4. Write a function rule for g(x) in terms of
f(x).
g(x) = f(x) + 7
✔
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3.7.2: Tranformations of Linear and Exponential Functions
Guided Practice: Example 3, continued
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3.7.2: Tranformations of Linear and Exponential Functions