IntroductionIt 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, continuedA 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 2Given 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

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3.7.2: Tranformations of Linear and Exponential Functions

Guided Practice: Example 2, continued

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Guided Practice

Example 3The 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 y-intercepts (–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

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3.7.2: Tranformations of Linear and Exponential Functions

Guided Practice: Example 3, continued