Introduction

1. 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. 3.7.2: Tranformations of Linear and Exponential Functions

2. 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. 3.7.2: Tranformations of Linear and Exponential Functions

3. 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. 3.7.2: Tranformations of Linear and Exponential Functions

4. 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. 3.7.2: Tranformations of Linear and Exponential Functions

5. 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 3.7.2: Tranformations of Linear and Exponential Functions

6. Guided Practice Example 2 Given f(x) = 2x + 1 and the graph of f(x) to the right, graph g(x) = f(x) – 5. 3.7.2: Tranformations of Linear and Exponential Functions

7. Guided Practice: Example 2, continued Graph g(x). 3.7.2: Tranformations of Linear and Exponential Functions

8. Guided Practice: Example 2, continued How are f(x) and g(x) related? g(x) is a vertical shift down 5 units of f(x). 3.7.2: Tranformations of Linear and Exponential Functions

9. Guided Practice: Example 2, continued 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. ✔ 3.7.2: Tranformations of Linear and Exponential Functions

10. Guided Practice: Example 2, continued 10 3.7.2: Tranformations of Linear and Exponential Functions

11. 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). 3.7.2: Tranformations of Linear and Exponential Functions

12. Guided Practice: Example 3, continued Write a function rule for the graph of f(x). f(x) = –x – 4 3.7.2: Tranformations of Linear and Exponential Functions

13. Guided Practice: Example 3, continued Write a function rule for the graph of g(x). g(x) = –x + 3 3.7.2: Tranformations of Linear and Exponential Functions

14. Guided Practice: Example 3, continued 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. 3.7.2: Tranformations of Linear and Exponential Functions

15. Guided Practice: Example 3, continued Write a function rule for g(x) in terms of f(x). g(x) = f(x) + 7 ✔ 3.7.2: Tranformations of Linear and Exponential Functions

16. Guided Practice: Example 3, continued 3.7.2: Tranformations of Linear and Exponential Functions