1. Introduction
You can change a function’s position or shape by adding or multiplying a constant to that function. This is called a transformation. When adding a constant, you can transform a function in two distinct ways. The first is a transformation on the independent variable of the function; that is, given a function f(x), we add some constant k to x: f(x) becomes f(x + k). The second is a transformation on the dependent variable; given a function f(x), we add some constant k to f(x): f(x) becomes f(x) + k.
5.8.1: Replacing f(x) with f(x) + k and f(x + k)

2. Introduction, continued
In this lesson, we consider the transformation on a function by a constant k, either when k is added to the independent variable, x, or when k is added to the dependent variable, f(x). Given f(x) and a constant k, we will observe the transformations f(x) + k and f(x + k), and examine how transformations affect the vertex of a quadratic equation.
5.8.1: Replacing f(x) with f(x) + k and f(x + k)

3. Key Concepts
To determine the effects of the constant on a graph, compare the vertex of the original function to the vertex of the transformed function.
Neither f(x + k) nor f(x) + k will change the shape of the function so long as k is a constant.
Transformations that do not change the shape or size of the function but move it horizontally and/or vertically are called translations.
Translations are performed by adding a constant to the independent or dependent variable.
5.8.1: Replacing f(x) with f(x) + k and f(x + k)

4. Key Concepts, continued
Vertical Translations—Adding a Constant to the Dependent Variable, f(x) + k
f(x) + k moves the graph of the function k units up or down depending on whether k is greater than or less than 0.
If k is positive in f(x) + k, the graph of the function will be moved up.
If k is negative in f(x) + k, the graph of the function will be moved down.
5.8.1: Replacing f(x) with f(x) + k and f(x + k)

5. Key Concepts, continued
Vertical translations: f(x) + k
When k is positive, k > 0, the graph moves up:
When k is negative, k < 0, the graph moves down:
5.8.1: Replacing f(x) with f(x) + k and f(x + k)

6. Key Concepts, continued
Horizontal Translations—Adding a Constant to the Independent Variable, f (x + k)
f(x + k) moves the graph of the function k units to the right or left depending on whether k is greater than or less than 0.
If k is positive in f(x + k), the function will be moved to the left.
If k is negative in f(x + k), the function will be moved to the right.
5.8.1: Replacing f(x) with f(x) + k and f(x + k)

7. Key Concepts, continued
Horizontal translations: f(x + k)
When k is positive, k > 0, the graph moves left:
When k is negative, k < 0, the graph moves right:
5.8.1: Replacing f(x) with f(x) + k and f(x + k)

8. Common Errors/Misconceptions
incorrectly moving the graph in the direction opposite that indicated by k, especially in horizontal shifts; for example, moving the graph left when it should be moved right
incorrectly moving the graph left and right versus up and down (and vice versa) when operating with f(x + k) and f(x) + k
5.8.1: Replacing f(x) with f(x) + k and f(x + k)

9. Guided Practice Example 1
Consider the function f(x) = x2 and the constant k = 2. What is f(x) + k? How are the graphs of f(x) and f(x) + k different?
5.8.1: Replacing f(x) with f(x) + k and f(x + k)

10. Guided Practice: Example 1, continued
Substitute the value of k into the function.
If f(x) = x2 and k = 2, then f(x) + k = x2 + 2.
5.8.1: Replacing f(x) with f(x) + k and f(x + k)

11. Guided Practice: Example 1, continued
Use a table of values to graph the functions on the same coordinate plane. x
f(x)
f(x) + 2 –2 4 6 –1 1 3 2
5.8.1: Replacing f(x) with f(x) + k and f(x + k)

12. Guided Practice: Example 1, continued
5.8.1: Replacing f(x) with f(x) + k and f(x + k)

13. ✔ Guided Practice: Example 1, continued
Compare the graphs of the functions.
Notice the shape and horizontal position of the two graphs are the same. The only difference between the two graphs is that the value of f(x) + 2 is 2 more than f(x) for all values of x. In other words, the transformed graph is 2 units up from the original graph. ✔
5.8.1: Replacing f(x) with f(x) + k and f(x + k)

14. Guided Practice: Example 1, continued
5.8.1: Replacing f(x) with f(x) + k and f(x + k)

15. Guided Practice Example 3
Consider the function f(x) = x2, its graph, and the constant k = 4. What is f(x + k)? How are the graphs of f(x) and f(x + k) different?
5.8.1: Replacing f(x) with f(x) + k and f(x + k)

16. Guided Practice: Example 3, continued
Substitute the value of k into the function.
If f(x) = x2 and k = 4, then f(x + k) = f(x + 4) = (x + 4)2.
5.8.1: Replacing f(x) with f(x) + k and f(x + k)

17. Guided Practice: Example 3, continued
Use a table of values to graph the functions on the same coordinate plane. x
f (x)
f (x + 4) –6 36 4 –4 16 –2 2 64
5.8.1: Replacing f(x) with f(x) + k and f(x + k)

18. Guided Practice: Example 3, continued
5.8.1: Replacing f(x) with f(x) + k and f(x + k)

19. ✔ Guided Practice: Example 3, continued
Compare the graphs of the functions.
Notice the shape and vertical position of the two graphs are the same. The only difference between the two graphs is that every point on the curve of
f(x) has been shifted 4 units to the left in the graph of f(x + 4). ✔
5.8.1: Replacing f(x) with f(x) + k and f(x + k)

20. Guided Practice: Example 3, continued
5.8.1: Replacing f(x) with f(x) + k and f(x + k)