introduction

IntroductionYou 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)

Introduction, continuedIn 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.

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.

Key Concepts, continuedVertical 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.

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:

Key Concepts, continuedHorizontal 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.

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:

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

Guided Practice

Example 1Consider 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?

Guided Practice: Example 1, continued

1. Substitute the value of k into the function. If f(x) = x2 and k = 2, then f(x) + k = x2 + 2.

2. 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

3. 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.

Guided Practice

Example 3Consider 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?

1. 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.

2. 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 0

–2 4 4

0 0 16

2 4 36

4 16 64

3. 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).

introduction

function fx

transformations fx

fx kwhen

graphs of fx

original function

transformed function

independent variable

key conceptsto

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