Replacing f(x) with f(x) + k and f(x + k)

Adapted from Walch EducationReplacing f(x) with f(x) + k and f(x + k)

Transformations5.8.1: Replacing f(x) with f(x) + k and f(x + k) 2Neither 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.

Vertical TranslationsAdding a Constant to the Dependent Variable, f(x) + k 5.8.1: Replacing f(x) with f(x) + k and f(x + k) 3 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. Vertical Translations5.8.1: Replacing f(x) with f(x) + k and f(x + k) 4Vertical translations: f(x) + k When k is positive, k > 0, the graph moves up: When k is negative, k < 0, the graph moves down:

Horizontal TranslationsAdding a Constant to the Independent Variable, f (x + k) 5.8.1: Replacing f(x) with f(x) + k and f(x + k) 5 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.Horizontal Translations5.8.1: Replacing f(x) with f(x) + k and f(x + k) 6Horizontal translations: f(x + k)When k is positive, k > 0, the graph moves left: When k is negative, k < 0, the graph moves right:

Practice # 15.8.1: Replacing f(x) with f(x) + k and f(x + k) 7Consider 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? Substitute the value of k into the function. 5.8.1: Replacing f(x) with f(x) + k and f(x + k) 8If f(x) = x2 and k = 2, then f(x) + k = x2 + 2. Use a table of values to graph the functions on the same coordinate plane. xf(x)f(x) + 2246113002113246Graph f(x) = x2 and f(x) + k = x2 + 2. 5.8.1: Replacing f(x) with f(x) + k and f(x + k) 9

Compare the graphs of the functions5.8.1: Replacing f(x) with f(x) + k and f(x + k) 10Notice 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.Your turn5.8.1: Replacing f(x) with f(x) + k and f(x + k) 11Consider 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?Ms. DambrevilleThanks for Watching!