Replacing f(x) with f(x) + k and f(x + k) Adapted from Walch Education

Transformations 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)

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)

Vertical Translations 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)

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)

Horizontal Translations 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)

Practice # 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)

Substitute the value of k into the function. If 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. 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)

Graph f(x) = x2 and f(x) + k = x2 + 2. 5.8.1: Replacing f(x) with f(x) + k and f(x + k)

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)

Your turn… 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)

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