All function rules can be described as a transformation of an original function rule. In the diagram below, f(x) was the original quadratic and g(x) is the quadratic after a series of transformations. When comparing the two graphs, you can see that it was reflected over the x-axis and translated to the right 4 units and translated down 1 unit.
When identifying transformations of functions, this original image is called the parent function. A parent function is the simplest function of a family of functions. The parent function of a quadratic is f(x) = x². Below you can see the graph and table of this function rule.
When a quadratic is written in vertex form, the transformations can easily be identified because you can pinpoint the vertex (h, k) as well as the value of a.
Function Notation
These transformations can also be written in function notation. Select the notes link to view example problems in function notation. Then complete the worksheet and check you answers.