In this lesson, our instructor Vincent Selhorst-Jones teaches about the Transformation of Functions. This lesson teaches vertical and horizontal shift, stretch, shrink, and flip. Youll get a summary of transformations and learn about how order matters when stacking transformations. The lesson ends with four examples for additional practice.
We often have to work with functions that are similar to ones we already know, but not precisely the same. Many times, this difference is the result of a transformation. A transformation is a shift, stretch, or flip of
A vertical shift moves a function up or down by some amount. If we want to shift a function f by k units, we use
f(x) + k.
[If k is positive, it moves up. If negative, down.]
A vertical stretch/shrink "pulls/pushes" the function away from/toward the x-axis by some multiplicative factor. If we want to vertically stretch/shrink a function by a multiplicative factor a, we use
[If a > 1, the function stretches. If 0 < a < 1, it shrinks. If a=1, nothing happens.]
A horizontal shift moves a function left or right by some amount. If we want to shift a function f by k units, we use
[If k is positive, the graph moves left. If k is negative, the graph moves right. (This may seem counter-intuitive, but remember that the shift is being caused by how f "sees" (x+k). Check out the video for an in-depth explanation
of what's going on.)]
A horizontal stretch/shrink changes how fast the function "sees" the x−axis. If we want to horizontally stretch/shrink a function by a multiplicative factor a, we use
[If a > 1, it shrinks horizontally ("speeds up"). If 0 < a < 1, it stretches horizontally ("slows down"). (This may seem counter-intuitive, but remember that the stretch/shrink is being caused by how f "sees" (a·x).
Check out the video for an in-depth explanation of what's going on.)]
To vertically flip a graph (mirror over the x-axis), we need to swap every output for the negative version. If we want to vertically flip, we use
To horizontally flip a graph (mirror over the y-axis), we need to "flip" how f "sees" the x-axis. We do this by plugging in −x (which is effectively a "flipped" x). If we want to horizontally flip a function, we use
If you want to do multiple transformations, just apply one transformation after another. However, order matters, so start by deciding on the order you want the transformations to occur in. Then apply them to the base function in that order.
Transformation of Functions
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.