Transformations of Graphs- Shifting, Reflecting, and Stretching

Shifting, Reflecting, and Stretching Functions and Their Graphs- Transformations

There are three types of transformations of functions which can be done to the graph of any function. By transforming the graphs, we end up with a new graph and a new function with some of the same properties of the old function. The three types of transformations are:

Shifts

Shifts are a type of transformation which does not alter the graph of a function in any way- it simply moves it along the x and/or y axis. The graph of the function can be shifted horizontally by adding or subtracting a constant to/from every x-coordinate, and vertically by doing the same to every y-coordinate.

Shifts are either added to the f(x) components of the function, which results in a vertical shift, and the (x) components of the graph, which results in a horizontal shift. 

For example:

Original Function: f(x)=x

Shifted Function f(x)=x+2

Functions can also be shifted in both the x and y axes by changing variables both in and out of the parentheses (or absolute value brackets)-

Reflections

A function can be reflected about an x or y axis by multiplying by negative one. Multiplying every x value by -1 results in a reflection about the y axis, and vice versa for the x axis.  Point (a,b), when reflected across the x axis, becomes point (a,-b), and it becomes (-a,b) when reflected about the y axis. 

Example:

Stretching and Compressing

Functions can be stretched or compressed, this is the only transformation which changes the function itself, as opposed to just moving it. 

Stretching and compressing alters the shape and size of the graph. A vertical stretch or compress either divides or multiplies each y-coordinate by a constant, while leaving the x-coordinates the same, and a horizontal stretch or compress does the opposite. 

Example:

For additional help, refer to the following video

-Jeff Levin