Graph transformations

Given the graph of a common function, (such as a simple polynomial, quadratic or trig function) you should be able to draw the graph of its related function.

The graph of the related function can be sketched without knowing the formula of the original function.

The following changes to a function will produce a similar effect on the graph regardless of the type of function involved. You should be familiar with the general effect of each change. You can also consider the effect on a few key points on each graph to help determine the related graph.

You must find the images of any given points and annotate them on your sketch.

This example uses the basic function y = f(x). This can then be uses to draw related functions.

Graph of y = f(x)

Notice that the main points on this graph are: x =  - 2,\,1,\,4

Graph of y = f(x) + k

Adding or subtracting a constant k to a function has the effect of shifting the graph up or down vertically by k units.

Graph of y = f(x)+k; shifts the graph up vertically by k

Graph of y=f(x)-k; shifts the graph down vertically by k

Graph of y = -f(x)

This has the effect of reflecting the graph about the x-axis.

Graph of y = -f(x); reflects the graph in the x-axis

Graph of y = f(x+k)

Adding or subtracting a constant k to the x term has the effect of shifting the graph left or right along the x-axis.

Graph of y = f(x+k); shifts the graph left horizontally by k

Graph of y = f(x-k); shifts the graph right horizontally by k

Graph of y = f(-x)

This has the effect of reflecting the graph about the x-axis.

Graph of y = f(-x); reflects the graph in the y-axis

Summary

Line graphs and related functions