Symmetry and Graphs

Types of Symmetry

1. Symmetry about the x-axis

· If (x,y) is on the graph, then (x,-y) is also on the graph

· To test for x-axis symmetry, replace y by –y. If the resulting equation is equivalent to the original then the graph has x-axis symmetry.

· Example: has x-axis symmetry

2. Symmetry about the y-axis

· If (x,y) is on the graph, then (-x,y) is also on the graph

· To test for y-axis symmetry, replace x by –x. If the resulting equation is equivalent to the original then the graph has y-axis symmetry.

· Example: has y-axis symmetry

3. Symmetry about the origin

· If (x,y) is on the graph, then (-x,-y) is also on the graph

· To test for x-axis symmetry, replace x by –x and y by –y. If the resulting equation is equivalent to the original then the graph has x-axis symmetry.

· Example: has origin symmetry

Quick and Dirty Method for Special Circumstances

IF you have a polynomial of the form y = ax+ ax+ + ax + a

a. AND IF all of the exponents on each of the terms of the polynomial are even, THEN the polynomial will have y-axis symmetry.

b. OR IF all of the exponents on each of the terms of the polynomial are odd, THEN the polynomial will have origin symmetry.