Math 41 Section 2.4 Symmetry and Graphs

Types of Symmetry

1. Symmetry about the x- axis. Eg. x = y
2. Symmetry about the y-axis. Eg. y = x
3. Symmetry about the origin. Eg. y = x
4. Symmetry about the line y = x.

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.

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.

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.

4. Symmetry about the line y = x.
• Two points P and Q are symmetric about the line y = x provided

(1) PQ is perpendicular to the line y = x. (This means the line y = x is the perpendicular bisector of the line segment PQ.)

(2) The points P and Q are equidistant from the line y = x.

• P and Q are reflections of one another through the line y =x, therefore making the line y = x the axis of symmetry.

Quick and Dirty Method for Special Circumstances

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

1. AND IF all of the exponents on each of the terms of the polynomial are even, THEN the polynomial will have y-axis symmetry.
2. OR IF all of the exponents on each of the terms of the polynomial are odd, THEN the polynomial will have origin symmetry.