Test for symmetry

This lesson will teach you how to test for symmetry. You can test the graph of a relation for symmetry with respect to the x-axis, y-axis, and the origin. In this lesson, we will confirm symmetry algebraically.

Test for symmetry with respect to the x-axis.

The graph of a relation is symmetric with respect to the x-axis if for every point (x,y) on the graph, the point (x, -y) is also on the graph. To check for symmetry with respect to the x-axis, just replace y with -y and see if you still get the same equation. If you do get the same equation, then the graph is symmetric with respect to the x-axis.

symmetry with respect to x-axis


Example #1:


is x = 3y4 - 2 symmetric with respect to the x-axis?

Replace y with -y in the equation.

X = 3(-y)4 - 2

X = 3y4 - 2

Since replacing y with -y gives the same equation, the equation x = 3y4 - 2 is symmetric with respect to the x-axis.


Test for symmetry with respect to the y-axis.

The graph of a relation is symmetric with respect to the y-axis if for every point (x,y) on the graph, the point (-x, y) is also on the graph.
To check for symmetry with respect to the y-axis, just replace x with -x and see if you still get the same equation. If you do get the same equation, then the graph is symmetric with respect to the y-axis.

Symmetry with respect to y-axis


Example #2:


is y = 5x2 + 4 symmetric with respect to the x-axis?

Replace x with -x in the equation.

Y = 5(-x)2 + 4

Y = 5x2 + 4

Since replacing x with -x gives the same equation, the equation y = 5x2 + 4 is symmetric with respect to the y-axis.

Test for symmetry with respect to the origin.

The graph of a relation is symmetric with respect to the origin if for every point (x,y) on the graph, the point (-x, -y) is also on the graph.
To check for symmetry with respect to the origin, just replace x with -x and y
with -y and see if you still get the same equation. If you do get the same equation, then the graph is symmetric with respect to the origin.

Symmetry with respect to the origin

Example #3:

is 2xy = 12 symmetric with respect to the origin?

Replace x with -x  and y with -y in the equation.

2(-x × -y) = 12

2xy = 12

Since replacing x with -x and y with -y gives the same equation, the equation 
2xy = 12  is symmetric with respect to the origin.

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