Test for symmetry


You can test the graph of a relation for symmetry with respect to the x-axis, y-axis, and the origin.

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.


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.

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.

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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