Determine the symmetries if any of the graph of the given re

Determine the symmetries (if any) of the graph of the given relation.

x² + y² = 7

Choose the correct symmetry or symmetries of the graph

a. x-axis only

b. x-axis, y-axis and origin

c. x-axis and y-axis only

d. origin only

Solution

For the graph symmetry we have the following rule:

1. A graph will have symmetry about the x-axis if we get an equivalent equation when all the y’s are replaced with -y.
2. A graph will have symmetry about the y-axis if we get an equivalent equation when all the x’s are replaced with -x.

3. A graph will have symmetry about the origin if we get an equivalent equation when all the y’s are replaced with -y and all the x’s are replaced with -x.

The equation of the graph is

x² + y² = 7

Replacing y by -y, we have

  x² + (-y)² = 7, or x² + y² = 7

Replacing x by -x, we have

(-x)² + y² = 7, or x² + y² = 7, and

Replacing x by -x and y by -y, we have

(-x)² + (-y)² = 7, or x² + y² = 7

Therefore, the graph is symmetric about x-axis, y-axis and origin.

Hence, option (b) is correct.