1. Checking an equation for symmetry

2. Types of symmetry Three types of symmetry
Symmetry with respect to the y-axis (vertical)
Symmetry with respect to the x-axis (horizontal)
Symmetry with respect to the origin

3. Checking for symmetry with respect to the y-axis
In the original equation, replace x with –x, and simplify. If the result is the original equation, there is symmetry about the y-axis.
Example
y = x2 – 2
y = (-x)2 – 2 Substitute –x for x
y = x2 – 2 Simplify
There is symmetry about the y-axis.

4. Checking for symmetry with respect to the y-axis
Try this one.
y = 3 – 2x
y = 3 - 2(-x)
y = 3 + 2x
There is not symmetry about the y-axis.

5. Check for symmetry about the x-axis
In the original equation, replace y with –y, solve for y. If the new equation is the same, there is symmetry about the x-axis.
y2 = x + 4
(-y)2 = x + 4
There is symmetry about the x-axis.

6. Check for symmetry about the x-axis
Try this one.
y = x3 – 2x
(-y) = x3 – 2x
y = -x3 + 2x
There is not symmetry about the x-axis.

7. Checking for symmetry about the origin
In the original equation, substitute –x for x and –y for y, then simplify the equation. If the new equation is the same as the original, there is symmetry about the origin.
y = x3 – 4x
-y = (-x)3 – 4(-x)
-y = -x3 + 4x
There is symmetry about the origin.

8. Checking for symmetry about the origin
Try this one.
y2 = x – 1
(-y)2 = (-x) – 1
y2 = -x – 1
There is no symmetry about the origin.