# Algebra Examples

Determine if the function is odd, even, or neither in order to find the symmetry.

1. If odd, the function is symmetric about the origin.

2. If even, the function is symmetric about the y-axis.

To check if the function is odd, substitute in for and check if the resulting function is the opposite of original function. In other words, determine if .

Simplify the expression.

Remove parentheses around .

Rewrite as .

Expand using the FOIL Method.

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Remove parentheses.

Simplify and combine like terms.

Simplify each term.

Move .

Use the power rule to combine exponents.

Add and to get .

Simplify .

Multiply by to get .

Multiply by to get .

Multiply by to get .

Multiply by to get .

Multiply by to get .

Add and to get .

Apply the distributive property.

Simplify.

Multiply by to get .

Multiply by to get .

The function is not odd because does not product the opposite function of . In other words, .

is not odd

is not odd

Since the function is not odd, it is not symmetric about the origin.

No origin symmetry

To check if a function is even, substitute in for and see if the resulting function is the same as the original. In other words, .

Simplify the expression.

Remove parentheses around .

Rewrite as .

Expand using the FOIL Method.

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Remove parentheses.

Simplify and combine like terms.

Simplify each term.

Move .

Use the power rule to combine exponents.

Add and to get .

Simplify .

Multiply by to get .

Multiply by to get .

Multiply by to get .

Multiply by to get .

Multiply by to get .

Add and to get .

Apply the distributive property.

Simplify.

Multiply by to get .

Multiply by to get .

The function is not even because the resulting function (after substituting in ) is not the same as the original.

is not even

is not even

Since the function is not even, it is not symmetric about the y-axis.

No y-axis symmetry

Since the function is neither odd nor even, there is no origin / y-axis symmetry.

Function is not symmetric