• We will investigate the symmetry of higher degree polynomial functions.
• We will generalize a rule that will assist us in recognizing even and odd symmetry, when it occurs in a polynomial function.

Recall, a function can be even, odd, or neither depending on its symmetry.

If a function is symmetric about the \(y\)-axis, then the function is an even function and \(f(-x)=f(x)\).

If a function is symmetric about the origin, that is \(f(-x)=-f(x)\), then it is an odd function.

The cubic function, \(y=x^3\), an odd degree polynomial function, is an odd function.

That is, the function is symmetric about the origin.

The cubic function, \(y=x^3\), an odd degree polynomial function, is an odd function.

That is, the function is symmetric about the origin.

If the graph of the function is reflected in the \(x\)-axis, followed by a reflection in the \(y\)-axis, it will map onto itself.

Algebraically,

Since \(f(-x)=-f(x)\), \(y=x^3\) is an odd function.

Is this the case for all cubic functions?

Consider the following cubic functions and their graphs.

What do these functions have in common?

They have a term of degree \(3\) and a term of degree \(1\); that is, an \(x^3\) term and an \(x\) term.

The other \(3\) functions defined by

are neither even nor odd.

Along with an odd degree term, \(x^3\), these functions also have terms of even degree; that is, an \(x^2\) term and/or a constant term of degree \(0\).

It appears an odd polynomial must have only odd degree terms.