A polynomial function is a function that involves variables raised to non-negative integer powers.

There are two special types of polynomial functions based on the degree of each term of the polynomial.

Note: If -x is substituted into the function and some but not all of the signs change, the function is neither even nor odd.

Depending on the leading coefficient and degree of the polynomial, we can identify some trends on the end behavior of the function, where the function values are increasing/decreasing, and where the function values are positive/negative.

Steps for graphing a polynomial function:

1. Identify the parent function and end behavior.

   Degree	Leading Coefficient	End Behavior	Graph of the function
   \text{even }(x^2)	\text{positive}	f(x) \to + \infty, \text{as } x \to -\infty \\ f(x) \to + \infty, \text{as } x \to +\infty	\text{rises to the left and} \\ \text{to the right}
   \text{even }(-x^2)	\text{negative}	f(x) \to - \infty, \text{as } x \to -\infty \\ f(x) \to - \infty, \text{as } x \to +\infty	\text{falls to the left and} \\ \text{to the right}
   \text{odd }(x^3)	\text{positive}	f(x) \to -\infty, \text{as } x \to -\infty \\ f(x) \to + \infty, \text{as } x \to +\infty	\text{falls to the left and} \\ \text{rises to the right}
   \text{odd }(-x^3)	\text{negative}	f(x) \to + \infty, \text{as } x \to -\infty \\ f(x) \to - \infty, \text{as } x \to + \infty	\text{rises to the left and} \\ \text{falls to the right}

2. Identify the transformations to shift the the point of inflection or vertex.
3. Find the y-intercept by evaluating the expression when x=0.
4. Find the x-intercepts by determining the zeros of the function (when y=0 or f\left(x\right)=0)

The parent functions for quadratic functions and cubic functions have some features in common and some different.

Worked examples