Polynomials are some of our favorite functions.

### What do the graphs look like?

We understand the graphs of polynomials of degrees and very well.

Polynomial functions with degree are referred to as linear polynomials. This is due to the fact that such a function can be written as . The graph of such a function is a straight line with slope and -intercept at .

Quadratic functions, written as with , have parabolas as their graphs.

The parabola we are looking for has vertex at , opens upward (since ), and is wider than the standard parabola.

In terms of graph transformations, we can think of this as the graph of , which has been shifted horizontally units to the left, vertically compressed by a factor of , then shifted vertically unit downward.

For polynomials of higher degree, we will have to take everything else we have been talking about and put it together. We need to understand the zeroes of the polynomial and its end behavior. The zeroes will correspond to -intercepts, and the end behavior will tell us what happens out past those intercepts.

One upshot of the Fundamental Theorem of Algebra (in terms of graphing) is that
when we plot a polynomial of degree , its graph will cross the -axis *at most* times.
Each crossing corresponds to a real root of that polynomial. (Complex roots do not
give crossings!)

We also see from the factorization that is a zero of multiplicity , and is a zero of multiplicity . That means our graph will have -intercepts at those locations.

At these -intercepts, will the graph pass through the -axis like a line does, or will it touch the axis and turn back around, like a parabola does at it’s vertex? To answer this, look at a sign-chart for .

Notice that around , only the factor changes signs. That means will change signs, so the graph will pass through the -axis at . At , however, the factor does NOT change signs (due to the even exponent). That means will turn around at that -intercept. Putting this together, we have the following graph.

More specifically: The graph of will resemble, at least locally near the point , of the graph of .

**even**or

**odd**degree, and if the leading coefficient (the one next to the highest power of ) of the polynomial is

**positive**or

**negative**.

- Curve is defined by an evenodd degree polynomial with a positivenegative leading term.
- Curve is defined by an evenodd degree polynomial with a positivenegative leading term.
- Curve is defined by an evenodd degree polynomial with a positivenegative leading term.
- Curve is defined by an evenodd degree polynomial with a positivenegative leading term.