Graphing Polynomials

For this section, it might be a good idea to recall the basic facts of the polynomial, p(x) = anxn + an-1xn-1 +...+ a1x + a0 in the Polynomials and Roots section. Here p(x) has at most n roots. What does that mean? Well, when you are graphing the polynomial, the number of different roots are the number of times the polynomial crosses the x-axis, where a root is the place where the polynomial meets the x-axis.

Let's start with a polynomial of degree 1. This is a straight line that will cross the x-axis 1 time. The sign of the coefficient of x tells us if the line will be sloping this way / or \. Let's take a look at some examples.

When we have a polynomial of degree 2, it's graph forms a parabola. The coefficient of the x2 term now determines if the parobola will point up (if the coefficient is positive) or down (a negative coefficient). Let's have a look at these types of graphs.

We will do one more specific example and then generalize. So, the next one is a third degree polynomial. Again, the leading coefficient (the coefficient of the highest power term - in this case the x3) determines which way the graph will point. If it is positive, then the graph will go from lower left to upper right, while a negative coefficient will give a graph going from upper left to lower right. Take a look at the graphs for these.

You might be noticing a pattern now. If the degree is odd, the polynomial will have at least one root and up to as many as the degree of the polynomial. The leading coefficient will determine whether the graph is tending from lower left to upper right or from upper left to lower right on the grand scale. When we have even degree polynomials, say of degree n, the graph has anywhere from 0 to n roots. Here the leading coefficient determines whether the graph points up (if it is positive) or down (if it is negative) in the big picture. So by plotting the roots and using these two "rules", we can get an idea of what the polynomial looks like.

These just give a sketch of the graph of the polynomials. To draw the polynomials accurately takes a lot more mathematics then what we are covering in this section. Now we will work through a few examples.

Introductory Exercises Moderate Exercises