How to Graph Polynomials

By Krystle Rose Forseth, Christopher Burger, Michelle Rose Gilman, and Deborah Rumsey

Finding the zeros is very important to graphing a polynomial, because they give you a general template for what your graph should look like. Remember that zeros are x-intercepts, and knowing where the graph crosses the x-axis is half the battle. The other half is knowing what the graph does in between these points. For example, graph the polynomial

Plot the critical points on the coordinate plane.

Mark the zeros: x = –3, x = –1/2, and x = 4 are all zeros.

Now plot the y-intercept of the polynomial. The y-intercept is always the constant term of the polynomial — in this case, y = 48. If no constant term is written, the y-intercept is 0 (because it’s understood).
Determine which way the ends of the graph will point.

You can use a handy test called the leading coefficient test, which helps you figure out how the polynomial begins and ends. The degree and leading coefficient of a polynomial always explain the end behavior of its graph:
- If the degree of the polynomial is even and the leading coefficient is positive, both ends of the graph point up.
- If the degree is even and the leading coefficient is negative, both ends of the graph point down.
- If the degree is odd and the leading coefficient is positive, the left side of the graph points down and the right side points up.
- If the degree is odd and the leading coefficient is negative, the left side of the graph points up and the right side points down.
The above figure displays this concept in correct mathematical terms.

The function

is even in degree and has a positive leading coefficient, so both ends of its graph point up (they go to positive infinity).
Figure out what happens between the critical points by picking any value to the left and right of each intercept and plugging it into the function.

You can either simplify each one or just figure out whether the end result is positive or negative. For now, you don’t really care about the exact look of the graph. (In calculus, you learn how to find additional values that lead to the most accurate graph you can get.)

A graphing calculator gives a very accurate picture of the graph. Calculus allows you to find the relative max and min exactly, using an algebraic process, but you can easily use the calculator to find them. You can use your graphing calculator to check your work and make sure the graph you’ve created looks like the one the calculator gives you.

Using the zeros for the function, set up a table to help you figure out whether the graph is above or below the x-axis between the zeros. See the following list.
- For the interval (–∞, –3), a test value (x) of –4 gives you a positive result for f (x) (above the x-axis)
- For the interval (–3, –1/2), a test value (x) of –2 gives you a negative result for f (x) (below the x-axis)
- For the interval (–1/2, 4), a test value (x) of 0 gives you a positive result for f (x) (above the x-axis)
- For the interval (4, ∞), a test value (x) of 5 gives you a positive result for f (x) (above the x-axis)
The first interval (–∞, –3) and the last interval (4, ∞) both confirm the leading coefficient test from Step 2 — this graph points up (to positive infinity) in both directions.
Plot the graph.

Now that you know where the graph crosses the x-axis, how the graph begins and ends, and whether the graph is positive (above the x-axis) or negative (below the x-axis), you can sketch out the graph of the function. Typically, in pre-calc, this is all the information you want or need when graphing. Calculus does show you how to get several other critical points that create an even better graph. If you’d like, you can always pick more points in the intervals and graph them to get a better idea of what the graph looks like. The above figure shows the completed graph.

Did you notice that the double root (with multiplicity two) causes the graph to “bounce” on the x-axis instead of actually crossing it? This is true for any root with even multiplicity. For any polynomial, if the root has an odd multiplicity at root c, the graph of the function crosses the x-axis at x = c. If the root has an even multiplicity at root c, the graph meets but doesn’t cross the x-axis at x = c.