How to Graph Polynomial Functions

By Linda Donahue; Updated April 24, 2017
The derivative of a function reveals how the original function's graph behaves.

In your Algebra 2 class, you'll learn how to graph polynomial functions of the form f(x) = x^2 + 5. The f(x), meaning function based on the variable x, is another way of saying y, as in the x-y coordinate graph system. Graph a polynomial function using a graph with an x and y axis. Of main interest is where either the x or the y value is zero, giving you the axis intercepts.

Draw your coordinate graph. Do this by drawing a horizontal line. This is the x axis. In the center, draw a vertical line to intercept (cross) it. This is the y, or f(x), axis. On each axis, mark several, evenly spaced hash marks for your integer values. Where the two lines intersect is (0,0). On the x axis, the positive numbers go on the right side and the negative on the left. On the y axis, the positive numbers go up, while the negative numbers go down.

Locate the y-intercept. Plug 0 into your function for x and see what you get. Say your function is: f(x) = x^3 - 5x^2 + 2x + 8. If you plug in 0 for x, you end up with 8, giving you the coordinate (0,8). Your y-intercept is at 8. Plot this point on your y axis.

Locate the x-intercepts, if possible. If you can, factor your polynomial function. (If it doesn't factor, it most likely means your x-intercepts are not integers.) For the given example, the function factors to: f(x) = (x+1)(x-2)(x-4). In this form, you can see if any of the parenthetical expressions equaled 0, then the whole function would equal 0. Therefore, the values -1, 2 and 4 would all produce a function value of 0, giving you three x-intercepts: (-1,0), (2,0) and (4,0). Plot these three points on your x axis. As a general rule of thumb, the degree of your polynomial indicates how many x-intercepts to expect. Since this is a third degree polynomial, it has three x intercepts.

Choose values of x to plug into the function that fall between and to the far sides of your x-intercepts. Typically, the curves of your function between intercept points will be fairly even and balanced so testing the mid-point will usually locate the top or bottom of a curve. At the two ends, past the outside x-intercepts, the line will continue off so you are finding points to determine the line's steepness. For instance, if you plug in the value 3, you'll get f(3) = -4. So the coordinate is (3,-4). Plug in several points, calculate and then plot.

Connect all your plotted points into a finished graph. Typically, for every degree, your polynomial function will have at most one fewer bend. So a second degree polynomial has 2-1 bends, or 1 bend, producing a U shaped graph. A third degree polynomial will most commonly have two bends. A polynomial has fewer than its maximum number of bends when it has a double root, meaning that two or more factors are the same. For example: f(x) = (x-2)(x-2)(x+5) has a double root at (2,0).