GRAPHING POLYNOMIAL FUNCTIONS

THE PROCESS

Polynomials can be complicated functions, but there is a process you can use to make it easier to graph them.

1. Use the leading coefficient test to determine what the graph does as its ends.

2. Find the zeroes of the equation.3. Find the y-intercept of the equation.4. Determine symmetry.5. Determine how many turning points there

can be.6. Find any additional points that you need and

graph the function.

EXAMPLE

Let’s go through the entire process and graph the function x3 – 2x2 – 5x + 6.

The first step is to find the end behavior of the function. The degree of the function is odd and its leading coefficient (1) is positive, so it increases without bound on the right side and decreases without bound on the left.

EXAMPLE

The second step is to find the zeroes of the function. There are several ways to do so – synthetic division, estimation using the intermediate value theorem, and simple guess and check. However you choose to do so, the zeroes of x3 – 2x2 – 5x + 6 are -2, 1, and 3. Let’s plot these on our axes.

EXAMPLE

Next, we find the y intercept of the function by setting x = 0. We obtain y = 6, so our y intercept is 6. Let’s add these to our plot.

EXAMPLE

Next, we test for symmetry. Because of the x and x3 term, f(-x) ≠ f(x), and because of the x2 and constant terms, -f(x) ≠ f(x). There is no symmetry in this function.

EXAMPLE

Next, we want to determine how many turning points there can be. We can have up to (degree – 1) turning points, so we can have up to 2.

EXAMPLE

Finally, we want to find whatever other points we need for our graph. We’re not sure what the function does between -2 and 0 or between 1 and 3, so we’ll find f(-1) and f(2).f(-1) = 8f(2) = -4Now we can graph the function using all the information we’ve gathered about it.

FIN

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Our result is shown on the left. Note the two turning points and its behavior at the ends.