7 Steps to Graph a Polynomial

by Yaneli Morales, CAPS Math Tutor

1. Find overall degree of function.
Example:

image

The overall degree of this function is 5. You get this number by adding all the exponents of the equation. In this case x has an exponent 2, the quantity (x+2) has an exponent of 2 as well, and the quantity (x-4) has an implied power of 1. When these exponents are added, the overall power is 5.

2. Find the leading sign of the graph.

To do this you first have to simplify the given function.
Example:

image

Notice that the leading sign of the graph is positive. Therefore, our function is positive.

3. Find the end behavior of your graph.

The end behavior of the graph is known by the overall power of the function. If the overall degree of the function is an odd number, the end behavior of the graph will be like a cubic graph. If the overall degree of the function is even then the graph will be like a parabola.

image

Keep in mind the leading sign of the graph.

4. Zeros/Multiplicity/Cross or Touch

To find the zeros of your function you will have to factor your function completely. After you have done that, set each factor equal to zero and solve for your variable.
Example:

image

The multiplicity of a zero is equal to the degree of factor it came from.

image

If the multiplicity of the zero is even then the line of the graph will touch at that point. If the multiplicity of the zero is odd then the line of the graph will cross at the point

image

5. Find your y intercept

To do this just set all your x equal to 0 and solve.
Example:

image

Therefore, the y intercept of this graph is 0.

6. Find other points of reference.

It would be really helpful if you find other points of reference so that you know how the graph is behaving at those points.

Some good points of reference would be numbers before and a number after your zeros.
Example:

image

Make a T-table with these reference points. You will find your plotting points by plugging your reference points into your original function. The T-table will just help you be more organized.

image

7. Graph polynomial

Now plot all your points, connect them (keeping in mind the behavior of the graph), and you are done!!
Example:

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