Graphing Polynomial Functions

Graphing polynomial functions can be an involved process that many students find daunting, but it isn't too bad if we take it one step at a time and utilize every trick we can. Remember that all polynomials have a degree that determines their general shape and a standard form that offers valuable information for graphing purposes if we know how to interpret it. In this article, we'll look at how to interpret it. Let's get going!

Five steps for graphing polynomial functions

While the exact process of graphing polynomial functions will vary according to the degree of the function, the following five steps will always prove useful:

1. Predict the end behavior of the function

You can use end behavior tests to determine how the function will act for large values in both directions using only the highest degree term, and whether the power on the term is even or odd.

2. Find the real zeros of the function

As a reminder, the zeros of a function are real numbers x such that $f\left(x\right)=0$ . These determine the x-intercepts of the graph, so finding them is immensely helpful when sketching any polynomial. Some polynomials can be easily factored, allowing us to find the real zeros quickly. Otherwise, we can use Descartes' Rule of Signs to at least determine how many real zeros there could be.

3. Make a table of values to find several points

Your graph will be more accurate the more specific points you know, so creating a table of values is always a good idea. Don't forget to include easy points such as the x and y-intercepts (when you have them) to save yourself some work. In general, we want to choose numbers that will make the underlying math as easy as possible.

4. Plot your points and draw a smooth, continuous curve to connect them

This is the step where we actually make our graph, and it's among the more straightforward ones. However, it's important to note that neatness counts. If you cannot read the sketch you've made, you'll have no idea whether it's accurate or not. Also, don't forget the arrows indicating that your curve extends indefinitely in either direction.

5. Double-check your work

Everybody makes mistakes, so it's always worth taking a little time to make sure what we did was right before moving on to the next problem. Does your curve exhibit the end behavior you predicted in Step 1? Can you read it? Do you notice anything that looks off?

Graphing polynomial functions: quadratic functions

Quadratic functions are frequently polynomial functions that students learn how to graph, and reviewing them here is a good idea since they illustrate many of the rules we apply to all polynomial functions. The standard form of a quadratic is:

${ax}^2+bx+c$

All three values (a, b, and c) must be real numbers and a cannot be zero. Since this is a polynomial with a degree of 2, we expect our graph to be a parabola.

Our leading coefficient is a. This is important because the leading coefficient helps determine the end behavior of any polynomial function. If a is positive, the parabola opens upward. If it's negative, the parabola opens downward. $f\left(x\right)$ constantly increases as $x\to -\infty$ and $x\to \infty$ when a is positive and constantly decreases as $x\to -\infty$ and $x\to \infty$ when a is negative, and that's because the degree of a quadratic function is even. All polynomial functions with an even degree will increase or decrease as the values for x become large, while polynomial functions with an odd degree will go in opposite directions instead. Here are two examples of parabolas:

There are several ways to factor quadratic equations, and we can use them to help us find the real zeros and x-intercepts for our graph. For example:

$f\left(x\right)=x^2+11x+30$

We can factor this to $f\left(x\right)=(x+5)(x+6)$ because it's the only factor pair with a product of 30 and a sum of 11. Setting each of these equal to zero gives us x values of -5 and -6, which are the x-intercepts for the corresponding graph. If we cannot find a factor pair for the numbers we have, the quadratic formula is also an option to find the real zeros. Factoring polynomials with a degree higher than 2 is more challenging, but the value it offers is the same.

Graphing polynomial functions: cubic functions

Cubic functions are the first polynomials with an odd degree that most students are exposed to, and comparing them to quadratic functions will help us identify constant rules. Cubic functions have a standard form of:

$f\left(x\right)=a{x}^3+b{x}^2+cx+d$

Once again, our leading coefficient is a, and whether it's positive or negative determines the end behavior of our function. If a is positive, $f\left(x\right)\to -\infty$ as $x\to -\infty$ and $f\left(x\right)\to \infty$ as $x\to \infty$ . If a is negative, the inverse is true: $f\left(x\right)\to \infty$ as $x\to -\infty$ and $f\left(x\right)\to -\infty$ as $x\to \infty$ . All other odd-degree polynomial functions follow the same rule.

The constant is represented by the letter d instead of c since cubic functions have an extra term, but it still tells us what our y-intercept will be. This is true for all polynomial functions regardless of degree.

Here are two examples of what cubic functions look like:

Note that both graphs cross the origin point since the d value of both graphs is 0.

We can factor cubic functions to find x-intercepts, but some cubic functions cannot be factored. In these circumstances, we need to create a table of values to ensure that we have points to work with.

Polynomials can have a degree higher than 3, but other than the shape of the curve they follow the rules for quadratic functions if even or cubic functions if odd. We can use that to our advantage.

Examples of graphing polynomial functions

a. Sketch a graph of the following polynomial function: $f\left(x\right)=x^3-2x^2-3x$

First, we want to guess what this graph will look like. It's a polynomial with a degree of 3, so it will look like a cubic function. The a value is 1, so it won't be especially wide or skinny, and $f\left(x\right)\to -\infty$ as $x\to -\infty$ and $f\left(x\right)\to \infty$ as $x\to \infty$ . The d value is 0, so its y-intercept will be the origin point. There should be three real zeros.

Next, we should try and figure out what those real zeros are. We can factor the polynomial to $f\left(x\right)=x(x+1)(x-3)$ , giving us three simple equations: one for each of the real zeros. One of them is $x=0$ , and the other 2 are:

$x+1=0$

$x=-1$

$x-3=0$

$x=3$

The real zeros are $x=0,x=-1$ , and $x=3$ . That means our x-intercepts will be $\left(0,0\right),\left(-1,0\right)$ , and $\left(3,0\right)$ .

That's a good start, but we need a few more points to graph this function. It's time to create a table of values by picking random values of x and solving for y:

$x$	$f\left(x\right)$
-3	-36
-2	-10
-1	0
0	0
1	-4
2	-6
3	0
4	20

Now, we can plot our points on the coordinate plane and connect them with a smooth, continuous curve. It should look something like this:

Double-checking our work, this graph fulfills all of the criteria we expected in step 1. We're done!

b. Sketch the following polynomial function: $f\left(x\right)=x^4-4x^3+3x+25$

First, we want to guess what the graph will look like. This polynomial has a degree of 4, which means that it's a quartic function. Quartic functions have a distorted W shape when graphed, so that's what we're expecting. Since the degree is even and a is positive, $f\left(x\right)$ should increase as x approaches both infinity and negative infinity. The constant is 25, so the y-intercept should be $\left(0,25\right)$ .

There's no way to find the real zeros of this function, and the reason why is that it doesn't have any. If we create a rough sketch with the information we have, we'll end up with an image like this:

We could make this graph a little more precise by calculating some specific points with a table of values.

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Varsity Tutors can help with graphing polynomial functions

Graphing polynomial functions is an essential skill in high school math and beyond, but that doesn't mean it's easy. Students need to memorize the shapes associated with different degrees of polynomials, solve complex algebra problems to create a table of values, and sketch their answers neatly. If you or your student need help with any of these things, a 1-on-1 math tutor can focus on your specific areas of opportunity to improve your study efficiency. Contact the friendly Educational Directors at Varsity Tutors for more information today. You'll be glad you did.