Using X-Intercepts to Graph Polynomial Functions

Instructor: Lauren Link

Lauren has taught high school Math and has a master's degree in education.

Graphing polynomials may seem daunting but when you can pinpoint where the function will cross or touch the x-axis, it's much easier. Learn how to find the x-intercepts of a given function and use them to sketch an accurate graph. Practice predicting what the graph should look like.


Let's start by reviewing the definition of a polynomial as a mathematical expression made up of more than one term. For example, 2x is a monomial (one term) but 2x + 3 is a polynomial (more specifically a 'binomial' since it only has two terms). We know that both of the functions f(x) = 2x and f(x) = 2x + 3 are linear since they each have a degree of 1. Also notice that they are in y = mx + b format. The degree of any monomial or polynomial is defined as the highest exponent. Usually polynomials are written in descending order of its exponents but if this is not the case, we will always take the first step of rearranging the terms appropriately.

Quadratic Functions

Okay, so how do we go about solving polynomial functions for their x-intercepts? Let's start with the common example of a quadratic. A quadratic equation is a specific type of polynomial that is in the form f(x) = ax2 + bx +c, where a, b, and c are constants > 0. We should be aware that a quadratic always has a degree of 2 and will always be in the shape of a parabola when graphed. A parabola looks like the shape of a 'U' and its direction will be evident in its equation. The base graph of a parabola is y = x2 and looks like this:

Basic parabola graph
Basic Parabola Graph

How many x-intercepts does this function have? Well, it touches the x-axis only one time at (0, 0), the origin. In order to find the zeros of the function algebraically we would set f(x) = 0 and solve for x. In this case we would take the square root of both sides and get x = 0. Once we solve the equation we have our zero(s), also known as x-intercept(s). Other terms that you might see used in place of x-intercepts are roots or solutions. A quadratic equation could have one, two, or zero solutions.

Example 1

Now let's try to algebraically solve for the x-intercepts of

f(x) = x2 - 4x - 12

By brainstorming the factors of -12 that give a sum of b value -4 we should choose factors -6 and 2. So the quadratic factors to

f(x) = (x - 6)(x + 2)

Then by setting f(x) = 0 we get x = 6 and x = -2. These are our zeros, aka x-intercepts. In order to sketch the graph of this function we would first plot our x-intercepts, coordinates (-2, 0) and (6, 0). But how do you know which direction the parabola will be facing?

In order to easily make this prediction without plotting other points we can just look at the leading coefficient (the a value in f(x)= ax2 + bx + c). If a > 0, then the parabola is an upright 'U' like we saw in the base graph. But if a < 0, then the parabola will open up downwards. So since the a value in our function is the implied coefficient of positive 1, the parabola will open upwards.

If we want to be more accurate when sketching we can find the turning point or vertex of the graph by first finding the axis of symmetry (AOS) with the equation x = -b/2a. In this case the AOS is x = 2. To find the y value of the vertex we must substitute x = 2 into the original equation. So, f(2) = 22 - 4(2) - 12 = -16, which is our y value. The vertex is (2, -16). Our graph should look like this:


Even and Odd Degrees

A quadratic function will always be in the shape of a parabola, but what if we have a polynomial function with a degree that is greater than 2? In these cases we will use some important facts to predict the shape of the graph.

EVEN Degree: If a polynomial function has an even degree (that is, the highest exponent is 2, 4, 6, etc.), then the graph will have two arms both facing the same direction. Our two examples so far followed this rule. We know that the sign of the leading coefficient or a value tells us which direction the arms will point.

ODD Degree: If a polynomial function has an odd degree greater than 1 (that is, the highest exponent is 3, 5, 7, etc.), then the graph will have two arms facing opposite directions. In these cases if a > 0 the graph will increase as it approaches positive infinity. If a < 0 the graph will decrease as it approaches positive infinity.

Here are some visuals:

EVEN Degree

ODD Degree

It is also helpful to know that a function with degree n can only have at most (n-1) turning points. For example, y = x3 can have at most two turning points.

Example 2

For polynomials with greater degrees we will solve for many zeros and sometimes the same zero will appear more than once. This is called multiplicity.

Suppose we are given the function

y = x^3+2x^2-7x+4

If we first test to see if either x = -1 or x = 1 are solutions to the equation we see that x = 1 is, in fact, a solution. This means that we need to use long division to divide the polynomial by factor (x -1) (since we now know that it must be in there). After performing long division we get

y = (x - 1)(x^2+3x-4)

and can further factor that to

y = (x - 1)(x+4)(x-1)

So this function shows that the solution x = -4 has a multiplicity of 1 and the solution x = 1 has a multiplicity of 2, since it appears twice. This lets us make an important prediction.


Given that a polynomial has x = s as a solution with multiplicity m:

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