Algebra Worksheets

Practice and master algebra concepts with our helpful walkthroughs and downloadable practice worksheets from our team of elite math educators.

Cubic Functions

By Ellen Rattin

Learn about what Cubic Function is and how to use it to solve problems. Find the definition, example problems, and practice problems at Thinkster Math

Why is this concept useful?

As we study further in algebra, we progress through different degrees of polynomials. First we study linear functions (degree 1), then quadratic functions (degree 2), and so naturally we progress to cubic functions (degree 3). At this point, we can start seeing how different families of functions act and appreciate their overall structure! On a practical level, cubic functions model real world situations, like volume.

Where does this concept fit into the curriculum?

Algebra 2

What is a Cubic Function?

A cubic function is a function in the form

y=ax3+bx2+cx+dy = ax^3 + bx^2 + cx + d

Note that the degree of this function (the highest exponent) is 3; hence why it is called a cubic function! Cubic functions have a very distinctive, curvy graph.

How to Use Cubic Functions?

The standard form of a cubic function is

y=ax3+bx2+cx+dy = ax^3+bx^2+cx+d

To graph a cubic function, choose several x values, plug them in, and find the y value at those points. Then plot those points and connect them with a line. All cubic functions will have a distinctive curvy shape (see example graph above).

As with quadratics, we can talk about the roots of the function (where it crosses the x-axis), as well as the y-intercept (where it crosses the y-axis).

Sample Math Problems

Problem

Which of the following are cubic functions?

(A)

y=x3+2x−4y=x^3+2x-4

(B)

y=x3+x5+2x y=x^3+x^5+2x

(C)

y=2x−1y=2x-1

Solution

A is a cubic function - the highest exponent is a 3. B is not a cubic function; although it includes an exponent that is 3, the highest one is 5. C is not a cubic function; it’s highest exponent is 1.

Problem

Where does the function

y=x3+2y=x^3+2
 cross the y-axis?

Solution

I can choose five or so x values to plug in, in order to get points to plot. Here is what a possible x, y chart would look like:

After I plot the points and connect them, I see that the graph looks like:

This shows me that the function crosses the y-axis at 2.

Problem

What are the zeros of

y=(x+3)(x−2)(x+5)y=(x+3)(x-2)(x+5)
?

Solution

As with quadratic functions, zeros are roots of the function, where it crosses the x-axis. Cubic functions will have 3 roots. Remembering back to quadratics, roots are where y equals 0 and are often found by factoring. This one is conveniently factored for us, so we can see that if x + 3 = 0, x - 2 = 0, or x + 5 = 0, then that would cause y to equal 0. So x = -3, 2, and -5 are the zeros of the function.

Problem

Construct a cubic function in the form 

y=ax3+bx2+cx+dy=ax^3+bx^2+cx+d
whose roots are 2, -4, and 0.

Solution

If the roots are 2, -4, and 0, this means that (x - 2), (x + 4), and (x) are factors of the cubic function. And, since they told me that it is a cubic function, I know that there are only three roots, so I have all the factors of the function. By multiplying all three factors together I get

(x−2)(x+4)(x)=(x2+4x−2x−8)(x)=(x2+2x−8)(x)=x3+2x2−8x(x-2)(x+4)(x)=(x^2+4x-2x-8)(x)=(x^2+2x-8)(x)=x^3+2x^2-8x
.

Download FREE Math Resources

Take advantage of our free downloadable resources and study materials for at-home learning.

8 Math Hacks and Tricks to Turn Your ‘Okay’ Math Student Into a Math Champion!

One thing we teach our students at Thinkster is that there are multiple ways to solve a math problem. This helps our students learn to think flexibly and non-linearly.

Get PDF

How to Make Sure Your Child is Highly Successful and Becomes a Millionaire

As a parent, you hope your child is extremely successful and likely become the next Gates, Zuckerberg, or Meg Whitman. To set your child on the right path, there are many skills and traits that you can start building and nurturing now. Doing so plants the seeds for future success.

Get PDF

Practice Math Problems

1. Which of the following are cubic functions?

y=4y=4
y=x3−2x2+3x−1y=x^3-2x^2+3x-1
y=2x−1+x3y=2x-1+x^3

2. Where does the function 

y=(x−3)3+2y={(x-3)}^3+2
cross the y-axis?

3. What are the zeros of

y=(x+1)(x)(x+7)y=(x+1)(x)(x+7)
?

4. Construct a cubic function in the form 

y=ax3+bx2+cx+dy=ax^3+bx^2+cx+d
whose roots are 1, 5, and -2.

Related Topics

By Puja Varude

Normal Distribution

By Patricia Martin

Tossing a Coin

By Mandi Elam

Properties of Exponents

By Debi DalPezzo

Solve for x

Math Tutoring to Boost Your Child’s Math Skills & Scores by 90% in Just 3 Months – Guaranteed!

Does your child struggle with math homework or understanding tricky math concepts? Do they do okay in math, but express excitement to learn new material or advanced math?

A Thinkster Math tutor provides one-to-one support to help elementary, middle school, and high school students build confidence and master math subjects like K-8 math, pre-algebra, algebra, geometry, calculus, and more.

Our expert math tutors customize math lessons to your child’s unique needs, making learning math fun and effective. We help students improve grades, develop strong critical thinking skills through solving word problems, excel in standardized tests, and develop strong problem-solving skills.

Our expert math tutors are ready to help make your child a champion and develop strong math mastery! Sign up for our 7-day free trial and get the best math tutor for your child today!

Start 7-Day Free Trial