Graphs of Cubic Functions

Mark Lewis, Benjamin Rosenthal
  • Author
    Mark Lewis

    Mark has taught college and university mathematics for over 8 years. He has a PhD in mathematics from Queen's University and previously majored in math and physics at the University of Victoria. He has extensive experience as a private tutor.

  • Instructor
    Benjamin Rosenthal
What is a cubic function? See examples of cubic functions and learn how to graph cubic functions. Learn the equation and properties of a standard cubic function. Updated: 10/22/2021

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What is a Cubic Function?

A cubic function is a polynomial of degree 3, meaning 3 is the highest power of {eq}x {/eq} which appears in the function's formula. The simplest example of such a function is the standard cubic function, which is simply

$$f(x) = x^3 $$

Cubic functions often arise in situations where three factors are multiplied together, such as the volume of a rectangular box, which is equal to length times width times height. If each dimension is related to the variable {eq}x{/eq}, the result could be a cubic function something like this:

$$V(x) = l \times w \times h = x(6-2x)(10 -2x) $$

The function has degree 3 because if we expand it, by multiplying all terms involved, the highest power is the result of multiplying 3 {eq}x {/eq}'s. The graph of this function is shown below; as we will see, the graphs of most cubic functions have several basic features in common.

The graph of y=x(6-2x)(10 -2x).

The graph of y=x(6-2x)(10 -2x).

The Equation of a Cubic Function

The equations of cubic functions can always be expressed in the standard form

$$y= ax^3+bx^2+cx+d $$

where {eq}a, b, c, d {/eq} are real-valued constants. Remember that a cubic function is a polynomial whose highest degree term must be {eq}x^3{/eq}. This means that the coefficient {eq}a {/eq} must not be equal to 0. Lower powers may or may not be present, without changing the degree of the polynomial, so {eq}b, c, d {/eq} are allowed to be 0. Indeed, the standard cubic function does have {eq}b=c=d=0 {/eq} and {eq}a= 1 {/eq}.

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How to Graph a Cubic Function

Cubic functions are more complicated than linear and quadratic functions, so it is often not easy to graph them based only on their equation in standard form. Creating a table of values and plotting points on the graph can be a good strategy, especially if we keep in mind one feature of cubic functions.

As an example, lets graph the cubic function

$$y=x^3 -8x^2 + 15 x + 1 $$

To create a table of values, we can choose a range of {eq}x {/eq} values, and calculate corresponding values of {eq}y{/eq}. Choosing values close to 0, including a few negative values, is often the best place to start.

x -2 -1 0 1 2 3 4 5 6
y -69 -23 1 9 7 1 -3 1 19

The points {eq}(x, y) {/eq} are plotted in the graph below.

Points on the graph of y=x^3-8x^2+15x+1.

Points on the graph of y=x^3-8x^2+15x+1.

Notice that the two graphs of cubic functions we have seen so far are very similar in shape. Both make a smooth, S-shaped curve that increases, then decreases, then increases again. Knowing this three-step pattern helps us accurately connect the dots to draw the graph of a cubic function. Also, seeing this pattern in the {eq}y {/eq} values in the table tells us that we have identified enough points to begin drawing the graph.

X-Intercepts and Y-Intercepts

The points where a graph crosses the horizontal and vertical axes are called {eq}x {/eq}-intercepts and {eq}y {/eq}-intercepts. These make great reference points when graphing a function, and conversely, recognizing them in a graph can help us to identify the function's equation.

Can you identify the intercepts of the cubic function graphed below?

The intercepts of an unknown cubic function...

The intercepts of an unknown cubic function...

If a cubic function has three {eq}x {/eq}-intercepts, or roots, {eq}r_1 {/eq}, {eq}r_2 {/eq}, and {eq}r_3 {/eq}, we can express its equation in the factored form

$$y = a (x-r_1)(x-r_2)(x-r_3) $$

The other constant {eq}a {/eq} can then be determined from the value of the {eq}y{/eq}-intercept.

The function in the graph above appears to cross the vertical axis at {eq}y=10 {/eq} and the horizontal axis at {eq}x=-2, 2, 5 {/eq}. Using these three roots, we can write the function as

$$y = a (x+2)(x-2)(x-5) $$

The graph reaches the {eq}y {/eq}-intercept of 10 when {eq}x=0 {/eq}, so it must be that

$$\begin{eqnarray} 10 &=& a (0+2)(0-2)(0-5) \\ 10& = & a (2)(-2)(-5) \\ 10 &=& 20 a \\ a &=& 0.5 \end{eqnarray} $$

The complete equation of the cubic function in factored form is

$$y = 0.5 (x+2)(x-2)(x-5) $$

It's important to note that not all cubic functions have three roots. Some may have two (including a double root), or even only one. For cubic functions with a single root, we won't be able to express the equation as a product with three linear factors.

Constants

The exact shape of a cubic function is completely determined from the values of the constants {eq}a, b, c, d {/eq} in its standard form equation. Two of these constants tell us particularly useful information about the shape of the graph.

The constant {eq}d {/eq} determines the {eq}y {/eq}-intercept, the value of the function when {eq}x=0 {/eq}.

The other important constant is the leading coefficient {eq}a {/eq}, which appears in both the standard and factored equations. This coefficient is a "stretch factor" which can make the graph taller and narrower if {eq}|a|>1 {/eq}, or wider and shallower if {eq}|a|<1 {/eq}.

If {eq}a {/eq} is a negative value, then the graph is flipped, or reflected across the {eq}x {/eq}-axis. In this case, the three-step pattern in the graph is reversed: the graph will decrease, then increase, then decrease again. Its important to identify the sign of {eq}a {/eq} and the corresponding three-step pattern when graphing a cubic function, or identifying its equation.

There is one group of cubic functions that actually don't show the three-step pattern at all, and the standard cubic, shown below, is one of them!

Graph of y=x^3.

Graph of y=x^3.

The standard cubic function passes through the origin {eq}(0,0) {/eq} and is always increasing, without forming the distinctive peak-and-valley shape of the previous graphs. Two adjacent points on the graph are easily identified at {eq}(1, 1) {/eq} and {eq}(-1, -1) {/eq}.

Some cubic equations consist of transformations of the standard function and will have the same basic shape. Their equations can be expressed in the form

$$y=a(x-h)^3+k $$

The constant {eq}a {/eq} is the same stretch/reflection as before, while {eq}h {/eq} and {eq}k {/eq} define horizontal and vertical translations of the parent function {eq}y= x^3 {/eq}.

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Frequently Asked Questions

How do you find the equation of a cubic graph?

If a cubic graph has three x-intercepts then it is possible to quickly express the equation in factored form. The leading coefficient can be determined from the y-intercept.

How do you translate a cubic function?

A translation of the standard cubic function, y=x^3, takes the form y=a(x-h)^3+k. The constant h is a horizontal translation to the right, and k is a vertical translation upwards.

What is the equation for a cubic function?

The equation of a cubic function can always be expressed in the standard form y=ax^3+bx^2+cx+d, where a, b, c, d are constants, with a non-zero.

How do you graph a cubic function?

In many cases, a cubic function is most easily graphed by creating a table of values and plotting the points. If the equation can be factored, it is instead possible to graph it by locating its x- and y-intercepts.

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