Cubic Function - Graphing | Cubic Graph

A cubic function is a polynomial function of degree 3. So the graph of a cubefunction may have a maximum of 3 roots. i.e., it may intersect the x-axis at a maximum of 3 points. Since complex roots always occur in pairs, a cubic function always has either 1 or 3 real zeros. It cannot have 2 real zeros.

Let us learn more about a cubic function along with its domain, range, and the process of graphing it. Let us also learn how to find the critical points and inflection points of a cubefunction and let us also see its end behavior.

1.	What is Cubic Function?
2.	Intercepts of a Cubic Function
3.	End Behavior of Cube Function
4.	Critical and Inflection Points of Cubic Function
5.	Cubic Graph
6.	FAQs on Cubic Function

What is Cubic Function?

A cubic function is a polynomial function of degree 3 and is of the form f(x) = ax³ + bx² + cx + d, where a, b, c, and d are real numbers and a ≠ 0. The basic cubic function (which is also known as the parent cubefunction) is f(x) = x³. Since a cubic function involves an odd degree polynomial, it has at least one real root. For example, there is only one real number that satisfies x³ = 0 (which is x = 0) and hence the cubic function f(x) = x³ has only one real root (the other two roots are complex numbers). Here are some examples of a cubic function.

A cubic function is an algebraic functionas all algebraic functions are polynomial functions.

Domain and Range of Cubic Function

Since a cubic function y = f(x) is a polynomial function, it is defined for all real values of x and hence its domain is the set of all real numbers (R). Also, if you observe the two examples (in the above figure), all y-values are being covered by the graph, and hence the range of a cubic function is the set of all numbers as well. Thus, we conclude that

The domain of a cubic function is R.
The range of a cubic function is R.

Asymptotes of CubeFunction

The asymptotes always correspond to the values that are excluded from the domain and range. Since both the domain and range of a cubic function is the set of all real numbers, no values are excluded from either the domain or the range. Hence a cubic function neither has vertical asymptotes nor has horizontal asymptotes.

Intercepts of a Cubic Function

As we know, there are two types of intercepts of a function: x-intercept(s) and y-intercept(s). Let us see how to find the intercepts of a cubic function.

X-Intercept of Cubic Function

The x-intercepts of a function are also known as roots (or) zeros. As the degree of a cubic function is 3, it can have a maximum of 3 roots. Since complex roots of any function always occur in pairs, a function will always have 0, 2, 4, ... complex roots. So a function can either have 0 or two complex roots. Thus, it has one or three real roots or x-intercepts. To find the x-intercept(s) of a cubic function, we just substitute y = 0 (or f(x) = 0) and solve for x-values.

Example: To find the x-intercept(s) of f(x) = x³ - 4x² + x - 4, substitute f(x) = 0. Then

x³ - 4x² + x - 4 = 0

x² (x - 4) + 1 (x - 4) = 0

Y-Intercept of Cubic Function

A cubic function always has exactly one y-intercept. To find the y-intercept of a cubic function, we just substitute x = 0 and solve for y-value.

Example: To find the y-intercept of f(x) = x³ - 4x² + x - 4, substitute x = 0. Then f(x) = 0³ - 4(0)² + (0) - 4 = -4.

Therefore, the y-intercept of the function is (0, -4).

End Behavior of CubeFunction

The end behavior of any function depends upon its degree and the sign of the leading coefficient. A cubefunction f(x) = ax³ + bx² + cx + d has an odd degree polynomial in it. So its end behavior is as follows:

When the leading coefficient is positive (a > 0):
f(x) → ∞ as x → ∞ and
f(x) → -∞ as x → -∞
In this case, the shape of the graph is from bottom to top.
When the leading coefficient is negative (a < 0):
f(x) → -∞ as x → ∞ and
f(x) → ∞ as x → -∞
In this case, the shape of the graph is from top to bottom.

We can better understand this from the figure below:

Critical and Inflection Points of Cubic Function

The critical points and inflection points play a crucial role in graphing a cubic function. Let us see how to find them.

Critical Points of Cubic Function

The critical points of a function are the points where the function changes from either "increasing to decreasing" or "decreasing to increasing". i.e., a function may have either a maximum or minimum value at the critical point. To find the critical points of a cubic function f(x) = ax³ + bx² + cx + d, we set the first derivative to zero and solve. i.e.,

f'(x) = 0

3ax² + 2bx + c = 0

This is a quadratic equation and we can solve it using the techniques of solving quadratic equations.

By quadratic formula,

x = \(\dfrac{-2b \pm \sqrt{4b^{2}-12 a c}}{6 a}\) (or)

x = \(\dfrac{-b \pm \sqrt{b^{2}-3 a c}}{3 a}\)

Hence:

f(x) has two critical points if b² - 3ac > 0
f(x) has only one critical point if b² - 3ac = 0
f(x) has no critical point if b² - 3ac <0

Inflection Points of Cubic Function

The inflection points of a function are the points where the function changes from either "concave up to concave down" or "concave down to concave up". To find the critical points of a cubic function f(x) = ax³ + bx² + cx + d, we set the second derivative to zero and solve. i.e.,

f''(x) = 0

6ax + 2b = 0

6ax = -2b

x = -b/3a

Thus, the cubic function f(x) = ax³ + bx² + cx + d has inflection point at (-b/3a, f(-b/3a)).

Cubic Graph

Here are the steps to graph a cubic function. The steps are explained with an example where we are going to graph the cubic function f(x) = x³ - 4x² + x - 4.

Step 1: Find the x-intercept(s).
We already found that the x-intercept of f(x) = x³ - 4x² + x - 4 is (4, 0).
Step 2: Find the y-intercept.
We already found that the y-intercept of f(x) = x³ - 4x² + x - 4 is (0, -4).
Step 3: Find the critical point(s) by setting f'(x) = 0.
3x² - 8x + 1 = 0.
By quadratic formula,
x = (-8 ± √64 - 12) / (6) ≈ 0.131 and 2.535
Step 4: Find the corresponding y-coordinate(s) of the critical points by substituting each of them in the given function.
f(0.131) ≈ -3.935
f(2.535) ≈ -10.879
Therefore, the critical points are (0.131, -3.935) and (2.535, -10.879).
Step 5: Find the end behavior of the function.
Since the leading coefficient of the function is 1 which is > 0, its end behavior is:
f(x) → ∞ as x → ∞ and
f(x) → -∞ as x → -∞
Step 6: Plot all the points from Step 1, Step 2, and Step 4. Join them by a curve (also extend the curve on both sides) keeping the end behavior from Step 5 in mind.

Note: We can compute a table of values by taking some random numbers for x and computing the corresponding y values to know the perfect shape of the graph. Also, we can find the inflection point and cross-check the graph.

Important Notes on Cubic Function:

A cubic function is of the form f(x) = ax³ + bx² + cx + d, where a, b, c, and d are constants and a ≠ 0.
The degree of a cubic function is 3.
A cubic function may have 1 or 3 real roots.
A cubic function may have 0 or 2 complex roots.
A cubic function is maximum or minimum at the critical points.

☛Related Topics:

Cubic Polynomials
Cubic Equation Solver
Graphing Functions
Graphing Calculator

FAQs on Cubic Function

What is the Standard Form of aCubeFunction?

A cubefunction is a third-degree polynomial function. It is of the form f(x) = ax³ + bx² + cx + d, where a ≠ 0.

What is Cubic Function Equation?

A cubic function equation is of the form f(x) = ax³ + bx² + cx + d, where a, b, c, and d are constants (or real numbers) and a ≠ 0.

How do You Determine a Cubic Function?

A function having an expression witha cube of the x variable can be a cubic function.If a function is of the form f(x) = ax³ + bx² + cx + d, then it is called a cubic function. Here, a, b, c, d can be any constants but take care that a ≠ 0. Any of the b, c, or d can be a zero.

How to Find the Intercepts of a Cubic Function?

For any cubic function y = f(x):

The x-intercepts are obtained by substituting y = 0.
The y-intercepts are obtained by substituting x = 0.

What is the Process of Graphing Cubic Function?

Here is the process of graphing a cubic function.

Find the intercepts.
Find the critical points.
Find the end behaviour.
Find some points on the curve using the given function.
Plot all the above information and join them by a smooth curve.

How Many Real Zeros a CubeFunction Have?

A cubefunction can have 1 or 3 real zeros. This is because

The degree of cubic function is 3.
Complex zeros occur in pairs.

How Many Complex Zeros a Cubic Function Have?

A cubic function can have 0 or 2 complex zeros. This is because

The degree of cubic function is 3 and so it has a maximum of 3 roots.
Complex zeros occur in pairs.

How to Sketch a CubeFunction?

To sketch a cubefunction:

Plot the x-intercepts.
Plot the y-intercept.
Plot the critical points.
Join them by all by taking care of the end behavior.

Cubic Function - Graphing | Cubic Graph | Cube Function (2024)