Cubic Function

A cubic function is a polynomial function of degree 3. So the graph of a cube function 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 cube function and let us also see its end behavior.

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 cube function) 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 function as 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 Cube Function

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.

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

(x - 4) (x² + 1) = 0

x - 4 = 0; x² + 1 = 0

x = 4 ; x² = -1

x = 4 ; x = ± i

Complex numbers cannot be the x-intercepts. Therefore, f(x) has only one x-intercept which is (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).

The end behavior of any function depends upon its degree and the sign of the leading coefficient. A cube function 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:

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)).

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

What is the Standard Form of a Cube Function?

A cube function 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 with a 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 Cube Function Have?

A cube function 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 Cube Function?

To sketch a cube function:

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