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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 xaxis 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.
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) = ax^{3} + bx^{2} + 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) = x^{3}. 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 x^{3} = 0 (which is x = 0) and hence the cubic function f(x) = x^{3} 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 yvalues 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.
Intercepts of a Cubic Function
As we know, there are two types of intercepts of a function: xintercept(s) and yintercept(s). Let us see how to find the intercepts of a cubic function.
XIntercept of Cubic Function
The xintercepts 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 xintercepts. To find the xintercept(s) of a cubic function, we just substitute y = 0 (or f(x) = 0) and solve for xvalues.
Example: To find the xintercept(s) of f(x) = x^{3}  4x^{2} + x  4, substitute f(x) = 0. Then
x^{3}  4x^{2} + x  4 = 0
x^{2} (x  4) + 1 (x  4) = 0
(x  4) (x^{2} + 1) = 0
x  4 = 0; x^{2} + 1 = 0
x = 4 ; x^{2} = 1
x = 4 ; x = ± i
Complex numbers cannot be the xintercepts. Therefore, f(x) has only one xintercept which is (4, 0).
YIntercept of Cubic Function
A cubic function always has exactly one yintercept. To find the yintercept of a cubic function, we just substitute x = 0 and solve for yvalue.
Example: To find the yintercept of f(x) = x^{3}  4x^{2} + x  4, substitute x = 0. Then f(x) = 0^{3}  4(0)^{2} + (0)  4 = 4.
Therefore, the yintercept of the function is (0, 4).
End Behavior of Cube Function
The end behavior of any function depends upon its degree and the sign of the leading coefficient. A cube function f(x) = ax^{3} + bx^{2} + 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) = ax^{3} + bx^{2} + cx + d, we set the first derivative to zero and solve. i.e.,
f'(x) = 0
3ax^{2} + 2bx + c = 0
This is a quadratic equation and we can solve it using the techniques of solving quadratic equations.
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 b^{2}  3ac > 0
 f(x) has only one critical point if b^{2}  3ac = 0
 f(x) has no critical point if b^{2}  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) = ax^{3} + bx^{2} + 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) = ax^{3} + bx^{2} + 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) = x^{3}  4x^{2} + x  4.
 Step 1: Find the xintercept(s).
We already found that the xintercept of f(x) = x^{3}  4x^{2} + x  4 is (4, 0).  Step 2: Find the yintercept.
We already found that the yintercept of f(x) = x^{3}  4x^{2} + x  4 is (0, 4).  Step 3: Find the critical point(s) by setting f'(x) = 0.
3x^{2}  8x + 1 = 0.
By quadratic formula,
x = (8 ± √64  12) / (6) ≈ 0.131 and 2.535  Step 4: Find the corresponding ycoordinate(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 crosscheck the graph.
Important Notes on Cubic Function:
 A cubic function is of the form f(x) = ax^{3} + bx^{2} + 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:
Examples on Cubic Function

Example 1: Find the x intercept(s) and y intercept of cubic function: f(x) = 3 (x  1) (x  2) (x  3).
Solution:
Xintercept(s): To find the xintercepts, substitute f(x) = 0.
Then 3 (x  1) (x  2) (x  3) = 0
x  1 = 0; x  2 = 0; x  3 = 0
x = 1; x = 2; x = 3
Yintercept: To find the yintercept, substitute x = 0.
Then y = 3 (0  1) (0  2) (0  3) = 18.
Answer: The xintercepts are (1, 0), (2, 0), and (3, 0); and the yintercept is (0, 18).

Example 2: Find the end behavior of the cubic function that is mentioned in Example 1.
Solution:
The given function is, f(x) = 3 (x  1) (x  2) (x  3).
Here, the leading coefficient is 3 > 0.
So its end behavior is:
f(x) → ∞ as x → ∞ and
f(x) → ∞ as x → ∞Answer: f(x) → ∞ as x → ∞ and f(x) → ∞ as x → ∞.

Example 3: Find the critical points of the cubic function that is mentioned in Example 1.
Solution:
Given that f(x) = 3 (x  1) (x  2) (x  3) = 3x^{3}  18x^{2 }+ 33x  18.
For critical points,
f'(x) = 0
9x^{2}  36x + 33 = 0
Dividing both sides by 3,
3x^{2}  12x + 11 = 0
Using the quadratic formula,
x = (12 ± √144  132) / (6) ≈ 1.423 and 2.577
Answer: The critical points are at x = 1.423 and x = 2.577.
FAQs on Cubic Function
What is the Standard Form of a Cube Function?
A cube function is a thirddegree polynomial function. It is of the form f(x) = ax^{3} + bx^{2} + cx + d, where a ≠ 0.
What is Cubic Function Equation?
A cubic function equation is of the form f(x) = ax^{3} + bx^{2} + 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) = ax^{3} + bx^{2} + 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 xintercepts are obtained by substituting y = 0.
 The yintercepts 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 xintercepts.
 Plot the yintercept.
 Plot the critical points.
 Join them by all by taking care of the end behavior.
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