How to graph cubic functions?

Traditionally a function in one variable, where in the highest power of the variable is 3 is called a cubic function. The general form of a cubic function is like this: f(x) = ax^3 + bx^2 + cx + d, where the leading coefficient ‘a’ cannot be equal to 0.

The parent function of a cubic function is f(x) = x^3. That means all cubic functions can be graphed by making suitable transformations of this parent function. Let us first try to understand this parent cubic function graph. For graphing this function we make a table of values of x and f(x) as follows:

X

-3

-2

-1

0

1

2

3

4

 

F(x)

-27

-8

-1

0

1

8

27

64

 

The sample calculation for the above table is as shown below:

For x = -3, f(x) = x^3 = (-3)^3 = -27.

Now using those points that we just found, we can graph the parent function as follows:

 

  

The points are (-2,-8), (-1,-1), (0,0), (1,1), (2,8) etc.

We can use this parent function to graph other cubic functions of the form f(x) + / - b, where b >0. The rule for graphing cubic functions of this type is:

(a) If the function is of the type f(x) + b, then the graph of f(x) = x^3 is moved b steps up.

(B) If the function is of the type f(x) – b, then the graph of f(x) = x^3 is moved b steps down.

Therefore the graphs of the following cubic functions:

g(x) = x^3 + 2

h(x) = x^3 + 5

k(x) = x^3 – 1

l(x) = x^3 – 3

 

Would be as shown in the figure below:

 

In the above figure, the blue curve is the graph of g(x) = f(x) + 5 = x^3 + 5. The green curve is the graph of h(x) = f(x) + 2 = x^3 + 2. The gold curve is the graph of k(x) = f(x) – 1 = x^3 – 1 and the purple curve is the graph of l(x) = f(x) – 3 = x^3 – 3. All these are the transformations of the parent function by moving along the y axis.

 

Similarly, there can be a generalized form of transformation along the x axis. If the cubic function is of the type (x +/- a)^3, then the following rules are to be followed:

 

(a) For f(x+a), the graph of x^3 is to be shifted a units to the left

(b) For f(x-a), the graph of x^3 is to be shifted a units to the right.

Where, a > 0.