Cubic functions have an equation with the highest power of variable to be 3, i.e. highest power of x is x³.

A function f(x) = x³ has

Domain: {x | }  or {x | all real x}

Domain: {y | }  or {y | all real y}

We first work out a table of data points, and use these data points to plot a curve:

x	-3	-2	-1	0	1	2	3
y	-27	-8	-1	0	1	4	27

The family of curves f(x) = x³ k can be translated along y-axis by ‘k’ units up or down. For example –

f(x) = x³ + k will be translated by ‘k’ units above the origin, and f(x) = x³ – k will be translated by ‘k’ units below the origin.

The family of curves f(x) = (x  k)³ translates the curve y = x³ along the x-axis by ‘k’ units left or right. For example –

f(x) = (x + k)³ will be translated by ‘k’ units towards the left of the origin along the x-axis, and f(x) = (x – k)³ will be translated by ‘k’ units towards the right of the origin along the x-axis.

The domain and range in a cubic graph is always real values.

 

What type of function is a cubic function?

The function f(x) = x³ increases for all real x, and hence it is a monotonic increasing function (a monotonic function either increases or decreases for all real values of x).

Similarly f(x) = -x³ is a monotonic decreasing function.

1. Applying the vertical line test, we can see that the vertical line cuts the curve at only one point. Hence a cubic graph/curve is a function.

   

2. To find out whether it is an odd or an even function, we find out f(-x).

   Given f(x) = x³, f'(-x) = (-x)³ =  -x³ = -f(x)

   f'(-x) = -f(x) means the cubic function f(x) = x³ is an odd function.

3. A cubic function of form f(x) = y = x³ has point symmetry.

   Example: Sketch the cubic function f(x) = y = x³ + 8.

   x-intercept when y = 0 – f(x) = x³ + 8 = 0

   x =  =  -2.   So (-2, 0) is the x-intercept point. 

   Note: If x³ has a negative value, then the cube root is also negative, because the odd power of negative number is negative.

   y-intercept when x = 0 –  f(x) = 0³ + 8 = 8.  So (0, 8) is the y-intercept.

   When x = , y = , and when x = , y = 

   

   Domain: {all real x}

   Range: {all real y}