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Cubic Functions

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 | x~in~R}  or {x | all real x}

Domain: {y | y~in~R}  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

cubic function

The family of curves f(x) = x3 pm 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 pm 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.

    vertical test for cubic 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 = root{3}{-8}  =  -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 = infty, y = infty, and when x = -infty, y = -infty

    translation of cubic function

    Domain: {all real x}

    Range: {all real y}