A step by step tutorial on how to determine the properties of the graph of cubic functions and graph them. Properties, of these functions, such as domain, range, x and y intercepts, zeros and factorization are used to graph this type of functions. Free graph paper is available.

Cubic Functions A quadratic function has the form

f (x) = a x^{3} + b x^{2} + c x + d

Where a, b, c and d are real numbers and a is not equal to 0.
The domain of this function is the set of all real numbers. The range of f is the set of all real numbers. The y intercept of the graph of f is given by f(0) = d. The x intercepts are found by solving the equation

a x^{3} + b x^{2} + c x + d = 0

The left hand side behavior of the graph of the cubic function is as follows:

If the leading coefficient is positive, as x increases the graph of f is up and as x decreases indefinitely the graph of f is down.

If the leading coefficient is negative, as x increases the graph of f is down and as x decreases indefinitely the graph of f is up.

Example 1: f is a cubic function given by

f (x) = x^{ 3}

Find the x and y intercepts of the graph of f.

Find the domain and range of f.

Sketch the graph of f.

Solution to Example 1

a - The y intercept is given by
(0 , f(0)) = (0 , 0)

The x coordinates of the x intercepts are the solutions to
x^{3} = 0

The x intercept are at the points (0 , 0).

b - The domain of f (x) is the set of all real numbers.

Since the leading coefficient (of x^{3}) is positive, the graph of f is up on the right and down on the left and hence the range of f is the set of all real numbers.

c - make a table of values and graph.

x

-2

-1

0

1

2

f(x) = x^{ 3}

-8

-1

0

1

8

Also since f(-x) = - f(x), function f is odd and its graph is symmetric with respect to the origin (0,0).

Example 2: f is a cubic function given by

f (x) = - (x - 2)^{ 3}

Find y intercepts of the graph of f.

Find all zeros of f and their mutliplicity.

Find the domain and range of f.

Use the y intercept, x intercepts and other properties of the graph of to sketch the graph of f.

Solution to Example 2

a - The y intercept is given by
(0 , f(0)) = (0 , 8)

b - The zeros of f are solutions to
- (x - 2)^{ 3} = 0

Function f has one zero at x = 2 of multiplicity 3 and therefore the graph of f cuts the x axis at x = 2.

c - The domain of f (x) is the set of all real numbers.

After expansion of f(x), we can see that the leading coefficient (of x^{3}) is negative, the graph of f is down on the right and up on the left and hence the range of f is the set of all real numbers.

d - Properties and graph.

At x = 2, the graph cuts the x axis. The y intercept is a point on the graph of f. Also the graph of f(x) = - (x - 2)^{ 3} is that of f(x) = x^{ 3} shifted 2 units to the right because of the term (x - 2) and reflected on the x axis because of the negative sign in f(x) = - (x - 2)^{ 3}. Adding to all these properties the left and right hand behavior of the graph of f, we have the follwoing graph.

Example 3: f is a cubic function given by

f (x) = x^{ 3} + 2 x^{ 2} - x - 2

Factor f(x).

Find all zeros of f and their mutliplicity.

Find the domain and range of f.

Use the y intercept, x intercepts and other properties of the graph of to sketch the graph of f.

b - The zeros of f are solutions to
(x + 2)(x^{ 2} - 1) = 0

Function f has zeros at x = - 2, at x = 1 and x = - 1. Therefore the graph of f cuts the x axis at all these x intercepts.

c - The domain of f (x) is the set of all real numbers.

c - The leading coefficient f(x) is positive, the graph of f is down on the left and up on the right and hence the range of f is the set of all real numbers.

d - Properties and graph.

The y intercept of the graph of f is at (0 , - 2). The graph cuts the x axis at x = -2, -1 and 1. Adding to all these properties the left and right hand behavior of the graph of f, we have the follwoing graph.

Example 4: f is a cubic function given by

f (x) = - x^{ 3} + 3 x + 2

Show that x - 2 is a factor of f(x) and factor f(x) completely.

Find all zeros of f and their mutliplicity.

Find the domain and range of f.

Use the y intercept, x intercepts and other properties of the graph of to sketch the graph of f.

Solution to Example 4

a - The division of f(x) by x - 2 gives a quotient equal to -x^{2} - 2x -1 and a remainder is equal to 0. Hence

b - The zeros of f are solutions to
-(x - 2)(x + 1)^{2} = 0

Function f has zeros at x = 2 and x = - 1 with multiplicity 2. Therefore the graph of f cuts the x axis at x = 2 and is tangent to the x axis at x = - 1 because the mutliplicity of this zero is even.

c - The domain of f (x) is the set of all real numbers.

The leading coefficient f(x) is negative, the graph of f is up on the left and down on the right and hence the range of f is the set of all real numbers.

d - Properties and graph.

The y intercept of the graph of f is at (0 , 2). The graph cuts the x axis at x = 2 and is tangent to it at x = - 1. Adding to all these properties the left and right hand behavior of the graph of f, we have the follwoing graph.