
Properties of Cubic Functions
Cubic functions have 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 y = 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 behaviour of the graph of the cubic function is as follows:
If the leading coefficient a is positive, as x increases f(x) increases and the graph of f is up and as x decreases indefinitely f(x) decreases and the graph of f is down.
If the leading coefficient is negative, as x increases f(x) decreases the graph of f is down and as x decreases indefinitely f(x) increases 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 multiplicity.
 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 multiplicity.
 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 3
 a  f (x) = x^{ 3} + 2 x^{ 2}  x  2
= x^{ 2} (x + 2)  (x + 2) = (x + 2)(x^{ 2}  1)
 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.
 d  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.
 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 behaviour of the graph of f, we have the following 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 multiplicity.
 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
f(x) = (x  2)(x^{2}  2x 1) =  (x  2)(x^{2} + 2x + 1) = (x  2)(x + 1)^{2}
 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 behaviour of the graph of f, we have the following graph.
More References and Links to Graphing
Graphing Functions.
