f(x) = a (x - c)^{ 3} + d , parameter a not equal to zero

as well as their properties such as
domain, range, x intercept, y intercept are explored interactively using an applet. Also, equations of the form a (x - c)^{ 3} + d = 0 are explored graphically.

The exploration is carried out by changing the parameters a, c and d defining the more general cubing function above. Answers to the questions in the tutorial are at the bottom of the page.

TUTORIAL

click on the button above "draw" to start.

The answers to the following questions are included in this page.

Use the sliders to set parameters a, b and c to different values (a not equal to zero) and determine the domain of the cubing function f.

What is the range of the cubing function f?

Set parameters a and c to some values and change d. What happens to the the graph when the value of parameter d changes? Give an analytical explanation.

Set parameters a and d to some values and change c. What happens to the the graph when the value of parameter c changes? Give an analytical explanation.

Use the sliders to set parameters b, c and d to some values and change parameters a. What happens to the graph when the value of parameter a changes? Give an analytical explanation.

How many x intercept the graph of the cubing function has?

How many solutions an equation of the form

a (x - c)^{ 3} + d = 0

has? (parameter a not equal to zero). Solve the above equation for x in terms of a, c and d and use it to check the solution (x intercept) given by the applet.

What is the y intercept of the graph of the cubing function? Use the result found to compare with the values of the y intercept displayed by the applet.

ANSWERS TO THE ABOVE QUESTIONS

The set of all real numbers.

The set of all real numbers.

When d increases, the graph is translated upward and when d decreases the graph is translated downward. The graph is shifted vertically because when d changes it is the y coordinates, given by f(x), of all points of the graph that changes.

Solve the equation y = a (x - c)^{ 3} + d for x to obtain.

x = [ (y - d)/a ]^{ 1/3} + c

When c increases, the x coordinate of all points of the graph increases and hence the translation to the right. When c decreases it is a translation to the left. This is horizontal shifting.

For a greater than zero, as gets larger than 1, the graph stretches (or expands) vertically. As a gets smaller than 1, the graph shrinks vertically. Parameter a is a multipicative factor for the y coordinate, hence the stretching and shrinking of the graph. When a changes sign, a reflection of the graph on the x axis occurs.

one x intercept.

One solution corresponding to the x intercept displayed by the applet. Let us solve the following equation to find an analytical solution to it.

a (x - c)^{ 3} + d = 0

x - c = (-d / a)^{ 1/3}

x = (-d / a)^{ 1/3} + c

Set x = 0 in the equation to obtain the y intercept.

y = - a * c^{ 3} + d

Substitute parameters a, c and d by some values and check with the y intercept given by the applet.