Cube root functions of the form
f(x) = a (x  c)^{ 1/3} + d
and the properties of their graphs such as
domain, range, x intercept, y intercept are explored interactively using an applet. Also cube root equations are explored graphically.
The exploration is carried out by changing the parameters a, c, and d defining the more general cube root function given above. Answers to the questions in the tutorial are at the bottom of the page.
TUTORIAL
click on the button above "click here to start" and MAXIMIZE the window obtained.
The answers to the following questions are included in this page.
 Use the sliders to 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.
 Use the sliders to 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 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.
 What is the domain of the cubing function f defined above?
 What is the range of the cubing function f defined above?
 How many x intercept the graph of f has?

How many solutions an equation of the form
a (x  c)^{ 1/3} + d = 0
has? (parameter a not equal to zero). Find the solution to this equation in terms of a, c and d and compare it to the x intercept given by the applet.
 Find the y intercept analytically and compare it to the one given by the applet.
ANSWERS TO THE ABOVE QUESTIONS

Changes in the parameter d translates (shifts) the graph vertically Shifting. When d increases, the graph is translated upward and when d decreases the graph is translated downward. With d = 0, the cube root function is given by
f(x) = a (x  c)^{ 1/3}
The graph is the graph of y = f(x) = a (x  c)^{ 1/3}. If d is not equal to zero, then y is given by y = a (x  c)^{ 1/3} + d and increases if d increases, hence the translation upward and decreases if d decreases, hence the translation downward.

When c increases, the graph is translated to the right and when c decreases, the graph is translated to the left. This is also called horizontal shifting.
The graph of the cube root function is the graph of the equation
y = a (x  c)^{ 1/3} + d
Solve the above equation for x to obtain
x = [ (y  d) / a ] ^{ 3} + c
If c increases, x increases hence the translation to the right. If c decreases, x decreases hence the translation to the left.

Let a be greater than zero. If a gets larger than 1, the graph stretches (or expands) vertically. If a gets smaller than 1, the graph shrinks vertically. This happens because a is a multipicative factor for the y coordinate. If a changes sign, a reflection of the graph on the x axis occurs.
 The set of all real numbers.
 The set of all real numbers.
 one x intercept.

The x intercept of the graph of f(x) = a (x  c)^{ 1/3} + d corresponds to one one solution of the equation a (x  c)^{ 1/3} + d = 0. Solve this equation analytically to obtain one solution.
x = ( d/a ) ^{ 3} + c
Compare the solution just obtained to the x intercept displayed by the applet.

To obtain the y intercept, let x = 0 in the equation
y = a (x  c)^{ 1/3} + d
Hence
y =  a*c^{ 1/3} + d
Calculate and compare to the y intercept displayed by the applet.
More on functions in this site.
