Cubing Functions

Graphs of the cubing functions of the form

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

a =
-10+10

c =
-10+10

d =
-10+10

>

click on the button above "draw" to start.

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

  1. 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.

  2. What is the range of the cubing function f?

  3. 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.

  4. 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.

  5. 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.

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

  7. 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.

  8. 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

  1. The set of all real numbers.

  2. The set of all real numbers.

  3. 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.

  4. 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.

  5. 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.

  6. one x intercept.

  7. 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

  8. 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.

More on functions in this site.



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