Graphs of the cubing functions of the form
f(x) = a (x - c), parameter a not equal to zero^{ 3} + d as well as their properties such as domain, range, x intercept, y intercept are explored interactively using an applet. Also, equations of the form are explored graphically.
a (x - c)^{ 3} + d = 0
The exploration is carried out by changing the parameters
## Tutorialclick 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.
## More References and Links to Functionsfunctions in this site. |