Cube root functions of the form
f(x) = a (x - c)^{ 1/3} + dand 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
## Tutorialclick on the button above "draw" and start exploring. You may enter values for a, c and d or use the sliders. 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.
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