Cube Root Functions

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

a =
-10+10

c =
-10+10

d =
-10+10

>


click 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.

  1. 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.

  2. 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.

  3. 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.

  4. What is the domain of the cubing function f defined above?

  5. What is the range of the cubing function f defined above?

  6. How many x intercept the graph of f has?

  7. 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.

  8. Find the y intercept analytically and compare it to the one given by the applet.

ANSWERS TO THE ABOVE QUESTIONS

  1. 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.

  2. 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.

  3. 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.

  4. The set of all real numbers.

  5. The set of all real numbers.

  6. one x intercept.

  7. 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.

  8. 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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