# 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 = 1 -10+10 c = 0 -10+10 d = 0 -10+10
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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.

More on functions in this site.