Cubing Functions

Graphs of the cubing functions of the form \[ f(x) = a (x - c)^3 + d \] with \(a \neq 0\), as well as their properties such as domain, range, x-intercept, y-intercept, are explored interactively using an applet. Equations of the form \[ a (x - c)^3 + d = 0 \] are also explored graphically.

The exploration is carried out by changing the parameters \(a\), \(c\), and \(d\) defining the general cubing function above. Answers to the tutorial questions are at the bottom of the page.

Tutorial


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Click on the "Draw" button to start.

The answers to the tutorial questions are included below.

  1. Use the sliders to set parameters \(a\), \(b\), and \(c\) to different values (\(a \neq 0\)) 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 graph when \(d\) changes? Give an analytical explanation.
  4. Set parameters \(a\) and \(d\) to some values and change \(c\). What happens to the graph when \(c\) changes? Give an analytical explanation.
  5. Use the sliders to set parameters \(b\), \(c\), and \(d\) to some values and change \(a\). What happens to the graph when \(a\) changes? Give an analytical explanation.
  6. How many x-intercepts does the graph of the cubing function have?
  7. How many solutions does an equation of the form \[ a (x - c)^3 + d = 0 \] have? Solve the 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? Compare with the values displayed by the applet.

Answers to the Above Questions

  1. The set of all real numbers: \(\mathbb{R}\).
  2. The set of all real numbers: \(\mathbb{R}\).
  3. When \(d\) increases, the graph is translated upward; when \(d\) decreases, the graph is translated downward. The graph is shifted vertically because the y-coordinates of all points change.
  4. Solve \(y = a (x - c)^3 + d\) for \(x\): \[ x = \sqrt[3]{\frac{y - d}{a}} + c \] When \(c\) increases, all x-coordinates increase (graph shifts right). When \(c\) decreases, the graph shifts left. See horizontal shifting.
  5. For \(a > 0\), as \(|a|>1\), the graph stretches vertically. As \(|a|<1\), it shrinks vertically. Parameter \(a\) multiplies y-coordinates. If \(a<0\), the graph reflects across the x-axis.
  6. Exactly one x-intercept.
  7. One solution corresponding to the x-intercept: \[ a (x - c)^3 + d = 0 \implies x = \sqrt[3]{- \frac{d}{a}} + c \]
  8. Set \(x = 0\) to find the y-intercept: \[ y = a(-c)^3 + d \] Substitute your chosen values of \(a\), \(c\), and \(d\) and compare with the applet.

More References and Links to Functions

See more functions on this site.