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, and 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.
Interactive Cubing Function Explorer
Tutorial Questions
- Use the sliders to set parameters \(a\), \(b\), and \(c\) to different values (\(a \neq 0\)) and determine the domain of the cubing function \(f\).
- What is the range of the cubing function \(f\)?
- Set parameters \(a\) and \(c\) to some values and change \(d\). What happens to the graph when \(d\) changes? Give an analytical explanation.
- Set parameters \(a\) and \(d\) to some values and change \(c\). What happens to the graph when \(c\) changes? Give an analytical explanation.
- 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.
- How many x-intercepts does the graph of the cubing function have?
- 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.
- What is the y-intercept of the graph of the cubing function? Compare with the values displayed by the applet.
Answer
The set of all real numbers: \(\mathbb{R}\) (or \((-\infty, \infty)\)).
Answer
The set of all real numbers: \(\mathbb{R}\) (or \((-\infty, \infty)\)).
Answer
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 by adding \(d\).
Answer
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.
Answer
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.
Answer
Exactly one x-intercept.
Answer
One solution corresponding to the x-intercept:
\[ a (x - c)^3 + d = 0 \implies x = \sqrt[3]{- \frac{d}{a}} + c \]Answer
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.