Activity 5.1.3.
Point your browser to the Desmos worksheet at http://gvsu.edu/s/0zu. In what follows, we explore the behavior of power functions of the form \(y = x^n\) where \(n \ge 1\text{.}\)
(a)
Press the “play” button next to the slider labeled “\(n\text{.}\)” Watch at least two loops of the animation and then discuss the trends that you observe. Write a careful sentence each for at least two different trends.
(b)
Click the icons next to each of the following 8 functions so that you can see all of \(y = x\text{,}\) \(y = x^2\text{,}\) \(\ldots\text{,}\) \(y = x^8\) graphed at once. On the interval \(0 \lt x \lt 1\text{,}\) how do the graphs of \(x^a\) and \(x^b\) compare if \(a \lt b\text{?}\)
(c)
Uncheck the icons on each of the 8 functions to hide their graphs. Click the settings icon to change the domain settings for the axes, and change them to \(-10 \le x \le 10\) and \(-10,000 \le y \le 10,000\text{.}\) Play the animation through twice and then discuss the trends that you observe. Write a careful sentence each for at least two different trends.
(d)
Click the icons next to each of the following 8 functions so that you can see all of \(y = x\text{,}\) \(y = x^2\text{,}\) \(\ldots\text{,}\) \(y = x^8\) graphed at once. On the interval \(x \gt 1\text{,}\) how do the graphs of \(x^a\) and \(x^b\) compare if \(a \lt b\text{?}\)
