Preview Activity 3.3.1.
Open a new Desmos worksheet and define the following functions: \(f(t) = 2^t\text{,}\) \(g(t) = 3^t\text{,}\) \(h(t) = (\frac{1}{3})^t\text{,}\) and \(p(t) = f(kt)\text{.}\) After you define \(p\text{,}\) accept the slider for \(k\text{,}\) and set the range of the slider to be \(-2 \le k \le 2\text{.}\)
(a)
By experimenting with the value of \(k\text{,}\) find a value of \(k\) so that the graph of \(p(t) = f(kt) = 2^{kt}\) appears to align with the graph of \(g(t) = 3^t\text{.}\) What is the value of \(k\text{?}\)
(b)
Similarly, experiment to find a value of \(k\) so that the graph of \(p(t) = f(kt) = 2^{kt}\) appears to align with the graph of \(h(t) = (\frac{1}{3})^t\text{.}\) What is the value of \(k\text{?}\)
(c)
(d)
(e)
Given any exponential function of the form \(b^t\text{,}\) do you think it’s possible to find a value of \(k\) to that \(p(t) = f(kt) = 2^{kt}\) is the same function as \(b^t\text{?}\) Why or why not?
