Activity 3.1.2.
In Desmos, define the function \(g(t) = ab^t\) and create sliders for both \(a\) and \(b\) when prompted. Click on the sliders to set the minimum value for each to \(0.1\) and the maximum value to \(10\text{.}\) Note that for \(g\) to be an exponential function, we require \(b \ne 1\text{,}\) even though the slider for \(b\) will allow this value.
(a)
What is the domain of \(g(t) = ab^t\text{?}\)
(b)
What is the range of \(g(t) = ab^t\text{?}\)
(c)
(d)
How does changing the value of \(b\) affect the shape and behavior of the graph of \(g(t) = ab^t\text{?}\) Write several sentences to explain.
(e)
For what values of the growth factor \(b\) is the corresponding growth rate positive? For which \(b\)-values is the growth rate negative?
(f)
Consider the graphs of the exponential functions \(p\) and \(q\) provided in the following figure. If \(p(t) = ab^t\) and \(q(t) = cd^t\text{,}\) what can you say about the values \(a\text{,}\) \(b\text{,}\) \(c\text{,}\) and \(d\) (beyond the fact that all are positive and \(b \ne 1\) and \(d \ne 1\))? For instance, can you say a certain value is larger than another? Or that one of the values is less than \(1\text{?}\)
ADD ALT TEXT TO THIS IMAGE
