Preview Activity 1.8.1.
Open a new Desmos graph and define the function \(f(x) = x^2\text{.}\) Adjust the window so that the range is for \(-4 \le x \le 4\) and \(-10 \le y \le 10\text{.}\)
(a)
In Desmos, define the function \(g(x) = f(x) + a\text{.}\) (That is, in Desmos on line 2, enter
g(x) = f(x) + a.) You will get prompted to add a slider for \(a\text{.}\) Do so.
Explore by moving the slider for \(a\) and write at least one sentence to describe the effect that changing the value of \(a\) has on the graph of \(g\text{.}\)
(b)
Next, define the function \(h(x) = f(x-b)\text{.}\) (That is, in Desmos on line 4, enter
h(x) = f(x-b) and add the slider for \(b\text{.}\))
Move the slider for \(b\) and write at least one sentence to describe the effect that changing the value of \(b\) has on the graph of \(h\text{.}\)
(c)
Now define the function \(p(x) = cf(x)\text{.}\) (That is, in Desmos on line 6, enter
p(x) = cf(x) and add the slider for \(c\text{.}\))
Move the slider for \(c\) and write at least one sentence to describe the effect that changing the value of \(c\) has on the graph of \(p\text{.}\) In particular, when \(c = -1\text{,}\) how is the graph of \(p\) related to the graph of \(f\text{?}\)
(d)
Finally, click on the icons next to \(g\text{,}\) \(h\text{,}\) and \(p\) to temporarily hide them, and go back to Line 1 and change your formula for \(f\text{.}\) You can make it whatever you’d like, but try something like \(f(x) = x^2 + 2x + 3\) or \(f(x) = x^3 - 1\text{.}\) Then, investigate with the sliders \(a\text{,}\) \(b\text{,}\) and \(c\) to see the effects on \(g\text{,}\) \(h\text{,}\) and \(p\) (unhiding them appropriately). Write a couple of sentences to describe your observations of your explorations.
