select the correct answer from each drop - down menu. which transformation causes the described change in…

Explanation:

Step1: Recall horizontal - compression rule

For the function $y = f(x)$, the transformation $y=f(bx)$ with $b> 1$ causes a horizontal compression. For $y = \cos x$, when $y=\cos(3x)$ ($b = 3>1$), it results in a horizontal compression.

Step2: Recall vertical - stretch rule

For the function $y = f(x)$, the transformation $y = af(x)$ with $a>1$ causes a vertical stretch. For $y=\cos x$, when $y = 9\cos(x)$ ($a = 9>1$), it results in a vertical stretch.

Step3: Recall horizontal - stretch rule

For the function $y = f(x)$, the transformation $y=f(bx)$ with $0 < b<1$ causes a horizontal stretch. For $y=\cos x$, when $y=\cos(\frac{x}{5})$ ($b=\frac{1}{5},0 < b<1$), it results in a horizontal stretch.

Step4: Recall vertical - compression rule

There is no correct option for vertical compression in the given list. A vertical compression for $y = f(x)$ is of the form $y=af(x)$ where $0 < a<1$.

Answer:

The transformation $\cos(3x)$ results in a horizontal compression. The transformation $9\cos(x)$ results in a vertical stretch. The transformation $\cos(\frac{x}{5})$ results in a horizontal stretch. There is no correct option for vertical compression.