consider the function f(x)=-\frac{1}{4}cos(\frac{1}{2}x).\na. the graph will have a horizontal…

Explanation:

Step1: Identify horizontal stretch - compression factor

For a function $y = A\cos(Bx)$, the horizontal stretch/compression factor is related to $B$. Given $f(x)=-\frac{1}{4}\cos(\frac{1}{2}x)$, here $B = \frac{1}{2}$. The horizontal stretch factor is $\frac{1}{|B|}$. So, $\frac{1}{\left|\frac{1}{2}\right|}=2$.

Step2: Identify vertical stretch - compression factor

The vertical stretch/compression factor of the function $y = A\cos(Bx)$ is given by $|A|$. For $f(x)=-\frac{1}{4}\cos(\frac{1}{2}x)$, $A =-\frac{1}{4}$, so the vertical compression factor is $\left|-\frac{1}{4}\right|=\frac{1}{4}$.

Step3: Calculate the period

The period of the cosine function $y = A\cos(Bx)$ is $T=\frac{2\pi}{|B|}$. Since $B=\frac{1}{2}$, then $T = \frac{2\pi}{\left|\frac{1}{2}\right|}=4\pi$.

Answer:

a. Stretch, 2 b. Compression, $\frac{1}{4}$ c. $4\pi$