12) for the equation $y = -\frac{3}{2}cos(\frac{x}{4})$ give the following values:\na) amplitude:\nb) period…

Explanation:

Step1: Recall amplitude formula

For $y = A\cos(Bx)$, amplitude is $|A|$. Given $y=-\frac{3}{2}\cos(\frac{x}{4})$, $A = -\frac{3}{2}$.

Step2: Calculate amplitude

$|A|=\left|-\frac{3}{2}\right|=\frac{3}{2}$

Step3: Recall period formula

For $y = A\cos(Bx)$, period $T=\frac{2\pi}{|B|}$. Here $B=\frac{1}{4}$.

Step4: Calculate period

$T = \frac{2\pi}{\left|\frac{1}{4}\right|}=8\pi$

Step5: Find key - points for graph

For $y =-\frac{3}{2}\cos(\frac{x}{4})$, when $\frac{x}{4}=0$, $x = 0$ and $y=-\frac{3}{2}$; when $\frac{x}{4}=\frac{\pi}{2}$, $x = 2\pi$ and $y = 0$; when $\frac{x}{4}=\pi$, $x=4\pi$ and $y=\frac{3}{2}$; when $\frac{x}{4}=\frac{3\pi}{2}$, $x = 6\pi$ and $y = 0$; when $\frac{x}{4}=2\pi$, $x = 8\pi$ and $y=-\frac{3}{2}$.

Answer:

a) $\frac{3}{2}$ b) $8\pi$ c) Key - points: $(0,-\frac{3}{2}),(2\pi,0),(4\pi,\frac{3}{2}),(6\pi,0),(8\pi,-\frac{3}{2})$. Sketch a cosine - type curve passing through these points over the interval $[0,8\pi]$, label the $x$ - axis with appropriate values ($0,2\pi,4\pi,6\pi,8\pi$) and the $y$ - axis with values ($-\frac{3}{2},0,\frac{3}{2}$).