write an equation of the form y = a sin bx or y = a cos bx to describe the graph below.

Explanation:

Step1: Determine the amplitude $a$

The amplitude is the maximum distance from the mid - line to the peak or trough. The mid - line is $y = 0$. The maximum value of the function is $2$ and the minimum is $- 2$. So, $a=\vert2\vert = 2$.

Step2: Determine the period $T$ and find $b$

The period $T$ is the distance between two consecutive peaks or troughs. From the graph, $T=\pi$. The formula for the period of $y = a\sin(bx)$ or $y=a\cos(bx)$ is $T=\frac{2\pi}{b}$. Since $T = \pi$, we have $\pi=\frac{2\pi}{b}$. Solving for $b$ gives $b = 2$.

Step3: Determine the function type

The graph passes through the point $(0, - 2)$. For $y=a\sin(bx)$, when $x = 0$, $y=a\sin(0)=0$. For $y=a\cos(bx)$, when $x = 0$, $y=a\cos(0)=a$. Since $a=- 2$ when $x = 0$, the function is $y=-2\cos(2x)$.

Answer:

$y=-2\cos(2x)$