find a function of the form y = a sin(kx)+c or y = a cos(kx)+c whose graph matches this one: (leave your…

Explanation:

Step1: Determine the amplitude $A$

The amplitude is half the vertical distance between the maximum and minimum values. The maximum value is $2$ and the minimum is $ - 4$. So, $A=\frac{2 - (-4)}{2}=\frac{6}{2}=3$.

Step2: Determine the vertical - shift $C$

The vertical - shift $C$ is the average of the maximum and minimum values. So, $C=\frac{2+( - 4)}{2}=\frac{-2}{2}=-1$.

Step3: Determine the period and $k$

The period $T$ is the horizontal distance between two consecutive maxima. Here, $T = 4$. Since the formula for the period of $y = A\sin(kx)+C$ or $y = A\cos(kx)+C$ is $T=\frac{2\pi}{k}$, and $T = 4$, we have $4=\frac{2\pi}{k}$, then $k=\frac{\pi}{2}$.

Step4: Choose the function form

The graph starts at the minimum value, which is the characteristic of a cosine function reflected about the $x$ - axis. So the function is $y=-3\cos(\frac{\pi}{2}x)-1$.

Answer:

$y=-3\cos(\frac{\pi}{2}x)-1$