find a function of the form y = asin (kx) + c or y = acos (kx) + c whose graph matches the function shown…

Explanation:

Step1: Determine the amplitude $A$

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

Step2: Determine the vertical - shift $C$

The vertical - shift $C$ is the average of the maximum and minimum values. So, $C=\frac{3+( - 3)}{2}=0$.

Step3: Determine the period $T$ and $k$

The period $T$ is the horizontal distance between two consecutive maxima or minima. From the graph, $T = 8$. Since the formula for the period of $y = A\sin(kx)+C$ or $y = A\cos(kx)+C$ is $T=\frac{2\pi}{k}$, then $k=\frac{2\pi}{T}$. Substituting $T = 8$ into the formula, we get $k=\frac{2\pi}{8}=\frac{\pi}{4}$.

Step4: Choose the function form

The graph passes through the origin $(0,0)$ and has a shape similar to $y = A\sin(kx)+C$. Substituting $A = 3$, $k=\frac{\pi}{4}$, and $C = 0$ into $y = A\sin(kx)+C$, we get $y = 3\sin(\frac{\pi}{4}x)$.

Answer:

$y = 3\sin(\frac{\pi}{4}x)$