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 of the vertical distance between the maximum and minimum values. The maximum value is $y = 6$ and the minimum value is $y=- 2$. So, $A=\frac{6 - (-2)}{2}=\frac{8}{2}=4$.

Step2: Determine the vertical - shift $C$

The vertical - shift $C$ is the mid - value between the maximum and minimum values. So, $C=\frac{6+( - 2)}{2}=\frac{4}{2}=2$.

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 $8=\frac{2\pi}{k}$, and $k=\frac{2\pi}{8}=\frac{\pi}{4}$.

Step4: Determine the function type

The graph passes through the point $(0,2)$ which is the mid - line value. A sine function $y = A\sin(kx)+C$ passes through the mid - line at $x = 0$. So the function is $y = 4\sin(\frac{\pi}{4}x)+2$.

Answer:

$y = 4\sin(\frac{\pi}{4}x)+2$