find a function of the form (y = asin(kx)+c) or (y = acos(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. From the graph, the maximum value is $y = 3$ and the minimum value is $y=-1$. So, $A=\frac{3 - (- 1)}{2}=\frac{4}{2}=2$.

Step2: Determine the vertical - shift $C$

The vertical - shift $C$ is the mid - value between the maximum and minimum values. So, $C=\frac{3+( - 1)}{2}=\frac{2}{2}=1$.

Step3: Determine the period and $k$

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

Step4: Choose the function form

The graph starts at the mid - value $y = 1$ when $x = 0$, so the function is of the form $y = A\sin(kx)+C$.

Answer:

$y = 2\sin\left(\frac{\pi}{2}x\right)+1$