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. For the first - graph: 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. For the first graph, $T = 6$. Since $T=\frac{2\pi}{k}$, then $k=\frac{2\pi}{T}=\frac{2\pi}{6}=\frac{\pi}{3}$. Since the graph starts at a maximum, we use the cosine form $y = A\cos(kx)+C$. For the second graph: The maximum value is $y = 8$ and the minimum value is $y=-2$. So, $A=\frac{8 - (-2)}{2}=\frac{10}{2}=5$. The vertical shift $C=\frac{8+( - 2)}{2}=3$. The period $T = 8$, and $k=\frac{2\pi}{T}=\frac{2\pi}{8}=\frac{\pi}{4}$. Since the graph starts at a mid - value and is increasing, we use the sine form $y = A\sin(kx)+C$.

For the first graph:

Answer:

$y = 4\cos(\frac{\pi}{3}x)+2$

For the second graph:

Answer:

$y = 5\sin(\frac{\pi}{4}x)+3$