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

Explanation:

Step1: Find the amplitude $A$

The amplitude is half the vertical distance between the maximum and minimum values. The maximum value is $y = - 1$ and the minimum value is $y=-5$. So, $A=\frac{(-1)-(-5)}{2}=\frac{-1 + 5}{2}=2$.

Step2: Find the vertical - shift $C$

The vertical - shift $C$ is the mid - value between the maximum and minimum values. $C=\frac{(-1)+(-5)}{2}=\frac{-6}{2}=-3$.

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

The period $T$ is the horizontal distance between two consecutive maxima or minima. From the graph, $T = 10$. 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 $T = 10$, we get $k=\frac{\pi}{5}$.

Step4: Determine the function type

The graph passes through the point $(0,-3)$ 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 $y = 2\sin(\frac{\pi}{5}x)-3$.

Answer:

$y = 2\sin(\frac{\pi}{5}x)-3$