the graph of a sinusoidal function has a maximum point at (0, 8) and then has a minimum point at (5, 2)…

Explanation:

Step1: Find the amplitude $A$

The amplitude is half the distance between the maximum and minimum values. $A=\frac{y_{max}-y_{min}}{2}=\frac{8 - 2}{2}=3$.

Step2: Find the vertical shift $D$

The vertical - shift is the average of the maximum and minimum values. $D=\frac{y_{max}+y_{min}}{2}=\frac{8 + 2}{2}=5$.

Step3: Find the period $T$

The distance between a maximum and a minimum is half of a period. So, $\frac{T}{2}=5-0 = 5$, then $T = 10$. The formula for the angular frequency $\omega=\frac{2\pi}{T}$, so $\omega=\frac{2\pi}{10}=\frac{\pi}{5}$.

Step4: Determine the phase shift $\varphi$

Since the maximum occurs at $x = 0$, for a cosine - type function $y=A\cos(\omega(x-\varphi))+D$, when $x = 0$, $y$ is maximum. For $y = A\cos(\omega(x-\varphi))+D$, when $x = 0$, $y=A\cos(-\varphi)+D$. Since $\cos(-\varphi)=\cos(\varphi)$ and $y$ is maximum at $x = 0$, $\varphi = 0$.

Answer:

$f(x)=3\cos\left(\frac{\pi}{5}x\right)+5$