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 the next - minimum is half of the period. So, $\frac{T}{2}=5$, then $T = 10$. Using the formula $T=\frac{2\pi}{B}$, we get $B=\frac{2\pi}{T}=\frac{2\pi}{10}=\frac{\pi}{5}$.

Step4: Determine the phase shift $C$

Since the maximum occurs at $x = 0$, for a cosine function of the form $y=A\cos(B(x - C))+D$, when $x = 0$, the function reaches its maximum. For $y = A\cos(Bx - BC)+D$, when $x = 0$, $y$ is maximum, and for the cosine function $y=\cos(u)$ which has a maximum at $u = 2k\pi,k\in\mathbb{Z}$. Here, with $A = 3,D = 5,B=\frac{\pi}{5}$, the function is of the form $y = A\cos(Bx)+D$ (because when $x = 0$, we have no phase - shift to account for to get the maximum), so $C = 0$.

Answer:

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