the graph of a sinusoidal function has a minimum point at (0,2) and then has a maximum point at (3π,6)…

Explanation:

Step1: Find the amplitude

The amplitude $A$ is half the distance between the maximum and minimum values. The maximum value is $y_{max}=6$ and the minimum value is $y_{min}=2$. So $A=\frac{y_{max}-y_{min}}{2}=\frac{6 - 2}{2}=2$.

Step2: Find the vertical shift

The vertical - shift $D$ is the average of the maximum and minimum values. So $D=\frac{y_{max}+y_{min}}{2}=\frac{6 + 2}{2}=4$.

Step3: Find the period

The distance between a minimum point $(0,2)$ and a maximum point $(3\pi,6)$ is half of the period. So $\frac{T}{2}=3\pi-0 = 3\pi$, then $T = 6\pi$. Since $T=\frac{2\pi}{B}$, we can solve for $B$: $B=\frac{2\pi}{T}=\frac{2\pi}{6\pi}=\frac{1}{3}$.

Step4: Determine the phase shift

The function has a minimum at $x = 0$. For a cosine - type function $y=A\cos(Bx - C)+D$, when $x = 0$, we want $A\cos(-C)+D$ to be at a minimum. For $y = A\cos(Bx - C)+D$, the general form of a cosine function with a minimum at $x = 0$ has $C = 0$. The general form of a sinusoidal function is $y = A\cos(Bx - C)+D$. Substituting $A = 2$, $B=\frac{1}{3}$, $C = 0$ and $D = 4$ into the formula, we get $f(x)=2\cos(\frac{1}{3}x)+4$.

Answer:

$f(x)=2\cos(\frac{1}{3}x)+4$