Question

The graph of a sinusoidal function has a minimum point at (0,2)

(0,2)

left parenthesis, 0, comma, 2, right parenthesis

 and then has a maximum point at (3\pi,6)

(3π,6)

left parenthesis, 3, pi, comma, 6, right parenthesis

.

Answer 1

We can write an equation of a sinusoidal function in the form

$$y = A\sin(B(x - C)) + D$$

The period of the graph is

$$T=2(3\pi-0)=6\pi$$

Then

$$B=\dfrac{2\pi}{T}=\dfrac{2\pi}{6\pi}=\dfrac{1}{3}$$

$$y = A\sin(\dfrac{1}{3}(x - C)) + D$$

$$A=\dfrac{y_{max}-y_{min}}{2}=\dfrac{6-2}{2}=2$$

$$y = 2\sin(\dfrac{1}{3}(x - C)) + D$$

$$x=0: \sin(\dfrac{1}{3}(0- C))=-1=>\sin \dfrac{C}{3}=1$$

$$-2+D=2=>D=4$$

$$x=3\pi: \sin(\dfrac{1}{3}(3\pi- C))=1=>\sin \dfrac{C}{3}=1$$

Let $\dfrac{C}{3}= \dfrac{\pi}{2}.$ Then $C= \dfrac{3\pi}{2}$

$$y = 2\sin(\dfrac{1}{3}(x - \dfrac{3\pi}{2})) + 4$$