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Find an equation for a sinusoidal function that has period $3\pi$ , amplitude $2$ , and contains the point $\left(\frac{\pi}{2}, 4\right)$ . Write your answer in the form $f(x)=A\sin(Bx+C)+D$ , where $A$ , $B$ , $C$ , and $D$ are real numbers.

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Write equations of sine and cosine functions using properties

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Q. Find an equation for a sinusoidal function that has period

3\pi

, amplitude

2

, and contains the point

\left(\frac{\pi}{2}, 4\right)

. Write your answer in the form

f(x)=A\sin(Bx+C)+D

, where

A

,

B

,

C

, and

D

are real numbers.

Find B using Period: Period = $3\pi$ , so find B using Period = $\frac{2\pi}{B}$ . $\newline$ $\frac{2\pi}{B} = 3\pi$ $\newline$ $B = \frac{2\pi}{3\pi}$ $\newline$ $B = \frac{2}{3}$
Determine Amplitude $A$ : Amplitude $A$ is given as $2$ . $\newline$ So, $A = 2$ .
General form substitution: Start with the general form $f(x) = A\sin(Bx + C) + D$ . Substitute $A$ and $B$ into the equation. $f(x) = 2\sin\left(\frac{2}{3}x + C\right) + D$
Solve for $C$ and $D$ : Plug in the point $(\frac{\pi}{2}, 4)$ to solve for $C$ and $D$ . $4 = 2\sin\left(\left(\frac{2}{3}\right)\left(\frac{\pi}{2}\right) + C\right) + D$
Simplify sine function: Simplify the inside of the sine function. $4 = 2\sin(\frac{\pi}{3} + C) + D$
Final equation with values: Since $\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$ , we have: $4 = 2(\frac{\sqrt{3}}{2} + C) + D$

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