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Find an equation for a sinusoidal function that has period $2\pi$ , amplitude $2$ , and contains the point $\left(-\frac{\pi}{2}, -2\right)$ . 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

2\pi

, amplitude

2

, and contains the point

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

. Write your answer in the form

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

, where

A

,

B

,

C

, and

D

are real numbers.

Determine B value: Determine the value of B based on the period. $\newline$ The period of a sinusoidal function in the form $f(x) = A\sin(Bx + C) + D$ is given by the formula $\text{Period} = \frac{2\pi}{B}$ . Since we are given that the period is $2\pi$ , we can set up the equation and solve for B. $\newline$ $\text{Period} = \frac{2\pi}{B}$ $\newline$ $2\pi = \frac{2\pi}{B}$ $\newline$ $B = 1$
Write general form: Write the general form of the sinusoidal function with the amplitude and B value. $\newline$ Since the amplitude $A$ is given as $2$ , and we have found $B$ to be $1$ , the general form of the function so far is: $\newline$ $f(x) = 2\sin(x + C) + D$ $\newline$ We still need to find the values of $C$ and $D$ .
Find phase shift and shift: Use the given point $(-\frac{\pi}{2}, -2)$ to find the phase shift $C$ and the vertical shift $D$ . Plugging the point into the equation, we get: $-2 = 2\sin(-\frac{\pi}{2} + C) + D$ Since $\sin(-\frac{\pi}{2}) = -1$ , the equation becomes: $-2 = 2(-1) + D$ $-2 = -2 + D$ $D = 0$
Find phase shift $C$ : Now that we have $D$ , we need to find $C$ , the phase shift. $\newline$ We know that $\sin(-\pi/2) = -1$ , and we want the function to pass through the point $(-\pi/2, -2)$ . Since the amplitude is $2$ , the function must reach its minimum at this point. This means that the phase shift $C$ must be such that the argument of the sine function is $-\pi/2$ when $x$ is $-\pi/2$ . $\newline$ So we set up the equation: $\newline$ $D$ $0$ $\newline$ Solving for $C$ gives us: $\newline$ $D$ $2$
Write final equation: Write the final equation of the sinusoidal function using the values of $A$ , $B$ , $C$ , and $D$ . Substituting the values we found into the general form, we get: $f(x) = 2\sin(x + 0) + 0$ Simplifying, we have: $f(x) = 2\sin(x)$

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