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Find an equation for a sinusoidal function that has period $2\pi$ , amplitude $1$ , and contains the point $(0,-1)$ . $\newline$ Write your answer in the form $f(x) = A\cos(Bx + C) + D$ , where $A$ , $B$ , $C$ , and $D$ are real numbers. $\newline$ $f(x) =$ _____

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Write equations of cosine functions using properties

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

2\pi

, amplitude

1

, and contains the point

(0,-1)

.

\newline

Write your answer in the form

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

, where

A

,

B

,

C

, and

D

are real numbers.

\newline

f(x) =

_____

Find Amplitude: Write the amplitude $A$ of the given problem. $\newline$ The amplitude is the absolute value of the coefficient in front of the cosine function. $\newline$ $A = 1$
Determine Period and B: Determine the period $T$ and find the value of $B$ . The period $T$ is given as $2\pi$ . The value of $B$ determines the period of the sinusoidal function according to the formula $T = \frac{2\pi}{B}$ . $B = \frac{2\pi}{T}$ $B = \frac{2\pi}{2\pi}$ $B = 1$
Solve for D: Solve for D. Substitute values of $x$ and $f(x)$ into $f(x) = A \cos(Bx + C) + D$ . Since the function contains the point $(0, -1)$ , we have: $-1 = A \cos(B(0) + C) + D$ $-1 = 1 \cos(0 + C) + D$ $-1 = \cos(C) + D$ Since $\cos(C)$ has a maximum value of $1$ , and we need $f(0)$ to be $f(x)$ $0$ , $\cos(C)$ must be at its maximum when $f(x)$ $2$ . This means $f(x)$ $3$ must be an even multiple of $f(x)$ $4$ . However, to get a negative value, we need to shift the cosine function by $f(x)$ $4$ . Therefore, $f(x)$ $6$ . $f(x)$ $7$ $f(x)$ $8$ $f(x)$ $9$
Write Sinusoidal Function: Write the equation of the sinusoidal function. $\newline$ We have found: $\newline$ $A = 1$ $\newline$ $B = 1$ $\newline$ $C = \pi$ $\newline$ $D = 0$ $\newline$ Substitute values of $A$ , $B$ , $C$ , and $D$ into $f(x) = A \cos(Bx + C) + D$ . $\newline$ $f(x) = 1 \cos(1x + \pi) + 0$ $\newline$ $B = 1$ $0$

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