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Find an equation for a sinusoidal function that has period $2\pi$ , amplitude $1$ , and contains the point $\left(\pi, -1\right)$ . $\newline$ 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 sine and cosine functions using properties

Full solution

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

2\pi

, amplitude

1

, and contains the point

\left(\pi, -1\right)

.

\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)=

Amplitude Given: Amplitude $A$ is given as $1$ . $\newline$ $A = 1$
Calculate B: Period is $2\pi$ , so B is found by dividing $2\pi$ by the period. $\newline$ $B = \frac{2\pi}{2\pi}$ $\newline$ $B = 1$
Find D: To find D, we use the point $(\pi, -1)$ . Since the amplitude is $1$ , the midline is at D, and the function value at $x=\pi$ should be at a minimum. $D = -1$
Determine Phase Shift: Now we need to find $C$ , the phase shift. Since the function is at a minimum at $x=\pi$ , and the cosine function has a minimum at $\pi$ , we can assume there's no horizontal shift. $C = 0$
Write Equation: Write the equation using the values of $A$ , $B$ , $C$ , and $D$ .
$f(x) = A\cos(Bx + C) + D$
$f(x) = 1\cos(1x + 0) - 1$
$f(x) = \cos(x) - 1$

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