"Find an equation for a sinusoidal function that has period 2𝜋, amplitude 1, and contains the point (𝜋, – 1). Write your answer in the form f(x)=Acos(Bx+C)+D, where A, B, C, and D are real numbers. f(x)="

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

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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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Algebra 2

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

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 $D$ .
$f(x) = A\cos(Bx + C) + D$
$f(x) = 1\cos(1x + 0) - 1$
$f(x) = \cos(x) - 1$