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

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Write equations of sine and cosine functions using properties

Full solution

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

2\pi

, amplitude

1

, and contains the point

(0, -1)

. Write your answer in the form

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

, where

A

,

B

,

C

, and

D

are real numbers.

Period Calculation: We are given the period of the sinusoidal function as $2\pi$ . To find the value of $B$ , which affects the period of the cosine function, we use the formula for the period of a cosine function, which is $(2\pi)/B$ .
Calculation of B: Since the period is $2\pi$ , we set up the equation $(2\pi)/B = 2\pi$ and solve for B. $\newline$ $(2\pi)/B = 2\pi$ $\newline$ $B = (2\pi)/(2\pi)$ $\newline$ $B = 1$
Amplitude and Shift: Next, we know the amplitude $A$ is given as $1$ . This means the function will have a maximum value of $1$ and a minimum value of $-1$ . Since the point $(0, -1)$ is on the graph, this indicates that the cosine function must be shifted vertically or horizontally to pass through this point.
General Cosine Function: The general form of the cosine function is $f(x) = A\cos(Bx + C) + D$ . Since we have $A = 1$ and $B = 1$ , we can substitute these values into the equation to get $f(x) = \cos(x + C) + D$ .
Determining Values of C and D: Now we need to determine the values of C and D. Since the function passes through the point $(0, -1)$ , we can substitute $x = 0$ and $f(x) = -1$ into the equation to find D. $\newline$ $-1 = \cos(0 + C) + D$ $\newline$ Since $\cos(0) = 1$ for any value of C, we have: $\newline$ $-1 = 1 + D$ $\newline$ $D = -1 - 1$ $\newline$ $D = -2$
Correction for Value of D: However, we have made a mistake in the previous step. The amplitude of the function is $1$ , so the maximum and minimum values of the function are $1$ and $-1$ , respectively. Since the point $(0, -1)$ is on the graph and the amplitude is $1$ , this means that the cosine function is at its minimum value when $x = 0$ . Therefore, the value of $D$ should be $-1$ , not $-2$ , because the vertical shift $D$ must bring the minimum value of the cosine function down to $-1$ .

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