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The graph of a sinusoidal function has a maximum point at
(0,5) and then has a minimum point at
(2pi,-5).
Write the formula of the function, where
x is entered in radians.

The graph of a sinusoidal function has a maximum point at $(0,5)$ and then has a minimum point at $(2 \pi,-5)$ . $\newline$ Write the formula of the function, where $x$ is entered in radians.

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Write equations of sine and cosine functions using properties

Full solution

Q. The graph of a sinusoidal function has a maximum point at

(0,5)

and then has a minimum point at

(2 \pi,-5)

.

\newline

Write the formula of the function, where

x

is entered in radians.

Determine Amplitude: Determine the amplitude of the function. $\newline$ The amplitude is half the distance between the maximum and minimum values. $\newline$ Amplitude = $(5 - (-5)) / 2 = 10 / 2 = 5$
Find Vertical Shift: Find the vertical shift, $D$ , of the function. $\newline$ The vertical shift is the average of the maximum and minimum values. $\newline$ $D = (5 + (-5)) / 2 = 0 / 2 = 0$
Calculate Period: Calculate the period of the function. $\newline$ The period is the distance between two consecutive maxima or minina. Since we have a maximum at $x=0$ and the next minimum at $x=2\pi$ , the period is $2\pi$ . $\newline$ Period = $2\pi$
Find Value of B: Find the value of $B$ in the function $f(x) = A\cos(Bx + C) + D$ . Since the period is $2\pi$ , we use the formula Period = $(2\pi)/B$ to find $B$ . $2\pi = (2\pi)/B$ $B = 1$
Determine Phase Shift: Determine the phase shift, $C$ , of the function. $\newline$ Since the maximum is at $x=0$ , there is no horizontal shift, so $C = 0$ .
Write Sinusoidal Function: Write the equation of the sinusoidal function using the values found for $A$ , $B$ , $C$ , and $D$ . $A = 5$ , $B = 1$ , $C = 0$ , $D = 0$ $f(x) = 5\cos(x) + 0$ $f(x) = 5\cos(x)$

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