The graph of a sinusoidal function has a maximum point at $\newline$ $(0,7)$ and then intersects its midline at $\newline$ $(3,3)$ . $\newline$ Write the formula of the function, where $\newline$ $x$ is entered in radians. $\newline$ $f(x)=$

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

\newline

(0,7)

and then intersects its midline at

\newline

(3,3)

\newline

Write the formula of the function, where

\newline

x

is entered in radians.

\newline

f(x)=

Determine Amplitude: Determine the amplitude $A$ of the sinusoidal function. $\newline$ The amplitude is the distance from the midline to the maximum point. $\newline$ Since the maximum point is at $(0,7)$ and the midline is at $y=3$ , the amplitude $A$ is $7 - 3 = 4$ .
Determine Vertical Shift: Determine the vertical shift $D$ . The vertical shift $D$ is the $y$ -coordinate of the midline. Since the midline is at $y=3$ , $D$ is $3$ .
Determine Period: Determine the period $T$ of the sinusoidal function. $\newline$ Since the function intersects its midline at $(3,3)$ after reaching its maximum, this point represents a quarter of the period. $\newline$ Therefore, the full period $T$ is $3 \times 4 = 12$ .
Determine Value of B: Determine the value of B in the function. $\newline$ The value of B is related to the period T by the formula $B = \frac{2\pi}{T}$ . $\newline$ Substitute $T = 12$ into the formula to find B. $\newline$ $B = \frac{2\pi}{12}$ $\newline$ $B = \frac{\pi}{6}$
Determine Phase Shift: Determine the phase shift $C$ . Since the maximum point is at $(0,7)$ , there is no horizontal shift, and thus $C = 0$ .
Write Sinusoidal Function: Write the equation of the sinusoidal function. $\newline$ Substitute the values of $A$ , $B$ , $C$ , and $D$ into the general form $f(x) = A \cos(Bx + C) + D$ . $\newline$ $f(x) = 4 \cos(\frac{\pi}{6} x + 0) + 3$