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

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

(0,7)

and then intersects its midline at

(3,3)

\newline

Write the formula of the function, where

x

is entered in radians.

\newline

f(x)=

Given Points Information: We are given a maximum point at $(0,7)$ and a midline intersection at $(3,3)$ . The maximum point gives us the amplitude and the vertical shift, while the midline intersection gives us the horizontal shift and the period.
Calculate Amplitude: The amplitude $A$ is the distance from the midline to the maximum point. Since the midline is at $y = 3$ and the maximum is at $y = 7$ , the amplitude is $7 - 3 = 4$ .
Calculate Vertical Shift: The vertical shift $D$ is the $y$ -value of the midline, which is $3$ .
Determine Sinusoidal Function Form: The sinusoidal function will have the form $f(x) = A \cdot \sin(B(x - C)) + D$ or $f(x) = A \cdot \cos(B(x - C)) + D$ . Since the maximum point is at $x = 0$ , we will use the cosine function, which has a maximum at $x = 0$ when there is no horizontal shift ( $C = 0$ ).
Find Period of Function: Now we need to find the period $T$ of the function. The graph intersects the midline at $(3,3)$ , which is a quarter of the period after the maximum. Therefore, the full period is $4$ times this value, so $T = 4 \times 3 = 12$ radians.
Calculate B Value: The value $B$ in the function $f(x) = A \cdot \cos(B(x - C)) + D$ is related to the period by the formula $B = \frac{2\pi}{T}$ . So $B = \frac{2\pi}{12} = \frac{\pi}{6}$ .
Final Sinusoidal Function: Putting all the values together, we get the sinusoidal function: $f(x) = 4 \cdot \cos\left(\frac{\pi}{6} \cdot (x - 0)\right) + 3$ .