The graph of a sinusoidal function intersects its midline at $(0,-6)$ and then has a minimum point at $(2.5,-9)$ . $\newline$ Write the formula of the function, where $x$ is entered in radians.

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Q. The graph of a sinusoidal function intersects its midline at

(0,-6)

and then has a minimum point at

(2.5,-9)

\newline

Write the formula of the function, where

x

is entered in radians.

Identify Midline: Identify the midline of the sinusoidal function. The midline is the horizontal line that the function oscillates around. Given that the function intersects the midline at $(0, -6)$ , the midline equation is $y = -6$ .
Determine Amplitude: Determine the amplitude of the function. $\newline$ The amplitude is the distance from the midline to a maximum or minimum point. Since we have a minimum point at $(2.5, -9)$ , and the midline is at $y = -6$ , the amplitude is $|-9 - (-6)| = 3$ .
Find Period: Find the period of the function. $\newline$ Since we only have information about one minimum point, we cannot directly find the period. However, we can assume that this minimum point is the first minimum after the function intersects the midline. For a sine function, this would occur a quarter period after the midline intersection. Therefore, the period $T$ is $4$ times the distance from the midline intersection to the minimum point, which is $4 \times 2.5 = 10$ radians.
Determine Phase Shift: Determine the phase shift of the function. The phase shift is the horizontal shift of the function from its standard position. Since the function intersects the midline at $(0, -6)$ , there is no horizontal shift, and the phase shift is $0$ .
Identify Vertical Shift: Identify the vertical shift of the function. The vertical shift is the value of the midline, which we have already determined to be $-6$ .
Choose Sinusoidal Function: Choose the correct sinusoidal function. $\newline$ Since the function has a minimum point at $(2.5, -9)$ , we should use the cosine function, which has a minimum point at the start of its period. However, because the minimum occurs after a quarter period, we will use the negative cosine function to reflect this.
Write Function Formula: Write the formula of the sinusoidal function. $\newline$ The general form of a sinusoidal function is $y = A \cdot \cos(B(x - C)) + D$ , where $A$ is the amplitude, $B$ is related to the period by $B = \frac{2\pi}{T}$ , $C$ is the phase shift, and $D$ is the vertical shift. Plugging in the values we have: $\newline$ $A = 3$ (amplitude) $\newline$ $B = \frac{2\pi}{T} = \frac{2\pi}{10} = \frac{\pi}{5}$ (since the period $T$ is $10$ radians) $\newline$ $A$ $0$ (phase shift) $\newline$ $A$ $1$ (vertical shift) $\newline$ So the formula is $A$ $2$ .