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The graph of a sinusoidal function intersects its midline at
(0,-7) and then has a minimum point at
((pi)/(4),-9).
Write the formula of the function, where
x is entered in radians.

The graph of a sinusoidal function intersects its midline at $(0,-7)$ and then has a minimum point at $\left(\frac{\pi}{4},-9\right)$ . $\newline$ 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 intersects its midline at

(0,-7)

and then has a minimum point at

\left(\frac{\pi}{4},-9\right)

.

\newline

Write the formula of the function, where

x

is entered in radians.

Determine Amplitude: Determine the amplitude of the function. $\newline$ Since the midline is at $y = -7$ and the minimum point is at $y = -9$ , the amplitude is the distance from the midline to the minimum, which is $2$ units. $\newline$ Amplitude ( $A$ ) = $|\text{midline} - \text{minimum}| = |-7 - (-9)| = 2$
Identify Midline: Identify the midline $D$ of the function. $\newline$ The midline is given as $y = -7$ . $\newline$ $D = -7$
Determine Period: Determine the period $T$ of the function. $\newline$ Since the function reaches its minimum at $\frac{\pi}{4}$ and the next similar point (maximum or minimum) would be after a full period, we can infer that the period is $4$ times the $x$ -coordinate of the minimum point. $\newline$ $T = 4 \times \left(\frac{\pi}{4}\right) = \pi$
Calculate Value of B: Calculate the value of B using the period. $\newline$ The period $T$ is related to $B$ by the formula $T = \frac{2\pi}{B}$ . $\newline$ $\pi = \frac{2\pi}{B}$ $\newline$ $B = 2$
Determine Phase Shift: Determine the phase shift $C$ . Since the function intersects its midline at $(0, -7)$ , there is no horizontal shift, so the phase shift $C$ is $0$ . $C = 0$
Choose Sinusoidal Function: Choose the correct sinusoidal function. $\newline$ Because the function has a minimum point at $(\pi/4, -9)$ , we should use a cosine function that starts at a maximum. However, since we have a minimum point, we need to reflect the cosine function over the midline. This means we will use a negative cosine function. $\newline$ $f(x) = -A\cos(Bx + C) + D$
Write Final Equation: Write the final equation of the sinusoidal function. $\newline$ Substitute the values of $A$ , $B$ , $C$ , and $D$ into the equation. $\newline$ $f(x) = -2\cos(2x) - 7$

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