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The graph of a sinusoidal function intersects its midline at $(0,-2)$ and then has a minimum point at $(\frac{3pi}{2},-7)$ . 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

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

(0,-2)

and then has a minimum point at

(\frac{3pi}{2},-7)

. Write the formula of the function, where

x

is entered in radians.

Determine Parameters: We need to determine the amplitude, midline, period, and phase shift of the sinusoidal function. The midline is given by the $y$ -coordinate of the point where the graph intersects the midline, which is $-2$ . This gives us the value of $D$ in the general equation $f(x) = A\cos(Bx + C) + D$ .
Find Amplitude: The minimum point of the function is at $(3\pi/2, -7)$ . Since the minimum point is $5$ units below the midline (from $-2$ to $-7$ ), the amplitude of the function is $5$ . This gives us the value of $A$ in the general equation.
Deduce Period: The period of the function is not directly given, but we can infer it from the fact that the minimum point occurs at $3\pi/2$ . Since a minimum point of a cosine function occurs at $3\pi/2 + 2\pi k$ , where $k$ is an integer, we can deduce that the period is $2\pi$ . This gives us the value of $B$ in the general equation, which is $1$ because the period $T$ is given by $T = 2\pi/B$ , so $B = 2\pi/T = 2\pi/2\pi = 1$ .
Calculate Phase Shift: The phase shift $C$ can be determined by the fact that the graph intersects its midline at $x = 0$ . For a cosine function, this would normally happen at $x = \frac{\pi}{2} + 2\pi k$ , but since it happens at $x = 0$ , we can deduce that there is a phase shift of $-\frac{\pi}{2}$ . This gives us the value of $C$ in the general equation.
Write Sinusoidal Function: Now we can write the equation of the sinusoidal function using the values of $A$ , $B$ , $C$ , and $D$ that we have found. The equation is: $\newline$ $f(x) = A\cos(Bx + C) + D$ $\newline$ $f(x) = 5\cos(1x - \frac{\pi}{2}) - 2$

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