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

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

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Identify Midline: Identify the midline of the sinusoidal function. The midline is given by the y-coordinate of the point where the graph intersects it, which is -2.Determine Amplitude: Determine the amplitude of the sinusoidal function. The amplitude is the distance from the midline to the maximum or minimum point of the function. Since we have a minimum point at $\left(\dfrac{3\pi}{2},-7\right)$, the amplitude is $|-7 - (-2)| = 5$.Find Period: Find the period of the sinusoidal function. Since we only have one point and the midline, we cannot determine the period directly from the given information. However, we can assume that the minimum point occurs at the end of one period for a cosine function, which typically starts at a maximum. This would mean the period is $2\pi$ because the minimum occurs at $\dfrac{3\pi}{2}$, which is $3/4$ of the way through the period of a typical cosine function.Write Equation: Write the equation of the sinusoidal function. Since the function has a minimum point at $\left(\dfrac{3\pi}{2},-7\right)$ and the midline is at -2, we can assume the function is a cosine function that has been reflected vertically. The general form of a cosine function is $y = A\cos(B(x - C)) + D$, where A is the amplitude, B is related to the period by $B = \dfrac{2\pi}{\text{period}}$, C is the phase shift, and D is the vertical shift (midline). We have A = 5, B = 1 (since the period is $2\pi$), C = 0 (since there is no horizontal shift mentioned), and D = -2. Therefore, the equation is $y = -5\cos(x) - 2$.

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

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