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The graph of a sinusoidal function intersects its midline at (0,-2) and then has a minimum point at ((3pi)/(2),-7). Write the formula of the function, where x is entered in radians. f(x)=

The graph of a sinusoidal function intersects its midline at (0,−2) (0,-2) and then has a minimum point at (3π2,−7) \left(\frac{3 \pi}{2},-7\right) .\newlineWrite the formula of the function, where x x is entered in radians.\newlinef(x)= f(x)=

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Q. The graph of a sinusoidal function intersects its midline at (0,−2) (0,-2) and then has a minimum point at (3π2,−7) \left(\frac{3 \pi}{2},-7\right) .\newlineWrite the formula of the function, where x x is entered in radians.\newlinef(x)= f(x)=
  1. Determine Midline: Determine the midline of the sinusoidal function. The midline is the horizontal line that the function oscillates around. Since the function intersects the midline at (0,−2)(0, -2), the midline is y=−2y = -2.
  2. Determine Amplitude: Determine the amplitude of the function.\newlineThe amplitude is the distance from the midline to the maximum or minimum point of the function. Since the minimum point is at (3π2,−7)\left(\frac{3\pi}{2}, -7\right), and the midline is at y=−2y = -2, the amplitude is ∣−7−(−2)∣=5|-7 - (-2)| = 5.
  3. Determine Period: Determine the period of the function.\newlineSince the minimum point occurs at x=3π2x = \frac{3\pi}{2}, and this is the first minimum point after the function intersects the midline, we can infer that the period is twice this value. Therefore, the period is 2×3π2=3π2 \times \frac{3\pi}{2} = 3\pi.
  4. Determine Value of B: Determine the value of BB in the function f(x)=Acos⁡(Bx+C)+Df(x) = A\cos(Bx + C) + D. The period of a sinusoidal function is given by 2πB\frac{2\pi}{B}. Since the period is 3π3\pi, we have 2πB=3π\frac{2\pi}{B} = 3\pi. Solving for BB gives B=2π3π=23B = \frac{2\pi}{3\pi} = \frac{2}{3}.
  5. Determine Phase Shift: Determine the phase shift, CC, of the function.\newlineSince the function intersects the midline at x=0x = 0, there is no horizontal shift, so C=0C = 0.
  6. Write Function Equation: Write the equation of the sinusoidal function.\newlineWe have determined the following values:\newlineAmplitude A=5A = 5\newlineB=23B = \frac{2}{3}\newlineC=0C = 0\newlineMidline D=−2D = -2\newlineThe function is a cosine function that has been vertically shifted down, so it will be of the form f(x)=Acos⁡(Bx+C)+Df(x) = A\cos(Bx + C) + D. Substituting the values we found, we get f(x)=5cos⁡(23x)−2f(x) = 5\cos\left(\frac{2}{3}x\right) - 2.

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