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The graph of a sinusoidal function has a minimum point at (0,2) and then has a maximum point at (3pi,6). Write the formula of the function, where x is entered in radians. f(x)=◻

The graph of a sinusoidal function has a minimum point at (0,2) (0,2) and then has a maximum point at (3π,6) (3 \pi, 6) .\newlineWrite the formula of the function, where x x is entered in radians.\newlinef(x)=□ f(x)=\square

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Q. The graph of a sinusoidal function has a minimum point at (0,2) (0,2) and then has a maximum point at (3π,6) (3 \pi, 6) .\newlineWrite the formula of the function, where x x is entered in radians.\newlinef(x)=□ f(x)=\square
  1. Calculate Amplitude: Determine the amplitude of the function.\newlineThe amplitude is half the distance between the maximum and minimum values of the function.\newlineAmplitude = (Maximum−Minimum)/2(\text{Maximum} - \text{Minimum}) / 2\newlineAmplitude = (6−2)/2(6 - 2) / 2\newlineAmplitude = 4/24 / 2\newlineAmplitude = 22
  2. Find Vertical Shift: Find the vertical shift, DD. The vertical shift is the average of the maximum and minimum values of the function. D=(Maximum+Minimum)/2D = (\text{Maximum} + \text{Minimum}) / 2 D=(6+2)/2D = (6 + 2) / 2 D=8/2D = 8 / 2 D=4D = 4
  3. Determine Period: Calculate the period of the function.\newlineThe period is the distance between two consecutive minimum or maximum points. Since we have a minimum at x=0x=0 and the next maximum at x=3πx=3\pi, the period is twice this distance.\newlinePeriod = 2×(3π−0)2 \times (3\pi - 0)\newlinePeriod = 6π6\pi
  4. Calculate Value of B: Find the value of B using the period formula for a sinusoidal function. \newlinePeriod = (2π)/B(2\pi) / B\newline6π=(2π)/B6\pi = (2\pi) / B\newlineB=(2π)/(6π)B = (2\pi) / (6\pi)\newlineB=1/3B = 1 / 3
  5. Apply Phase Shift: Since the function has a minimum at x=0x=0, we will use a cosine function with a phase shift to reflect this. A cosine function normally has a maximum at x=0x=0, so we need to shift it by π\pi to make it a minimum.\newlineC=πC = \pi
  6. Write Function Equation: Write the equation of the function using the values found for AA, BB, CC, and DD.
    f(x)=A⋅cos⁡(Bx+C)+Df(x) = A \cdot \cos(Bx + C) + D
    f(x)=2⋅cos⁡(13x+π)+4f(x) = 2 \cdot \cos(\frac{1}{3}x + \pi) + 4

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