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The graph of a sinusoidal function has a minimum point at (0,3) and then intersects its midline at (5pi,5). 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 (0,3) (0,3) and then intersects its midline at (5π,5) (5 \pi, 5) .\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 (0,3) (0,3) and then intersects its midline at (5π,5) (5 \pi, 5) .\newlineWrite the formula of the function, where x x is entered in radians.\newlinef(x)=□ f(x)=\square
  1. Determine Amplitude: Determine the amplitude AA of the function.\newlineSince the graph has a minimum point at (0,3)(0,3) and intersects its midline at a higher value (5π,5)(5\pi,5), the amplitude is half the distance between the midline and the minimum point.\newlineMidline value = 55\newlineMinimum point value = 33\newlineAmplitude AA = Midline−Minimum point2\frac{\text{Midline} - \text{Minimum point}}{2}\newlineA=5−32A = \frac{5 - 3}{2}\newlineA=22A = \frac{2}{2}\newlineA=1A = 1
  2. Find Vertical Shift: Find the vertical shift DD of the function.\newlineThe midline value gives us the vertical shift.\newlineD=D = Midline value\newlineD=5D = 5
  3. Determine Period: Determine the period TT of the function.\newlineSince the function intersects its midline at (5π,5)(5\pi,5), and knowing that this happens at a quarter of the period for a sinusoidal function, we can find the period.\newlineT=4×(x-value at midline intersection)T = 4 \times (\text{x-value at midline intersection})\newlineT=4×5πT = 4 \times 5\pi\newlineT=20πT = 20\pi
  4. Calculate Value of B: Calculate the value of B, which is related to the period TT by the formula T=2πBT = \frac{2\pi}{B}.
    B=2πTB = \frac{2\pi}{T}
    B=2π20πB = \frac{2\pi}{20\pi}
    B=110B = \frac{1}{10}
  5. Determine Function Type: Since the function has a minimum at x=0x = 0, we can determine that the function is a cosine function shifted by π/2\pi/2 to the right, making it a sine function. Therefore, C=π/2C = \pi/2. However, because the minimum is at x=0x = 0, we actually need to use a negative cosine function to represent the graph correctly. This means that C=0C = 0 for a negative cosine function.
  6. Write Equation: Write the equation of the function using the values of AA, BB, CC, and DD.
    f(x)=A⋅cos⁡(Bx+C)+Df(x) = A \cdot \cos(Bx + C) + D
    f(x)=−1⋅cos⁡(110x+0)+5f(x) = -1 \cdot \cos(\frac{1}{10}x + 0) + 5
    f(x)=−cos⁡(110x)+5f(x) = -\cos(\frac{1}{10}x) + 5

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