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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.

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.

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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.
  1. Determine Amplitude: Determine the amplitude of the function.\newlineSince the graph has a minimum point at (0,3)(0,3) and intersects its midline at a higher value, the midline value is the average of the maximum and minimum. The given midline value is 55, and the minimum value is 33, so the amplitude (AA) is the difference between the midline and the minimum value.\newlineA=Midline−MinimumA = \text{Midline} - \text{Minimum}\newlineA=5−3A = 5 - 3\newlineA=2A = 2
  2. Identify Vertical Shift: Identify the vertical shift DD. The midline value also represents the vertical shift of the function. Since the midline is at y=5y = 5, the vertical shift DD is 55. D=5D = 5
  3. Calculate Period: Calculate the period of the function.\newlineThe function intersects its midline at (5π,5)(5\pi,5), which is a quarter of the period away from the minimum point at (0,3)(0,3). Therefore, the period TT is four times the x-value of the midline intersection.\newlineT=4×(5π)T = 4 \times (5\pi)\newlineT=20πT = 20\pi
  4. Find Value of B: Find the value of B using the period.\newlineThe period TT is related to BB by the formula T=2πBT = \frac{2\pi}{B}. We can solve for BB using the period we found.\newlineT=2πBT = \frac{2\pi}{B}\newline20π=2πB20\pi = \frac{2\pi}{B}\newlineB=2π20πB = \frac{2\pi}{20\pi}\newlineB=110B = \frac{1}{10}
  5. Determine Phase Shift: Determine the phase shift CC. Since the minimum point is at (0,3)(0,3), and we know that the cosine function starts at a maximum, we will use a sine function which starts at a minimum when there is no phase shift. Therefore, C=0C = 0. C=0C = 0
  6. Write Equation: Write the equation of the function.\newlineWe have determined the amplitude AA, the vertical shift DD, the value of BB, and the phase shift CC. The general form of a sinusoidal function is f(x)=A⋅cos⁡(Bx+C)+Df(x) = A \cdot \cos(Bx + C) + D or f(x)=A⋅sin⁡(Bx+C)+Df(x) = A \cdot \sin(Bx + C) + D. Since we are using a sine function that starts at a minimum, we will use the sine form.\newlinef(x)=A⋅sin⁡(Bx+C)+Df(x) = A \cdot \sin(Bx + C) + D\newlinef(x)=2⋅sin⁡(110x+0)+5f(x) = 2 \cdot \sin(\frac{1}{10}x + 0) + 5

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