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The graph of a sinusoidal function has a maximum point at (0,5) and then has a minimum point at (2pi,-5). Write the formula of the function, where x is entered in radians. f(x)=

The graph of a sinusoidal function has a maximum point at (0,5) (0,5) and then has a minimum point at (2π,−5) (2 \pi,-5) .\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 has a maximum point at (0,5) (0,5) and then has a minimum point at (2π,−5) (2 \pi,-5) .\newlineWrite the formula of the function, where x x is entered in radians.\newlinef(x)= f(x)=
  1. Amplitude Calculation: The amplitude AA of the sinusoidal function is half the distance between the maximum and minimum values. The maximum value is 55 and the minimum value is −5-5, so the amplitude is (5−(−5))/2=10/2=5(5 - (-5)) / 2 = 10 / 2 = 5.
  2. Period Calculation: The period (T) of the sinusoidal function is the distance between a maximum point and the next minimum point. Since the maximum is at x = 00 and the minimum is at x = 2π2\pi, the period is 2π−0=2π2\pi - 0 = 2\pi.
  3. Value of B Calculation: The value of B in the function f(x)=Acos⁡(Bx+C)+Df(x) = A\cos(Bx + C) + D is related to the period by the formula B=2πTB = \frac{2\pi}{T}. Since T=2πT = 2\pi, B=2π2π=1B = \frac{2\pi}{2\pi} = 1.
  4. Vertical Shift Calculation: Since the maximum point is at (0,5)(0,5), we know that the vertical shift (D)(D) is equal to the maximum value, which is 55.
  5. Horizontal Shift Calculation: The function starts at a maximum point when x=0x = 0, which means there is no horizontal shift (C=0C = 0) and the cosine function is appropriate to use because cos⁡(0)=1\cos(0) = 1.
  6. Final Equation: Substituting A=5A = 5, B=1B = 1, C=0C = 0, and D=5D = 5 into the f(x)=Acos⁡(Bx+C)+Df(x) = A\cos(Bx + C) + D gives us the final equation f(x)=5cos⁡(x)+5f(x) = 5\cos(x) + 5.

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