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The graph of a sinusoidal function has a maximum point at (0,7) and then intersects its midline at (3,3). 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 \newline(0,7)(0,7) and then intersects its midline at \newline(3,3)(3,3).\newlineWrite the formula of the function, where \newlinexx is entered in radians.\newlinef(x)=f(x)=

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Q. The graph of a sinusoidal function has a maximum point at \newline(0,7)(0,7) and then intersects its midline at \newline(3,3)(3,3).\newlineWrite the formula of the function, where \newlinexx is entered in radians.\newlinef(x)=f(x)=
  1. Determine Amplitude: Determine the amplitude AA of the sinusoidal function.\newlineThe amplitude is the distance from the midline to the maximum point.\newlineSince the maximum point is at (0,7)(0,7) and the midline is at y=3y=3, the amplitude AA is 7−3=47 - 3 = 4.
  2. Determine Vertical Shift: Determine the vertical shift DD. The vertical shift DD is the yy-coordinate of the midline. Since the midline is at y=3y=3, DD is 33.
  3. Determine Period: Determine the period TT of the sinusoidal function.\newlineSince the function intersects its midline at (3,3)(3,3) after reaching its maximum, this point represents a quarter of the period.\newlineTherefore, the full period TT is 3×4=123 \times 4 = 12.
  4. Determine Value of B: Determine the value of B in the function.\newlineThe value of B is related to the period T by the formula B=2πTB = \frac{2\pi}{T}.\newlineSubstitute T=12T = 12 into the formula to find B.\newlineB=2π12B = \frac{2\pi}{12}\newlineB=π6B = \frac{\pi}{6}
  5. Determine Phase Shift: Determine the phase shift CC. Since the maximum point is at (0,7)(0,7), there is no horizontal shift, and thus C=0C = 0.
  6. Write Sinusoidal Function: Write the equation of the sinusoidal function.\newlineSubstitute the values of AA, BB, CC, and DD into the general form f(x)=Acos⁡(Bx+C)+Df(x) = A \cos(Bx + C) + D.\newlinef(x)=4cos⁡(π6x+0)+3f(x) = 4 \cos(\frac{\pi}{6} x + 0) + 3

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