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The graph of a sinusoidal function has a maximum point at (0,10) and then intersects its midline at ((pi)/(4),4). 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 (0,10) (0,10) and then intersects its midline at (π4,4) \left(\frac{\pi}{4}, 4\right) .\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 (0,10) (0,10) and then intersects its midline at (π4,4) \left(\frac{\pi}{4}, 4\right) .\newlineWrite the formula of the function, where x x is entered in radians.\newlinef(x)= f(x)=
  1. Identify Maximum Point: The maximum point of the sinusoidal function is at (0,10)(0,10), which means the amplitude (A)(A) is the distance from the midline to the maximum, and the vertical shift (D)(D) is the midline value. Since the maximum value is 1010 and it occurs at x=0x=0, we can infer that the function is a cosine function that has been shifted up.
  2. Determine Midline and Amplitude: The midline of the function is at y=4y=4, which is the average of the maximum and minimum values. Since we have a maximum at y=10y=10, the amplitude is 10−4=610 - 4 = 6. Therefore, A=6A = 6.
  3. Calculate Period: The function intersects its midline at (π4,4)\left(\frac{\pi}{4},4\right). Since the midline is at y=4y=4 and this is a cosine function, this point represents a quarter of the period of the function. Therefore, the full period (P)(P) is 4×(π4)=π4 \times \left(\frac{\pi}{4}\right) = \pi. The period of a cosine function is given by 2π/B2\pi/B, so B=2πP=2ππ=2B = \frac{2\pi}{P} = \frac{2\pi}{\pi} = 2.
  4. Find Horizontal Shift: Since the function is a cosine function with a maximum at x=0x=0, there is no horizontal shift, so C=0C = 0.
  5. Finalize Function Equation: We have determined that A=6A = 6, B=2B = 2, C=0C = 0, and the midline D=4D = 4. The equation of the function is therefore f(x)=Acos⁡(Bx+C)+Df(x) = A\cos(Bx + C) + D.
  6. Finalize Function Equation: We have determined that A=6A = 6, B=2B = 2, C=0C = 0, and the midline D=4D = 4. The equation of the function is therefore f(x)=Acos⁡(Bx+C)+Df(x) = A\cos(Bx + C) + D.Substituting the values of AA, BB, CC, and DD into the equation, we get f(x)=6cos⁡(2x)+4f(x) = 6\cos(2x) + 4.

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