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The graph of a sinusoidal function has a minimum point at (0,-10) and then has a maximum point at (2,-4). Write the formula of the function, where x is entered in radians.

The graph of a sinusoidal function has a minimum point at (0,−10) (0,-10) and then has a maximum point at (2,−4) (2,-4) .\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,−10) (0,-10) and then has a maximum point at (2,−4) (2,-4) .\newlineWrite the formula of the function, where x x is entered in radians.\newlinef(x)=□ f(x)=\square
  1. Calculate Amplitude: We need to determine the amplitude, period, phase shift, and vertical shift of the sinusoidal function. The amplitude AA is half the distance between the maximum and minimum values. \newlineCalculation: A=(Max−Min)2=(−4−(−10))2=62=3A = \frac{(\text{Max} - \text{Min})}{2} = \frac{(-4 - (-10))}{2} = \frac{6}{2} = 3
  2. Calculate Vertical Shift: The vertical shift DD is the average of the maximum and minimum values.\newlineCalculation: D=(Max+Min)2=(−4+(−10))2=−142=−7D = \frac{(\text{Max} + \text{Min})}{2} = \frac{(-4 + (-10))}{2} = -\frac{14}{2} = -7
  3. Calculate Period: The period TT of the function is the distance between two consecutive minimum or maximum points. Since we only have one minimum and one maximum, we can infer that the period is twice the distance between these points.\newlineCalculation: T=2×(2−0)=4T = 2 \times (2 - 0) = 4 radians
  4. Calculate Phase Shift: The phase shift CC is the x-coordinate of the minimum point, which is 00 in this case. So, there is no phase shift.\newlineCalculation: C=0C = 0
  5. Determine Cosine Function: Since the minimum occurs at (0,−10)(0, -10) and not the maximum, we know that the sinusoidal function is a cosine function reflected over the xx-axis.\newlineThe general form of a reflected cosine function is:\newlinef(x)=−Acos⁡(B(x−C))+Df(x) = -A\cos(B(x - C)) + D,\newlinewhere BB is related to the period by the formula B=2πTB = \frac{2\pi}{T}.
    Calculation: B=2πT=2π4=π2B = \frac{2\pi}{T} = \frac{2\pi}{4} = \frac{\pi}{2}
  6. Write Function Formula: Now we can write the formula of the function using the values we have found for AA, BB, CC, and DD.\newlineThe formula is f(x)=−3cos⁡(π2×(x−0))−7f(x) = -3\cos(\frac{\pi}{2} \times (x - 0)) - 7.

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