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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). Write the formula of the function, where XX is entered in radians.

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Full solution

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). Write the formula of the function, where XX is entered in radians.
  1. Identify Amplitude: Identify the amplitude of the sinusoidal function.\newlineThe amplitude AA is half the distance between the maximum and minimum values of the function. The maximum value is −4-4 and the minimum value is 1010, so the amplitude is (10−(−4))/2=14/2=7(10 - (-4))/2 = 14/2 = 7.
  2. Determine Vertical Shift: Determine the vertical shift DD of the sinusoidal function.\newlineThe vertical shift is the average of the maximum and minimum values. So, D=(10+(−4))/2=6/2=3D = (10 + (-4))/2 = 6/2 = 3.
  3. Calculate Period: Calculate the period TT of the sinusoidal function.\newlineSince the function goes from a minimum at x=0x=0 to a maximum at x=2x=2, half of the period is 22. Therefore, the full period TT is 2×2=42 \times 2 = 4 radians.
  4. Find Horizontal Shift: Find the horizontal shift CC of the sinusoidal function. Since the minimum occurs at x=0x=0, the horizontal shift CC is 00.
  5. Determine Reflection: Determine the reflection of the sinusoidal function.\newlineSince the function goes from a minimum to a maximum as xx increases, it is a reflection of the standard sine function. This means we will use a negative amplitude.
  6. Write Sinusoidal Function: Write the formula of the sinusoidal function.\newlineThe general form of a sinusoidal function is y=A⋅sin⁡(B(x−C))+Dy = A \cdot \sin(B(x - C)) + D, where B=2πTB = \frac{2\pi}{T}. Since we have a reflection, AA is negative. Plugging in the values we have:\newlineA=−7A = -7, B=2π4=π2B = \frac{2\pi}{4} = \frac{\pi}{2}, C=0C = 0, and D=3D = 3.\newlineSo the formula is y=−7⋅sin⁡(π2x)+3y = -7 \cdot \sin\left(\frac{\pi}{2}x\right) + 3.

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