Find an equation for a sinusoidal function that has period 3𝜋, amplitude 2, and contains the point 3𝜋2,–2.Write your answer in the form f(x)=Acos(Bx+C)+D, where A, B, C, and D are real numbers.
Question
Find an equation for a sinusoidal function that has period 3𝜋, amplitude 2, and contains the point 3𝜋2,–2.Write your answer in the form f(x)=Acos(Bx+C)+D, where A, B, C, and D are real numbers.
Solution
The general form of a sinusoidal function is f(x) = A cos(Bx + C) + D, where:
- A is the amplitude
- B determines the period
- C is the phase shift
- D is the vertical shift
Given that the amplitude is 2, we have A = 2.
The period of the function is given by 2π/B. We know the period is 3π, so we can set up the equation 2π/B = 3π and solve for B. This gives us B = 2/3.
The function contains the point (3π/2, -2). We can use this information to find C and D.
Substitute x = 3π/2 and f(x) = -2 into the equation:
-2 = 2 cos[(2/3)*(3π/2) + C] + D -2 = 2 cos[π + C] + D -2 = -2 cos[C] + D
This tells us that D = 0 and C = π.
So, the equation of the sinusoidal function is f(x) = 2 cos[(2/3)x + π].
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