Find an equation for a sinusoidal function that has period 2π, amplitude 1, and contains the point (−2π,−1). Write your answer in the form f(x)=Asin(Bx+C)+D, where A, B, C, and D are real numbers. `f(x)` =______
Q. Find an equation for a sinusoidal function that has period 2π, amplitude 1, and contains the point (−2π,−1). Write your answer in the form f(x)=Asin(Bx+C)+D, where A, B, C, and D are real numbers. `f(x)` =______
Period formula: The period of a sinusoidal function is given by the formula 2π/B, where B is the frequency. Since we are given that the period is 2π, we can find B by setting 2π/B equal to 2π and solving for B.
Finding B: Solving 2π/B=2π gives us B=1 because any number divided by itself is 1.
Amplitude: The amplitude of the function is given as 1, which means A=1.
Phase shift: To find the phase shift C, we need to consider the point (−2π,−1). Since the amplitude is 1, the sinusoidal function at its minimum value (−1) occurs at −2π for the sine function. Normally, the sine function has its minimum value at −23π, −2π, 25π, etc. Since −2π is already a standard point for the minimum of the sine function, we do not need a phase shift, so C=0.
Vertical shift: The vertical shift D can be determined by looking at the value of the function at the given point. Since the amplitude is 1 and the function value at x=−2π is −1, which is the minimum value of the sine function, there is no vertical shift. Therefore, D=0.
Final equation: Putting all the values together, we get the equation of the sinusoidal function: f(x)=Asin(Bx+C)+D, which becomes f(x)=1sin(1x+0)+0, or simply f(x)=sin(x).
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