Find an equation for a sinusoidal function that has period 2π, amplitude 1, and contains the point (0,−1). Write your answer in the form f(x)=Acos(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 (0,−1). Write your answer in the form f(x)=Acos(Bx+C)+D, where A, B, C, and D are real numbers. f(x)= _____
Find Amplitude: Write the amplitude A of the given problem.The amplitude is the absolute value of the coefficient in front of the cosine function.A=1
Determine Period and B: Determine the period T and find the value of B. The period T is given as 2π. The value of B determines the period of the sinusoidal function according to the formula T=B2π. B=T2πB=2π2πB=1
Solve for D: Solve for D. Substitute values of x and f(x) into f(x)=Acos(Bx+C)+D. Since the function contains the point (0,−1), we have: −1=Acos(B(0)+C)+D−1=1cos(0+C)+D−1=cos(C)+D Since cos(C) has a maximum value of 1, and we need f(0) to be f(x)0, cos(C) must be at its maximum when f(x)2. This means f(x)3 must be an even multiple of f(x)4. However, to get a negative value, we need to shift the cosine function by f(x)4. Therefore, f(x)6. f(x)7f(x)8f(x)9
Write Sinusoidal Function: Write the equation of the sinusoidal function.We have found:A=1B=1C=πD=0Substitute values of A, B, C, and D into f(x)=Acos(Bx+C)+D.f(x)=1cos(1x+π)+0B=10
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