Final answer:
The function f(x) = 2(cos(x−π/4))² has an amplitude of 2, a period of 2π, and a midline at y = 0, not π/4 as none of the provided choices directly match these correct values.
Step-by-step explanation:
The function given is f(x) = 2(cos(x−π/4))². To determine its amplitude, period, and midline, we analyze the form of the cosine term. The function can be rewritten as f(x) = 2[cos²(x - π/4)]. This represents a cosine function that is squared and then scaled in amplitude by a factor of 2.The amplitude of a function in the form A*cos(kx) is given by the absolute value of A. However, since our function is squared, the output of the cosine term will always be non-negative, so the amplitude is the maximal value that the squared term can reach, which would be when cos(x - π/4) = ± 1. So, the amplitude, in this case, is 2.The period of a function in the form A*cos(kx) is given by 2π/k. As the inside of the cosine function isn't scaled in our example (no k value to influence the period), the period remains the standard period of the cosine function, which is 2π.
The midline of the function is the horizontal line that the function oscillates around. Since the cosine function is squared and the minimum value is 0 (the graph does not dip below the x-axis), the midline doesn't deviate from y=0. Therefore, the midline is y = 0, not π/4 as suggested in the question choices.The given wave function can be written in the form y(x, t) = A sin(kx - wt + p), where A represents the amplitude, k represents the wave number, w represents the angular frequency, and p represents the phase shiftIn this case, the amplitude of the wave is the coefficient in front of the sine function. So, the amplitude is 2.The period of the wave can be calculated using the formula T = 2π/w. From the given function, we can see that w is equal to 1, so the period T is 2π/1 = 2π.The midline of the wave can be determined by finding the average value of the wave function, which is the y-value that the graph oscillates around. In this case, the midline is given by the constant term in the function, which is π/4.Therefore, the correct answer is option a) Amplitude: 2, Period: 2π, Midline: π/4.So, the correct choice is Amplitude: 2, Period: 2π, Midline: y = 0.