In Exercises 43–52, determine the amplitude, period, and phase shift of each function. Then graph one period of the function.y = cos(x − π/2)

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1. Identify the standard form of the cosine function: \( y = a \cos(bx - c) + d \).
2. Determine the amplitude by identifying the coefficient \( a \). In this case, \( a = 1 \), so the amplitude is \( |1| = 1 \).
3. Find the period of the function using the formula \( \frac{2\pi}{b} \). Here, \( b = 1 \), so the period is \( 2\pi \).
4. Calculate the phase shift using \( \frac{c}{b} \). Here, \( c = \pi/2 \) and \( b = 1 \), so the phase shift is \( \frac{\pi/2}{1} = \pi/2 \). The shift is to the right.
5. Graph one period of the function by starting at the phase shift \( \pi/2 \) and using the period \( 2\pi \) to determine the end of one cycle. Plot key points such as maximum, minimum, and intercepts based on the amplitude and phase shift.

Key Concepts

Amplitude

Amplitude refers to the maximum height of a wave from its midline. In the context of trigonometric functions like cosine, it indicates how far the graph reaches above and below the midline (y=0). For the function y = cos(x - π/2), the amplitude is 1, as the coefficient of the cosine function is 1, meaning the graph oscillates between 1 and -1.

Period

The period of a trigonometric function is the length of one complete cycle of the wave. For the cosine function, the standard period is 2π. In the function y = cos(x - π/2), there is no coefficient affecting the x variable, so the period remains 2π, indicating that the graph will repeat every 2π units along the x-axis.

Phase Shift

Phase shift refers to the horizontal shift of the graph of a function. It is determined by the value subtracted from the x variable inside the function. In y = cos(x - π/2), the phase shift is π/2 units to the right, as the function is shifted from the standard cosine function, which starts at (0, 1), to the new starting point at (π/2, 1).