In Exercises 17–30, determine the amplitude, period, and phase shift of each function. Then graph one period of the function.y = 1/2 sin(x + π/2)

Verified step by step guidance

1. Identify the standard form of the sine function: \( y = a \sin(bx + c) + d \).
2. Determine the amplitude by identifying the coefficient \( a \). In this case, \( a = \frac{1}{2} \), so the amplitude is \( \frac{1}{2} \).
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 \( -\frac{c}{b} \). With \( c = \frac{\pi}{2} \) and \( b = 1 \), the phase shift is \( -\frac{\pi}{2} \).
5. Graph one period of the function by starting at the phase shift \( -\frac{\pi}{2} \) and ending at \( -\frac{\pi}{2} + 2\pi \), marking key points such as maximum, minimum, and intercepts based on the amplitude and period.

Key Concepts

Amplitude

Amplitude refers to the maximum height of a wave from its midline. In the context of sine functions, it is determined by the coefficient in front of the sine term. For the function y = 1/2 sin(x + π/2), the amplitude is 1/2, indicating that the wave oscillates between 1/2 and -1/2.

Period

The period of a trigonometric function is the length of one complete cycle of the wave. For sine functions, the standard period is 2π. However, if the function includes a coefficient affecting the x variable, the period is calculated as 2π divided by that coefficient. In this case, the period remains 2π since there is no coefficient affecting x.

Phase Shift

Phase shift refers to the horizontal displacement of a wave from its standard position. It is determined by the value added or subtracted from the x variable inside the function. For y = 1/2 sin(x + π/2), the phase shift is -π/2, meaning the graph is shifted to the left by π/2 units.