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)

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1. Identify the general form of the sine function: \(y = A \sin(B(x - C))\), where \(A\) is the amplitude, \(\frac{2\pi}{B}\) is the period, and \(C\) is the phase shift.
2. Rewrite the given function \(y = \frac{1}{2} \sin(x + \frac{\pi}{2})\) in the form \(y = A \sin(B(x - C))\). Notice that \(x + \frac{\pi}{2}\) can be written as \(x - (-\frac{\pi}{2})\), so \(C = -\frac{\pi}{2}\).
3. Determine the amplitude \(A\) by looking at the coefficient in front of the sine function. Here, \(A = \frac{1}{2}\), which means the graph oscillates between \(\frac{1}{2}\) and \(-\frac{1}{2}\).
4. Find the period by identifying \(B\). Since the function is \(\sin(x)\), \(B = 1\), so the period is \(\frac{2\pi}{B} = 2\pi\).
5. Determine the phase shift \(C\), which is \(-\frac{\pi}{2}\). This means the graph is shifted to the left by \(\frac{\pi}{2}\). Use this information to sketch one full period of the sine wave starting at \(x = -\frac{\pi}{2}\).

Key Concepts

Amplitude of a Trigonometric Function

Amplitude is the maximum absolute value of a sine or cosine function, representing the height from the midline to the peak. For y = (1/2) sin(x + π/2), the amplitude is 1/2, indicating the wave oscillates between -1/2 and 1/2.

Period of a Sine Function

The period is the length of one complete cycle of the sine wave, calculated as 2π divided by the coefficient of x inside the function. Since the coefficient of x is 1 here, the period is 2π, meaning the function repeats every 2π units.

Phase Shift in Trigonometric Functions

Phase shift is the horizontal translation of the graph, determined by solving inside the function for zero. For y = (1/2) sin(x + π/2), the phase shift is -π/2, meaning the graph shifts π/2 units to the left.