Problem 34 - Precalculus Mathematics for Calculus

Question

Find the amplitude, period, and phase shift of the function, and graph one complete period. $$y=\sin \frac{1}{2}\left(x+\frac{\pi}{4}\right)$$

Step-by-Step Solution

Answer

Amplitude: 1, Period: \(4\pi\), Phase Shift: \(-\frac{\pi}{4}\).

Step 1: Identify the Amplitude

The amplitude of a sinusoidal function \( y = a \sin(b(x - c)) + d \) or \( y = a \cos(b(x - c)) + d \) is given by \(|a|\). For the given function \( y = \sin\left( \frac{1}{2} (x + \frac{\pi}{4}) \right) \), the amplitude \( a = 1 \). Thus, the amplitude is \( 1 \).

Step 2: Determine the Period

The period of a sine or cosine function \( y = \sin(b(x-c)) \) or \( y = \cos(b(x-c)) \) is calculated using the formula \( \frac{2\pi}{|b|} \). In this function, \( b = \frac{1}{2} \). Therefore, the period is \( \frac{2\pi}{\frac{1}{2}} = 4\pi \).

Step 3: Calculate the Phase Shift

The phase shift of the function is given by \( -\frac{c}{b} \). In our function, we can rewrite it as \( y = \sin\left( \frac{1}{2}(x + \frac{\pi}{4}) \right) = \sin\left( \frac{1}{2}x + \frac{1}{2} \cdot \frac{\pi}{4} \right) = \sin\left( \frac{1}{2}x + \frac{\pi}{8} \right) \). This gives \( c = -\frac{\pi}{8} \), where the phase shift is \( -\frac{c}{\frac{1}{2}} = -\frac{\pi}{8} \cdot 2 = -\frac{\pi}{4} \). The negative sign indicates a shift to the left.

Step 4: Graph One Complete Period

To graph one complete period of \( y = \sin\left( \frac{1}{2}x + \frac{\pi}{8} \right) \), note that the period is \( 4\pi \) and the phase shift is \( -\frac{\pi}{4} \). Begin one period before the phase shift at \(-\frac{\pi}{4}\) and end at \[-\frac{\pi}{4} + 4\pi = \frac{15\pi}{4}\]. The key points are at \(-\frac{\pi}{4}\) (start at 0), \(\frac{3\pi}{4}\) (peak at 1), \(\frac{7\pi}{4}\) (zero crossing), \(\frac{11\pi}{4}\) (trough at -1), and \(\frac{15\pi}{4}\) (end at 0).

Key Concepts

Understanding Amplitude

In trigonometric functions, the amplitude represents the maximum displacement of the function's wave from its central axis. It's essentially a measure of the wave's height and is always a non-negative number. For functions in the form \[ y = a \sin(b(x - c)) + d \] or \[ y = a \cos(b(x - c)) + d \], the amplitude is given by \(|a|\), which is the absolute value of the coefficient in front of the sine or cosine function.

• A larger amplitude indicates a taller wave, whereas a smaller amplitude indicates a shorter wave.
• The amplitude affects how far up or down the graph stretches from the centerline of the graph, which is determined by the vertical translation \(d\).

For the given function \( y = \sin\left( \frac{1}{2} (x + \frac{\pi}{4}) \right) \), the amplitude is \(1\) since the coefficient in front of the sine function is \(1\). This means the wave oscillates 1 unit above and below the center line.

The Period of a Function

The period of a trigonometric function indicates the length of one complete cycle of the wave. For the sine and cosine functions of the form \[ y = \sin(b(x-c)) \] or \[ y = \cos(b(x-c)) \], the period is calculated using the formula \( \frac{2\pi}{|b|} \). This tells us how far along the horizontal axis the graph must travel to repeat itself.

• The period is important for understanding how frequently the wave pattern repeats over the x-axis.
• A smaller \(|b|\) results in a longer period, meaning the wave stretches more horizontally, while a larger \(|b|\) results in a shorter period, causing the wave to compress and repeat more frequently.

In the function \( y = \sin\left( \frac{1}{2}(x + \frac{\pi}{4}) \right) \), \(b = \frac{1}{2}\). Plugging this into the formula gives \( \frac{2\pi}{\frac{1}{2}} = 4\pi \). Hence, the wave repeats every \(4\pi\) units along the x-axis, meaning the graph must travel \(4\pi\) units before the pattern starts over again.

Exploring Phase Shift

Phase shift refers to the horizontal displacement of the graph of a trigonometric function. It's the amount by which the entire graph is moved left or right from its usual starting point. For a function of the form \[ y = \sin(b(x-c)) \] or \[ y = \cos(b(x-c)) \], the phase shift is calculated using \(-\frac{c}{b} \), which determines how far and in what direction the graph is shifted horizontally.

• A positive phase shift \((-\frac{c}{b} > 0)\) indicates a move to the right, whereas a negative phase shift \((-\frac{c}{b} < 0)\) indicates a leftward move.
• Understanding phase shift is crucial for correctly placing critical points such as maximums, minimums, and zero crossings on a graph.

For the function \( y = \sin\left( \frac{1}{2}(x + \frac{\pi}{4}) \right) \), rewriting it gives \( y = \sin\left( \frac{1}{2}x + \frac{\pi}{8} \right) \). Thus, \(c = -\frac{\pi}{8}\), leading to a phase shift of \(-(-\frac{\pi}{8} \times 2) = -\frac{\pi}{4} \). This reflects a shift to the left by \(\frac{\pi}{4}\) units, altering where the wave starts relative to the vertical axis.