Problem 45 - Precalculus with Calculus Previews

Question

Find the amplitude, period, and phase shift of the given function. Sketch at least one cycle of the graph. $$ y=3 \sin \left(\frac{x}{2}-\frac{\pi}{3}\right) $$

Step-by-Step Solution

Answer

Amplitude: 3; Period: \(4\pi\); Phase Shift: \(\frac{2\pi}{3}\) right.

Step 1: Identify the Standard Form

The standard form for a sine function is \(y = a \sin(bx + c)\), where \(a\) is the amplitude, \(b\) relates to the period, and \(c\) affects the phase shift. For the function given, \(y = 3 \sin \left(\frac{x}{2} - \frac{\pi}{3}\right)\), we have \(a = 3\), \(b = \frac{1}{2}\), and \(c = -\frac{\pi}{3}\).

Step 2: Find the Amplitude

The amplitude of a sine function \(y = a \sin(bx + c)\) is the absolute value of \(a\). In our case, \(a = 3\), so amplitude \(= |3| = 3\).

Step 3: Determine the Period

The period of the sine function \(y = a \sin(bx + c)\) is given by \(\frac{2\pi}{|b|}\). Here \(b = \frac{1}{2}\), so the period is \(\frac{2\pi}{\frac{1}{2}} = 4\pi\).

Step 4: Calculate the Phase Shift

The phase shift of a sine function \(y = a \sin(bx + c)\) is found by solving \(-\frac{c}{b}\). Here, \(c = -\frac{\pi}{3}\) and \(b = \frac{1}{2}\), so phase shift \(= -\left(-\frac{\pi}{3} \times 2\right) = \frac{2\pi}{3}\) to the right.

Step 5: Sketch the Graph

To sketch the function, start at the phase shift \(x = \frac{2\pi}{3}\). The amplitude is 3, so the graph oscillates from -3 to 3. One complete cycle will take \(4\pi\) units in the x-direction, ending at \(x = \frac{2\pi}{3} + 4\pi\). Sketch a regular sine wave starting from \(x = \frac{2\pi}{3}\), peaking at \(x = \frac{4\pi}{3}\), crossing the x-axis at \(x = 2\pi\), going to a minimum at \(x = \frac{8\pi}{3}\), and completing the cycle at \(x = \frac{14\pi}{3}\).

Key Concepts

Amplitude in Trigonometric Functions

Amplitude is a key characteristic of trigonometric functions, particularly sine and cosine. It tells us how far the graph of the function stretches above and below the horizontal axis. The amplitude is the absolute value of the coefficient in front of the sine or cosine function. For our function, \[ y = 3 \sin \left( \frac{x}{2} - \frac{\pi}{3} \right) \] the amplitude is determined by the coefficient of 3. This means:

• The graph's peaks reach up to 3 units above the horizontal axis.
• Its troughs go down to -3 units.

In real-world terms, amplitude can represent the strength or intensity of a wave, such as a sound wave. It's vital in applications like physics and engineering to understand signal strength.
In our sine function, the maximum and minimum values will always be offset by the amplitude, giving it a defined range of values.

Periodicity and Trigonometric Functions

Periodicity in trigonometric functions refers to the repeating nature of these functions over a set interval. The period is the length of one complete cycle of a wave. It's easily found using the formula: \[ \text{Period} = \frac{2\pi}{|b|} \] where \(b\) is the coefficient of \(x\) inside the sine or cosine function.
For our function:

• \( b = \frac{1}{2} \), thus \( \text{Period} = \frac{2\pi}{\frac{1}{2}} = 4\pi \).

This means the sine wave completes one full cycle every \(4\pi\) units along the x-axis. Understanding periodicity is crucial in scenarios like signal processing, where waves repeat regularly. When sketching the function, watch how the sine wave starts repeating its pattern after covering a distance of \(4\pi\).

Phase Shift in Sine Waves

The phase shift of a trigonometric function tells us how far horizontally the entire graph is moved from its usual position. This shift is calculated using: \[ \text{Phase Shift} = -\frac{c}{b} \] where \(c\) is a constant added or subtracted inside the function's argument. For our function:

• \( c = -\frac{\pi}{3} \) and \( b = \frac{1}{2} \).

This results in the phase shift:

• \( \text{Phase Shift} = -(-\frac{\pi}{3}) \times 2 = \frac{2\pi}{3} \) units to the right.

Phase shifts are crucial when synchronizing different waves or adjusting their positions. In our graph, this means the sine wave starts \( \frac{2\pi}{3} \) units to the right of the origin, affecting where each peak and trough occurs. By appreciating the phase shift, one can better predict changes in wave behavior.