Problem 19 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 5-40: Find the amplitude, the period, and the phase shift and sketch the graph of the equation. $$ y=\sin \left(\frac{1}{2} x-\frac{\pi}{3}\right) $$

Step-by-Step Solution

Answer

Amplitude: 1, Period: 4π, Phase Shift: 2π/3 units right.

Step 1: Identify the Amplitude

For the function \(y = a \sin(bx - c)\), the amplitude is determined by the absolute value of the coefficient \(a\). In this case, the coefficient of the sine function is 1, so the amplitude of \(y = \sin\left(\frac{1}{2}x - \frac{\pi}{3}\right)\) is 1.

Step 2: Calculate the Period

The period of a sine function is given by \(\frac{2\pi}{b}\), where \(b\) is the coefficient of \(x\). Here, \(b = \frac{1}{2}\). So the period is \(\frac{2\pi}{\frac{1}{2}} = 4\pi\).

Step 3: Determine the Phase Shift

The phase shift can be calculated using \(\frac{c}{b}\), where \(c\) is the horizontal shift and \(b\) is the coefficient of \(x\). Here, \(c = \frac{\pi}{3}\) and \(b = \frac{1}{2}\). Thus, the phase shift is \(\frac{\frac{\pi}{3}}{\frac{1}{2}} = \frac{2\pi}{3}\) units to the right.

Step 4: Sketch the Graph

To sketch the graph of \(y = \sin\left(\frac{1}{2}x - \frac{\pi}{3}\right)\), start by plotting one full cycle of a standard sine wave from 0 to \(4\pi\) since the period is \(4\pi\). Shift this graph \(\frac{2\pi}{3}\) units to the right to account for the phase shift. Finally, since the amplitude is 1, ensure the graph oscillates between -1 and 1 along the y-axis.

Key Concepts

Amplitude

Amplitude refers to the maximum extent of a vibration or oscillation, measured from the position of equilibrium. In simple terms, it indicates how 'tall' or how 'short' a wave is along the y-axis as it oscillates. For trigonometric functions like sine or cosine, amplitude measures the peak (highest point) and trough (lowest point) height of the wave.

• The amplitude is always a positive number, as it represents an absolute distance from the center line of the wave to its peak.
• For a function in the form of \(y = a \sin(bx - c)\), \(a\) is the amplitude.
• By changing the value of \(a\), you alter how far up and down the graph extends.
• In the equation \(y = \sin(\frac{1}{2}x - \frac{\pi}{3})\), the amplitude is 1 because the coefficient of the sine function, \(a\), is 1.

A useful mnemonic is to remember that "A" in amplitude stands for "Altitude," connecting to the height of the wave.

Period

The period of a trigonometric function refers to the horizontal length of one complete cycle of the wave. It tells you how long it takes for the function to repeat itself. Essentially, it's the width of one wave on the x-axis.

• The period is calculated with the formula \(\frac{2\pi}{b}\) for the function \(y = a \sin(bx - c)\).
• This means that a smaller \(b\) value results in a longer period, spreading the wave further on the x-axis.
• Conversely, a larger \(b\) compresses the wave, making it cycle more frequently.
• In the given function \(y = \sin(\frac{1}{2}x - \frac{\pi}{3})\), our \(b\) value is \(\frac{1}{2}\), meaning the period is \(4\pi\).

This longer period means the wave takes twice as long to complete a cycle compared to a "traditional" sine wave, which has a period of \(2\pi\).

Phase Shift

Phase shift refers to the horizontal shift left or right for the graph of a sine or cosine function. It's like sliding the whole wave forward or backward along the x-axis without changing its shape.

• In the function \(y = a \sin(bx - c)\), the phase shift is given by \(\frac{c}{b}\).
• If \(c\) is positive, the graph shifts to the right.
• If \(c\) is negative, the graph goes to the left.
• Our equation \(y = \sin(\frac{1}{2}x - \frac{\pi}{3})\) involves a phase shift of \(\frac{2\pi}{3}\) units to the right.

This means that where the wave would typically start at \(x = 0\), it's now starting at \(x = \frac{2\pi}{3}\). Think of this as the starting line of a race being moved further down the track, with the whole race swaying along with it.