Problem 8 - 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. $$ y=2 \sin \left(x-\frac{\pi}{3}\right) $$

Step-by-Step Solution

Answer

Amplitude is 2, period is \(2\pi\), and phase shift is \(\frac{\pi}{3}\) to the right.

Step 1: Identify the Amplitude

The amplitude of a sine function of the form \(y = a \sin(bx + c)\) is given by \(|a|\). For the function \(y = 2 \sin(x - \frac{\pi}{3})\), the amplitude \(|a|\) is \(2\).

Step 2: Determine the Period

The period of a sine function is calculated using the formula \(\frac{2\pi}{|b|}\). In the given equation \(y = 2 \sin(x - \frac{\pi}{3})\), \(b = 1\). Thus, the period is \(\frac{2\pi}{1} = 2\pi\).

Step 3: Find the Phase Shift

The phase shift is determined by \(-\frac{c}{b}\). For \(y = 2 \sin(x - \frac{\pi}{3})\), \(c = -\frac{\pi}{3}\) and \(b = 1\). Thus, the phase shift is \(-\frac{-\frac{\pi}{3}}{1} = \frac{\pi}{3}\). This indicates a shift to the right by \(\frac{\pi}{3}\).

Step 4: Sketch the Graph

To sketch the graph, start with the basic sine curve of one period from \(0\) to \(2\pi\). Apply the amplitude by scaling the y-values, resulting in a max of \(2\) and a min of \(-2\). Shift the whole graph to the right by \(\frac{\pi}{3}\) due to the phase shift.

Key Concepts

Understanding Amplitude in Trigonometric Functions

When dealing with trigonometric functions like sine and cosine, the amplitude plays a crucial role in determining the height of the wave. Amplitude is essentially the measure of how much the function stretches or compresses on the vertical axis. It's the maximum distance from the horizontal axis to the peak or trough of the wave.
For functions in the format \(y = a \sin(bx + c)\), or similar with cosine, the amplitude is given by \(|a|\). This is because it defines how far the wave goes up and down from its central position, which is line \(y = 0\) for sine. The center line might be different if there's a shift, but for sine without vertical shifts, it's 0. For example, in the function \(y = 2 \sin(x - \frac{\pi}{3})\), the amplitude is \(|2| = 2\).

• This value tells us that the wave reaches its highest point at 2 and its lowest at -2.
• The amplitude captures the intensity or strength of the sine wave's oscillation.
• It dictates how 'tall' or 'short' a sine wave appears on a graph.

Understanding amplitude helps visualize how dynamic or subdued a function can appear in real-life applications, making it a key element in graphing trigonometric functions.

Grasping the Period of Trigonometric Functions

The period of a trigonometric function is the length it takes for the function to complete one full cycle and start repeating itself. This concept is especially visible in the repetitive nature of functions like sine and cosine.
In general, for a function given by \(y = a \sin(bx + c)\), the period can be calculated by the formula \(\frac{2\pi}{|b|}\). This formula arises from the property that sine functions have a standard period of \(2\pi\), and \(b\) acts as a stretching or compressing factor along the x-axis.
Taking the function \(y = 2 \sin(x - \frac{\pi}{3})\) into consideration, \(b = 1\). Therefore, the period of this function is \(\frac{2\pi}{1} = 2\pi\).

• This means every \(2\pi\) units along the x-axis, the sine wave starts to repeat its pattern.
• The period influences how many waves you see over a specific interval.
• Every wave segment between 0 and \(2\pi\) mirrors the entire behavior of the function.

This concept is vital for predicting the behavior of waves in a given domain and is essential for applications requiring periodicity understanding, like signal processing or acoustics.

Exploring Phase Shift in Trigonometric Functions

Phase shift in trigonometric functions indicates how the function moves horizontally on a graph. It's essentially the horizontal shift of the sine or cosine wave along the x-axis.
For a function in the form \(y = a \sin(bx + c)\) or \(y = a \cos(bx + c)\), the phase shift is determined by the formula \(-\frac{c}{b}\). This formula helps identify the `starting point` or initial horizontal adjustment made to the graph of the function.

• In our specific example, \(y = 2 \sin(x - \frac{\pi}{3})\), \(c = -\frac{\pi}{3}\) and \(b = 1\).
• Using the formula, the phase shift comes out as \(-\frac{-\frac{\pi}{3}}{1} = \frac{\pi}{3}\).
• This means the entire graph of the sine function is shifted to the right by \(\frac{\pi}{3}\) units.

Understanding phase shift is crucial, especially in fields such as physics and engineering, because it allows prediction of wave behavior at various points.