Problem 25 - 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. $$ y=\frac{1}{2} \sin 2 \pi x $$

Step-by-Step Solution

Answer

Amplitude: \(\frac{1}{2}\); Period: 1; Phase shift: 0.

Step 1: Understand the Equation Structure

The general form of the sine function is \ y = A \sin(Bx - C) + D \, where \(A\) is the amplitude, \(B\) affects the period, \(C\) causes a phase shift, and \(D\) is the vertical shift. In this equation, \(y = \frac{1}{2} \sin(2\pi x)\), it can be observed that \(A = \frac{1}{2}\), \(B = 2\pi\), \(C = 0\), and \(D = 0\).

Step 2: Determine the Amplitude

The amplitude of a sine function is the absolute value of \(A\) in the general equation form and dictates the vertical stretch or compression of the graph. For the given function, the amplitude is \ |\frac{1}{2}| = \frac{1}{2}. \

Step 3: Calculate the Period

The period of the sine function is given by the formula \ \frac{2\pi}{B} \. In this case, \(B = 2\pi\), so the period is \ \frac{2\pi}{2\pi} = 1. \ This means the function repeats its pattern every 1 unit along the x-axis.

Step 4: Analyze the Phase Shift

The phase shift is calculated using \ \frac{C}{B} \. Since \(C = 0\) in the equation, the phase shift is \ \frac{0}{2\pi} = 0. \ Hence, there is no horizontal shift in the graph.

Step 5: Sketch the Graph

Plot the sine curve considering the amplitude, period, and phase shift determined. Since the amplitude is \(\frac{1}{2}\), the highest point of the graph is \( \frac{1}{2} \) and the lowest is \(-\frac{1}{2}\). The period of 1 implies that one complete wave occurs from \(x = 0\) to \(x = 1\), with no horizontal shift. Hence, one cycle completes as x moves from 0 to 1.

Key Concepts

Amplitude

The amplitude of a sine function reveals how tall or short the waves appear on the graph. In general, the amplitude is determined by the absolute value of the coefficient in front of the sine, which is denoted as \( A \) in the standard form \( y = A \sin(Bx - C) + D \).

For the equation \( y = \frac{1}{2} \sin(2 \pi x) \), the amplitude is \( \left | \frac{1}{2} \right | = \frac{1}{2} \). This means the function's peaks and troughs will reach \( \frac{1}{2} \) above and below its equilibrium position, or zero line.

In practical terms:

• The wave's maximum height is at \( \frac{1}{2} \),
• The lowest point is at \(-\frac{1}{2}\).

Amplitude provides insight into the intensity or strength of the wave's oscillations. This is a key feature in identifying how stretched the wave looks vertically compared to a plain sine wave.

Period

The period of a sine wave indicates the horizontal length needed for the wave to complete one full cycle. In the general sine function \( y = A \sin(Bx - C) + D \), the period is calculated as \( \frac{2\pi}{B} \).

In our function \( y = \frac{1}{2} \sin(2 \pi x) \), \( B = 2 \pi \). Let's find the period:

The formula is \( \frac{2\pi}{2\pi} = 1 \).
This result tells us that the wave pattern repeats every 1 unit along the x-axis.

This implies:

• The wave completes a complete cycle from crest to crest or trough to trough within this distance of 1.
• When graphing, you'll notice one entire oscillation divided into four equal parts from 0 to 1 along the x-axis.

Understanding the period of a sine wave assists with predicting where the cycles begin and end on the graph, making them a handy way to visualize repetitive patterns.

Phase Shift

Phase shift helps to detect any horizontal displacement that occurs in a graph of a sine wave. It shows how the graph is shifted left or right from its usual starting position.

Using the general sine equation \( y = A \sin(Bx - C) + D \), we find the phase shift with \( \frac{C}{B} \).

In this scenario, \( C = 0 \) and \( B = 2\pi \), leading to a phase shift calculation of:

\( \frac{0}{2\pi} = 0 \).

This reveals:

• The graph is not moved left or right along the x-axis.
• It commences its oscillation at the origin point \((0,0)\)

Phase shift is crucial because it gives clues about where the wave starts on the graph. Even when everything else remains the same, changing the phase shift modifies where each cycle begins. The absence of a phase shift indicates a standard start point without any additional modifications.