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Chapter 6: Problem 24

$$\text {Graph each function over a two-period interval. Give the period and amplitude.}$$ $$y=\sin \frac{2}{3} x$$

Short Answer

Expert verified
Period: 3蟺, Amplitude: 1. Graph over [0, 6蟺].

Step by step solution

01

- Identify the Basic Function

The given function is a sine function: \(y = \frac{2}{3} x\). Understand that the basic sine function is \(y = \text{sin}(x)\).
02

- Determine the Period

The period of the sine function \(y = \text{sin}(Bx)\) is given by \( \frac{2\text{蟺}}{B} \). Here, \(B = \frac{2}{3}\). Thus, the period is \[\text{Period} = \frac{2\text{蟺}}{\frac{2}{3}} = 3\text{蟺}\].
03

- Determine the Amplitude

The amplitude of a sine function \(y = A \text{sin}(Bx)\) is the absolute value of the coefficient A in front of the sine function. For \(y = \text{sin}(\frac{2}{3}x)\), the amplitude is 1, since there is no coefficient A visible (which implies A = 1).
04

- Graph Over Two Periods

Since the period is \(3 \text{蟺}\), the function should be graphed over the interval from 0 to \(2 \times 3\text{蟺} = 6 \text{蟺}\). Plotting the function from 0 to 6\text{蟺} will encompass two full periods.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sine Function
A sine function is one of the essential functions in trigonometry. It is represented by the equation \(y = \text{sin}(x)\). The sine function describes the relationship between the angle of a right triangle and the ratio of the length of the opposite side to the hypotenuse. When plotted on a graph, it produces a smooth, wave-like pattern that repeats every \(2\text{蟺}\) units. Knowing how to identify and manipulate sine functions is crucial for understanding periodic phenomena in various fields like physics and engineering.
Period of a Sine Function
The period of a sine function determines how long it takes for the function to complete one full cycle. For a basic sine function \(y = \text{sin}(x)\), the period is \(2\text{蟺}\). This means that every \(2\text{蟺}\) units, the sine wave starts to repeat itself.
The period \(T\) of a sine function \(y = \text{sin}(Bx)\) is given by \[ T = \frac{2\text{蟺}}{B} \]
In the example provided, since \(B = \frac{2}{3}\), we calculate the period as follows:
  • First, identify \(B\): \frac{2}{3}
  • Next, plug it into the period formula: \[ T = \frac{2\text{蟺}}{\frac{2}{3}} = 3\text{蟺} \]
This means the sine function \(y = \text{sin}\frac{2}{3}x\) has a period of \(3\text{蟺}\).
Amplitude of a Sine Function
The amplitude of a sine function is the height from the centerline of the graph to its peak. It represents the maximum value of the sine wave. The equation for a sine function with amplitude is \[ y = A \text{sin}(Bx) \]
The amplitude is represented by the coefficient \(A\). If there is no visible coefficient, it means the amplitude is 1, as the default value.
  • In the provided sine function, \(y = \text{sin}\frac{2}{3}x\), there is no coefficient before the sine term,
    so the amplitude is 1.
The amplitude tells you how 'tall' or 'short' the wave appears on the graph.
Graphing Functions
Graphing functions, especially trigonometric ones, allows you to visualize the behavior of the function over a specified interval. For the function \(y = \text{sin}\frac{2}{3}x\), the plotting should include multiple key characteristics:
  • Identifying key points like maximum, minimum, and zero-crossings.
  • Knowing the period and amplitude.
  • Plotting accurately over two periods for better analysis.
To graph \(y = \text{sin}\frac{2}{3}x\) over two periods:First, determine the interval: If one period is \(3\text{蟺}\), then two periods will span from 0 to \(6 \text{蟺}\).
It's essential to keep these intervals straight, so the graph is accurate and meaningful. This approach enables a comprehensive understanding and visual representation to solidify learning.

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