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Chapter 6: Problem 18

$$\text {Graph each function over the interval }[-2 \pi, 2 \pi] . \text { Give the amplitude.}$$ $$y=-\sin x$$

Short Answer

Expert verified
The amplitude is 1. The graph covers the interval from \(-2\pi\) to \(2\pi\) with reflections over the x-axis.

Step by step solution

01

- Identify the function

The function to be graphed is given by: \[y = -\sin(x)\]
02

- Determine the interval

The interval to consider for this graph is \[-2 \pi \leq x \leq 2 \pi\].
03

- Graph the basic sine function

First, recall the graph of the basic sine function, \[y = \sin(x)\], which passes through the following key points: \(0,0\), \(((\pi/2),(1))\), \(\pi,0\), \(((3\pi/2),(-1))\), \(2\pi,0\).
04

- Reflect the sine function over the x-axis

Since the function is \(y = - \sin(x)\), we need to reflect the graph of \(y = \sin(x)\) across the x-axis. This changes the positive values of \sin(x) to negative values, and vice versa. So, the function's key points become: \(0,0\), \(((\pi/2),(-1))\), \(\pi,0\), \(((3\pi/2),(1))\), \(2\pi,0\).
05

- Sketch the graph

Plot these points and draw a smooth curve to complete one period of the function from \(0 \text{ to } 2\pi\). Since the interval is larger, extend the graph to cover the entire interval \([-2 \pi, 2 \pi]\). For \(-\pi < x < 0\), the graph is the same as for \(0 < x < \pi\), and similarly extend from \(-2 \pi\) to \(2\pi\).
06

- Identify the amplitude

The amplitude of \(y = -\sin(x)\) is the same as \(y = \sin(x)\), which is 1. The amplitude is the absolute value of the coefficient of \sin(x). So, the amplitude is \(|-1| = 1\).

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Key Concepts

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

amplitude of sine function
The amplitude of a sine function represents the height from the centerline of the graph to the peak or valley. In the function \(y = -\sin(x)\), the amplitude can be determined by looking at the coefficient in front of the sine function.

For the given function, \(y = -\sin(x)\), the coefficient is -1. The amplitude is always a positive value, as it measures distance. Thus, the amplitude is the absolute value of the coefficient:

\[| -1 | = 1\]

Therefore, the amplitude of the function \(y = -\sin(x)\) is 1. This means that the highest point and the lowest point on the graph are 1 unit away from the centerline, which in this case is the x-axis.
reflection across x-axis
Reflection across the x-axis inverts the values of the function. For the sine function, this means changing positive values to negative and vice versa.

To reflect the sine function \(y = \sin(x)\) to obtain \(y = -\sin(x)\), follow these steps:

  • Identify key points of \(y = \sin(x)\). For example, when \(x = \pi/2\), \(\sin(\pi/2) = 1\).
  • Apply the negative sign to each sine value. So, for \(x = \pi/2\), \(\sin(\pi/2) = -1\).
  • Repeat this for all key points to complete one period, then extend to the full interval.


This will transform key points from \(\sin(x)\) as follows: \( (\pi/2, 1) \rightarrow (\pi/2, -1)\) and \( (3\pi/2, -1) \rightarrow (3\pi/2, 1)\). Overall, when graphed, \(y = -\sin(x)\) creates a mirror image of \(y = \sin(x)\) across the x-axis.
graph intervals
The graph interval determines the segment of the x-axis we are considering for plotting. For the function \(y = -\sin(x)\), we need to graph within \([-2\pi, 2\pi]\).

This means:

  • Starting from \(-2\pi\), moving to \(-\pi\), reaching \(0\), continuing to \(\pi\), and finishing at \(2\pi\).
  • The interval covers two full periods of the sine function (one from \(-2\pi\) to \(0\) and another from \(0\) to \(2\pi\)).
  • Plot key points of the function within this interval and draw a smooth curve connecting them.
  • Remember to consider the reflected values due to the negative sign.


By using the interval \([-2\pi, 2\pi]\), we ensure that all critical points and the overall shape of the sine wave are included, giving a comprehensive visualization of the function.

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