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Magazine Chapter 4: Problem 18
Graph one complete cycle of each of the following. In each case, label the axes accurately and identify the amplitude for each graph. $$ y=-4 \sin x $$
Short Answer
Expert verified
The amplitude is 4, and the graph completes one cycle from 0 to \(2\pi\).
Step by step solution
01
Understand the Basic Sine Wave
The basic sine function is given by the equation \( y = \sin x \). It completes one cycle from 0 to \(2\pi\) radians, and has an amplitude of 1, reaching its maximum at 1 and minimum at -1.
02
Identify the Amplitude
The given equation is \( y = -4 \sin x \). The general form of a sine wave is \( y = A \sin Bx \), where \( A \) is the amplitude. Here, the amplitude is \(|-4| = 4\). This means the wave will reach a maximum of 4 and a minimum of -4.
03
Determine the Period
The period of a sine function \( y = A \sin Bx \) is given by \( \frac{2\}{B} \). In this case, since \( B = 1 \), the period is \( 2\pi \). This means one complete cycle of the wave is between 0 and \( 2\pi \).
04
Graph the Wave
To graph \( y = -4 \sin x \), start by plotting the sine wave over one full cycle from 0 to \( 2\pi \). The negative sign indicates a reflection over the x-axis, so rather than starting at 0 and going upwards, the graph starts at 0 and goes down. Label the x-axis from 0 to \( 2\pi \) and the y-axis from -4 to 4.
05
Mark Key Points
In one cycle from 0 to \( 2\pi \), the key points for \( -4 \sin x \) are: starting at 0, reaching its minimum at \( -4 \) when \( x = \frac{\pi}{2}\), crossing zero again at \( x = \pi \), reaching maximum at 4 when \( x = \frac{3\pi}{2}\), and returning to zero at \( x = 2\pi \). Plot these points accurately to form one cycle of the wave.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Sine Wave Properties
The sine wave is a fundamental wave shape that you will see in mathematics and natural sciences. It arises from the function \( y = \sin x \) and is known for its smooth periodic oscillations. Key properties to know include:
- Cycle: A sine wave's pattern is repetitive. This repetition occurs over any interval of \( 2\pi \) radians, which means if you began at 0 radians, you'd end up at the same state after moving through the circle by \( 2\pi \) radians.
- Oscillation: The classic sine wave oscillates between -1 and 1, meaning it reaches these points at its peaks and troughs.
- Contour: It smoothly varies without any abrupt jumps or curves, a signature feature of its wave nature.
Amplitude of a Sine Wave
Amplitude is a critical property that tells us how "tall" or "deep" a wave is, essentially measuring its height from the central axis (usually the x-axis) to its peak. In the equation \( y = A \sin x \), the value of \( A \) indicates the amplitude.
- Positive Amplitude: For the function \( y = 4\sin x \), the amplitude is straightforwardly 4. This means maximum value is 4 and minimum is -4.
- Negative Coefficients: In our case, the equation is \( y = -4 \sin x \). Despite the negative sign, amplitude is always a positive number, so it is still 4. The negative indicates a reflection over the x-axis which we'll explore later.
Period of a Sine Wave
The period of a sine function describes how "long" it takes for the sine wave to complete one cycle.Typically given as \( \frac{2\pi}{B} \) for a function \( y = A \sin Bx \), the period is the length of one full cycle of the wave. For our example, \( B = 1 \) in \( y = -4 \sin x \), so the period is \( 2\pi \), confirming that the wave returns to its starting point after \( 2\pi \) radians. Important aspects to remember include:
- Full Completion: A cycle happens from any starting point back to the same orientation.
- Frequency Relationship: The longer the period, the less frequent the wave cycles repeat, and vice versa.
Reflection of Functions
Reflection in mathematics often involves flipping a graph over one of the axes, like creating a mirror image. For sine waves, this effect is achieved by multiplying the sine equation by -1.When \( y = -4 \sin x \) instead of \( y = 4 \sin x \), the minus sign inverts the wave across the x-axis. Thus, instead of beginning with the usual arch upward, the wave dives downward first. Quickly identifying reflections helps in graphically representing modifications to standard forms. Key aspects include:
- Visual Change: Starting above vs. below the x-axis significantly changes the appearance.
- Practical Representation: Reflections might represent physical phenomena like phase shifts or directional changes in waves.
