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Magazine Chapter 10: Problem 18
Give the amplitude and sketch the graphs of the given functions. Check each using a calculator. $$y=-0.2 \sin x$$
Short Answer
Expert verified
The amplitude of \( y=-0.2 \sin x \) is 0.2.
Step by step solution
01
Identify the Standard Sine Function
The standard sine function is given by \( y = A \, \sin(Bx + C) + D \), where \( A \) is the amplitude, \( B \) affects the period, \( C \) is the phase shift, and \( D \) is the vertical shift.
02
Extract the Amplitude
For the given function \( y = -0.2 \sin x \), compare it to the standard form. Here, \( A = -0.2 \). The amplitude is the absolute value of \( A \), which is \( |A| = |-0.2| = 0.2 \).
03
Sketch the Basic Sine Wave
The function \( y = \sin x \) oscillates between -1 and 1 with a period of \( 2\pi \). We'll base our sketch on this wave.
04
Adjust Amplitude and Reflect
Multiply the y-values of the sine wave by \( -0.2 \) to adjust the amplitude and reflect. This will compress the wave to oscillate between -0.2 and 0.2 and flip it upside down.
05
Plot Key Points for One Cycle
For one period, plot important points: \( (0, 0) \), \( (\frac{\pi}{2}, -0.2) \), \( (\pi, 0) \), \( (\frac{3\pi}{2}, 0.2) \), and \( (2\pi, 0) \).
06
Sketch the Graph
Connect these points smoothly with a sinusoidal curve to complete the graph of the function over one period. Continue the wave pattern for additional cycles as needed.
07
Verify Using a Calculator
Using a graphing calculator, input the function \( y = -0.2 \sin x \) and verify that the graph oscillates between -0.2 and 0.2 and reflects the sketch you've drawn.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Standard Sine Function
The sine function is a fundamental concept in trigonometry and is represented in its most simple form as \( y = \sin x \). This foundational wave is characterized by its smoothly repeating oscillation between -1 and 1 with a period of \( 2\pi \).
The standard form of any sine function can be expressed as \( y = A \sin(Bx + C) + D \). In this formula:
The standard form of any sine function can be expressed as \( y = A \sin(Bx + C) + D \). In this formula:
- \( A \) is the amplitude, dictating the wave's height.
- \( B \) affects the wave's period, or the distance needed for a full cycle.
- \( C \) shifts the graph horizontally (phase shift).
- \( D \) vertically shifts the graph (vertical shift).
Graphing Trigonometric Functions
Graphing a sine function involves understanding how changes in the equation affect the wave's appearance. Starting with the standard function \( y = \sin x \), the graph naturally oscillates from -1 to 1. The amplitude, a crucial component, modifies this behavior.
When you encounter a function like \( y = -0.2 \sin x \), first identify the amplitude, which in this case is the absolute value of \(-0.2\), or 0.2. This compresses the wave's height, making it only reach values from -0.2 to 0.2. Here are some steps to follow:
When you encounter a function like \( y = -0.2 \sin x \), first identify the amplitude, which in this case is the absolute value of \(-0.2\), or 0.2. This compresses the wave's height, making it only reach values from -0.2 to 0.2. Here are some steps to follow:
- Plot key points for the basic sine wave over one period, focusing on the start, the maximum, crossing, minimum, and end points.
- Adjust these points by the given amplitude, which changes their y-values accordingly.
- Connect these adjusted points to sketch the sine curve.
Wave Reflection
Wave reflection in trigonometry, especially for sine functions, refers to flipping the graph over the x-axis. This occurs when the amplitude is negative, as seen in functions like \( y = -0.2 \sin x \).
To visualize this, consider the standard sine wave which rises to positive values first. When reflected, it will invert, starting with a descent. This changes the graph's appearance significantly while still following the sine pattern.
Here's how to manage reflection in your sketches:
To visualize this, consider the standard sine wave which rises to positive values first. When reflected, it will invert, starting with a descent. This changes the graph's appearance significantly while still following the sine pattern.
Here's how to manage reflection in your sketches:
- Identify the negative amplitude, indicating the need for reflection.
- Invert the graph's rise and fall based on the amplitude. For instance, instead of rising to \(0.2\), it descends to \(-0.2\).
- Maintain smooth transitions between plotted points, following the inverted pattern.
