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Magazine Chapter 6: Problem 169
For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes. $$ f(x)=\frac{1}{4} \sin x $$
Short Answer
Expert verified
Amplitude: \( \frac{1}{4} \), Period: \( 2\pi \), Midline: \( y = 0 \), Asymptotes: None.
Step by step solution
01
Identify the function type and basic features
The given function is \( f(x) = \frac{1}{4} \sin x \), which is a sine function. We will identify the amplitude, period, and midline from the function's form. The function represents a sine wave with modifications to the amplitude.
02
Determine the amplitude
The amplitude of a sine function \( f(x) = a \sin x \) is given by \( |a| \). Here, the amplitude is \( \left| \frac{1}{4} \right| = \frac{1}{4} \). This means the wave's maximum displacement from the midline is 0.25.
03
Calculate the period
The period of a standard sine function \( \sin x \) is \( 2\pi \). Since there is no horizontal scaling factor in \( f(x) = \frac{1}{4} \sin x \), the period remains \( 2\pi \). The function completes one full cycle in an interval width of \( 2\pi \).
04
Determine the midline equation
The midline of a sine function \( f(x) = a \sin x + d \) is \( y = d \). The function \( f(x) = \frac{1}{4} \sin x \) has no vertical shift \((d = 0)\), so the midline equation is \( y = 0 \).
05
Identify asymptotes for a sine function
Sine functions like \( \frac{1}{4} \sin x \) do not have vertical asymptotes because the sine function is continuous and periodic without undefined values.
06
Graph the function over two periods
To graph \( f(x) = \frac{1}{4} \sin x \) over two periods (\( 0 \leq x \leq 4\pi \)), plot points at key positions like 0, \( \frac{\pi}{2} \), \( \pi \), \( \frac{3\pi}{2} \), and \( 2\pi \). The graph oscillates between \( -\frac{1}{4} \) and \( \frac{1}{4} \), with a midline at \( y = 0 \) and repeating every \( 2\pi \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Sine Function
The sine function is a fundamental trigonometric function that describes a smooth, periodic oscillation. When graphing the sine function, you can visualize waves, which is why it is often associated with cycles in nature, like sound or water waves. For a function in the form of \( f(x) = a \sin(bx + c) + d \), the sine function holds key characteristics that define its shape: amplitude, period, phase shift, and vertical shift.
- **Amplitude** determines how tall or short the waves appear.
- **Period** indicates how long it takes to complete a single cycle of the wave.
- **Midline** or vertical shift changes the center line of the wave up or down.
Amplitude
Amplitude is a measure of how far the peaks and troughs of the wave are from the midline. In mathematical terms, it is expressed as the absolute value of the coefficient \( a \) in the sine function \( f(x) = a \sin(x) \).
For example, in the function \( f(x) = \frac{1}{4} \sin(x) \), the amplitude is \( |\frac{1}{4}| = 0.25 \). This shows that:
For example, in the function \( f(x) = \frac{1}{4} \sin(x) \), the amplitude is \( |\frac{1}{4}| = 0.25 \). This shows that:
- The wave oscillates between \( -0.25 \) and \( 0.25 \).
- Amplitude affects the maximum and minimum points of the graph, providing a clear boundary for where the wave will extend.
Period
The period of a sine function defines the length of one complete wave cycle. It is determined by the coefficient \( b \) in \( f(x) = \sin(bx) \), calculated using the formula \( \frac{2\pi}{b} \).
For the function \( f(x) = \frac{1}{4} \sin(x) \), which has no coefficient other than 1 for \( x \), the period remains the standard \( 2\pi \). This implies that:
For the function \( f(x) = \frac{1}{4} \sin(x) \), which has no coefficient other than 1 for \( x \), the period remains the standard \( 2\pi \). This implies that:
- The wave repeats itself every \( 2\pi \) units along the x-axis.
- You would see identically shaped cycles every \( 2\pi \) distance, such as from 0 to \( 2\pi \) and again from \( 2\pi \) to \( 4\pi \).
Midline
The midline of a sine function represents the horizontal line that runs through the middle of the waveform, around which the wave oscillates. This is given by \( y = d \) in the function \( f(x) = a \sin(bx) + d \). For the function \( f(x) = \frac{1}{4} \sin(x) \), the midline is \( y = 0 \) because there is no vertical shift added to the function.
- The midline serves as a baseline for measuring amplitude.
- In applications, it's often a reference level, like the average level in alternating electrical currents.
Graphing
Graphing sine functions entails plotting their oscillatory behavior over a specified range or period. For \( f(x) = \frac{1}{4} \sin(x) \), we want to plot it over two periods, or from 0 to \( 4\pi \).
Key points to plot:
Key points to plot:
- Start at \( (0, 0) \), indicating the wave starts on the midline.
- The maximum at \( \frac{\pi}{2} \) gives \( (\frac{\pi}{2}, \frac{1}{4}) \).
- Return to midline at \( \pi \), giving \( (\pi, 0) \).
- The minimum at \( \frac{3\pi}{2} \) is \( (\frac{3\pi}{2}, -\frac{1}{4}) \).
- Finally, at \( 2\pi \), back to \( (2\pi, 0) \).
