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Magazine Chapter 10: Problem 3
Determine the amplitude, period, and displacement for each function. Then sketch the graphs of the ficnctions. Check each using a calculator. $$y=\sin \left(x-\frac{\pi}{6}\right)$$
Short Answer
Expert verified
Amplitude: 1, Period: \(2\pi\), Phase Shift: \(\frac{\pi}{6}\) right. Vertical shift: 0.
Step by step solution
01
Understand the Standard Form
The standard form of a sine function used to determine amplitude, period, and phase shift is \(y = a \sin(bx - c) + d\). Here \(a\) is the amplitude, \(\frac{2\pi}{b}\) is the period, \(\frac{c}{b}\) is the phase shift, and \(d\) is the vertical shift. For the given function, \(y = \sin(x - \frac{\pi}{6})\), \(a = 1\), \(b = 1\), \(c = \frac{\pi}{6}\), and \(d = 0\).
02
Determine the Amplitude
The amplitude is derived from the coefficient \(a\) of the sine function. For the function \(y = \sin(x - \frac{\pi}{6})\), the amplitude is \(|a| = |1| = 1\). This means the graph will oscillate 1 unit above and below the midline.
03
Calculate the Period
The period of a sine function is calculated as \(\frac{2\pi}{b}\). Given that \(b = 1\), the period is \(\frac{2\pi}{1} = 2\pi\). This indicates that the cycle of the sine wave repeats every \(2\pi\) units along the x-axis.
04
Determine Phase Shift
Phase shift is found using the formula \(\frac{c}{b}\). Here, \(c = \frac{\pi}{6}\) and \(b = 1\), so the phase shift is \(\frac{\pi}{6} / 1 = \frac{\pi}{6}\). This represents a horizontal translation of the graph by \(\frac{\pi}{6}\) units to the right.
05
Identify Vertical Shift
The vertical shift, \(d\), is the constant added to the sine function. In this case, \(d = 0\), indicating there is no vertical shift. The midline of the graph remains the x-axis.
06
Sketch the Graph
Using the identified amplitude (1), period \(2\pi\), phase shift \(\frac{\pi}{6}\), and vertical shift (0), sketch the sine function. Start by drawing a sinusoidal wave oscillating between 1 and -1, initiating its cycle from \(x = \frac{\pi}{6}\) due to the phase shift.
07
Verify Using a Calculator
Use a graphing calculator to input the function \(y = \sin(x - \frac{\pi}{6})\) and verify the sketch. Ensure the amplitude reaches 1, the period spans \(2\pi\), and the graph origin is shifted by \(\frac{\pi}{6}\) along the x-axis.
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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 concept in trigonometry and can be represented by the equation \( y = a \sin(bx - c) + d \). In this form:
- \( a \) is the amplitude that determines how far up and down the graph stretches from the midline.
- \( b \) affects the period, which is the length of one complete cycle of the wave.
- \( c \) influences the phase shift, or how far the wave moves horizontally.
- \( d \) adjusts the vertical shift, balancing the graph's position up or down the y-axis.
Amplitude
Amplitude refers to the maximum height of the wave from its central axis or midline in a sine function. It is calculated by the absolute value of \( a \) in \( y = a \sin(bx - c) + d \). For our exercise, the function is \( y = \sin(x - \frac{\pi}{6}) \), where \( a = 1 \). Thus, the amplitude is \( |1| = 1 \).
This implies the graph oscillates 1 unit above and below its midline, which in this case, coincides with the x-axis. Understanding amplitude helps to visualize how stretched or compressed the sine wave appears on the graph.
This implies the graph oscillates 1 unit above and below its midline, which in this case, coincides with the x-axis. Understanding amplitude helps to visualize how stretched or compressed the sine wave appears on the graph.
Period
The period of a sine function describes how long it takes for the function to complete one full cycle. It is controlled by \( b \) in the equation \( y = a \sin(bx - c) + d \), using the formula \( \frac{2\pi}{b} \). For \( y = \sin(x - \frac{\pi}{6}) \), we have \( b = 1 \).
The period is therefore \( \frac{2\pi}{1} = 2\pi \). This means every complete sine wave repeats itself every \( 2\pi \) units along the x-axis. Visualizing the period helps in understanding how frequently waves pass a given point on the graph.
The period is therefore \( \frac{2\pi}{1} = 2\pi \). This means every complete sine wave repeats itself every \( 2\pi \) units along the x-axis. Visualizing the period helps in understanding how frequently waves pass a given point on the graph.
Phase Shift
Phase shift in trigonometric functions describes the horizontal translation of the graph along the x-axis. In our function, \( y = \sin(x - \frac{\pi}{6}) \), the phase shift is derived from \( \frac{c}{b} \) in \( y = a \sin(bx - c) + d \) with \( c = \frac{\pi}{6} \) and \( b = 1 \).
The phase shift is consequently \( \frac{\pi}{6} \), representing a movement of the sine wave \( \frac{\pi}{6} \) units to the right. Understanding phase shift is crucial as it indicates the starting point of the sine wave cycle on the graph, affecting how the wave aligns with the x-axis.
The phase shift is consequently \( \frac{\pi}{6} \), representing a movement of the sine wave \( \frac{\pi}{6} \) units to the right. Understanding phase shift is crucial as it indicates the starting point of the sine wave cycle on the graph, affecting how the wave aligns with the x-axis.
