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Chapter 5: Problem 39

Find the amplitude, period, and phase shift of the function, and graph one complete period. $$y=\sin (\pi+3 x)$$

Short Answer

Expert verified
Amplitude = 1, Period = \(\frac{2\pi}{3}\), Phase Shift = \(-\frac{\pi}{3}\).

Step by step solution

01

Identify Amplitude

The amplitude of a sine function of the form \(y = A \sin(Bx + C)\) is given by \(|A|\). In the function \(y = \sin(\pi + 3x)\), the coefficient of \(\sin\) is 1. Thus, the amplitude is \(|1| = 1\).
02

Determine the Period

The period of a sine function is given by \(\frac{2\pi}{|B|}\). Here, \(B = 3\), so the period is \(\frac{2\pi}{3}\).
03

Calculate the Phase Shift

The phase shift of a sine function \(y = A \sin(Bx + C)\) is given by \(-\frac{C}{B}\). The function \(y = \sin(\pi + 3x)\) can be rewritten as \(y = \sin(3x + \pi)\), where \(C = \pi\). Thus, the phase shift is \(-\frac{\pi}{3}\).
04

Graph One Complete Period

To graph one complete period, we start at the phase shift, \(-\frac{\pi}{3}\), and complete one full cycle within a distance of \(\frac{2\pi}{3}\). This means the function will complete one full cycle at \(-\frac{\pi}{3} + \frac{2\pi}{3} = \frac{\pi}{3}\). Plot the points starting from \(-\frac{\pi}{3}\), peaking at \(-\frac{\pi}{6}\), crossing the x-axis at 0, hitting the minimum at \(\frac{\pi}{6}\), and returning to the x-axis at \(\frac{\pi}{3}\).

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Key Concepts

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

Amplitude in Trigonometric Functions
The amplitude of a trigonometric function, such as a sine wave, represents the height of its peaks from the center line of the wave. It's a measure of how far the wave oscillates above or below the central horizontal axis.
  • In mathematical terms, for a function like \(y = A \sin(Bx + C)\), the amplitude is \(|A|\), the absolute value of the coefficient \(A\).
  • In the given problem \(y = \sin(\pi + 3x)\), the coefficient \(A\) is 1.
  • Therefore, the amplitude is simply \(1\).
The amplitude affects how "tall" or "short" your wave appears, but does not influence where it starts or the width of each cycle.
  • Higher amplitude means the wave reaches higher peaks and lower valleys.
  • An amplitude of 1 means a standard sine or cosine wave height in its usual [-1, 1] interval.
Understanding amplitude is critical, as it helps us determine the extent of variation in periodic functions, which is particularly useful in fields like sound, light, and radio waves.
Understanding the Period of Sine Waves
The period of a sine function refers to the distance along the x-axis required for the function to complete one full cycle. This is an essential characteristic for understanding how frequently the wave repeats.
  • For the function \(y = A \sin(Bx + C)\), period is calculated using the formula \(\frac{2\pi}{|B|}\).
  • In the exercise \(y = \sin(\pi + 3x)\), the value of \(B\) is 3.
  • So, the period is \(\frac{2\pi}{3}\), indicating the wave takes this length on the x-axis to loop through its full cycle.
The period tells us how "squeezed" or "stretched" the wave is along the x-axis.
  • A shorter period means more cycles are packed into the same space.
  • A longer period means the cycles are elongated, spreading apart.
Grasping the concept of periods is crucial in graphing and interpreting trigonometric functions effectively, particularly when dealing with applications in physics and engineering.
Phase Shift: Shifting Trigonometric Waves
Phase shift describes how a sine or cosine wave moves horizontally from its standard position. It's the shift along the x-axis from the pattern where the function would typically start.
  • For \(y = A \sin(Bx + C)\), the phase shift is given by \(-\frac{C}{B}\).
  • In our function \(y = \sin(\pi + 3x)\), note that the function can be rearranged to \(y = \sin(3x + \pi)\), where \(C = \pi\).
  • This gives a phase shift of \(-\frac{\pi}{3}\).
The negative sign indicates a shift to the left.
  • A negative phase shift takes the wave to the left.
  • A positive phase shift moves it to the right.
Understanding phase shifts enable you to determine starting points and predict wave behavior accurately, which is essential when modeling real-world periodic phenomena such as tides, seasons, and oscillatory motion.

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Most popular questions from this chapter

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\) $$\left(-\frac{6}{7}, \frac{\sqrt{13}}{7}\right)$$

Graph the three functions on a common screen. How are the graphs related? $$y=x^{2}, \quad y=-x^{2}, \quad y=x^{2} \sin x$$

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57-58 - Graph \(f, g,\) and \(f+g\) on a common screen to illustrate graphical addition. $$f(x)=x, \quad g(x)=\sin x$$

Write the first expression in terms of the second if the terminal point determined by \(t\) is in the given quadrant. $$\sin t, \cos t ; \quad \text { quadrant II }$$
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