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Chapter 5: Problem 38

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. \(y=3 \cos (x+3 \pi)-2\)

Short Answer

Expert verified
Amplitude: 3, Period: \(2\pi\), Phase Shift: \(-3\pi\) left.

Step by step solution

01

Identify the amplitude

The general form of a cosine function is given by \( y = a \cos(b(x + c)) + d \). In this function, the amplitude is the coefficient \(a\) in front of the cosine term. For the function \( y = 3 \cos(x + 3\pi) - 2 \), the amplitude \( a = 3 \). Thus, the amplitude is 3.
02

Determine the period

The period of a cosine function \( y = a \cos(b(x + c)) + d \) is calculated by \( \frac{2\pi}{|b|} \). In our function, \( b = 1 \), which is the coefficient of \( x \).Therefore, the period is \( \frac{2\pi}{1} = 2\pi \).
03

Find the phase shift

The phase shift is determined by the expression \( -\frac{c}{b} \), where \( c \) is inside the cosine function's argument \( (x + c) \). Here, \( c = 3\pi \) and \( b = 1 \).Thus, the phase shift is \( -3\pi \) (shifting the graph to the left by \( 3\pi \)).
04

Determine the vertical shift

The vertical shift is given by \( d \) in the function \( y = a \cos(b(x + c)) + d \). For the function \( y = 3 \cos(x + 3\pi) - 2 \), \( d = -2 \).This means the graph is shifted downward by 2 units.
05

Sketch the graph

To sketch \( y = 3 \cos(x + 3\pi) - 2 \):1. Start with the basic cosine graph.2. Apply the amplitude: peaks at 3 and troughs at -3.3. Apply the period: one cycle spans from \(0\) to \(2\pi\).4. Apply the phase shift: shift the graph \(3\pi\) to the left.5. Apply the vertical shift: move the entire graph down by 2.

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Key Concepts

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

Amplitude
When discussing trigonometric functions like cosine and sine, amplitude is a crucial aspect. Amplitude refers to the height of the wave, measured from the center line to a peak. For a function in the form \( y = a \cos(b(x + c)) + d \), the amplitude is represented by the absolute value of \( a \). This value determines how tall and deep the wave oscillates. In the given function \( y = 3 \cos(x + 3\pi) - 2 \), the amplitude is 3. This means the highest point of the cosine wave is 3 units above the midline, and the lowest point is 3 units below the midline. Amplitude does not affect the width of the wave; it only stretches or compresses it vertically.
Period
The period of a trigonometric function tells us the length of one full cycle of the wave. In cosine or sine functions like \( y = a \cos(b(x + c)) + d \), the period is determined by \( \frac{2\pi}{|b|} \). This formula shows how the factor \( b \) compresses or stretches the wave horizontally. For \( y = 3 \cos(x + 3\pi) - 2 \), with \( b = 1 \), the period is \( 2\pi \). So, the cycle repeats every \( 2\pi \) units along the x-axis. Knowing the period is useful in graphing as it indicates where the wave starts and ends, helping to predict its pattern.
Phase Shift
Phase shift refers to the horizontal movement of the wave along the x-axis. In the general cosine function \( y = a \cos(b(x + c)) + d \), the phase shift is calculated using \( -\frac{c}{b} \). Essentially, it's how much the entire graph of the function is shifted left or right. For our function \( y = 3 \cos(x + 3\pi) - 2 \), the value \( c = 3\pi \) and \( b = 1 \) results in a phase shift of \( -3\pi \). This indicates that the graph is moved 3\pi units to the left. Understanding phase shift is important for predicting where the wave starts its cycle, especially in real-world applications like sound waves or tides.
Graphing Functions
Graphing trigonometric functions like the cosine involves several steps to ensure accuracy and understanding of the graph's behavior. To graph \( y = 3 \cos(x + 3\pi) - 2 \), follow these key adjustments:
  • Start with a basic cosine wave, which usually begins at its highest point at zero.
  • Apply the amplitude by adjusting the wave to peak at 3 and trough at -3, both measured from the midline.
  • Introduce the period of \( 2\pi \) to map one complete cycle from 0 to \( 2\pi \).
  • Shift the entire wave to the left by \( 3\pi \) units due to the phase shift.
  • Finally, move the whole graph downward by 2 units for the vertical shift."
By following these adjustments, you can accurately portray the behavior of the trigonometric function. Recognize that graphing is a skill that may take some practice to master, and each step builds on the previous ones to reflect the function's many transformations.

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