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Chapter 2: Problem 16

Determine amplitude, period, and phase shift for each function. $$ y=4 \cos (3 x-2 \pi) $$

Short Answer

Expert verified
Amplitude is 4, period is \( \frac{2 \pi}{3} \), and phase shift is \( \frac{2 \pi}{3} \) to the right.

Step by step solution

01

Identify the Amplitude

The general form of the cosine function is given as \(y = A \cos(Bx + C)\).Here, the amplitude \(A\) is the coefficient of the cosine term. In the function \(y = 4 \cos (3x - 2 \pi)\), the amplitude is \(A = 4\).
02

Determine the Period

The period of a cosine function is determined by the formula \(\text{Period} = \frac{2 \pi}{B}\). In the given function \(y = 4 \cos (3x - 2 \pi)\), \(B = 3\). Thus, the period is \(\text{Period} = \frac{2 \pi}{3}\).
03

Calculate the Phase Shift

The phase shift is determined by solving for \(C\) in the inner function of the cosine. The general form is \(y = A \cos(B(x - C))\).In the given function, \(3(x - \frac{2 \pi}{3})\). Therefore, \(C = \frac{2 \pi}{3}\) which represents a phase shift to the right by \( \frac{2 \pi}{3}\).

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Key Concepts

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

Amplitude of the cosine function
The amplitude of a cosine function represents the peak value of the wave. It measures how far the wave stretches vertically from its mean position.

The general form of a cosine function is expressed as \( y = A \cos(Bx + C) \) where \( A \) indicates the amplitude.

In the function \( y = 4 \cos(3x - 2 \pi) \), the coefficient of the cosine function is 4. Thus, the amplitude is 4.

This means that the highest point on the wave will be 4 units above the mean position and the lowest point will be 4 units below.
Period of the cosine function
The period of a cosine function tells us how long it takes for the wave to complete one full cycle. It's a measure of the horizontal length of one cycle of the wave.

You can calculate the period using the formula \( \text{Period} = \frac{2\pi}{B} \) where \( B \) is the coefficient of \( x \) in the cosine function.

In our given function \( y = 4 \cos(3x - 2 \pi) \), \( B = 3 \). Therefore, the period is \( \frac{2\pi}{3} \).

This means that the wave repeats every \( \frac{2\pi}{3} \) units along the x-axis.
Phase Shift of the cosine function
The phase shift of a cosine function is the horizontal displacement from the usual position of the wave. Essentially, it tells us how much the entire wave is shifted to the left or right.

In general form \( y = A \cos(B(x - C)) \), the phase shift is given by \( \frac{C}{B} \).

For our function \( y = 4 \cos(3x - 2 \pi) \), rewriting the inner function as \( 3(x - \frac{2\pi}{3})\), it becomes easy to identify that \( C = \frac{2\pi}{3} \) and \( B = 3 \). Consequently, the phase shift is \( \frac{2\pi}{3} \) units to the right.

Thus, the entire wave is moved \( \frac{2\pi}{3} \) units to the right along the x-axis.

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Most popular questions from this chapter

Use a calculator to find the acute angle \(\alpha\) (to the nearest tenth of a degree) that satisfies \(\sin (\alpha)=0.36\).

Find the period and range for the function \(y=5 \sec (\pi x)\).

Sketch at least one cycle of the graph of each function. Determine the period and the equations of the vertical asymptotes. $$ y=\cot (x / 3) $$

Find \(\sin (\alpha)\) given that \(\cos (\alpha)=1 / 3\) and \(\alpha\) lies in quadrant IV.

Determine the amplitude, phase shift, and range for each function. Sketch at least one cycle of the graph and label the five key points on one cycle as done in the examples. $$ y=-\cos x $$
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