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Chapter 7: Problem 464

For the following exercises, prove the identity. Plot the points and find a function of the form \(y=A \cos (B x+C)+D\) that fits the given data. $$ \begin{array}{|c|c|c|c|c|c|c|}\hline x & {0} & {1} & {2} & {3} & {4} & {5} \\\ \hline y & {-2} & {2} & {-2} & {2} & {-2} & {2} \\ \hline\end{array} $$

Short Answer

Expert verified
The function is \(y = 2 \cos(\pi x + \pi)\).

Step by step solution

01

Analyze the given data

Observe the function data table given as: \(x = \{0, 1, 2, 3, 4, 5\}\) and \(y = \{-2, 2, -2, 2, -2, 2\}\). Notice that \(y\) alternates between two values: \(-2\) and \(2\). This indicates a cosine function because cosine alternates in a waveform. The data also indicates a wave that repeats every 2 units, suggesting a period of 2.
02

Determine the amplitude (A)

Amplitude \(A\) is the distance from the midline of the wave to its peak. The maximum value of \(y\) is 2, and the minimum is -2, so the amplitude \(A = \frac{2 - (-2)}{2} = 2\).
03

Calculate the period to find B

The period of the cosine function determines \(B\) in \(y = A \cos(Bx + C) + D\). The period \(T\) is given by the repeating cycle, which is 2 intervals here. We know that for cosine \(T = \frac{2\pi}{|B|}\), so \(2 = \frac{2\pi}{|B|}\). Therefore, \(|B| = \pi\). Since the function is increasing from x = 0, \(B = \pi\).
04

Determine phase shift C

Since the given points show a cosine wave starting at its minimum at \(x=0\), there is a vertical shift but no noticeable horizontal shift (phase shift \(C\)). Hence, we can assume \(C = \pi\).
05

Find the vertical shift D

The vertical shift \(D\) is the average of the maximum and minimum values, which is the midline of the cosine function. \(D = \frac{2 + (-2)}{2} = 0\).
06

Compile the function

By substituting all calculated values into the function form, we get \(y = 2 \cos(\pi x + \pi) + 0\). Simplified, \(y = 2 \cos(\pi x + \pi)\).

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Key Concepts

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

Amplitude
The amplitude of a function depicts the height of its peaks from the baseline or midline of the wave. In a cosine or sine wave, the amplitude is measured as the maximum value minus the minimum value, divided by two.
Using the provided data, you see that the values for y oscillate between -2 and 2. Therefore, the maximum value is 2 and the minimum value is -2.
You calculate the amplitude using the formula:
  • Amplitude, \( A = \frac{2 - (-2)}{2} = 2 \)
The amplitude of 2 indicates that from the midline of this cosine function, the peaks stretch 2 units upwards and downwards. It is important because the amplitude defines the extent of variation and vibrancy of the wave.
Period of a function
The period of a function tells you how long it takes for the function to complete one full cycle of its pattern. You can think of it as the horizontal length of one complete wave. For trigonometric functions like cosine, the normal period is \(2\pi\), but it can be modified by the coefficient \(B\) in the formula \(y = A \cos(Bx + C) + D\).
  • The period \(T\) is given by the formula: \(T = \frac{2\pi}{|B|} \)
In this exercise, the wave repeats every 2 units along the x-axis. Thus, the period is 2. By setting \(2 = \frac{2\pi}{|B|}\), you determine that \(|B| = \pi\). Recognizing the period helps you understand the frequency of oscillation, where a smaller period suggests rapid oscillations.
Phase shift
Phase shift refers to the horizontal movement of a wave along the x-axis. It is denoted by \(C\) in the equation \(y = A \cos(Bx + C) + D\). The purpose of identifying a phase shift is to determine where the wave begins on the coordinate plane relative to its standard position. In this exercise, since the cosine wave starts at its minimum at \(x=0\), a shift occurs. However, a careful assessment does imply a phase shift of \(C = \pi\) to align with the given start point by adjusting where the cycle starts.
Phase shifts are significant when mapping out waves to match specific points or patterns in data, making it easier for you to predict or analyze future behavior of the wave.
Vertical shift
The vertical shift in trigonometric functions represents a shift up or down on the y-axis and is accounted for by the constant \(D\) in the equation \(y = A \cos(Bx + C) + D\).
In practice, it indicates how far the midline of the wave has been moved from its standard position of \(y = 0\). You can determine the vertical shift by averaging the maximum and minimum values of the function.
  • Calculate it as follows: \(D = \frac{2 + (-2)}{2} = 0\)
In this particular exercise, there's no vertical shift, as \(D = 0\). Understanding vertical shift is essential, as it directly affects where the wave's midpoint sits, relevantly affecting how you perceive the entire wave form.

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Most popular questions from this chapter

For the following exercises, create a function modeling the described behavior. Then, calculate the desired result using a calculator. Whitefish populations are currently at 500 in a lake. The population naturally oscillates above and below by 25 each year. If humans overfish, taking 4% of the population every year, in how many years will the lake first have fewer than 200 whitefish?

For the following exercises, s simplify the equation algebraically as much as possible. Then use a calculator to find the solutions on the interval \([0,2 \pi)\) . Round to four decimal places. $$ \csc ^{2} x-3 \csc x-4=0 $$

For the following exercises, find all exact solutions to the equation on \([0,2 \pi)\) Rewrite the expression \(\sin ^{4} x\) with no powers greater than 1

For the following exercises, find all exact solutions on the interval \([0,2 \pi)\) $$ 2 \sin ^{2} x+5 \sin x+3=0 $$

For the following exercises, find all solutions exactly that exist on the interval \([0,2 \pi)\) $$ 1-2 \tan (\omega)=\tan ^{2}(\omega) $$
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