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Chapter 7: Problem 340

For the following exercises, find a possible formula for the trigonometric function represented by the given table of values. $$ \begin{array}{|c|c|c|c|c|c|c|}\hline x & {0} & {1} & {2} & {3} & {4} & {5} & {6} \\ \hline y & {-2} & {4} & {10} & {4} & {-2} & {4} & {10} \\\ \hline\end{array} $$

Short Answer

Expert verified
The function is \(y = -6 \cos\left(\frac{\pi}{2} x\right) - 4\).

Step by step solution

01

Recognizing a Pattern

Examine the values of \(y\) given for each corresponding \(x\). Notice that the values repeat every period of 4 units as \(-2, 4, 10, 4\). This suggests that the function could be periodic.
02

Identifying a Potential Trigonometric Function

Since the function is periodic, consider a cosine function which commonly starts at a maximum or minimum when shifted. The repeating pattern at \(x = 0\) suggests that the function is a cosine function reflecting vertically, possibly starting from a minimum at \(y = -2.\)
03

Determining Amplitude, Midline, and Vertical Shift

Recognize the symmetry in the \(y\) values. Calculate the midline \(y = \ rac{10 + (-2)}{2} = 4\). However, since the sequences drop below 4, the amplitude is \(A = 6\), and the midline is \(y = 4\). The graph should be vertically shifted down by \(4\) units.
04

Determine the Period

The values of the function repeat every 4 units, so the period of the function is 4. This implies that inside the cosine function, we have \(2\pi\) divided by the period, giving \(b = \ rac{2\pi}{4} = \ rac{\pi}{2}\).
05

Deriving the Formula

Construct the cosine function based on these parameters: vertical shift \(-4\), amplitude 6 (reflecting), and period \( \frac{\pi}{2}\). This yields \(y = -6\cos\left(\frac{\pi}{2} x\right) - 4\).
06

Verification with Given Values

Check the derived function against the table values: \(x = 0, y = -6\cos(0) - 4 = -2\), \(x = 1, y = -6\cos\left(\frac{\pi}{2}\right) - 4 = 4\), and so on, confirming the function accurately models the provided data.

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Key Concepts

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

Periodicity
Understanding periodicity is crucial when dealing with trigonometric functions. A function is periodic if it repeats values at regular intervals, known as the period.
Periodic functions have repeating cycles, allowing them to model wave-like patterns. This property is essential for identifying the trigonometric nature of a given function.
In our exercise, the values of the function repeat every 4 units:
  • At \(x = 0\), \(-2\)
  • At \(x = 4\), \(-2\) again
This clearly illustrates that the function is periodic with a period of 4. Recognizing this repetition is the first step in determining the type of trigonometric function at play.
Cosine Function
The cosine function is one of the fundamental trigonometric functions, characterized by its wave shape. Unlike the sine function, the cosine graph typically starts at a maximum or a minimum point, making it ideal for modeling functions that exhibit such behavior from the start.
In our problem, the sequence starts at a minimum value of \(-2\) when \(x = 0\), suggesting that our function is a cosine function that has been vertically reflected. This initial observation helps in making the decision on which trigonometric function to use for building the formula.
Amplitude
Amplitude represents the height from the midline to the peak (or trough) of the function. It shows the extent of variation of the function values. Calculating amplitude is a straightforward process:
  • Find the maximum value: here it's 10.
  • Find the minimum value: here it's \(-2\).
Amplitude is the difference between the maximum and minimum values divided by 2. Hence, \[A = \frac{|10 - (-2)|}{2} = 6\] In this exercise, our amplitude is 6, and since the function dips below the midline, it implies a reflection is involved.
Vertical Shift
A vertical shift moves the entire function up or down on the coordinate plane. It is influenced by the midline of the function, which is calculated as the average of the maximum and minimum function values.
Here, the midline is:\[y = \frac{10 + (-2)}{2} = 4\]This indicates that the values are symmetric around this line.
The cosine function has been vertically shifted down by 4 units, adjusting the whole wave down from the usual center at zero. This shift is crucial for accurately aligning the function with the symmetrical repeating behavior of the data points.
Function Formula Derivation
To derive the cosine function formula, consider all previously identified parameters: amplitude, period, and vertical shift. Here's a quick breakdown of the formula derivation:
  • The amplitude, which is 6, indicates a reflection.
  • A period of 4 gives a frequency factor \(b = \frac{\pi}{2}\).
  • A vertical shift of -4 moves the midline down.
Combining these, the cosine function is expressed as:\[y = -6\cos\left(\frac{\pi}{2} x\right) - 4\]This formula effectively captures the data behavior, showing how trigonometric concepts translate into function representation. Confirm accuracy by checking the calculated values against the data points, ensuring they align perfectly with the expected results.

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