Answered

Which transformations are needed to change the parent cosine function to

$y=3 \cos (10(x-\pi))$?


AVertical compression of 3, horizontal stretch to a period of $5\pi$, phase shift of $\pi$ units to the left
BVertical stretch of 3, horizontal compression to a period of $\frac{\pi}{5}$, phase shift of $\pi$ units to the right
CVertical compression of 3, horizontal stretch to a period of $10\pi$, phase shift of $\pi$ units to the right
DVertical stretch of 3, horizontal compression to a period of $\frac{\pi}{5}$, phase shift of $\pi$ units to the left

Verified Answer :

  • The parameter \(A=3\) causes a vertical stretch by a factor of 3.
  • The parameter \(B=10\) compresses the period of the cosine wave to \(\frac{2\pi}{10} = \frac{\pi}{5}\).
  • The term \((x-z)\) shifts the graph to the right by \(z\) units.
  • These transformations match the second option: vertical stretch of 3, horizontal compression to a period of \(\frac{\pi}{5}\), phase shift of \(z\) units to the right.

Explanation

  1. Analyze the Parameters

To identify the transformations from \(y = \cos(x)\) to \(y = 3 \cos(10(x - z))\), we represent the target function using the general form \(y = A \cos(B(x - C)) + D\). By comparing this with \(y = 3 \cos(10(x - z))\), we identify the parameters:
\(A = 3\)
\(B = 10\)
\(C = z\)
\(D = 0\)

  1. Evaluate the Transformations

The parameter \(A = 3\) represents the vertical transformation. Since \(|A| > 1\), this indicates a vertical stretch by a factor of 3. Next, we examine the horizontal transformation controlled by \(B = 10\). The period of the parent function \(y = \cos(x)\) is \(2\pi\). The period of the transformed function \(y = \cos(10x)\) is calculated as \(T = \frac{2\pi}{|B|} = \frac{2\pi}{10} = \frac{\pi}{5}\). A change in period from \(2\pi\) to \(\frac{\pi}{5}\) is a horizontal compression. Finally, the parameter \(C = z\) in the expression \((x - z)\) indicates a horizontal phase shift. Since \(C = z\) is subtracted from \(x\), this corresponds to a phase shift of \(z\) units to the right.

  1. Conclusion

Comparing our findings with the given options:

  1. A vertical stretch of 3.
  2. A horizontal compression to a period of \(\frac{\pi}{5}\).
  3. A phase shift of \(z\) units to the right.

This corresponds exactly to the second option provided in the problem.

Examples

Understanding transformations of trigonometric functions is essential in fields like audio engineering and signal processing. For instance, if you are modeling a sound wave, a vertical stretch might represent an increase in volume (amplitude), while a horizontal compression could represent an increase in the pitch (frequency) of the sound. Being able to shift these waves horizontally allows engineers to synchronize signals or account for delays in transmission, ensuring that the sound is perfectly tuned and timed.

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