- Identify the transformation parameters: \(A=3\), \(B=10\), and \(C=\pi\).
- Recognize that \(A=3\) causes a vertical stretch of 3.
- Calculate the new period \(P = \frac{2\pi}{10} = \frac{\pi}{5}\), indicating a horizontal compression.
- Identify the phase shift as \(\pi\) units to the right due to \(C=\pi\).
- Select the option: \(\boxed{\text{vertical stretch of 3, horizontal compression to a period of } \frac{\pi}{5}, \text{ phase shift of } \pi \text{ units to the right}}\).
Explanation
- Identifying Transformation Parameters
The transformation of the parent cosine function \(y = \cos(x)\) to the target function \(y = 3 \cos(10(x - \pi))\) can be understood by comparing the latter to the general sinusoidal form:
\[y = A \cos(B(x - C)) + D\]
By comparing \(y = 3 \cos(10(x - \pi))\) with the general form, we identify the following parameters:
- \(A = 3\)
- \(B = 10\)
- \(C = \pi\)
- \(D = 0\)
These constants dictate how the parent function is stretched, compressed, or shifted in the coordinate plane.
- Analyzing Vertical Transformation
The parameter \(A\) represents the amplitude, which controls the vertical scaling of the function. Because \(|A| = 3\), the graph of the function is vertically stretched by a factor of \(3\). This means that the peaks and troughs of the cosine wave, originally at \(1\) and \(-1\), are now at \(3\) and \(-3\), respectively.
- Analyzing Horizontal Transformation and Period
The parameter \(B\) influences the frequency and period of the function. The period of the parent cosine function is \(P = \frac{2\pi}{1} = 2\pi\). For the transformed function, the new period \(P_{new}\) is calculated using the formula:
\[P_{new} = \frac{2\pi}{|B|} = \frac{2\pi}{10} = \frac{\pi}{5}\]
Since the period is reduced from \(2\pi\) to \(\frac{\pi}{5}\), the function undergoes a horizontal compression.
- Analyzing Phase Shift
The parameter \(C\) represents the phase shift, which controls horizontal translation. In the form \(B(x - C)\), a positive value for \(C\) indicates a shift to the right, while a negative value indicates a shift to the left. Since \(C = \pi\), the graph is shifted \(\pi\) units to the right.
- Final Conclusion
Combining all observations:
- Vertical stretch of 3.
- Horizontal compression to a period of \(\frac{\pi}{5}\).
- Phase shift of \(\pi\) units to the right.
Comparing these results to the given options, we identify the second option as the correct choice.
Final Answer: The transformations are a vertical stretch of 3, horizontal compression to a period of \(\frac{\pi}{5}\), and a phase shift of \(\pi\) units to the right. Therefore, the correct option is the second one:
\(\boxed{\text{vertical stretch of 3, horizontal compression to a period of } \frac{\pi}{5}, \text{ phase shift of } \pi \text{ units to the right}}\)
Examples
In engineering and physics, transformations of sinusoidal functions are essential for analyzing waves. For example, if you are designing a sound system or analyzing signal processing, changing the amplitude (vertical stretch) affects the volume of the sound, while changing the period (horizontal compression) alters the pitch or frequency of the signal. Mastering these transformations allows engineers to precisely adjust mechanical or electronic signals to meet specific requirements, such as tuning a radio to a specific frequency or calibrating audio output.