Question
The graph of y = sin 4x is shown
What is different anout the graph when compared to its parent function y = sin x?
A. The amplitude
B. The period
C. The midline
D. The starting point
What is the correct description of the graph as jt compares to its parent function.
A. The graph is stretched vertically so it is taller
B. The graph is compressed vertically so that it is shorter
C. The graph is stretched horizontally so that there are less waves in each section
D. The graph is compressed horizontally so there are more waves in each section
What is different anout the graph when compared to its parent function y = sin x?
A. The amplitude
B. The period
C. The midline
D. The starting point
What is the correct description of the graph as jt compares to its parent function.
A. The graph is stretched vertically so it is taller
B. The graph is compressed vertically so that it is shorter
C. The graph is stretched horizontally so that there are less waves in each section
D. The graph is compressed horizontally so there are more waves in each section
Answers
GPT-4o mini
To determine the differences between the graph of \( y = \sin(4x) \) and its parent function \( y = \sin(x) \), let's analyze each option based on the transformations involved.
1. **Amplitude**: The amplitude of \( y = \sin(x) \) is 1, and the amplitude of \( y = \sin(4x) \) remains 1. Therefore, there is no change in amplitude.
2. **Period**: The period of \( y = \sin(x) \) is \( 2\pi \). The period of \( y = \sin(kx) \) is given by \( \frac{2\pi}{k} \). In this case, \( k = 4 \), thus the period of \( y = \sin(4x) \) is \( \frac{2\pi}{4} = \frac{\pi}{2} \). This means the period is decreased, resulting in more waves in the same horizontal space.
3. **Midline**: Both functions have a midline of \( y = 0 \), so there is no change in the midline.
4. **Starting Point**: The starting point (or phase shift) of \( y = \sin(x) \) is at the origin (0,0). The graph of \( y = \sin(4x) \) also starts at (0,0), so there is no change in the starting point.
### Summary:
- The changes that occur are primarily related to the period of the function.
### Correct Answer for the first part:
- The correct description of the graph compared to its parent function is **B. The period**.
### Correct Answer for the second part:
Given that the period is reduced, resulting in more waves (or cycles) within the same horizontal length, we pick:
- **D. The graph is compressed horizontally so there are more waves in each section**.
Combining these observations:
- **B** and **D** are the correct answers respectively for your questions about differences in the graph.
1. **Amplitude**: The amplitude of \( y = \sin(x) \) is 1, and the amplitude of \( y = \sin(4x) \) remains 1. Therefore, there is no change in amplitude.
2. **Period**: The period of \( y = \sin(x) \) is \( 2\pi \). The period of \( y = \sin(kx) \) is given by \( \frac{2\pi}{k} \). In this case, \( k = 4 \), thus the period of \( y = \sin(4x) \) is \( \frac{2\pi}{4} = \frac{\pi}{2} \). This means the period is decreased, resulting in more waves in the same horizontal space.
3. **Midline**: Both functions have a midline of \( y = 0 \), so there is no change in the midline.
4. **Starting Point**: The starting point (or phase shift) of \( y = \sin(x) \) is at the origin (0,0). The graph of \( y = \sin(4x) \) also starts at (0,0), so there is no change in the starting point.
### Summary:
- The changes that occur are primarily related to the period of the function.
### Correct Answer for the first part:
- The correct description of the graph compared to its parent function is **B. The period**.
### Correct Answer for the second part:
Given that the period is reduced, resulting in more waves (or cycles) within the same horizontal length, we pick:
- **D. The graph is compressed horizontally so there are more waves in each section**.
Combining these observations:
- **B** and **D** are the correct answers respectively for your questions about differences in the graph.