Question
Periodic Functions Online Practice Part 2
Complete this assessment to review what you’ve learned. It will not count toward your grade.
2 of 112 of 11 Items
Question
Which function has a graph that overlaps the graph of sinx
?(1 point)
Responses
f(x)=sin(x−π)
sine left parenthesis x minus pi right parenthesis
f(x)=sin(x+π)
sine left parenthesis x plus pi right parenthesis
f(x)=sin(x+2π)
sine left parenthesis x plus 2 pi right parenthesis
f(x)=sin(x+π2)
sine left parenthesis x plus Start Fraction pi over 2 End Fraction right parenthesis
Remaining Attempts : 3
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Answers
GPT-4o mini
To determine which function has a graph that overlaps with the graph of \( \sin(x) \), we need to analyze each of the given options:
1. **\( f(x) = \sin(x - \pi) \)**:
- This is a phase shift to the right by \( \pi \). The graph will be shifted and will not overlap with \( \sin(x) \).
2. **\( f(x) = \sin(x + \pi) \)**:
- This is a phase shift to the left by \( \pi \). The sine function \( \sin(x + \pi) \) is equal to \( -\sin(x) \), so it is not the same as \( \sin(x) \).
3. **\( f(x) = \sin(x + 2\pi) \)**:
- This represents a phase shift to the left by \( 2\pi \). However, since the sine function has a periodicity of \( 2\pi \), \( \sin(x + 2\pi) = \sin(x) \). Thus, this function overlaps with \( \sin(x) \).
4. **\( f(x) = \sin\left(x + \frac{\pi}{2}\right) \)**:
- This is a phase shift to the left by \( \frac{\pi}{2} \). The graph will not overlap with \( \sin(x) \) as it represents a cosine function shift.
Based on this analysis, the function that has a graph that overlaps with the graph of \( \sin(x) \) is:
**\( f(x) = \sin(x + 2\pi) \)**.
1. **\( f(x) = \sin(x - \pi) \)**:
- This is a phase shift to the right by \( \pi \). The graph will be shifted and will not overlap with \( \sin(x) \).
2. **\( f(x) = \sin(x + \pi) \)**:
- This is a phase shift to the left by \( \pi \). The sine function \( \sin(x + \pi) \) is equal to \( -\sin(x) \), so it is not the same as \( \sin(x) \).
3. **\( f(x) = \sin(x + 2\pi) \)**:
- This represents a phase shift to the left by \( 2\pi \). However, since the sine function has a periodicity of \( 2\pi \), \( \sin(x + 2\pi) = \sin(x) \). Thus, this function overlaps with \( \sin(x) \).
4. **\( f(x) = \sin\left(x + \frac{\pi}{2}\right) \)**:
- This is a phase shift to the left by \( \frac{\pi}{2} \). The graph will not overlap with \( \sin(x) \) as it represents a cosine function shift.
Based on this analysis, the function that has a graph that overlaps with the graph of \( \sin(x) \) is:
**\( f(x) = \sin(x + 2\pi) \)**.