Asked by Jaz
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Question
How do the graphs of f left parenthesis x right parentheis equals sine x and g left parenthesis x right parenthesis equals sine 2 x plus 3 compare? Select all that apply.
(1 point)
Responses
The amplitude of g left parenthesis x right parenthesis is double the amplitude of f left parenthesis x right parenthesis.
The amplitude of Image with alt text: g left parenthesis x right parenthesis is double the amplitude of Image with alt text: f left parenthesis x right parenthesis .
g left parenthesis x right parenthesis and f left parenthesis x right parenthesis have the same zeroes.
Image with alt text: g left parenthesis x right parenthesis and Image with alt text: f left parenthesis x right parenthesis have the same zeroes.
g left parenthesis x right parenthesis is shifted 3 units higher than f left parenthesis x right parenthesis.
Image with alt text: g left parenthesis x right parenthesis is shifted 3 units higher than Image with alt text: f left parenthesis x right parenthesis .
g left parenthesis x right parenthesis is shifted 3 units to the left of f left parenthesis x right parenthesis.
Image with alt text: g left parenthesis x right parenthesis is shifted 3 units to the left of Image with alt text: f left parenthesis x right parenthesis .
g left parenthesis x right parenthesis has a period that is half the period of f left parenthesis x right parenthesis.
Image with alt text: g left parenthesis x right parenthesis has a period that is half the period of Image with alt text: f left parenthesis x right parenthesis .
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Answers
Answered by
GPT-4o mini
To analyze the functions \( f(x) = \sin(x) \) and \( g(x) = \sin(2x) + 3 \), let's examine their properties:
1. **Amplitude**:
- The amplitude of \( f(x) = \sin(x) \) is \( 1 \) (since the sine function oscillates between -1 and 1).
- The amplitude of \( g(x) = \sin(2x) + 3 \) is also \( 1 \). The amplitude remains unchanged regardless of vertical shifts.
- Therefore, **the amplitude of \( g(x) \)** is NOT double the amplitude of \( f(x) \).
2. **Zeroes**:
- The zeroes of \( f(x) = \sin(x) \) are at \( n\pi \) for \( n \in \mathbb{Z} \).
- To find the zeroes of \( g(x) = \sin(2x) + 3 \), we set it to zero: \( \sin(2x) + 3 = 0\) implies \( \sin(2x) = -3 \), which is not possible since the sine function can only output values in the range [-1, 1]. Thus, \( g(x) \) has no zeroes.
- Therefore, **\( g(x) \) and \( f(x) \) do NOT have the same zeroes**.
3. **Vertical Shift**:
- The function \( g(x) = \sin(2x) + 3 \) is shifted **3 units upward** compared to \( f(x) = \sin(x) \).
- Therefore, **\( g(x) \) is shifted 3 units higher than \( f(x) \)**.
4. **Horizontal Shift**:
- Neither \( g(x) \) nor \( f(x) \) exhibits a horizontal shift. Instead, \( g(x) \) includes a change in the frequency (due to the factor of 2 in \( \sin(2x) \)) which does not equate to a horizontal shift.
5. **Period**:
- The period of \( f(x) = \sin(x) \) is \( 2\pi \).
- The period of \( g(x) = \sin(2x) + 3 \) is \( \frac{2\pi}{2} = \pi \).
- Therefore, **the period of \( g(x) \) is half the period of \( f(x) \)**.
Based on this analysis, the correct responses are:
- **g(x) is shifted 3 units higher than f(x).**
- **g(x) has a period that is half the period of f(x).**
These are the applicable comparisons between the functions \( f(x) \) and \( g(x) \).
1. **Amplitude**:
- The amplitude of \( f(x) = \sin(x) \) is \( 1 \) (since the sine function oscillates between -1 and 1).
- The amplitude of \( g(x) = \sin(2x) + 3 \) is also \( 1 \). The amplitude remains unchanged regardless of vertical shifts.
- Therefore, **the amplitude of \( g(x) \)** is NOT double the amplitude of \( f(x) \).
2. **Zeroes**:
- The zeroes of \( f(x) = \sin(x) \) are at \( n\pi \) for \( n \in \mathbb{Z} \).
- To find the zeroes of \( g(x) = \sin(2x) + 3 \), we set it to zero: \( \sin(2x) + 3 = 0\) implies \( \sin(2x) = -3 \), which is not possible since the sine function can only output values in the range [-1, 1]. Thus, \( g(x) \) has no zeroes.
- Therefore, **\( g(x) \) and \( f(x) \) do NOT have the same zeroes**.
3. **Vertical Shift**:
- The function \( g(x) = \sin(2x) + 3 \) is shifted **3 units upward** compared to \( f(x) = \sin(x) \).
- Therefore, **\( g(x) \) is shifted 3 units higher than \( f(x) \)**.
4. **Horizontal Shift**:
- Neither \( g(x) \) nor \( f(x) \) exhibits a horizontal shift. Instead, \( g(x) \) includes a change in the frequency (due to the factor of 2 in \( \sin(2x) \)) which does not equate to a horizontal shift.
5. **Period**:
- The period of \( f(x) = \sin(x) \) is \( 2\pi \).
- The period of \( g(x) = \sin(2x) + 3 \) is \( \frac{2\pi}{2} = \pi \).
- Therefore, **the period of \( g(x) \) is half the period of \( f(x) \)**.
Based on this analysis, the correct responses are:
- **g(x) is shifted 3 units higher than f(x).**
- **g(x) has a period that is half the period of f(x).**
These are the applicable comparisons between the functions \( f(x) \) and \( g(x) \).
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