Question
Describe the transformation of the graph f of x equals sin x + 2 on the graph labeled option 2 option two is the graph of f of x - 7 option 2 is a graph of negative f of x option 2 is a graph of f of negative X option two is graph of negative f of negative X
Answers
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GPT-4o mini
To understand the transformation of the graph of the function \( f(x) = \sin x + 2 \) in relation to the options given, let's analyze what each transformation entails:
1. **Original Function**: \( f(x) = \sin x + 2 \)
- This is the sine function shifted vertically upward by 2 units. So, the midline of the sine wave is at \( y = 2 \) instead of the usual \( y = 0 \).
2. **Option 1: \( f(x) - 7 \)**:
- This would be \( \sin x + 2 - 7 = \sin x - 5 \). It represents a vertical shift downward by 5 units from the original function. The midline would be at \( y = -5 \).
3. **Option 2: \( -f(x) \)**:
- This would be \( -(\sin x + 2) = -\sin x - 2 \). This represents a reflection about the x-axis and then a vertical shift downward by 2 units. The midline moves to \( y = -2 \).
4. **Option 3: \( f(-x) \)**:
- This would be \( \sin(-x) + 2 = -\sin x + 2 \). This is the sine function reflected about the y-axis, with a vertical shift upward by 2 units.
5. **Option 4: \( -f(-x) \)**:
- This would be \( -(\sin(-x) + 2) = -(-\sin x) - 2 = \sin x - 2 \). This represents a reflection about the x-axis and then a vertical shift downward by 2 units. The midline moves to \( y = -2 \).
Based on the comparisons above, you can determine the relationship between the graph of \( f(x) = \sin x + 2 \) and each of these graphs by identifying the transformations applied. Each option represents a different transformation or combination of transformations.
If you are looking for which option transforms \( f(x) \) to match \( -f(x) \), or embeds transformations to create a different function altogether, please specify or clarify so we can better identify and describe it!
1. **Original Function**: \( f(x) = \sin x + 2 \)
- This is the sine function shifted vertically upward by 2 units. So, the midline of the sine wave is at \( y = 2 \) instead of the usual \( y = 0 \).
2. **Option 1: \( f(x) - 7 \)**:
- This would be \( \sin x + 2 - 7 = \sin x - 5 \). It represents a vertical shift downward by 5 units from the original function. The midline would be at \( y = -5 \).
3. **Option 2: \( -f(x) \)**:
- This would be \( -(\sin x + 2) = -\sin x - 2 \). This represents a reflection about the x-axis and then a vertical shift downward by 2 units. The midline moves to \( y = -2 \).
4. **Option 3: \( f(-x) \)**:
- This would be \( \sin(-x) + 2 = -\sin x + 2 \). This is the sine function reflected about the y-axis, with a vertical shift upward by 2 units.
5. **Option 4: \( -f(-x) \)**:
- This would be \( -(\sin(-x) + 2) = -(-\sin x) - 2 = \sin x - 2 \). This represents a reflection about the x-axis and then a vertical shift downward by 2 units. The midline moves to \( y = -2 \).
Based on the comparisons above, you can determine the relationship between the graph of \( f(x) = \sin x + 2 \) and each of these graphs by identifying the transformations applied. Each option represents a different transformation or combination of transformations.
If you are looking for which option transforms \( f(x) \) to match \( -f(x) \), or embeds transformations to create a different function altogether, please specify or clarify so we can better identify and describe it!