To determine the transformation of the graph of \( f(x) = \sin(x) + 2 \) to the graph labeled Option 2, we need to analyze what kind of transformations have occurred.
- Start from \( f(x) = \sin(x) + 2 \):
- This graph of \( f(x) \) is a sine wave that has been vertically shifted up by 2 units.
Given this, let’s look at the options:
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Option 2 is the graph of \( f(x-7) \): This represents a horizontal shift to the right by 7 units.
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Option 2 is the graph of \( f(x - 7) \): This is just the same as the previous option written differently and also indicates a shift to the right.
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Option 2 is the graph of \( -f(x) \): This represents a reflection over the x-axis and would result in a downward shift.
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Option 2 is the graph of negative \( f \times x \): This is not a standard transformation notation but seems to imply a reflection, which doesn’t apply here.
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Option 2 is the graph of \( -f(-x) \): This would reflect over the y-axis and then the x-axis, altering the wave significantly.
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Option 2 is the graph of \( f(-x) \): This reflects the graph over the y-axis, flipping its direction horizontally.
To solve, you need to compare the visual characteristics that reflect those transformations in Option 2.
Assuming Option 2 indicates simple transformations such as shifts, if Option 2 represents a sine function that's been reflected or shifted but still follows the basic oscillation of the sine function, it could likely be related to a transformation like \( -f(x) \) or \( f(-x) \).
If you recognize that Option 2 mirrors the original function's oscillation yet appears below, the best match (often used in coordinate plane analysis) would generally be:
Option 2 is the graph of \( -f(x) \).
But without the visual, please select according to the reflected transformation observed.
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