To identify the transformations for the function \( f(x) = 3 \cos(-x + \frac{\pi}{3}) - 2 \), we can break it down into components based on the standard cosine function \( \cos(x) \).
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Start with the base function: The base function is \( \cos(x) \).
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Horizontal reflection: The term \( -x \) indicates a reflection across the vertical axis. So, \( \cos(-x) = \cos(x) \) reflects the function horizontally.
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Horizontal shift: The term \( +\frac{\pi}{3} \) in \( -x + \frac{\pi}{3} \) indicates a horizontal shift. Since it's part of the argument of the cosine function, this means the function is shifted to the left by \( \frac{\pi}{3} \).
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Vertical stretch: The coefficient \( 3 \) in front of the cosine function indicates a vertical stretch of the graph. The amplitude of the cosine function is stretched from \( 1 \) to \( 3 \).
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Vertical shift: The term \( -2 \) at the end indicates a vertical shift downward by \( 2 \) units.
Combining all these transformations, we can summarize them as:
- Reflect across the vertical axis (due to the negative sign in front of \( x \)).
- Shift left by \( \frac{\pi}{3} \) units.
- Stretch vertically by a factor of \( 3 \).
- Shift down by \( 2 \) units.
In conclusion, the transformations of the function \( f(x) = 3 \cos(-x + \frac{\pi}{3}) - 2 \) are:
- Reflect in the vertical axis.
- Shift left by \( \frac{\pi}{3} \).
- Stretch vertically by a factor of \( 3 \).
- Shift down by \( 2 \) units.
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