Question
Identify the transformations for the function f(x)=3cos(-x+π/3)-2.
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Answered by
GPT-4o mini
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 \( \cos(x) \).
1. **Start with the base function**: The base function is \( \cos(x) \).
2. **Horizontal reflection**: The term \( -x \) indicates a reflection across the vertical axis. So, \( \cos(-x) = \cos(x) \) reflects the function horizontally.
3. **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 \( \frac{\pi}{3} \).
4. **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 \( 3 \).
5. **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:
1. Reflect in the vertical axis.
2. Shift left by \( \frac{\pi}{3} \).
3. Stretch vertically by a factor of \( 3 \).
4. Shift down by \( 2 \) units.
1. **Start with the base function**: The base function is \( \cos(x) \).
2. **Horizontal reflection**: The term \( -x \) indicates a reflection across the vertical axis. So, \( \cos(-x) = \cos(x) \) reflects the function horizontally.
3. **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 \( \frac{\pi}{3} \).
4. **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 \( 3 \).
5. **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:
1. Reflect in the vertical axis.
2. Shift left by \( \frac{\pi}{3} \).
3. Stretch vertically by a factor of \( 3 \).
4. Shift down by \( 2 \) units.