Question
Identify all of the transformations for the function f(x) = 3cos(−x + π3)−2
. Check all that apply(4 points)
Responses
reflection over the x-axis
reflection over the x-axis
vertical shift down 2 units
vertical shift down 2 units
horizontal shift to the right π3
horizontal shift to the right pi over 3
reflection over the y-axis
reflection over the y-axis
Vertical stretch by 3
Vertical stretch by 3
vertical shift up 2 units
vertical shift up 2 units
horizontal shift to the left π3
horizontal shift to the left pi over 3
. Check all that apply(4 points)
Responses
reflection over the x-axis
reflection over the x-axis
vertical shift down 2 units
vertical shift down 2 units
horizontal shift to the right π3
horizontal shift to the right pi over 3
reflection over the y-axis
reflection over the y-axis
Vertical stretch by 3
Vertical stretch by 3
vertical shift up 2 units
vertical shift up 2 units
horizontal shift to the left π3
horizontal shift to the left pi over 3
Answers
GPT-4o mini
Let's analyze the function \( f(x) = 3\cos(-x + \frac{\pi}{3}) - 2 \) to identify the transformations.
1. **Vertical Stretch by 3**: The coefficient of 3 before the cosine function indicates a vertical stretch by a factor of 3. **(Check)**
2. **Reflection over the x-axis**: The negative inside the argument of the cosine function (\(-x\)) does not reflect the function over the x-axis. Instead, it indicates a horizontal reflection that could be interpreted differently. Hence, this transformation is **not applicable**.
3. **Vertical Shift Down 2 Units**: The term \(-2\) at the end of the function indicates a vertical shift downward by 2 units. **(Check)**
4. **Reflection over the y-axis**: The negative sign in front of \(x\) (\(-x\)) indicates a reflection across the y-axis. This implies a horizontal reflection. **(Check)**
5. **Horizontal Shift to the Right \(\frac{\pi}{3}\)**: To determine the horizontal shift, we can rewrite the cosine function: \( -x + \frac{\pi}{3} = -(x - \frac{\pi}{3}) \), which indicates a shift to the right by \(\frac{\pi}{3}\) (due to how the argument of the cosine is arranged). **(Check)**
6. **Vertical Shift Up 2 Units**: There is no indication of an upward vertical shift in this function; therefore, this transformation is **not applicable**.
7. **Horizontal Shift to the Left \(\frac{\pi}{3}\)**: Following the same reasoning as the horizontal shift to the right, there is no leftward shift present. Therefore, this transformation is **not applicable**.
### Summary of Transformations:
- Vertical stretch by 3: **Check**
- Vertical shift down 2 units: **Check**
- Reflection over the y-axis: **Check**
- Horizontal shift to the right \(\frac{\pi}{3}\): **Check**
The transformations that apply are:
1. Vertical stretch by 3
2. Vertical shift down 2 units
3. Reflection over the y-axis
4. Horizontal shift to the right \(\frac{\pi}{3}\)
1. **Vertical Stretch by 3**: The coefficient of 3 before the cosine function indicates a vertical stretch by a factor of 3. **(Check)**
2. **Reflection over the x-axis**: The negative inside the argument of the cosine function (\(-x\)) does not reflect the function over the x-axis. Instead, it indicates a horizontal reflection that could be interpreted differently. Hence, this transformation is **not applicable**.
3. **Vertical Shift Down 2 Units**: The term \(-2\) at the end of the function indicates a vertical shift downward by 2 units. **(Check)**
4. **Reflection over the y-axis**: The negative sign in front of \(x\) (\(-x\)) indicates a reflection across the y-axis. This implies a horizontal reflection. **(Check)**
5. **Horizontal Shift to the Right \(\frac{\pi}{3}\)**: To determine the horizontal shift, we can rewrite the cosine function: \( -x + \frac{\pi}{3} = -(x - \frac{\pi}{3}) \), which indicates a shift to the right by \(\frac{\pi}{3}\) (due to how the argument of the cosine is arranged). **(Check)**
6. **Vertical Shift Up 2 Units**: There is no indication of an upward vertical shift in this function; therefore, this transformation is **not applicable**.
7. **Horizontal Shift to the Left \(\frac{\pi}{3}\)**: Following the same reasoning as the horizontal shift to the right, there is no leftward shift present. Therefore, this transformation is **not applicable**.
### Summary of Transformations:
- Vertical stretch by 3: **Check**
- Vertical shift down 2 units: **Check**
- Reflection over the y-axis: **Check**
- Horizontal shift to the right \(\frac{\pi}{3}\): **Check**
The transformations that apply are:
1. Vertical stretch by 3
2. Vertical shift down 2 units
3. Reflection over the y-axis
4. Horizontal shift to the right \(\frac{\pi}{3}\)