Question
The function y=x2+2
undergoes a reflection. Which of the following could not represent f′(x)
, the image of f
after the reflection? (1 point)
Responses
y=−x2
y equals negative x squared
y=−x2−2
y equals negative x squared minus 2
y=x2+2
y equals x squared plus 2
y=x2−4x+5
undergoes a reflection. Which of the following could not represent f′(x)
, the image of f
after the reflection? (1 point)
Responses
y=−x2
y equals negative x squared
y=−x2−2
y equals negative x squared minus 2
y=x2+2
y equals x squared plus 2
y=x2−4x+5
Answers
GPT-4o mini
To understand how reflecting the function \( f(x) = x^2 + 2 \) affects its derivative, \( f'(x) \), we first need to determine what a reflection means in this context.
1. **Reflection across the x-axis**: This would produce \( f(x) = - (x^2 + 2) = -x^2 - 2 \)
2. **Reflection across the y-axis**: This would produce \( f(x) = (−x)^2 + 2 = x^2 + 2 \), which is the same as the original function.
3. **Reflection across the line \( y = x \)**: This changes the equation in a way that doesn't directly apply here, but primarily affects how we think of inputs and outputs.
Given the choices provided:
- **Option A**: \( y = -x^2 \) (reflection across x-axis without the vertical shift)
- **Option B**: \( y = -x^2 - 2 \) (reflection across x-axis with the vertical shift)
- **Option C**: \( y = x^2 + 2 \) (no reflection; same as original function)
- **Option D**: \( y = x^2 - 4x + 5 \) (a transformed version of the original function but not a reflection)
Since we are asked which could NOT represent \( f'(x) \) after a reflection, we will evaluate these candidates:
1. **For \( y = -x^2 \)**: This reflects the graph across the x-axis but does not maintain vertical alignment.
2. **For \( y = -x^2 - 2 \)**: This also reflects across the x-axis and moves the entire graph down.
3. **For \( y = x^2 + 2 \)**: This does not show reflection; it's identical to the original.
4. **For \( y = x^2 - 4x + 5 \)**: This represents a different quadratic expression by shifting and possibly changing vertex but does not match up as a reflection of the original function.
Given these observations, **the function that could not possibly represent \( f'(x) \) after a reflection is**:
**y = x^2 + 2** (Option C), as it represents the original function with no reflective change.
1. **Reflection across the x-axis**: This would produce \( f(x) = - (x^2 + 2) = -x^2 - 2 \)
2. **Reflection across the y-axis**: This would produce \( f(x) = (−x)^2 + 2 = x^2 + 2 \), which is the same as the original function.
3. **Reflection across the line \( y = x \)**: This changes the equation in a way that doesn't directly apply here, but primarily affects how we think of inputs and outputs.
Given the choices provided:
- **Option A**: \( y = -x^2 \) (reflection across x-axis without the vertical shift)
- **Option B**: \( y = -x^2 - 2 \) (reflection across x-axis with the vertical shift)
- **Option C**: \( y = x^2 + 2 \) (no reflection; same as original function)
- **Option D**: \( y = x^2 - 4x + 5 \) (a transformed version of the original function but not a reflection)
Since we are asked which could NOT represent \( f'(x) \) after a reflection, we will evaluate these candidates:
1. **For \( y = -x^2 \)**: This reflects the graph across the x-axis but does not maintain vertical alignment.
2. **For \( y = -x^2 - 2 \)**: This also reflects across the x-axis and moves the entire graph down.
3. **For \( y = x^2 + 2 \)**: This does not show reflection; it's identical to the original.
4. **For \( y = x^2 - 4x + 5 \)**: This represents a different quadratic expression by shifting and possibly changing vertex but does not match up as a reflection of the original function.
Given these observations, **the function that could not possibly represent \( f'(x) \) after a reflection is**:
**y = x^2 + 2** (Option C), as it represents the original function with no reflective change.