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=x2−4x+5
y equals x squared minus 4 x plus 5
y=−x2−2
y equals negative x squared minus 2
y=−x2
y equals negative x squared
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=x2−4x+5
y equals x squared minus 4 x plus 5
y=−x2−2
y equals negative x squared minus 2
y=−x2
y equals negative x squared
y=x2+2
Answers
GPT-4o mini
To determine which function \( f'(x) \) could not represent the image of the function \( f(x) = x^2 + 2 \) after a reflection, we need to consider the types of reflections that can happen to a function.
1. **Reflection across the x-axis**: This changes \( f(x) \) to \( -f(x) \).
- For \( f(x) = x^2 + 2 \), this becomes \( f'(x) = -x^2 - 2 \).
2. **Reflection across the y-axis**: This changes \( f(x) \) to \( f(-x) \).
- For \( f(x) = x^2 + 2 \), this becomes \( f'(-x) = (-x)^2 + 2 = x^2 + 2 \).
Now let's analyze each of the given options to see if they can represent a reflection of \( f(x) \):
1. **Option 1**: \( y = x^2 - 4x + 5 \)
- This does not represent a reflection of the original function as it contains a linear term and does not correspond to a simple reflection.
2. **Option 2**: \( y = -x^2 - 2 \)
- This represents a reflection across the x-axis.
3. **Option 3**: \( y = -x^2 \)
- This also represents a reflection across the x-axis.
4. **Option 4**: \( y = x^2 + 2 \)
- This is the original function and does not represent any reflection.
The question asks which of the options could **not** represent the image of \( f \) after reflection. Both Option 1 and Option 4 are not simple reflections of the function. However, upon inspection, Option 1 is the more complex polynomial and could not be derived from basic reflection processes of the original function.
Thus, the answer is **Option 1: \( y = x^2 - 4x + 5 \)**.
1. **Reflection across the x-axis**: This changes \( f(x) \) to \( -f(x) \).
- For \( f(x) = x^2 + 2 \), this becomes \( f'(x) = -x^2 - 2 \).
2. **Reflection across the y-axis**: This changes \( f(x) \) to \( f(-x) \).
- For \( f(x) = x^2 + 2 \), this becomes \( f'(-x) = (-x)^2 + 2 = x^2 + 2 \).
Now let's analyze each of the given options to see if they can represent a reflection of \( f(x) \):
1. **Option 1**: \( y = x^2 - 4x + 5 \)
- This does not represent a reflection of the original function as it contains a linear term and does not correspond to a simple reflection.
2. **Option 2**: \( y = -x^2 - 2 \)
- This represents a reflection across the x-axis.
3. **Option 3**: \( y = -x^2 \)
- This also represents a reflection across the x-axis.
4. **Option 4**: \( y = x^2 + 2 \)
- This is the original function and does not represent any reflection.
The question asks which of the options could **not** represent the image of \( f \) after reflection. Both Option 1 and Option 4 are not simple reflections of the function. However, upon inspection, Option 1 is the more complex polynomial and could not be derived from basic reflection processes of the original function.
Thus, the answer is **Option 1: \( y = x^2 - 4x + 5 \)**.