Question
Transformations Online Practice
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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=−x2−2
y equals negative x squared minus 2
y=x2−4x+5
y equals x squared minus 4 x plus 5
y=x2+2
y equals x squared plus 2
y=−x2
Answers
GPT-4o mini
To determine which function could not represent \( f'(x) \), the image of \( f(x) = x^2 + 2 \) after a reflection, we need to understand what a reflection means in this context.
A reflection can happen across the x-axis or the y-axis:
1. **Reflection across the x-axis**: This changes the sign of the entire function. For example, reflecting \( f(x) = x^2 + 2 \) across the x-axis would yield \( f'(x) = - (x^2 + 2) = -x^2 - 2 \).
2. **Reflection across the y-axis**: This changes the sign of the x but leaves the value of the function unchanged. For instance, reflecting \( f(x) = x^2 + 2 \) across the y-axis would yield \( f'(-x) = (-x)^2 + 2 = x^2 + 2 \).
Let's analyze the given options:
- **Option 1**: \( y = -x^2 - 2 \)
- This is a reflection across the x-axis and is valid.
- **Option 2**: \( y = x^2 - 4x + 5 \)
- This is a different quadratic function and does not represent a direct reflection of the original function.
- **Option 3**: \( y = x^2 + 2 \)
- This is the original function itself and represents a reflection across the y-axis (which is not a typical reflection in this context).
- **Option 4**: \( y = -x^2 \)
- This is also a reflection across the x-axis but without the vertical shift.
Based on this analysis, the option that **could not** represent \( f'(x) \) after a reflection of \( y = x^2 + 2 \) is:
**y = x^2 - 4x + 5**.
A reflection can happen across the x-axis or the y-axis:
1. **Reflection across the x-axis**: This changes the sign of the entire function. For example, reflecting \( f(x) = x^2 + 2 \) across the x-axis would yield \( f'(x) = - (x^2 + 2) = -x^2 - 2 \).
2. **Reflection across the y-axis**: This changes the sign of the x but leaves the value of the function unchanged. For instance, reflecting \( f(x) = x^2 + 2 \) across the y-axis would yield \( f'(-x) = (-x)^2 + 2 = x^2 + 2 \).
Let's analyze the given options:
- **Option 1**: \( y = -x^2 - 2 \)
- This is a reflection across the x-axis and is valid.
- **Option 2**: \( y = x^2 - 4x + 5 \)
- This is a different quadratic function and does not represent a direct reflection of the original function.
- **Option 3**: \( y = x^2 + 2 \)
- This is the original function itself and represents a reflection across the y-axis (which is not a typical reflection in this context).
- **Option 4**: \( y = -x^2 \)
- This is also a reflection across the x-axis but without the vertical shift.
Based on this analysis, the option that **could not** represent \( f'(x) \) after a reflection of \( y = x^2 + 2 \) is:
**y = x^2 - 4x + 5**.