Question
The funcation 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 = x2−4x + 5
y = x2
y = x2
y =− x2
y =− x2
160
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 = x2−4x + 5
y = x2
y = x2
y =− x2
y =− x2
160
Answers
GPT-4o mini
To determine which of the given functions could NOT represent \( f'(x) \) after the reflection of the function \( f(x) = x^2 + 2 \), we need to consider what a reflection of the function would look like.
The original function \( f(x) = x^2 + 2 \) opens upwards (as the coefficient of \( x^2 \) is positive).
1. **Reflection over the x-axis**: When a function is reflected over the x-axis, the resulting function changes the sign of all its output values. Thus, the reflection would become:
\[
f'(x) = - (x^2 + 2) = -x^2 - 2
\]
2. **Reflection over the y-axis**: Reflecting over the y-axis does not change the shape of the parabola, and it remains \( f'(x) = x^2 + 2 \).
With this in mind, let's analyze the options:
- \( y = x^2 - 4x + 5 \): This is a quadratic function that opens upwards (as the coefficient of \( x^2 \) is positive) and can represent a shifted version of \( f \); it does not represent a reflection.
- \( y = x^2 \): This is also a quadratic function that opens upwards, so it does not represent a reflection.
- \( y = -x^2 \): This function opens downwards and represents a reflection of the original function over the x-axis. Therefore, this function could represent \( f'(x) \) after the reflection over the x-axis.
From this analysis, the function that **could NOT** represent \( f'(x) \) after the reflection is:
- \( y = x^2 - 4x + 5 \)
So the answer is \( y = x^2 - 4x + 5 \).
The original function \( f(x) = x^2 + 2 \) opens upwards (as the coefficient of \( x^2 \) is positive).
1. **Reflection over the x-axis**: When a function is reflected over the x-axis, the resulting function changes the sign of all its output values. Thus, the reflection would become:
\[
f'(x) = - (x^2 + 2) = -x^2 - 2
\]
2. **Reflection over the y-axis**: Reflecting over the y-axis does not change the shape of the parabola, and it remains \( f'(x) = x^2 + 2 \).
With this in mind, let's analyze the options:
- \( y = x^2 - 4x + 5 \): This is a quadratic function that opens upwards (as the coefficient of \( x^2 \) is positive) and can represent a shifted version of \( f \); it does not represent a reflection.
- \( y = x^2 \): This is also a quadratic function that opens upwards, so it does not represent a reflection.
- \( y = -x^2 \): This function opens downwards and represents a reflection of the original function over the x-axis. Therefore, this function could represent \( f'(x) \) after the reflection over the x-axis.
From this analysis, the function that **could NOT** represent \( f'(x) \) after the reflection is:
- \( y = x^2 - 4x + 5 \)
So the answer is \( y = x^2 - 4x + 5 \).