Question

10.

Select all the transformations undergone from f(x) = x2
to g(x)= −(2x−2)2+2

(4 points)
Responses

reflected across the y-axis
reflected across the y-axis

reflected across the x-axis
reflected across the x-axis

reflected about the origin
reflected about the origin

compressed horizontally
compressed horizontally

stretched horizontally
stretched horizontally

translated to the right
translated to the right

translated to the left
translated to the left

translated up
translated up

translated down

Answer 1

To analyze the transformations from \( f(x) = x^2 \) to \( g(x) = -(2x - 2)^2 + 2 \), let's break it down step-by-step.

1. **Start with \( f(x) = x^2 \)**.
2. **Horizontal transformation**: The term \( 2x - 2 \) indicates a factor of 2, which will compress the function horizontally by a factor of \( \frac{1}{2} \) (since it’s in the form of \( kx \), where \( k = 2 \), compression occurs).
3. **Translation**: The term \( -2 \) in the expression \( 2x - 2 \) indicates a translation to the right by 1 unit (since \( x - 1 = 2 \) gives us \( x = 1 \)).
4. **Reflection**: The negative sign outside the squared term (the leading negative in \( g(x) \)) indicates that the function is reflected across the x-axis.
5. **Vertical translation**: The \( +2 \) at the end means we translate the function up by 2 units.

Collating the transformations, we have:
- Compressed horizontally
- Translated to the right
- Reflected across the x-axis
- Translated up

Based on these explanations, the selected transformations are:
- **Compressed horizontally**
- **Translated to the right**
- **Reflected across the x-axis**
- **Translated up**