Select all the transformations undergone from f(x) = x2

Question

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
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 from \( f(x) = x^2 \).
2. Horizontal stretching/compression:
   • The expression \( 2x \) indicates a horizontal compression by a factor of \( \frac{1}{2} \). This means that the graph will be compressed horizontally.
3. Horizontal translation:
   • The expression \( 2x - 2 \) means the function is translated to the right by 1 unit (since setting \( 2x - 2 = 0 \) gives \( x = 1 \)).
4. Reflection across the x-axis:
   • The negative sign in front of the squared term indicates that the graph is reflected across the x-axis.
5. Vertical translation:
   • The addition of \( +2 \) indicates that the graph is translated up by 2 units.

Now, summarizing these transformations:

• Compressed horizontally (due to the coefficient of \( x \) being 2)
• Translated to the right (by 1 unit)
• Reflected across the x-axis (due to the negative sign)
• Translated up (by 2 units)

Thus, the selected transformations are:

• Compressed horizontally
• Translated to the right
• Reflected across the x-axis
• Translated up