Select all the transformations undergone from f(x) = x^2 to g(x) = -(2x -2)^2+2

Question

Reflected across the y-axis
Reflected across the x-axis
Reflected about the origin
Compressed horizontally
Stretched horizontally
Translated to the right
Translated to the left
Translated up
Translated down

Answer 1

To analyze the transformations from \( f(x) = x^2 \) to \( g(x) = -(2x - 2)^2 + 2 \), we will break down the transformations step-by-step.

1. Starting Function: \[ f(x) = x^2 \]

2. Horizontal Adjustment: \[ g(x) = -(2(x - 1))^2 + 2 \] This indicates a transformation where:

   • The term \( (x - 1) \) suggests a translation to the right by 1 unit.

3. Horizontal Compression: The factor of 2 in the term \( 2(x - 1) \) indicates a compression horizontally by a factor of 2.

4. Reflection across the x-axis: The negative sign in front of the square indicates that the function is reflected across the x-axis.

5. Vertical Translation: The \( +2 \) at the end of the function indicates a translation up by 2 units.

Now, summarizing the transformations:

• Reflected across the x-axis
• Compressed horizontally (by a factor of 2)
• Translated to the right (by 1 unit)
• Translated up (by 2 units)

The following transformations are NOT applicable:

• Reflected across the y-axis
• Reflected about the origin
• Stretched horizontally
• Translated to the left
• Translated down

Thus, the selected transformations are:

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