Question

The graph f(x)=x2 was transformed to create the graph g(x)=−(x2)2.

Which statement describes this transformation?

Responses

a reflection over the x-axis and a horizontal stretch
a reflection over the x -axis and a horizontal stretch

a reflection over the x-axis and a vertical stretch
a reflection over the x -axis and a vertical stretch

a reflection over the y-axis and a horizontal stretch
a reflection over the y -axis and a horizontal stretch

a reflection over the y-axis and a vertical stretch

Answer 1

To analyze the transformation from the function \( f(x) = x^2 \) to \( g(x) = -(x^2)^2 \), we can break it down:

1. Start with \( f(x) = x^2 \), which is a parabola opening upwards.
2. The term \( (x^2)^2 \) can be simplified to \( x^4 \). Therefore, we have \( g(x) = -x^4 \).
3. The negative sign in front of \( x^4 \) indicates a reflection over the x-axis, which will flip the graph upside down.
4. Since \( x^4 \) has a different shape due to the exponent being greater than 2, this implies a vertical stretch compared to the original quadratic \( x^2 \).

Putting this together, the transformation involves reflecting over the x-axis and stretching it vertically due to the exponent changing from 2 to 4.

Thus, the correct answer is:

**a reflection over the x-axis and a vertical stretch.**