The text in the image is a math problem about a transformation of a parabola. Here's a translation of the problem into English:

"If a parabola with the equation $y = x^2 - 2$ is rotated around the origin (0, 0) by 90°, and then shifted downward by 3 units, what is the equation of the new parabola?"

Step-by-Step Solution:

1. Initial Equation: The original equation is $y = x^2 - 2$.

2. Rotation by 90°:

   • Rotating a point $(x, y)$ counterclockwise by 90° around the origin transforms the coordinates as follows: \quad \text{and} \quad y' = x$$
   • Substituting these into the equation $y = x^2 - 2$, we get: \quad \text{or} \quad x' = -x^2 + 2$$ and $$y' = x$$.

   So the new equation after rotation is:

   \quad \text{or} \quad x = -y^2 + 2$$

3. Shifting Down by 3 Units:

   • A vertical shift by 3 units downward affects the $y'$-coordinate. If we translate the equation downward by 3 units, we subtract 3 from $y'$, yielding: $y' = x - 3$
   • So, the new equation becomes: $y' = -x^2 + 2 - 3 = -x^2 - 1$

Thus, after a 90° rotation and shifting downward by 3 units, the new equation is:

$x = -y^2 - 1$

Let me know if you'd like further clarification!

Related Questions:

1. What would happen to the equation if the rotation was 180° instead of 90°?
2. How would the equation change if the parabola was shifted upward by 3 units instead of downward?
3. What are the general transformation rules for rotating conic sections?
4. Can this transformation be done using matrix operations? How?
5. What is the new vertex of the parabola after the transformation?

Tip:

Always consider transformations step-by-step (like rotation, translation) in the correct order to avoid confusion.