The problem shown involves using transformations of the basic quadratic function $f(x) = x^2$ to match the provided graph. Let's break it down based on what is visible in the image:

1. Basic function: The graph of $f(x) = x^2$ is a standard parabola opening upwards with its vertex at the origin (0,0).

2. Given graph: The new graph is also a parabola, but it appears to have undergone a vertical reflection and a vertical shift. Specifically:

   • Reflection: The parabola is upside down, indicating that the function has been reflected across the x-axis. This transformation is represented by multiplying the function by $-1$, giving $-x^2$.
   • Vertical shift: The vertex of the given parabola seems to have moved downward by 3 units (this is based on the general position of the vertex). This is a vertical translation, which means we subtract 3 from the function, giving $f(x) = -x^2 - 3$.

Thus, the transformation rule that would produce the given graph is: $f(x) = -x^2 - 3$

Would you like a more detailed explanation of the steps or have any questions about this?

Here are 5 related questions to deepen your understanding:

1. How do horizontal shifts affect the graph of a quadratic function?
2. What happens when you apply a horizontal stretch to $f(x) = x^2$?
3. How can you identify whether a graph has undergone a vertical stretch or compression?
4. How does a vertical shift affect the vertex of a parabola?
5. Can a quadratic function have both horizontal and vertical transformations?

Tip: Always check both vertical and horizontal shifts carefully when transforming graphs. Vertical changes affect the "up and down" movement, while horizontal changes adjust the graph "left and right."