The equation $y = -(x - 3)^2 - 4$ represents a transformation of the parent graph of $y = x^2$, which is a parabola.

Here’s a step-by-step description of the transformations applied to the parent graph:

1. Horizontal Translation:

   • The expression $(x - 3)$ inside the square indicates a horizontal shift.
   • Specifically, the graph is shifted 3 units to the right.

2. Reflection:

   • The negative sign in front of the squared term $-(x - 3)^2$ reflects the graph over the x-axis.
   • This means the parabola, which normally opens upwards, will now open downwards.

3. Vertical Translation:

   • The term $-4$ outside the squared expression shifts the graph 4 units downward.

Summary of the Transformation:

• Start with the parent graph $y = x^2$.
• Shift the graph 3 units to the right.
• Reflect it across the x-axis (so it opens downward).
• Finally, move the graph 4 units down.

The vertex of the transformed parabola will be at the point $(3, -4)$, and it opens downward.

Would you like more details or have any questions about this transformation? Here are some related questions to consider:

1. What would be the vertex if the equation were $y = (x + 2)^2 + 5$?
2. How does the width of the parabola change if the equation were $y = -2(x - 3)^2 - 4$?
3. What is the axis of symmetry for the given equation?
4. How does a vertical stretch affect the transformation of the parabola?
5. How would the transformation differ if the equation were $y = (x - 3)^2 + 4$?

Tip: Always identify horizontal shifts from the expression inside the parentheses and vertical shifts from the constant added or subtracted outside.