**Describe the transformation from the parent graph.  Y = -(x – 3)**2 **– 4**


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.

Explore the transformation of the quadratic function y = -(x - 3)^2 - 4 with a detailed explanation. Learn about horizontal translations, reflections, and vertical translations. Suitable for high school students studying quadratic functions.

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