To analyze the equation \( y = -2x(x + 3) \), let's first rewrite it in standard form:

1. **Expand the expression**:
   \[
   y = -2x^2 - 6x
   \]

2. **Identify characteristics**:
   - **Type**: This is a quadratic function.
   - **Opening direction**: Since the coefficient of \( x^2 \) is negative (-2), the parabola opens downwards.
   - **Vertex**: The vertex can be found using the formula \( x = -\frac{b}{2a} \):
     - Here, \( a = -2 \) and \( b = -6 \):
     \[
     x = -\frac{-6}{2 \times -2} = \frac{6}{-4} = -\frac{3}{2}
     \]
     Substitute \( x \) back into the equation to find \( y \):
     \[
     y = -2\left(-\frac{3}{2}\right)\left(-\frac{3}{2} + 3\right) = -2\left(-\frac{3}{2}\right)\left(\frac{3}{2}\right) = -\frac{9}{2}
     \]
     So, the vertex is \( \left(-\frac{3}{2}, -\frac{9}{2}\right) \).

3. **Intercepts**:
   - **Y-intercept**: Set \( x = 0 \):
     \[
     y = -2(0)(0 + 3) = 0 \quad \Rightarrow \quad (0, 0)
     \]
   - **X-intercepts**: Set \( y = 0 \):
     \[
     -2x(x + 3) = 0 \quad \Rightarrow \quad x = 0 \text{ or } x + 3 = 0 \Rightarrow x = -3
     \]
     So, the x-intercepts are \( (0, 0) \) and \( (-3, 0) \).

### Summary:
- The equation represents a downward-opening parabola with vertex at \( \left(-\frac{3}{2}, -\frac{9}{2}\right) \).
- Intercepts are at \( (0, 0) \) and \( (-3, 0) \).

Would you like further details or have any questions? Here are some related questions you might consider:

1. How can I graph this quadratic function?
2. What transformations can be applied to this parabola?
3. How can I find the maximum value of this function?
4. What are the real-life applications of quadratic functions?
5. Can you explain the significance of the vertex and intercepts in graphing? 

**Tip**: When graphing quadratics, always identify the vertex and intercepts first for a clearer understanding of the shape.

Learn how to analyze and graph the quadratic function y = -2x(x + 3). This solution provides a detailed breakdown, including vertex calculation, direction of the parabola, and intercepts. Suitable for students in Grades 9-10.

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