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The given equation is:

\[
2x^2 + 9y = 0
\]

### Step 1: Rewrite the equation in standard form
We can express \( y \) in terms of \( x \) to rewrite the equation in a more familiar form.

\[
2x^2 + 9y = 0 \implies 9y = -2x^2 \implies y = -\frac{2}{9}x^2
\]

This is the equation of a parabola in the form \( y = ax^2 \), where \( a = -\frac{2}{9} \). The negative sign indicates that the parabola opens **downwards**.

### Step 2: Key features of the parabola
- **Vertex**: Since the equation is in the form \( y = ax^2 \), the vertex of the parabola is at the origin \((0, 0)\).
- **Direction of Opening**: Since \( a = -\frac{2}{9} \) is negative, the parabola opens downwards.
- **Stretch Factor**: The value \( |a| = \frac{2}{9} \) indicates that the parabola is wider than the standard parabola \( y = x^2 \), meaning it opens more slowly.

### Step 3: Sketch of the parabola
The parabola has a vertex at the origin and opens downward, with the graph being relatively wide due to the small value of \( |a| = \frac{2}{9} \).

### Step 4: Discussion
- The parabola is symmetric about the y-axis because there is no linear term in \( x \), meaning that it’s a vertical parabola.
- The vertex at \( (0, 0) \) represents the highest point of the parabola because it opens downward.
- As \( x \) increases or decreases, \( y \) decreases because of the negative coefficient \( -\frac{2}{9} \).

Would you like me to generate a sketch of the parabola or provide any more details?

### Follow-up Questions:
1. How does the coefficient of \( x^2 \) affect the width of a parabola?
2. What happens to the shape and direction of the parabola if \( a \) were positive?
3. How would the vertex shift if a constant term were added to the equation?
4. Can you derive the focus and directrix of the parabola?
5. How does the symmetry of the parabola affect its graph?

### Tip:
When dealing with parabolas in the form \( y = ax^2 \), always check the sign of \( a \) to determine the direction of the opening, and the magnitude of \( a \) to understand how "wide" or "narrow" the parabola will be.

Sketch and discuss the equation of the parabola 2x^2 + 9y = 0.

Explore how to sketch and analyze the parabola defined by 2x^2 + 9y = 0. Learn about the vertex, direction of opening, and how the coefficient of x^2 affects the graph. Suitable for high school students in Grades 10-12.

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