Sketch and discuss the equation of the parabola 2x²+9y=0

### Equation of the Parabola:  
Given equation:  
\[
2x^2 + 9y = 0
\]

To write this in standard form, solve for \(y\):

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

This is the equation of a **parabola** with the form:

\[
y = ax^2
\]

where \(a = -\frac{2}{9}\). This means the parabola opens **downward** because \(a < 0\), and it is **vertically oriented**.

### Key Features of the Parabola:
1. **Vertex**: The vertex of the parabola is at \((0,0)\) since there is no horizontal or vertical shift.
2. **Axis of Symmetry**: The axis of symmetry is the vertical line \(x = 0\) (the y-axis).
3. **Direction**: The parabola opens downward due to the negative sign in the equation.
4. **Shape**: The coefficient \(-\frac{2}{9}\) controls the "width" of the parabola. Since \(\frac{2}{9}\) is a small value, the parabola will be relatively wide compared to a parabola with a larger absolute value of the coefficient (like \(-2\)).

### Sketch:
- The vertex is at the origin \((0,0)\).
- The parabola is symmetric about the y-axis.
- It opens downward and is relatively wide.

I can provide a sketch of this parabola as well. Would you like me to generate the sketch?

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#### Related Questions:
1. How would the graph change if the coefficient of \(x^2\) were positive?
2. What is the effect of shifting the vertex horizontally or vertically?
3. How do you calculate the focus and directrix of this parabola?
4. What happens to the parabola if you multiply the equation by a constant?
5. How does the parabola behave when the coefficient of \(x^2\) becomes larger or smaller?

**Tip**: The sign of the coefficient in a quadratic equation determines whether the parabola opens upward or downward.

Understand the equation of the parabola 2x² + 9y = 0, how to write it in standard form, and sketch its graph. Learn about its key features including vertex, axis of symmetry, and direction of opening. Suitable for students in Grades 9-11.

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