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

To solve and discuss the equation of the parabola given by:

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

### Step 1: Rearranging the equation
First, we can rearrange the equation into a more familiar form, isolating \( y \) on one side:

\[
9y = -2x^2
\]

Now, divide both sides by 9 to solve for \( y \):

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

This is the equation of a **parabola** in its standard form. 

### Step 2: Identifying key features
The general form of a parabola is:

\[
y = ax^2 + bx + c
\]

In this case, \( a = -\frac{2}{9} \), \( b = 0 \), and \( c = 0 \). Let’s now discuss the properties of this parabola.

1. **Direction of the parabola**:
   - Since the coefficient of \( x^2 \) (i.e., \( a \)) is negative (\( a = -\frac{2}{9} \)), the parabola opens **downwards**.
  
2. **Vertex**:
   - The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. In this equation, since there are no terms involving \( x \) or constants added, the vertex is at the origin \( (0, 0) \).
   
3. **Axis of symmetry**:
   - The axis of symmetry is the vertical line that passes through the vertex. For this parabola, the axis of symmetry is the line \( x = 0 \).

4. **Focus and Directrix**:
   The general formula for the focus and directrix of a parabola of the form \( y = ax^2 \) is derived from the fact that the distance from the vertex to the focus (or directrix) is \( \frac{1}{4|a|} \).
   
   In our case, \( a = -\frac{2}{9} \), so the distance is:

   \[
   \text{Distance} = \frac{1}{4|a|} = \frac{1}{4 \times \frac{2}{9}} = \frac{9}{8}
   \]

   - **Focus**: Since the parabola opens downward, the focus is \( \frac{9}{8} \) units below the vertex. The focus is at \( \left( 0, -\frac{9}{8} \right) \).
   - **Directrix**: The directrix is \( \frac{9}{8} \) units above the vertex, so the equation of the directrix is \( y = \frac{9}{8} \).

### Summary of key features:
- **Equation**: \( y = -\frac{2}{9}x^2 \)
- **Vertex**: \( (0, 0) \)
- **Axis of symmetry**: \( x = 0 \)
- **Focus**: \( \left( 0, -\frac{9}{8} \right) \)
- **Directrix**: \( y = \frac{9}{8} \)
- **Direction**: Opens downward.

Would you like further details or have any questions?

Here are 5 related questions for further exploration:
1. How would the equation change if the parabola opened upwards?
2. What happens to the focus and directrix if \( a \) becomes more negative (e.g., \( a = -1 \))?
3. Can you convert this equation to parametric form?
4. How would you graph this parabola based on its equation?
5. What is the importance of the directrix in defining a parabola?

**Tip**: When determining the focus and directrix of a parabola, always remember that the focus lies on the same side as the parabola's opening, while the directrix is on the opposite side.

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

Discover how to solve and analyze the parabola equation 2x² + 9y = 0. This detailed step-by-step solution explores key concepts in quadratic equations, parabolas, and coordinate geometry. Suitable for high school students in Grades 9-12.

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