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

The equation given is:

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

### Step 1: Simplify the Equation
To express this in the standard form of a parabola, let's solve for \( y \):

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

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

### Step 2: Identify the Type of Parabola
The equation \( y = -\frac{2}{9}x^2 \) is a parabola that opens downwards because the coefficient of \( x^2 \) is negative.

### Step 3: Key Features of the Parabola

- **Vertex**: The vertex is the point where the parabola changes direction. Since the equation is in the form \( y = ax^2 + bx + c \), the vertex is at the origin \((0, 0)\).
- **Axis of Symmetry**: The axis of symmetry is the vertical line through the vertex. Here, it is the y-axis, so the axis of symmetry is \( x = 0 \).
- **Direction**: The parabola opens downwards because the coefficient of \( x^2 \) is negative.
- **Width**: The coefficient \( \frac{2}{9} \) affects the width of the parabola. Since it's less than 1, the parabola is wider compared to the standard parabola \( y = x^2 \).

### Step 4: Sketch the Parabola
- The vertex is at the origin \((0, 0)\).
- Since the parabola opens downward and is relatively wide, you would plot a few points on either side of the vertex to get a sense of the shape.

### Plotting the Points
To sketch, choose values of \( x \) and calculate the corresponding \( y \) values:

- For \( x = 1 \), \( y = -\frac{2}{9}(1)^2 = -\frac{2}{9} \).
- For \( x = -1 \), \( y = -\frac{2}{9}(-1)^2 = -\frac{2}{9} \).
- For \( x = 2 \), \( y = -\frac{2}{9}(2)^2 = -\frac{8}{9} \).
- For \( x = -2 \), \( y = -\frac{8}{9} \).

The sketch would show a symmetric parabola opening downwards with these points plotted along with the vertex.

### Discussion
- **Vertex**: \( (0,0) \)
- **Direction**: Opens downward.
- **Wideness**: The factor \( \frac{2}{9} \) makes the parabola wider than the standard \( y = -x^2 \).
- **Axis of Symmetry**: The line \( x = 0 \) (the y-axis).

### Graphical Representation (Text-Based)
```
        |
        |          *
        |        *   *
        |      *       *
        |     *         *
--------*----------------*-------
        |     *         *
        |      *       *
        |        *   *
        |          *
        |
```

This is a text-based representation of the graph.

Would you like a detailed explanation on how to sketch this graph by hand or have any other questions? Here are some additional questions:

1. How do different coefficients affect the shape and position of the parabola?
2. What is the general form of a parabola, and how can it be transformed?
3. How do you find the focus and directrix of a parabola?
4. How does the sign of the coefficient of \( x^2 \) determine the direction of the parabola?
5. How would the equation change if the parabola were shifted up or down?

**Tip:** Understanding how the coefficients in a quadratic equation affect the graph will help you visualize parabolas more easily.

Explore the equation 2x² + 9y = 0 and understand its properties through a detailed explanation. Learn how to sketch the parabola, identify its vertex, axis of symmetry, and understand the impact of coefficients on its shape. Suitable for Grades 9-12.

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