The equation given is \( y^2 = 20x \). This is a type of equation known as a **parabola**. Specifically, it is a parabola that opens to the right. The general form of a parabola that opens horizontally is given by:

\[
y^2 = 4px
\]

Where \( p \) is the distance from the vertex to the focus, and also from the vertex to the directrix.

### Analyzing the Given Equation:

1. **Comparing with the standard form:**
   \[
   y^2 = 20x \implies 4p = 20
   \]
   So, \( p = \frac{20}{4} = 5 \).

2. **Vertex:** The vertex of this parabola is at the origin \((0, 0)\).

3. **Focus:** Since \( p = 5 \), the focus is at \((5, 0)\).

4. **Directrix:** The directrix is a vertical line, given by \( x = -5 \).

5. **Axis of Symmetry:** The axis of symmetry is the y-axis.

### Summary:
- **Equation:** \( y^2 = 20x \)
- **Vertex:** \((0, 0)\)
- **Focus:** \((5, 0)\)
- **Directrix:** \( x = -5 \)
- **Orientation:** Opens to the right

Would you like more details or have any questions?

### Follow-up Questions:
1. How does the focus of the parabola influence its shape?
2. Can you describe the difference between a parabola that opens left versus right?
3. What happens to the graph if we change the equation to \( y^2 = 10x \)?
4. How do you find the length of the latus rectum for this parabola?
5. What are the properties of a parabola that opens upward or downward?

### Tip:
For parabolas, the sign of the coefficient in front of the \( x \) or \( y \) term determines the direction in which the parabola opens.

Explore the parabola equation y^2 = 20x, identifying its vertex, focus, and directrix. Learn how the focus influences the shape of the parabola and its orientation. Suitable for high school students studying analytic geometry.

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