Question 1170335
Let's solve this problem step-by-step.

**1. Determine the Orientation of the Parabola:**

* The vertex is at the origin (0, 0).
* The focus is at (3, 0).
* The directrix is x = -3.
* Since the focus is to the right of the vertex and the directrix is to the left, the parabola opens to the right.

**2. Determine the Value of 'p':**

* The distance between the vertex and the focus is 'p'.
* The distance between the vertex (0, 0) and the focus (3, 0) is 3.
* Therefore, p = 3.

**3. Find the Equation of the Parabola:**

* Since the parabola opens to the right and the vertex is at the origin, the equation is of the form:
   $y^2 = 4px$
* Substitute p = 3 into the equation:
   $y^2 = 4(3)x$
   $y^2 = 12x$

**4. Find the Length of the Latus Rectum:**

* The length of the latus rectum is 4p.
* Since p = 3, the length of the latus rectum is 4(3) = 12.

**5. Sketch the Graph:**

* **Vertex:** (0, 0)
* **Focus:** (3, 0)
* **Directrix:** x = -3
* **Latus Rectum:** The latus rectum passes through the focus and is perpendicular to the axis of symmetry. Its endpoints are at a distance of 2p from the focus.
   * Since 2p = 6, the endpoints of the latus rectum are at (3, 6) and (3, -6).

**Graph:**

```
      ^ y-axis
      |
    6 |      * (3, 6)
      |     /
      |    /
      |   /
      |  /
    0 +-----------+---> x-axis
      | \
      |  \
      |   \
   -6 |      * (3, -6)
      |
  -3  |   Directrix x=-3
```

**Summary:**

* **Length of the Latus Rectum:** 12
* **Equation of the Parabola:** $y^2 = 12x$