Given the information about the parabola, we can deduce the following:

1. Form of the Equation:

Since the vertex is $(-2, 3)$ and the focus is $(-1, 3)$, the parabola opens horizontally. For a horizontally oriented parabola, the general equation is:

$(y - k)^2 = 4p(x - h)$

where $(h, k)$ is the vertex, and $p$ is the distance from the vertex to the focus.

2. Finding the Value of $p$:

The focus is $(-1, 3)$ and the vertex is $(-2, 3)$. The distance $p$ is the horizontal distance between the vertex and focus:

$p = -1 - (-2) = 1$

So, $p = 1$.

3. Equation of the Parabola:

Using the vertex $(-2, 3)$ and $p = 1$, we can plug these into the equation:

$(y - 3)^2 = 4(1)(x + 2)$

Simplifying, the equation of the parabola is:

$(y - 3)^2 = 4(x + 2)$

4. Directrix:

The directrix is a line perpendicular to the axis of symmetry and at the same distance from the vertex as the focus, but on the opposite side. Since $p = 1$, the directrix is one unit to the left of the vertex:

$\text{Directrix: } x = -3$

5. Length of the Latus Rectum (LR):

The length of the latus rectum (LR) is given by $|4p|$:

$\text{Length of LR} = |4 \times 1| = 4$

6. Endpoints of the Latus Rectum:

The endpoints of the latus rectum are at a distance of $2p$ vertically from the focus:

$\text{Endpoints of LR: } \left(-1, 3 + 2\right) = (-1, 5) \text{ and } \left(-1, 3 - 2\right) = (-1, 1)$

7. Axis of Symmetry:

The axis of symmetry is the vertical line passing through the vertex:

$\text{Axis of Symmetry: } y = 3$

Summary:

• Directrix: $x = -3$
• Length of LR: 4
• Endpoints of LR: $(-1, 5)$ and $(-1, 1)$
• Axis of Symmetry: $y = 3$

Would you like a graph of this parabola?