Graph the parabola (pre calculus)  1. Vertex: (-2,3) Focus: (-1,3) give the Directrix, Length of LR, Endpoints and axis of symmetry

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?

Learn how to graph a parabola given the vertex at (-2, 3) and the focus at (-1, 3). Understand the directrix, length of the latus rectum (LR), endpoints, and axis of symmetry. Suitable for pre-calculus students.

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