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We are tasked with finding the equation of a parabola with the focus at \((4, -2)\) and the directrix as \(x = -2\).

### Key Information:
- Focus: \((4, -2)\)
- Directrix: \(x = -2\)

### Step-by-Step Approach:
This parabola opens horizontally since the directrix is vertical (\(x = -2\)).

The general form of the equation of a horizontally opening parabola is:
\[
(y - k)^2 = 4p(x - h)
\]
Where:
- \((h, k)\) is the vertex of the parabola,
- \(p\) is the distance between the vertex and the focus (or the directrix).

#### Step 1: Find the Vertex
The vertex lies halfway between the focus and the directrix. The \(x\)-coordinate of the vertex is the average of the \(x\)-coordinates of the focus and the directrix:
\[
x_{\text{vertex}} = \frac{4 + (-2)}{2} = \frac{2}{2} = 1
\]
Since the \(y\)-coordinate of the focus is \(-2\), the \(y\)-coordinate of the vertex is also \(-2\) (as the vertex is horizontally aligned with the focus).

Thus, the vertex is at \((1, -2)\).

#### Step 2: Find \(p\)
The distance \(p\) is the distance between the vertex and the focus (or the directrix). Here, the distance between the vertex \((1, -2)\) and the focus \((4, -2)\) is:
\[
p = 4 - 1 = 3
\]
Thus, \(p = 3\).

#### Step 3: Write the Equation
Now we can plug the vertex \((1, -2)\) and \(p = 3\) into the standard form of the equation:
\[
(y + 2)^2 = 4(3)(x - 1)
\]
Simplifying:
\[
(y + 2)^2 = 12(x - 1)
\]

This is the equation of the parabola.

If you have any questions or need further clarifications, feel free to ask! Here are some related questions for further exploration:

1. How do you find the vertex of a parabola given the focus and directrix?
2. What is the significance of the parameter \(p\) in the equation of a parabola?
3. How does the orientation of the parabola change with the position of the directrix?
4. Can the equation of a parabola be derived using the distance formula between the focus and any point on the parabola?
5. What is the difference between parabolas that open horizontally and those that open vertically?

**Tip:** Always double-check if the parabola opens horizontally or vertically by looking at the focus and directrix!

Hallar una ecuación de la parábola cuyo foco es (4, -2) y cuya directriz es x = -2.

Learn how to find the equation of a parabola with the focus at (4, -2) and directrix at x = -2. This solution covers key concepts in conic sections, using the vertex formula and the standard equation of a parabola. Suitable for Grades 10-12.

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