getting the general for (process how to get the p) Vertex (1,-2), F(1,0)

To find the equation of a parabola given the vertex \((1, -2)\) and the focus \(F(1, 0)\), we follow these steps:

### General Form of a Parabola:
The general form of a parabola that opens vertically is:

\[
(y - k) = \frac{1}{4p}(x - h)^2
\]

where:
- \((h, k)\) is the vertex of the parabola
- \(p\) is the distance from the vertex to the focus (positive if the parabola opens upwards, negative if it opens downwards).

### Step-by-Step Process:

1. **Identify Vertex Coordinates:**
   The vertex is \((h, k) = (1, -2)\).

2. **Identify Focus Coordinates:**
   The focus is \(F(1, 0)\). Since the x-coordinate of the focus is the same as the x-coordinate of the vertex, the parabola opens **vertically**.

3. **Calculate the Distance \(p\):**
   The distance between the vertex and the focus, \(p\), is the difference in the y-coordinates:
   
   \[
   p = 0 - (-2) = 2
   \]
   
   Since the focus is above the vertex (i.e., the parabola opens upwards), \(p = +2\).

4. **Plug Values into the General Form:**
   Substitute the values of the vertex \((h, k) = (1, -2)\) and \(p = 2\) into the general form equation:

   \[
   (y - (-2)) = \frac{1}{4(2)}(x - 1)^2
   \]

   Simplify:

   \[
   (y + 2) = \frac{1}{8}(x - 1)^2
   \]

   This is the equation of the parabola in standard form.

### Final Answer:
The equation of the parabola with vertex \((1, -2)\) and focus \(F(1, 0)\) is:

\[
(y + 2) = \frac{1}{8}(x - 1)^2
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you derive the general formula for a horizontally-opening parabola?
2. What is the directrix of the parabola from the given vertex and focus?
3. How would the equation change if the parabola opened downwards instead of upwards?
4. How do you find the length of the latus rectum of a parabola?
5. How can you convert the equation of a parabola from vertex form to standard form?

### Tip:
Remember that the value \(p\) determines the direction and "width" of the parabola; a larger \(p\) makes the parabola wider, while a smaller \(p\) makes it narrower.

Learn how to find the equation of a parabola with vertex (1, -2) and focus (1, 0) using the general formula. This step-by-step guide covers key concepts in coordinate geometry and conic sections, suitable for students in Grades 10-12.

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