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

To find the general form of a parabola given the vertex \((1, -2)\) and the focus \((1, 0)\), let's go through the process step by step.

### Step 1: Identify Key Components
- **Vertex** \((h, k)\): The vertex of the parabola is \((1, -2)\), which means \(h = 1\) and \(k = -2\).
- **Focus** \((1, 0)\): The focus is located above the vertex on the same vertical line, meaning the parabola opens **upward**.

Since the parabola opens vertically (up or down), we will use the general form of the equation for a vertical parabola:

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

where:
- \((h, k)\) is the vertex,
- \(p\) is the distance from the vertex to the focus.

### Step 2: Calculate \(p\)
The distance \(p\) represents the vertical distance between the vertex and the focus. Here, the vertex is at \((1, -2)\) and the focus is at \((1, 0)\). The vertical distance is:

\[
p = 0 - (-2) = 2
\]

### Step 3: Substitute Values
We now have:
- \(h = 1\),
- \(k = -2\),
- \(p = 2\).

Substitute these values into the general equation:

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

Simplify:

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

This is the **vertex form** of the parabola.

### Step 4: Convert to General Form
To convert the equation to the general form \(Ax^2 + Bx + Cy + D = 0\), expand the equation:

1. Multiply out the right-hand side:
\[
y + 2 = \frac{1}{8}(x^2 - 2x + 1)
\]
\[
y + 2 = \frac{1}{8}x^2 - \frac{1}{4}x + \frac{1}{8}
\]

2. Subtract 2 from both sides to isolate \(y\):
\[
y = \frac{1}{8}x^2 - \frac{1}{4}x + \frac{1}{8} - 2
\]
\[
y = \frac{1}{8}x^2 - \frac{1}{4}x - \frac{15}{8}
\]

3. Multiply through by 8 to eliminate fractions:
\[
8y = x^2 - 2x - 15
\]

This is the **general form** of the parabola:
\[
x^2 - 2x - 8y - 15 = 0
\]

### Summary
- The **vertex form** of the parabola is:
  \[
  y + 2 = \frac{1}{8}(x - 1)^2
  \]
- The **general form** of the parabola is:
  \[
  x^2 - 2x - 8y - 15 = 0
  \]

Would you like any further details or have any questions?

### Related Questions:
1. What is the general equation for a horizontal parabola?
2. How can you find the directrix of this parabola?
3. How do you convert between vertex form and general form of a parabola?
4. What does the value of \(p\) represent in a parabola's equation?
5. How do you determine if a parabola opens upward or downward?

### Tip:
Remember, the distance \(p\) always tells you how far the focus is from the vertex, and whether the parabola opens upward, downward, left, or right depends on the sign of \(p\).

getting the general form of the parabola with Vertex (1,-2), Focus (1,0)

Learn how to find the general form of a parabola given the vertex at (1, -2) and the focus at (1, 0). This solution covers key geometric concepts, including the vertex form of a parabola and the conversion to general form. Suitable for high school students in Grades 9-12.

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