The interior of a satellite TV antenna is a dish having the shape of a paraboloid. Its receiver is placed 2m 
meters from the vertex. If the vertex is at (1,-2) and it opens to the right, find the equation (in general 
form) of the parabola, and graph it completely. 

To solve the problem, we'll determine the equation of the parabola based on the given information.

### Step 1: Identify the Standard Form of the Parabola
The problem states that the parabola opens to the right, and the vertex is at \((1, -2)\). For a parabola that opens horizontally (to the right or left), the standard form is:

\[
(y - k)^2 = 4p(x - h)
\]

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

### Step 2: Substitute the Given Information
Given:
- Vertex \((h, k) = (1, -2)\)
- Distance from the vertex to the focus, \(p = 2\) (since the focus is placed 2 meters from the vertex and the parabola opens to the right).

So, substituting into the standard form:

\[
(y + 2)^2 = 4(2)(x - 1)
\]

Simplifying:

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

### Step 3: Expand and Write in General Form
To convert the equation to the general form, expand and rearrange terms:

\[
y^2 + 4y + 4 = 8x - 8
\]

Now, rearrange the equation:

\[
y^2 + 4y - 8x + 12 = 0
\]

This is the equation of the parabola in general form.

### Step 4: Graph the Parabola
To graph the parabola, we use the following key points:
- **Vertex**: \((1, -2)\)
- **Focus**: \((1 + 2, -2) = (3, -2)\)
- **Directrix**: Since the parabola opens to the right, the directrix is a vertical line given by \(x = 1 - 2 = -1\).

The parabola is symmetric about the line \(y = -2\) and opens to the right.

Would you like me to generate a graph of this parabola?

The interior of a satellite TV antenna is a dish having the shape of a paraboloid. Its receiver is placed 2 meters from the vertex. If the vertex is at (1,-2) and it opens to the right, find the equation (in general form) of the parabola, and graph it completely.

This tutorial explains how to find the equation of a parabola representing a satellite dish. Given that the vertex is at (1, -2) and the parabola opens to the right, learn the steps to derive the equation in both standard and general forms. The problem covers concepts in analytic geometry and parabolas, suitable for high school students in Grades 10-12.

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