Find the equation of the parabola with vertex (2, 0) and focus at (6,0)

a. Find the vertex

b.) Find the focus from the vertex

c.) Find the directrix from the vertex

d.) Graph the parabola equation

To find the equation of the parabola with the given vertex \((2, 0)\) and focus at \((6, 0)\), let's go through each step.

### a.) Find the Vertex
The vertex of the parabola is already given as \((2, 0)\).

### b.) Find the Focus from the Vertex
The focus is given as \((6, 0)\). Since the focus lies on the x-axis and the vertex also lies on the x-axis, this indicates that the parabola opens horizontally.

### c.) Find the Directrix from the Vertex
The distance between the vertex and the focus is \( |6 - 2| = 4 \).

For a parabola, the directrix is the same distance from the vertex as the focus, but on the opposite side. Since the focus is 4 units to the right of the vertex, the directrix will be 4 units to the left.

- Vertex: \((2, 0)\)
- Focus: \((6, 0)\)
- Directrix: \(x = 2 - 4 = -2\)

### d.) Find the Equation of the Parabola

For a horizontally oriented parabola with a vertex at \((h, k)\) and focus at \((h + p, k)\), the equation is given by:

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

Here, the vertex is \((h, k) = (2, 0)\), and \(p\) is the distance from the vertex to the focus, which is 4.

Substituting into the equation:

\[
(y - 0)^2 = 4(4)(x - 2)
\]

\[
y^2 = 16(x - 2)
\]

This is the equation of the parabola.

### Graph the Parabola
I'll describe the graph for you:

1. **Vertex:** \((2, 0)\)
2. **Focus:** \((6, 0)\)
3. **Directrix:** \(x = -2\)

The parabola opens to the right because the value of \(p\) is positive. It is symmetric about the x-axis with its vertex at \((2, 0)\).

Would you like to see a visual graph or need further explanation on any of these steps?

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**Relative Questions:**
1. How do you find the equation of a parabola with a vertical orientation?
2. What changes if the focus of the parabola is at a point above the vertex?
3. How can you derive the directrix from the focus of a parabola?
4. What is the significance of the value \(p\) in the parabola's equation?
5. How does the distance between the vertex and focus affect the shape of the parabola?

**Tip:** Remember that the vertex form of a parabola is a powerful tool to quickly determine its properties, including direction, focus, and directrix.

Discover how to find the equation of a parabola with a vertex at (2, 0) and a focus at (6, 0). This detailed solution covers key geometric concepts, including the vertex form of a parabola. Suitable for high school students in Grades 10-12.

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