1. 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

Let's go through the problem step by step to find the equation of the parabola with vertex \((2, 0)\) and focus at \((6, 0)\).

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

### b. Find the Focus from the Vertex
The focus is given as \((6, 0)\). The distance between the vertex and the focus is:

\[
p = \text{distance between the vertex and focus} = 6 - 2 = 4
\]

This tells us that the focus is 4 units to the right of the vertex along the x-axis.

### c. Find the Directrix from the Vertex
The directrix of a parabola is a line that is equidistant from the vertex as the focus but lies on the opposite side of the vertex. Since the focus is at \((6, 0)\), the directrix will be a vertical line to the left of the vertex.

To find the equation of the directrix, we move 4 units to the left of the vertex \((2, 0)\):

\[
\text{Directrix} = x = 2 - 4 = -2
\]

So, the equation of the directrix is \(x = -2\).

### d. Graph the Parabola Equation
For a horizontal parabola with the vertex at \((h, k)\) and the focus \(p\) units to the right, the equation of the parabola is given by:

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

Substituting the given values:

- Vertex: \((h, k) = (2, 0)\)
- Focus: \((6, 0)\) which gives \(p = 4\)

The equation of the parabola is:

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

Simplifying:

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

This is the equation of the parabola.

### Graph
To graph the parabola, we plot the vertex at \((2, 0)\) and the focus at \((6, 0)\). The parabola opens to the right since the focus is to the right of the vertex. The directrix is the vertical line \(x = -2\).

Would you like me to create a graph of this parabola? Also, feel free to ask for more details or further explanations.

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Here are 5 relative questions:
1. How would the equation change if the parabola opened to the left instead of the right?
2. What is the significance of the distance between the vertex and the focus?
3. How do you find the equation of the parabola if the vertex and focus are aligned vertically?
4. What happens to the directrix if the focus is shifted upwards instead of horizontally?
5. How do you find the length of the latus rectum for this parabola?

**Tip:** When graphing parabolas, always identify the direction it opens (left, right, up, or down) based on the position of the focus relative to the vertex.

Learn how to find the equation of a parabola with a given vertex at (2, 0) and focus at (6, 0). Understand the concepts of vertex, focus, directrix, and how to graph the parabola. Suitable for high school students studying parabolas and algebraic geometry.

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