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 solve the problem step by step to find the equation of the parabola and the other required components.

### Given:
- **Vertex**: \( (2, 0) \)
- **Focus**: \( (6, 0) \)

### a.) **Find the Vertex**

The vertex is already given as \( (2, 0) \).

### b.) **Find the Focus from the Vertex**

The focus is given as \( (6, 0) \).

### c.) **Find the Directrix from the Vertex**

For a parabola, the distance from the vertex to the focus \( p \) is also the distance from the vertex to the directrix. 

Given:
- Vertex \( (2, 0) \)
- Focus \( (6, 0) \)

The distance \( p \) between the vertex and the focus is:

\[
p = 6 - 2 = 4
\]

Since the parabola opens horizontally and the focus is to the right of the vertex, the directrix will be a vertical line to the left of the vertex. The equation of the directrix is:

\[
x = 2 - 4 = -2
\]

### d.) **Graph the Parabola Equation**

The standard form of the equation for a parabola with a horizontal axis is:

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

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

Given:
- \( h = 2 \)
- \( k = 0 \)
- \( p = 4 \)

Substitute these into the equation:

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

Simplifying:

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

This is the equation of the parabola.

### **Summary of Results:**

- **Vertex**: \( (2, 0) \)
- **Focus**: \( (6, 0) \)
- **Directrix**: \( x = -2 \)
- **Equation**: \( y^2 = 16(x - 2) \)

Would you like a graphical representation of this parabola, or do you have any further questions? 

Here are 5 relative questions that build on this information:
1. How does the distance between the vertex and the focus affect the shape of the parabola?
2. What would the equation be if the parabola opened downward instead of horizontally?
3. How do you find the focal length of a parabola from its equation?
4. What changes in the equation if the vertex is moved to another point on the coordinate plane?
5. How do the directrix and focus relate to each other geometrically in a parabola?

**Tip:** For parabolas, the sign of \( p \) determines the direction of the opening (positive for right/up, negative for left/down).

Learn how to find the equation of a parabola with a given vertex and focus. Understand the relationship between the vertex, focus, and directrix in a parabola. Suitable for high school students studying analytic geometry.

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