Graph the parabola with focus at (2, 2) and whose directrix is given by the equation x = 8.

To graph the parabola, we need to understand the relationship between the focus, directrix, and the vertex of the parabola. The standard equation of a parabola that opens horizontally (left or right) is:

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

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

### Step 1: Identify the given elements
- **Focus**: \((2, 2)\)
- **Directrix**: \(x = 8\)

### Step 2: Calculate the vertex
The vertex lies exactly midway between the focus and the directrix. Since the directrix is a vertical line \(x = 8\), and the focus has the x-coordinate \(2\), the x-coordinate of the vertex \((h, k)\) can be calculated as:

\[
h = \frac{2 + 8}{2} = 5
\]

The y-coordinate \(k\) of the vertex is the same as the y-coordinate of the focus:

\[
k = 2
\]

So, the vertex is \((5, 2)\).

### Step 3: Determine the distance \(p\)
The distance \(p\) between the vertex and the focus is:

\[
p = 2 - 5 = -3
\]

The negative sign indicates that the parabola opens to the left.

### Step 4: Write the equation of the parabola
Substitute the values of \(h\), \(k\), and \(p\) into the standard equation:

\[
(y - 2)^2 = 4(-3)(x - 5)
\]

Simplifying, we get:

\[
(y - 2)^2 = -12(x - 5)
\]

### Step 5: Graph the parabola
- **Vertex**: \((5, 2)\)
- **Focus**: \((2, 2)\)
- **Directrix**: \(x = 8\)
- **Parabola opens leftward**.

This equation represents a parabola that opens to the left, with its vertex at \((5, 2)\).

Would you like me to generate the graph for this parabola, or do you have any other questions?

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Here are five related questions:

1. How do you find the vertex of a parabola given its standard equation?
2. What is the significance of the value of \(p\) in the equation of a parabola?
3. How does the directrix influence the orientation of a parabola?
4. Can you derive the equation of a parabola if given the vertex and the focus?
5. How would the equation change if the parabola opened upwards instead of leftwards?

**Tip:** Always remember that the vertex is equidistant from the focus and the directrix. This helps in identifying the correct form of the parabola’s equation.

Learn how to graph a parabola with focus at (2, 2) and directrix x = 8 using the standard equation of a parabola. Understand the relationship between focus, directrix, and vertex. Suitable for Grades 10-12 students studying geometry and parabolas.

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