Determine the focus and directrix of the parabola with the given equation. Sketch the graph, and indicate the focus, directrix, and vertex if the equation is x^2- 6x - 2y + 9 = 0

To determine the focus and directrix of the parabola, we first need to rewrite the given equation in the standard form of a parabola. The given equation is:

\[
x^2 - 6x - 2y + 9 = 0
\]

### Step 1: Complete the Square

Start by isolating the \(x\) terms on one side of the equation:

\[
x^2 - 6x = 2y - 9
\]

Now, complete the square on the left side. To do this, take half the coefficient of \(x\), square it, and add it to both sides of the equation. The coefficient of \(x\) is \(-6\), so:

\[
\left(\frac{-6}{2}\right)^2 = 9
\]

Add 9 to both sides:

\[
x^2 - 6x + 9 = 2y - 9 + 9
\]

This simplifies to:

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

### Step 2: Convert to Standard Form

Now, the equation is in the standard form of a parabola that opens upwards or downwards:

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

where \((h, k)\) is the vertex, and \(p\) is the distance from the vertex to the focus and directrix. Comparing with the standard form:

\[
(x - 3)^2 = 4\left(\frac{1}{2}\right)(y - 0)
\]

Here, \(h = 3\), \(k = 0\), and \(4p = 2\), so \(p = \frac{1}{2}\).

### Step 3: Determine the Focus and Directrix

- **Vertex**: \((h, k) = (3, 0)\)
- **Focus**: The focus lies \(p\) units above the vertex, since the parabola opens upwards. Thus, the focus is at \((3, \frac{1}{2})\).
- **Directrix**: The directrix is a horizontal line \(p\) units below the vertex, so the directrix is \(y = -\frac{1}{2}\).

### Step 4: Sketch the Graph

- The vertex is at \((3, 0)\).
- The focus is at \((3, \frac{1}{2})\).
- The directrix is the line \(y = -\frac{1}{2}\).

Let me know if you want a sketch of the graph or if you have any questions.

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Here are some related questions you might find useful:

1. How would you determine the vertex, focus, and directrix for a parabola that opens sideways?
2. What are the differences in the equations of parabolas that open up, down, left, and right?
3. How can you derive the equation of a parabola given the vertex and focus?
4. How does changing the value of \(p\) affect the shape of the parabola?
5. What is the significance of the directrix in relation to the parabola's shape?

**Tip:** Completing the square is a useful technique not just for conic sections but also for solving quadratic equations and optimization problems.

Discover how to find the focus and directrix of the parabola given by the equation x^2 - 6x - 2y + 9 = 0. This problem involves completing the square and understanding parabolas in coordinate geometry. Suitable for high school students.

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