3. From the given equation (x-4)² = -6 (y-4)

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 for the given parabola equation:

\[
(x - 4)^2 = -6(y - 4)
\]

This is a vertical parabola, as the \(x\) term is squared and the \(y\) term is linear.

### a. Find the Vertex
The standard form of a vertical parabola is:

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

Comparing it with the given equation \((x - 4)^2 = -6(y - 4)\), we can identify the vertex \((h, k)\) as:

\[
h = 4, \quad k = 4
\]

So, the vertex is \((4, 4)\).

### b. Find the Focus from the Vertex
In the standard form \((x - h)^2 = 4p(y - k)\), \(p\) represents the distance from the vertex to the focus. Here, \(4p = -6\), so:

\[
p = \frac{-6}{4} = -\frac{3}{2} = -1.5
\]

Since \(p\) is negative, the parabola opens downwards, meaning the focus is located \(1.5\) units below the vertex.

The focus is therefore at:

\[
(4, 4 - 1.5) = (4, 2.5)
\]

### c. Find the Directrix from the Vertex
The directrix is a line that is \(p\) units away from the vertex, but in the opposite direction from the focus. Since the focus is \(1.5\) units below the vertex, the directrix will be \(1.5\) units above the vertex.

The equation of the directrix is:

\[
y = 4 + 1.5 = 5.5
\]

So, the directrix is the horizontal line \(y = 5.5\).

### d. Graph the Parabola Equation
To graph the parabola, we need to plot the vertex at \((4, 4)\), the focus at \((4, 2.5)\), and the directrix at \(y = 5.5\). The parabola will open downward since the focus is below the vertex.

Would you like me to create a graph of this parabola?

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Here are 5 related questions:
1. How does the value of \(p\) affect the width and orientation of the parabola?
2. What would be the equation of the parabola if the focus were above the vertex instead of below it?
3. How do you determine whether a parabola opens up or down from the given equation?
4. What is the significance of the directrix in relation to the parabola's shape?
5. How would you find the equation of the parabola if you only knew the focus and directrix?

**Tip:** The sign of \(p\) in the parabola equation determines whether the parabola opens upward/downward or left/right. Negative \(p\) indicates a downward or leftward opening.

Learn how to find the vertex, focus, and directrix of the parabola given by equation (x-4)^2 = -6(y-4). Detailed step-by-step solution covering the vertex form, focus, directrix, and graphing of a vertical parabola. Suitable for high school students studying parabolas and conic sections.

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