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.

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

This equation is in the form of a parabola. The general form of a parabola that opens vertically is:

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

where \((h, k)\) is the **vertex** of the parabola, and \(p\) is the distance from the vertex to the focus (and also from the vertex to the directrix).

### a.) Find the Vertex

The vertex form of a parabola is given by \((h, k)\). By comparing the given equation \((x - 4)^2 = -6(y - 4)\) with the standard form \((x - h)^2 = 4p(y - k)\), we can identify:
- \(h = 4\)
- \(k = 4\)

**Vertex**: \((4, 4)\)

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

To find the focus, we need to determine the value of \(p\). The equation \((x - 4)^2 = -6(y - 4)\) can be written in the form:

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

Comparing the equation, \(4p = -6\), so:
\[
p = \frac{-6}{4} = -\frac{3}{2}
\]

Since \(p\) is negative, the parabola opens downward. The focus lies at a distance \(p\) from the vertex along the axis of symmetry (which is vertical here).

The focus is located at \((h, k + p)\):
\[
\text{Focus} = (4, 4 - \frac{3}{2}) = (4, \frac{5}{2}) = (4, 2.5)
\]

### 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 of the focus. Since the parabola opens downward, the directrix will be a horizontal line above the vertex.

The equation of the directrix is \(y = k - p\):
\[
\text{Directrix} = y = 4 + \frac{3}{2} = \frac{11}{2} = 5.5
\]

### d.) Graph the Parabola Equation

To graph the parabola:

1. **Vertex** at \((4, 4)\).
2. **Focus** at \((4, 2.5)\).
3. **Directrix** is the line \(y = 5.5\).
4. Since the parabola opens downward, the curve will be below the vertex.

I can plot the graph for you. Would you like me to proceed with that?

---

### Would you like more details or have any questions?

### Here are 5 related questions:
1. How do you find the axis of symmetry for a parabola?
2. What is the significance of the focus and directrix in a parabola?
3. How do you convert a parabola equation from standard form to vertex form?
4. What happens to the parabola if the value of \(p\) changes?
5. Can a parabola open sideways instead of up or down? How would that change the equation?

### Tip:
When graphing parabolas, always start by identifying the vertex, as it serves as a critical point that helps in understanding the parabola's orientation and position.

Learn how to find the vertex, focus, and directrix of the parabola equation (x-4)^2 = -6(y-4) step-by-step. Understand the significance of these elements in describing the shape and orientation of the parabola. Suitable for high school students studying algebra and geometry.

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