a. Coordinates of the vertex
b. Equation of axis of symmetry
c. Equation of directrix
d. Coordinates of the focus
e. Opening of the parabola

1. (x - 2) = - 16(y - 2)

The given equation is:

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

This is the equation of a parabola in **vertex form**: \((x - h)^2 = 4p(y - k)\), where:
- \((h, k)\) is the vertex of the parabola.
- \(p\) is the distance from the vertex to the focus and directrix.
- The parabola opens **upward** if \(p > 0\), **downward** if \(p < 0\), **rightward** if the equation involves \(y^2\), and **leftward** if \(p < 0\) with \(x^2\).

### Let's extract the needed information:

1. **Coordinates of the vertex:**

The vertex form is \((x - h)^2 = 4p(y - k)\). From the given equation, we can identify:
- \(h = 2\)
- \(k = 2\)

So, the **vertex** is at \((2, 2)\).

2. **Equation of the axis of symmetry:**

For a vertical parabola, the axis of symmetry is the vertical line passing through the vertex, so it's given by:
\[
x = h = 2
\]

Thus, the **axis of symmetry** is:
\[
x = 2
\]

3. **Equation of the directrix:**

The equation is in the form \((x - h)^2 = 4p(y - k)\), where \(4p = -16\), so:
\[
p = \frac{-16}{4} = -4
\]

The directrix is a horizontal line \(p\) units below the vertex since the parabola opens downwards. Therefore, the equation of the directrix is:
\[
y = k - p = 2 - (-4) = 6
\]

Thus, the **equation of the directrix** is:
\[
y = 6
\]

4. **Coordinates of the focus:**

The focus is \(p\) units below the vertex because the parabola opens downward. The focus is located at:
\[
(2, 2 + (-4)) = (2, -2)
\]

So, the **coordinates of the focus** are:
\[
(2, -2)
\]

5. **Opening of the parabola:**

Since the coefficient of \(y\) is negative, the parabola opens **downward**.

---

### Summary:
- a. **Vertex**: \((2, 2)\)
- b. **Axis of symmetry**: \(x = 2\)
- c. **Directrix**: \(y = 6\)
- d. **Focus**: \((2, -2)\)
- e. **Opening**: Downward

Would you like any further details or clarifications?

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Here are 5 related questions to expand on the topic:
1. How would the equation change if the parabola opened upward?
2. What is the general form of a parabola that opens horizontally?
3. How can you find the length of the latus rectum for this parabola?
4. What happens to the parabola if the coefficient 16 is doubled?
5. How would you determine the equation of a parabola given the vertex and the focus?

**Tip**: The value of \(p\) in a parabola helps you find the distance between the vertex and the focus or directrix, making it essential in understanding its shape and direction.

Explore the properties of the parabola given by the equation (x - 2)^2 = -16(y - 2). This step-by-step solution explains how to find the vertex, axis of symmetry, directrix, focus, and the opening of the parabola. Suitable for high school students studying conic sections in Grades 10-12.

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