To find the **directrix** and **axis of symmetry** of a parabola given its vertex and focus, let's use the details we've already calculated:

### Recap of Parabola's Properties
- **Vertex**: \((-1.5, 4)\)
- **Focus**: \((-4, 4)\)
- **Equation of the parabola**: \((y - 4)^2 = -10(x + 1.5)\)
- **Distance \(p\)** from the vertex to the focus: \(p = -2.5\) (since the parabola opens to the left)

### 1. Finding the Directrix

The **directrix** of a parabola is a line perpendicular to the axis of symmetry and is located the same distance \(p\) from the vertex as the focus, but on the opposite side.

Since the parabola opens horizontally to the left, the directrix will be a vertical line.

- **Distance from the vertex to the focus**: \(p = -2.5\)
- **Vertex**: \((-1.5, 4)\)

To find the directrix, we add the absolute value of \(p\) to the x-coordinate of the vertex (because the parabola opens left, opposite to the focus):

\[
x = -1.5 + |p| = -1.5 + 2.5 = 1
\]

So, the **directrix** is the vertical line:

\[
x = 1
\]

### 2. Finding the Axis of Symmetry

The **axis of symmetry** of a parabola is a line that passes through the vertex and the focus. For a parabola that opens horizontally (either to the left or the right), the axis of symmetry is a horizontal line at the \(y\)-coordinate of the vertex and focus.

- Since the vertex and the focus share the same \(y\)-coordinate (\(4\)), the axis of symmetry is a horizontal line at:

\[
y = 4
\]

### Summary

- **Directrix**: \(x = 1\)
- **Axis of Symmetry**: \(y = 4\)

Would you like more details on how these were calculated, or do you have any further questions?

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Here are 5 related questions to consider:

1. How does the equation of the directrix change for parabolas that open vertically?
2. Can you find the focus of a parabola given its vertex and directrix?
3. How does the position of the focus affect the shape and direction of the parabola?
4. What are the differences between the focus and directrix of a parabola?
5. How would you find the length of the latus rectum of this parabola?

**Tip:** For parabolas, the directrix and focus are always equidistant from any point on the parabola, which helps in understanding its geometric properties!

Learn how to find the directrix and axis of symmetry of a parabola using its vertex and focus. Detailed step-by-step explanation with equations and geometric reasoning included. Suitable for high school students studying conic sections and geometry.

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