Find the vertex, focus, length of latus rectum, opening, axis of symmetry, directrix, ends of latus rectum, and the general form of the equation (x+1/4)² = -18(y–1/2)

The given equation is:

\[
(x + \frac{1}{4})^2 = -18\left(y - \frac{1}{2}\right)
\]

This is a standard form of the equation of a parabola that opens either upwards or downwards.

### 1. **Vertex**:
The vertex form of the equation of a parabola is:

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

Comparing this with the given equation, \((x + \frac{1}{4})^2 = -18\left(y - \frac{1}{2}\right)\), we can identify:
- \(h = -\frac{1}{4}\)
- \(k = \frac{1}{2}\)

So, the **vertex** is \(\left(-\frac{1}{4}, \frac{1}{2}\right)\).

### 2. **Focus**:
The parameter \(p\) is related to the coefficient of the equation as follows: \(4p = -18\), which gives \(p = -\frac{9}{2}\).

For a parabola that opens vertically, the focus is located \(p\) units from the vertex along the axis of symmetry. Since \(p = -\frac{9}{2}\) and the parabola opens downwards, the focus is:

\[
\text{Focus} = \left(-\frac{1}{4}, \frac{1}{2} - \frac{9}{2}\right) = \left(-\frac{1}{4}, -4\right)
\]

### 3. **Length of the Latus Rectum**:
The length of the latus rectum is given by \(|4p|\). Since \(4p = -18\), the **length of the latus rectum** is 18.

### 4. **Opening**:
Since the square term involves \((x + \frac{1}{4})^2\) and the coefficient of \(y\) is negative, the parabola **opens downwards**.

### 5. **Axis of Symmetry**:
The axis of symmetry is the vertical line passing through the vertex. The equation for this line is:

\[
x = -\frac{1}{4}
\]

### 6. **Directrix**:
The directrix is a horizontal line that is \(p\) units away from the vertex, in the direction opposite to the focus. Since \(p = -\frac{9}{2}\) and the parabola opens downwards, the directrix is:

\[
y = \frac{1}{2} - \left(-\frac{9}{2}\right) = \frac{1}{2} + \frac{9}{2} = 5
\]

### 7. **Ends of the Latus Rectum**:
The ends of the latus rectum are points on the parabola that are equidistant from the focus. They can be found by moving \(9\) units (half of the latus rectum length) left and right from the focus along the axis of symmetry. The coordinates of the ends are:

\[
\left(-\frac{1}{4} - 9, -4\right) = \left(-\frac{37}{4}, -4\right)
\]

\[
\left(-\frac{1}{4} + 9, -4\right) = \left(\frac{35}{4}, -4\right)
\]

### 8. **General Form of the Equation**:
To convert the given equation to the general form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), we expand the equation:

\[
(x + \frac{1}{4})^2 = -18y + 9
\]

Expanding:

\[
x^2 + \frac{1}{2}x + \frac{1}{16} = -18y + 9
\]

To eliminate the fraction, multiply the entire equation by 16:

\[
16x^2 + 8x + 1 = -288y + 144
\]

Finally, bring all terms to one side:

\[
16x^2 + 8x + 288y - 143 = 0
\]

This is the **general form** of the equation.

---

Would you like more details or have any questions? Here are some related questions you might find useful:

1. How do you determine the direction of opening for a parabola given its equation?
2. What is the significance of the directrix in the context of a parabola?
3. How does the value of \(p\) affect the shape and position of a parabola?
4. Can you describe how the ends of the latus rectum are derived for parabolas that open horizontally?
5. What happens to the equation of a parabola if the focus is moved closer or further from the vertex?

**Tip:** The sign of the coefficient in front of the \(y\) term in the equation of a parabola helps determine whether the parabola opens upwards or downwards (for a vertical parabola).

Explore the equation (x + 1/4)^2 = -18(y - 1/2) in detail. Learn about its vertex, focus, axis of symmetry, directrix, length of latus rectum, and the general form. Suitable for high school students studying parabolas and quadratic equations.

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