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–1/2)² = 16(y+3/2)

The given equation is:

\[
(x - \frac{1}{2})^2 = 16\left(y + \frac{3}{2}\right)
\]

This is a standard equation of a parabola that opens vertically. Let's analyze each component.

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

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

Comparing this with the given equation, \((x - \frac{1}{2})^2 = 16(y + \frac{3}{2})\), we can identify:
- \(h = \frac{1}{2}\)
- \(k = -\frac{3}{2}\)

So, the **vertex** is \(\left(\frac{1}{2}, -\frac{3}{2}\right)\).

### 2. **Focus**:
The parameter \(p\) is related to the coefficient of the equation as follows: \(4p = 16\), which gives \(p = 4\).

For a parabola that opens upwards or downwards, the focus is located \(p\) units from the vertex along the axis of symmetry. Since \(p = 4\) and the parabola opens upwards, the focus is:

\[
\text{Focus} = \left(\frac{1}{2}, -\frac{3}{2} + 4\right) = \left(\frac{1}{2}, \frac{5}{2}\right)
\]

### 3. **Length of the Latus Rectum**:
The length of the latus rectum is given by \(|4p|\). Since \(4p = 16\), the **length of the latus rectum** is 16.

### 4. **Opening**:
Since the square term involves \((x - h)^2\) and the coefficient of \(y\) is positive, the parabola **opens upwards**.

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

\[
x = \frac{1}{2}
\]

### 6. **Directrix**:
The directrix is a horizontal line that is \(p\) units away from the vertex, in the direction opposite to the focus. Since \(p = 4\) and the parabola opens upwards, the directrix is:

\[
y = -\frac{3}{2} - 4 = -\frac{11}{2}
\]

### 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 \(8\) units (half of the latus rectum length) left and right from the focus along the axis of symmetry. The coordinates of the ends are:

\[
\left(\frac{1}{2} - 8, \frac{5}{2}\right) = \left(-\frac{15}{2}, \frac{5}{2}\right)
\]

\[
\left(\frac{1}{2} + 8, \frac{5}{2}\right) = \left(\frac{17}{2}, \frac{5}{2}\right)
\]

### 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:

\[
(x - \frac{1}{2})^2 = 16y + 24
\]

Expanding:

\[
x^2 - x + \frac{1}{4} = 16y + 24
\]

To get rid of the fraction, multiply the entire equation by 4:

\[
4x^2 - 4x + 1 = 64y + 96
\]

Finally, bring all terms to one side:

\[
4x^2 - 4x - 64y - 95 = 0
\]

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 derive the equation of a parabola from its vertex, focus, and directrix?
2. What are the key differences between parabolas that open upwards and those that open downwards?
3. How do you find the length of the latus rectum from the equation of a parabola?
4. Can a parabola have a horizontal axis of symmetry? How would its equation change?
5. What is the significance of the latus rectum in understanding the shape of a parabola?

**Tip:** The focus of a parabola always lies on its axis of symmetry and is crucial in determining the direction in which the parabola opens.

Explore the details of a vertical parabola given by (x - 1/2)^2 = 16(y + 3/2). Learn about its vertex, focus, length of latus rectum, axis of symmetry, directrix, and the general form of its equation. Suitable for high school students studying analytical geometry.

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