Find the vertex, focus, length of latus rectum, opening, axis of symmetry, directrix, ends of latus rectum, and the general form of the equation (y+5/2)² = 24(x+7/2)

Given the equation of the parabola:

\[
(y + \frac{5}{2})^2 = 24\left(x + \frac{7}{2}\right)
\]

This is a standard form of a parabola equation, which is written as:

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

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

### 1. Vertex
The vertex \((h, k)\) can be directly identified from the equation:

\[
h = -\frac{7}{2}, \quad k = -\frac{5}{2}
\]

So, the vertex is:

\[
\text{Vertex: } \left(-\frac{7}{2}, -\frac{5}{2}\right)
\]

### 2. Focus
The standard form of the equation \((y - k)^2 = 4p(x - h)\) allows us to identify \(4p\):

\[
4p = 24 \implies p = \frac{24}{4} = 6
\]

The parabola opens horizontally (to the right) since the \(x\) term is isolated. The focus is located at \((h + p, k)\):

\[
\text{Focus: } \left(-\frac{7}{2} + 6, -\frac{5}{2}\right) = \left(\frac{5}{2}, -\frac{5}{2}\right)
\]

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

\[
\text{Length of Latus Rectum: } |24| = 24
\]

### 4. Opening
The parabola opens horizontally to the right because the equation is in the form \((y - k)^2 = 4p(x - h)\) and \(p > 0\).

### 5. Axis of Symmetry
The axis of symmetry is a horizontal line passing through the vertex, which is \(y = k\):

\[
\text{Axis of Symmetry: } y = -\frac{5}{2}
\]

### 6. Directrix
The directrix is a vertical line that is \(p\) units away from the vertex in the opposite direction of the focus:

\[
\text{Directrix: } x = h - p = -\frac{7}{2} - 6 = -\frac{19}{2}
\]

### 7. Ends of Latus Rectum
The ends of the latus rectum can be found by moving \(2p\) units vertically from the focus in both directions:

\[
\text{Ends of Latus Rectum: } \left(\frac{5}{2}, -\frac{5}{2} + 6\right) = \left(\frac{5}{2}, \frac{7}{2}\right) \quad \text{and} \quad \left(\frac{5}{2}, -\frac{5}{2} - 6\right) = \left(\frac{5}{2}, -\frac{17}{2}\right)
\]

### 8. General Form of the Equation
To convert the equation into the general form, expand and simplify the given equation:

\[
(y + \frac{5}{2})^2 = 24\left(x + \frac{7}{2}\right)
\]
\[
y^2 + 5y + \frac{25}{4} = 24x + 84
\]
\[
4y^2 + 20y + 25 = 96x + 336
\]
\[
4y^2 + 20y - 96x - 311 = 0
\]

Thus, the general form of the equation is:

\[
4y^2 + 20y - 96x - 311 = 0
\]

---

Would you like further details or have any questions about these calculations?

### 5 Related Questions:
1. How can you derive the focus and directrix for a parabola in general form?
2. What are the implications of the value of \(p\) in the equation of a parabola?
3. Can you determine the orientation of the parabola by looking at its equation?
4. What are the steps to convert the vertex form of a parabola equation to its general form?
5. How does the axis of symmetry relate to the vertex in parabolic equations?

### Tip:
For parabolas, the value of \(p\) directly controls the width of the parabola and the distance between the focus and directrix. A larger \(p\) results in a wider parabola.

Learn how to find the vertex, focus, directrix, and other key properties of a parabola with the equation (y + 5/2)^2 = 24(x + 7/2). Understand the standard form of a parabola, its orientation, and how to convert its equation to general form. Suitable for high school students and anyone interested in conic sections and algebraic geometry.

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