• Find the standard form of the equation of the parabola given its parts.

a) V (-4,2); F (-4,1)

b) V (-10,5); directrix: x = -1

c) V (-6, 7); LR = 32; the axis of symmetry is horizontal.

Let's break down each problem and find the standard form of the equation of the parabola. 

### a) **Vertex** \((-4, 2)\); **Focus** \((-4, 1)\)

- The vertex form of a parabola's equation is given by:
  \[
  (x - h)^2 = 4p(y - k)
  \]
  where \((h, k)\) is the vertex, and \(p\) is the distance from the vertex to the focus.
  
- The distance between the vertex and the focus \(p = 1\) (since the y-coordinate decreases by 1).
  
- The parabola opens **downward** because the focus is below the vertex.

Thus, the equation is:
\[
(x + 4)^2 = -4(y - 2)
\]

### b) **Vertex** \((-10, 5)\); **Directrix**: \(x = -1\)

- The vertex form for a horizontally-oriented parabola is:
  \[
  (y - k)^2 = 4p(x - h)
  \]
  where \((h, k)\) is the vertex, and \(p\) is the distance from the vertex to the directrix.

- The distance between the vertex \((-10, 5)\) and the directrix \(x = -1\) is \(p = 9\).

- Since the directrix is to the right of the vertex, the parabola opens **left**.

Thus, the equation is:
\[
(y - 5)^2 = -36(x + 10)
\]

### c) **Vertex** \((-6, 7)\); **Length of the Latus Rectum (LR)** = 32; axis of symmetry is horizontal.

- The vertex form of a horizontally-oriented parabola is:
  \[
  (y - k)^2 = 4p(x - h)
  \]

- The length of the latus rectum (LR) is given by \( |4p| = 32 \), so \( p = 8 \) (since it's positive, the parabola opens to the right).

Thus, the equation is:
\[
(y - 7)^2 = 32(x + 6)
\]

### Summary of the Standard Forms:

a) \((x + 4)^2 = -4(y - 2)\)

b) \((y - 5)^2 = -36(x + 10)\)

c) \((y - 7)^2 = 32(x + 6)\)

Would you like more details or have any questions?

Here are some additional questions related to parabolas:

1. How do you determine the direction in which a parabola opens?
2. What is the significance of the latus rectum in a parabola?
3. How can you find the focus of a parabola given its equation?
4. What are the key differences between vertical and horizontal parabolas?
5. How does changing the value of \(p\) affect the shape of the parabola?

**Tip:** The latus rectum's length directly relates to the parabola's width; a larger \(p\) value results in a wider parabola.

Learn how to find the standard form of a parabola's equation given its parts such as vertex, focus, directrix, and length of Latus Rectum. Detailed solutions provided for problems involving parabolas. Suitable for high school students and those studying algebra and geometry.

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