find the standard equation of the parabola if v (2,-4) and f is (5,-4)

To find the standard equation of the parabola given the vertex \(V\) and the focus \(F\), we can follow these steps:

### Step 1: Identify the coordinates of the vertex and the focus.
- **Vertex \(V\):** \((2, -4)\)
- **Focus \(F\):** \((5, -4)\)

### Step 2: Determine the direction in which the parabola opens.
Since the x-coordinates of the vertex and focus differ (the y-coordinates are the same), the parabola opens horizontally. Specifically, because the focus is to the right of the vertex, the parabola opens to the right.

### Step 3: Calculate the distance between the vertex and the focus.
The distance \(p\) between the vertex and the focus is:
\[
p = |x_{\text{focus}} - x_{\text{vertex}}| = |5 - 2| = 3
\]
Since the parabola opens to the right, \(p = 3\).

### Step 4: Write the standard equation of the parabola.
For a parabola that opens horizontally, the standard form of the equation is:
\[
(x - x_0)^2 = 4p(y - y_0)
\]
Substituting the values:
\[
(x - 2)^2 = 4(3)(y + 4)
\]
Simplify the equation:
\[
(x - 2)^2 = 12(y + 4)
\]

### Final Answer:
The standard equation of the parabola is:
\[
(x - 2)^2 = 12(y + 4)
\]

Would you like any further details or have any questions about this solution?

### Related Questions:
1. What is the equation of a parabola with a vertex at \((3, 2)\) and a focus at \((3, 5)\)?
2. How can you find the directrix of a parabola given its vertex and focus?
3. What happens to the equation of the parabola if the vertex and focus have the same y-coordinate but different x-coordinates?
4. How do you determine if a parabola opens left or right from its equation?
5. Can you derive the equation of a parabola if only the vertex and directrix are given?
6. How does the distance \(p\) between the vertex and focus affect the width of the parabola?
7. What is the relationship between the latus rectum and the parameter \(p\) in the parabola's equation?
8. How can you convert a parabola's standard equation to its general form?

### Tip:
When dealing with parabolas, always check the orientation (horizontal or vertical) before writing the standard equation. This depends on whether the difference between the vertex and focus is in the x-coordinates (horizontal) or y-coordinates (vertical).

Learn how to find the standard equation of a parabola given the vertex (2, -4) and the focus (5, -4). This problem involves determining the direction in which the parabola opens, calculating the distance between the vertex and the focus, and writing the standard form of the parabola's equation. Suitable for high school students studying coordinate geometry and quadratic equations.

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