https://file.aiquickdraw.com/mathbot/file/17266245090142ppz69nk.jpg

Given the information provided, we can determine the equation of the parabola. Let's break down the details step-by-step:

1. **Vertex**: The vertex of the parabola is \((-1, 4)\).
2. **Focus**: The focus is at \((2, 4)\).
3. **Directrix**: The directrix is the vertical line \(x = -4\).
4. **Opening direction**: The parabola opens to the right, indicating it is a horizontal parabola.

For a horizontal parabola with vertex \((h, k)\) and opening to the right, the equation is of the form:
\[
(y - k)^2 = 4p(x - h),
\]
where \( (h, k) \) is the vertex, and \( p \) is the distance between the vertex and the focus. 

### Finding \(p\)
Since the focus is \((2, 4)\) and the vertex is \((-1, 4)\), the distance \(p\) is:
\[
p = 2 - (-1) = 3.
\]

Since the parabola opens to the right, \(p\) is positive. Thus, the equation becomes:
\[
(y - 4)^2 = 12(x + 1).
\]

### Final Equation
The equation of the parabola is:
\[
(y - 4)^2 = 12(x + 1).
\]

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How do you derive the equation of a parabola that opens upwards or downwards?
2. What changes in the equation of a parabola if it opens to the left?
3. How can you find the directrix of a parabola given its equation?
4. How does the value of \(p\) affect the shape of the parabola?
5. What is the relationship between the focus, vertex, and directrix in a parabola?

**Tip**: For any parabola, the distance between the vertex and the focus is the same as the distance between the vertex and the directrix. This is a key property when deriving equations!

The graph of the parabola is a curve that opens to the right, with vertex at (-1, 4), focus at (2, 4), and directrix at x = -4.

Learn how to derive the equation of a parabola that opens to the right with a vertex at (-1, 4), focus at (2, 4), and directrix at x = -4. This solution breaks down the core concepts of conic sections, including the relationship between the vertex, focus, and directrix.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation