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The problem describes a parabola that opens to the right, with the following characteristics:

- **Vertex**: \((-1, 4)\)
- **Focus**: \((2, 4)\)
- **Directrix**: \(x = -4\)

### Standard Form of a Parabola Opening Horizontally
For a parabola that opens horizontally (to the right or left), 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 (or vertex and the directrix).

### Step-by-Step Solution:

1. **Find the distance \(p\)**:
   The vertex is \((-1, 4)\) and the focus is \((2, 4)\), so the distance \(p\) is the horizontal distance between these two points:
   \[
   p = 2 - (-1) = 3
   \]
   Since the parabola opens to the right, \(p\) is positive.

2. **Substitute the vertex and \(p\) into the equation**:
   The vertex \((h, k)\) is \((-1, 4)\), and \(p = 3\). Plug these values into the standard form of the equation:
   \[
   (y - 4)^2 = 4(3)(x + 1)
   \]
   Simplifying:
   \[
   (y - 4)^2 = 12(x + 1)
   \]

### Final Equation of the Parabola:
\[
(y - 4)^2 = 12(x + 1)
\]

Would you like further details or have any questions?

Here are some related questions:
1. How is the directrix used in finding the equation of a parabola?
2. What changes in the equation if the parabola opened to the left?
3. How would the equation change if the vertex were different?
4. What is the general form for vertical parabolas?
5. How can you find the axis of symmetry from the equation?

**Tip**: The value of \(p\) determines the "width" of the parabola — larger values of \(p\) make it wider, while smaller values make it narrower.

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.

Explore the equation of a parabola that opens to the right with vertex (-1, 4), focus (2, 4), and directrix x = -4. Learn how to derive the equation (y - 4)^2 = 12(x + 1) and understand the key concepts of conic sections and quadratic equations.

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