https://file.aiquickdraw.com/mathbot/file/1726662788115m4t51q1n.jpg

The problem shows a parabola with its vertex at \((-4, 5)\) and another point on the parabola at \((5, -1)\). The task is to write the equation of this parabola.

### Steps to find the equation of the parabola:

1. **Standard form of a parabola**: Since the vertex is provided, we can use the vertex form of a parabola:
   \[
   y = a(x - h)^2 + k
   \]
   where \((h, k)\) is the vertex of the parabola. From the image, we know the vertex is \((-4, 5)\), so:
   \[
   y = a(x + 4)^2 + 5
   \]

2. **Use the point \((5, -1)\)**: Substitute the point \((5, -1)\) into the equation to find the value of \(a\):
   \[
   -1 = a(5 + 4)^2 + 5
   \]
   \[
   -1 = a(9)^2 + 5
   \]
   \[
   -1 = 81a + 5
   \]
   \[
   -1 - 5 = 81a
   \]
   \[
   -6 = 81a
   \]
   \[
   a = \frac{-6}{81} = -\frac{2}{27}
   \]

3. **Write the final equation**: Now that we have \(a = -\frac{2}{27}\), the equation of the parabola is:
   \[
   y = -\frac{2}{27}(x + 4)^2 + 5
   \]

Would you like further clarification or a more detailed explanation of any step?

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### Related Questions:
1. How do you determine the direction in which the parabola opens based on the value of \(a\)?
2. Can you convert this vertex form to standard form \(y = ax^2 + bx + c\)?
3. How do you find the axis of symmetry for this parabola?
4. What is the process for determining the focus and directrix of the parabola?
5. How would the equation change if the parabola were opening horizontally instead of vertically?

### Tip:
When identifying a parabola's vertex from its equation, focus on the signs inside the squared term and the constant outside—these directly give the coordinates of the vertex.

A continuación se muestra una parábola. Su vértice y otro punto de la parábola están indicados. Escribir una ecuación de la parábola. (The vertex is (-4, 5) and another point on the parabola is (5, -1).)

Learn how to find the equation of a parabola given its vertex at (-4, 5) and another point at (5, -1). This problem involves key concepts in algebra and quadratic functions, including using the vertex form of a parabola and solving for 'a'.

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