https://file.aiquickdraw.com/mathbot/file/1725712415244swhw9bsg.jpg

The image mentions a vertex at \( (-5, -7) \) and states that there is a vertical axis of symmetry through the point \( P(7, 11) \).

Let me break this down:

1. **Vertex form of a parabola**: The equation of a parabola in vertex form is typically written as:
   \[
   y = a(x - h)^2 + k
   \]
   where \( (h, k) \) is the vertex, and \( a \) controls the "width" and direction of the parabola.

2. **Given vertex**: The vertex is at \( (-5, -7) \), so the equation starts as:
   \[
   y = a(x + 5)^2 - 7
   \]

3. **Axis of symmetry**: A vertical axis of symmetry implies the parabola opens either upwards or downwards, meaning the symmetry lies along the line \( x = -5 \). However, the equation must also satisfy the condition that the parabola passes through the point \( P(7, 11) \).

To find the constant \( a \), substitute \( P(7, 11) \) into the equation:

\[
11 = a(7 + 5)^2 - 7
\]
\[
11 = a(12)^2 - 7
\]
\[
11 = 144a - 7
\]
\[
18 = 144a
\]
\[
a = \frac{1}{8}
\]

So, the equation of the parabola is:
\[
y = \frac{1}{8}(x + 5)^2 - 7
\]

This is the equation of the parabola with a vertex at \( (-5, -7) \) and a vertical axis of symmetry passing through \( P(7, 11) \).

Would you like a deeper explanation or have any questions?

Here are some related questions:

1. How do we determine if a parabola opens upwards or downwards?
2. Can the axis of symmetry ever be horizontal in this kind of problem?
3. What does the value of \( a \) in the vertex form tell us about the shape of the parabola?
4. How would the equation change if the vertex was different?
5. What happens if the axis of symmetry is not vertical?

**Tip:** When solving parabolic equations, always use the given points to solve for unknowns like \( a \) to ensure the equation fits all conditions.

The user provided an image that mentions a vertex at (-5, -7) and states there is a vertical axis of symmetry through the point P(7, 11). How is the equation of the parabola formed?

Explore how to derive the equation of a parabola given the vertex and a point on the parabola. This page includes a comprehensive explanation of the vertex form, the role of the vertex, and the axis of symmetry.

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