Question

Which equation has a vertex of `(-4,5)`?

Answer 1

To find an equation of a parabola with a given vertex, we can use the vertex form of a quadratic equation, which is:

\[
y = a(x - h)^2 + k
\]

where \((h, k)\) is the vertex of the parabola and \(a\) determines the direction and width of the parabola.

Given the vertex \((-4, 5)\), we can substitute \(h = -4\) and \(k = 5\) into the equation:

\[
y = a(x + 4)^2 + 5
\]

At this point, \(a\) can be any non-zero constant:

1. If \(a > 0\), the parabola opens upwards.
2. If \(a < 0\), the parabola opens downwards.

For example:
- If we choose \(a = 1\), the equation becomes:
\[
y = (x + 4)^2 + 5
\]

- If we choose \(a = -1\), the equation becomes:
\[
y = -(x + 4)^2 + 5
\]

You can choose any value for \(a\) to create various equations with the vertex \((-4, 5)\).