A parabola has a vertex at (-2,-8) and a y-intercept at 4, what is the equation of this parabola in vertex form? 

The equation of a parabola in vertex form is given by:

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

Where:
- \((h, k)\) is the vertex of the parabola.
- \(a\) is a constant that affects the "width" and direction of the parabola (opens up or down).

Given that the vertex is \((-2, -8)\), we can substitute \(h = -2\) and \(k = -8\) into the vertex form:

\[
y = a(x + 2)^2 - 8
\]

Now, we need to determine the value of \(a\). We can use the fact that the parabola passes through the point \((0, 4)\), since the y-intercept is 4.

Substitute \(x = 0\) and \(y = 4\) into the equation:

\[
4 = a(0 + 2)^2 - 8
\]

Simplify:

\[
4 = a(2)^2 - 8
\]
\[
4 = 4a - 8
\]

Solve for \(a\):

\[
4a = 12
\]
\[
a = 3
\]

Thus, the equation of the parabola is:

\[
y = 3(x + 2)^2 - 8
\]

Would you like more details on this, or do you have any other questions?

Here are five related questions:
1. How do we derive the standard form from the vertex form of a parabola?
2. What is the significance of the parameter \(a\) in the equation of a parabola?
3. How do we find the axis of symmetry of a parabola given in vertex form?
4. Can you find the focus and directrix of this parabola?
5. How does the graph of a parabola change when \(a\) becomes negative?

**Tip**: The vertex form of a parabola is especially useful because it directly provides the vertex, which is key to understanding the graph's shape and direction.

A parabola has a vertex at (-2,-8) and a y-intercept at 4, what is the equation of this parabola in vertex form?

Find the equation of a parabola with vertex at (-2, -8) and a y-intercept of 4. This problem explores key quadratic concepts such as the vertex form of a parabola and solving for 'a'. Suitable for Grades 9-11.

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