Vertex, the original, opens upward,  through the point (-4,-2). Find the equation of a parabola 

Given that the vertex of the parabola is the point \((-4, -2)\) and that the parabola opens upward, the equation of the parabola can be written in **vertex form** as:

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

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

In your case:
- The vertex is \((-4, -2)\), so \( h = -4 \) and \( k = -2 \).

Thus, the equation becomes:
\[
y = a(x + 4)^2 - 2
\]

To fully determine the equation, we need the value of \(a\). If you're given another point that lies on the parabola, we can substitute that point's coordinates into the equation to solve for \(a\).

Without any additional points, \(a\) can remain as an unknown parameter. So, the general form of the equation with an upward-opening parabola is:

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

### Do you have any additional points or values for \(a\) that I should consider?

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### Expanding understanding:
1. How do you determine the value of \(a\) in a parabola equation?
2. What is the significance of the vertex in the equation of a parabola?
3. Can the parabola open in other directions besides upward?
4. What are the effects of changing \(a\) on the shape of the parabola?
5. What happens to the parabola when the vertex is shifted to other points?

#### Tip:
To find \(a\), use another point on the parabola! Substituting that point into the vertex form equation helps determine the value of \(a\).

Vertex, the original, opens upward, through the point (-4,-2). Find the equation of a parabola.

Discover how to find the equation of a parabola that opens upward through the vertex (-4, -2) using the vertex form equation y = a(x - h)^2 + k. This guide covers essential algebraic concepts, vertex form, and the significance of the 'a' parameter. Suitable for students in Grades 9-11.

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