determine the equation of the parabola with vertex at (-2,5) and that passes through (4,-8)

To determine the equation of the parabola, we use the vertex form of a parabola's equation:

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

Where:
- \( (h, k) \) is the vertex of the parabola,
- \( a \) is a constant that determines the width and direction of the parabola.

### Step 1: Substitute the vertex
Given that the vertex is \( (-2, 5) \), the equation becomes:
\[
y = a(x + 2)^2 + 5
\]

### Step 2: Use the point (4, -8) to find \( a \)
Now, substitute \( (x, y) = (4, -8) \) into the equation to solve for \( a \):
\[
-8 = a(4 + 2)^2 + 5
\]
\[
-8 = a(6)^2 + 5
\]
\[
-8 = 36a + 5
\]
\[
36a = -8 - 5
\]
\[
36a = -13
\]
\[
a = \frac{-13}{36}
\]

### Step 3: Write the equation of the parabola
Now that we have \( a = \frac{-13}{36} \), we can substitute it back into the vertex form:
\[
y = \frac{-13}{36}(x + 2)^2 + 5
\]

This is the equation of the parabola.

Would you like any further details or explanations? Here are some additional questions to extend this concept:

1. What is the significance of the constant \(a\) in a parabola's equation?
2. How would the equation change if the parabola opened upwards instead of downwards?
3. Can you express the same equation in standard form by expanding it?
4. How do we find the x-intercepts of the parabola from this equation?
5. What is the axis of symmetry of this parabola?

**Tip:** The sign of \(a\) in the vertex form of a parabola determines whether it opens upwards (\(a > 0\)) or downwards (\(a < 0\)).

Find the equation of a parabola with vertex (-2, 5) that passes through point (4, -8). This solution explains how to use the vertex form of a quadratic equation, substitute a known point to find 'a', and provides a detailed step-by-step approach. Suitable for Grades 9-11 algebra students.

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