https://file.aiquickdraw.com/mathbot/file/1727059791952uukrkdzi.jpg

The image shows the graph of a parabola, and we are tasked with writing the function in standard form.

The parabola has the following important points:
- The x-intercepts are \((-4, 0)\) and \((-0.5, 0)\).
- The vertex of the parabola is \((-2.25, -6)\), which is the minimum point of the parabola.

To write the equation of a parabola in **standard form**, we use the vertex form:
\[
y = a(x - h)^2 + k
\]
where:
- \((h, k)\) is the vertex of the parabola.
- \(a\) is a coefficient that determines the direction and width of the parabola.

### Step 1: Use the vertex \((-2.25, -6)\)
Substitute \(h = -2.25\) and \(k = -6\) into the vertex form:
\[
y = a(x + 2.25)^2 - 6
\]

### Step 2: Use another point on the parabola to find \(a\)
Using the x-intercept \((-4, 0)\), substitute \(x = -4\) and \(y = 0\) into the equation:
\[
0 = a(-4 + 2.25)^2 - 6
\]
\[
0 = a(-1.75)^2 - 6
\]
\[
0 = a(3.0625) - 6
\]
\[
6 = 3.0625a
\]
\[
a = \frac{6}{3.0625} \approx 1.96 \quad \text{(approximately 2)}
\]

Thus, the equation of the parabola is approximately:
\[
y = 2(x + 2.25)^2 - 6
\]

Would you like more details or have any further questions?

### Relative Questions:
1. How do you convert from vertex form to standard form?
2. What does the coefficient \(a\) represent in the parabola?
3. How would the graph change if \(a\) was negative?
4. How do you find the x-intercepts from the standard form?
5. Can you use a different point on the graph to confirm \(a\)'s value?

### Tip:
Always double-check the points you use for accuracy when determining the parameters of a function.

Learn how to convert the vertex form of a quadratic function y = a(x - h)^2 + k into standard form. This problem involves finding the equation of a parabola with specific x-intercepts and a vertex, focusing on key algebraic concepts like quadratic functions and parabolas. Suitable for Grades 9-11.

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