https://file.aiquickdraw.com/mathbot/file/17282645428781j9pwt8j.jpg

The graph provided shows a parabola, and the task is to write the quadratic function in standard form. The standard form of a quadratic function is:

\[
y = ax^2 + bx + c
\]

From the graph, we can identify key points to find the equation:

- The vertex is at \((-2, -3)\), which suggests using the vertex form of a parabola:
\[
y = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex. In this case, \((h, k) = (-2, -3)\).

Thus, we start with:
\[
y = a(x + 2)^2 - 3
\]

- We also know the parabola passes through the point \((0, 1)\), which can be used to solve for \(a\). Plugging in this point \((x = 0, y = 1)\):

\[
1 = a(0 + 2)^2 - 3
\]
\[
1 = a(4) - 3
\]
\[
4a = 4 \implies a = 1
\]

So, the equation of the parabola is:
\[
y = (x + 2)^2 - 3
\]

Expanding this to standard form:
\[
y = (x^2 + 4x + 4) - 3
\]
\[
y = x^2 + 4x + 1
\]

Thus, the standard form of the quadratic function is:

\[
y = x^2 + 4x + 1
\]

Would you like further details or have any questions?

Here are some related questions:
1. How do we convert a quadratic function from vertex form to standard form?
2. How can we find the vertex of a parabola if it's given in standard form?
3. What are other methods to determine the quadratic equation from a graph?
4. How does the value of \(a\) affect the shape of the parabola?
5. How can we determine the axis of symmetry from the standard form of a quadratic?

**Tip:** The coefficient \(a\) in a quadratic equation controls the "width" and direction of the parabola: if \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards.

Write the standard form of the quadratic function whose graph is shown in the parabola, with the vertex (-2, -3) and passing through the point (0,1).

Learn how to write the quadratic equation in standard form from a graph with the vertex (-2, -3) and passing through the point (0, 1). Includes solving for 'a' and expanding the vertex form to standard form. Suitable for students in Grades 9-11.

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