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To determine the quadratic function from the given graph, we need to consider key points and the general form of a quadratic function. The standard form is:

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

From the graph, we can identify the following important points:

1. The vertex of the parabola is \( (0, -3) \).
2. The x-intercepts (roots) are \( (-1, 0) \) and \( (3, 0) \).

### Step 1: Use the vertex form of a quadratic function

The vertex form of a quadratic function is given by:

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

Where \( (h, k) \) is the vertex of the parabola. From the graph, the vertex is \( (0, -3) \), so we substitute \( h = 0 \) and \( k = -3 \):

\[
y = a(x - 0)^2 - 3 = a x^2 - 3
\]

### Step 2: Use a known point to find \( a \)

Now, we can use one of the x-intercepts, such as \( (3, 0) \), to find the value of \( a \). Substitute \( x = 3 \) and \( y = 0 \) into the equation:

\[
0 = a(3)^2 - 3
\]

Solve for \( a \):

\[
0 = 9a - 3
\]
\[
9a = 3
\]
\[
a = \frac{1}{3}
\]

### Step 3: Write the quadratic function

Now that we know \( a = \frac{1}{3} \), the quadratic function is:

\[
y = \frac{1}{3}x^2 - 3
\]

Thus, the quadratic function that represents the given graph is:

\[
y = \frac{1}{3}x^2 - 3
\]

Would you like further details or have any questions?

### Related questions:

1. What is the role of the vertex in determining the quadratic function?
2. How do we determine the axis of symmetry from a quadratic graph?
3. Can the quadratic function be written in factored form? How?
4. How does the value of \( a \) affect the direction and width of the parabola?
5. What is the significance of the x-intercepts in finding the quadratic function?

### Tip:
To quickly estimate the vertex form of a quadratic equation, start by identifying the vertex and using any other point on the graph to solve for \( a \).

Learn how to determine the quadratic function from a given graph by identifying the vertex and using the x-intercepts. This example uses the vertex form of the quadratic equation and provides a step-by-step solution. Suitable for students in Grades 9-11.

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