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 $a$.