To determine the quadratic function depicted in the graph, let's analyze the key features of the parabola:

1. Vertex: The graph shows that the vertex of the parabola is at the origin (0, 0).
2. Direction: The parabola opens upwards, which means the coefficient of the $x^2$ term is positive.
3. Points on the Graph: It appears that the parabola passes through points $(0,0)$, $(1, 4)$, and $(-1, 4)$.

Given that the vertex form of a parabola is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex, and here $(h, k) = (0, 0)$, the equation simplifies to:

$y = ax^2$

To find the value of $a$, use the point $(1, 4)$:

$4 = a(1)^2 \implies a = 4$

Thus, the quadratic function for this parabola is:

$y = 4x^2$

Would you like to explore further or have any questions?

Here are some related questions:

1. What is the effect of changing the value of $a$ in the quadratic function $y = ax^2$?
2. How do you determine if a parabola opens upwards or downwards based on its equation?
3. What are the key differences between the vertex form and standard form of a quadratic equation?
4. How can you find the x-intercepts of a parabola from its quadratic function?
5. How does the vertex form help in easily identifying the vertex of a parabola?

Tip: To quickly determine the equation of a parabola from a graph, identify the vertex and another point on the parabola to solve for the coefficient $a$.