Decipher the Graph's Story: Which Equation Represents This Function?

The given graph is a parabola that opens downwards and is symmetrical with respect to the y-axis, with its vertex at the origin. This is characteristic of a standard quadratic function of the form $y = -ax^2$, where $a$ is positive, indicating the parabola opens downwards because $a$ is negative in the form $y = -x^2$.

Let's compare the graph with the choices given:

• The first choice $y = x^2$ opens upwards, so it doesn't match.
• The second choice $y = -x^2$ opens downwards, matching the graph.
• The third choice $y = 3x^2$ opens upwards and is more compressed, so it doesn’t match.
• The fourth choice $y = \frac{1}{2}x^2$ opens upwards and is wider, so it doesn’t match.

Therefore, the equation that corresponds to the graph is $y = -x^2$.