The graph in the image shows three parabolas, each labeled $y_1$, $y_2$, and $y_3$. The question asks which function best represents $y_3$, the black parabola.

Let's analyze the options provided and the behavior of $y_3$:

1. $y = x^2$: This is the basic parabola with its vertex at $(0, 0)$, opening upwards. This matches $y_1$ (green graph), which passes through the origin and has this shape.

2. $y = 2x^2$: This is a vertically stretched parabola, still with its vertex at $(0, 0)$. This could correspond to $y_2$ (blue graph), which is narrower than $y_1$, suggesting it has been vertically stretched.

3. $y = x^2 - 2$: This is a parabola that opens upwards but is shifted downward by 2 units. This doesn't seem to match any of the graphs.

4. $y = (x - 2)^2$: This is a parabola with its vertex shifted 2 units to the right. Looking at $y_3$ (black graph), its vertex is at $(2, 0)$, matching the transformation described by this equation.

Therefore, the function that best represents $y_3$ is $y = (x - 2)^2$, as it correctly shifts the parabola horizontally to the right by 2 units.

Would you like further details on this solution or have any questions?

Here are 5 follow-up questions:

1. How can we describe the transformation of a parabola from $y = x^2$ to $y = (x - h)^2$?
2. What does vertical stretching or compressing mean for parabolas?
3. How do we identify a parabola’s vertex from its equation?
4. Why does $y = x^2 - 2$ shift a parabola downward?
5. How can we determine the width of a parabola from its equation?

Tip: The vertex form of a parabola, $y = (x - h)^2 + k$, is useful for easily identifying shifts and the vertex of the graph.