The graph of the function $y = f(x)$ is shown - HSC - SSCE Mathematics Extension 2 - Question 4 - 2017 - Paper 1

To find the equation that generates the second graph from the original graph of $y = f(x)$, we analyze the transformations applied to the function. The original graph shows a function with critical points where it crosses the x-axis at certain points.

Among the options:

• Option A: $y = \sqrt{f(x)}$ - This would only represent the positive values of $f(x)$. The graph would not have negative parts mirrored but truncated, which does not match the provided graph.

• Option B: $y^2 = f(x)$ - This option suggests that $f(x)$ can take both positive and negative values of $y$. This transformation would reflect the graph across the x-axis and might better fit transformations, but it doesn't maintain vertex structure.

• Option C: $y = [f(x)]^2$ - Squaring a function retains the shape while reflecting all negative values into positive ones. This changes all parts of the graph that are negative into positive, which occurs in the second graph. This matches the provided second graph, maintaining the original critical points.

• Option D: $y = f(x^2)$ - This transformation would compress the graph horizontally but might distort the original function’s symmetry as x gets larger.

Thus, after analyzing these options, the equation that best represents the second graph is:

C. $y = [f(x)]^2$