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

To determine which equation best represents the second graph, we need to analyze how each equation affects the original graph of $y = f(x)$:

• Option A: $y = \sqrt{f(x)}$ - This transforms the graph by taking the square root. This would eliminate any portions of the graph below the x-axis, potentially altering the original shape too much.

• Option B: $y^2 = f(x)$ - This would imply that the positive and negative values of $y$ could replicate the original graph, but it could also create extraneous intersections, complicating the graph without matching the original clearly.

• Option C: $y = [f(x)]^2$ - Squaring the function means that all values will be non-negative. This would create a graph where originally negative values of $f(x)$ would reflect above the x-axis, thus altering the graph significantly. This matches the second graph observed, which does cross the x-axis only at essential points.

• Option D: $y = f(x^2)$ - This would imply that the input value x is squared, which modifies the function but may not yield a similar graph in behavior, especially since it modifies the x-values available.

Given these transformations, the equation that best represents the second graph is C. $y = [f(x)]^2$, as it captures the essential features without misrepresenting the original function's characteristics.