To determine which graph represents the function \(y = f(3x)\) derived from the parent function \(f(x) = x^2\), we need to understand that a horizontal compression occurs when the input \(x\) is scaled by a factor greater than 1. In this case, since we have \(f(3x)\), \(x\) is being multiplied by 3, which compresses the graph horizontally by a factor of \(\frac{1}{3}\).
The effects of this horizontal compression on the graph of \(f(x) = x^2\) include that:
- The vertex remains at the same point \((0, 0)\).
- The width of the parabola decreases; it will appear narrower compared to the original \(f(x) = x^2\).
- The x-values will be scaled down (compressed), meaning fewer x-values out to the same y-values compared to the original parabola.
Given the options, we will look for a graph that displays a compressed version of the standard upward-opening parabola.
Considering the descriptions of the graphs:
- Since a compression effect means the graph will reach a certain height (y-value) at lower x-values than the original \(f(x)\), we would expect the range of x-values shown in the graph to be less than in the original graph.
- The original graph of \(f(x) = x^2\) intersects at \(x = -1, 0, 1\) (with corresponding y-values of \(1, 0, 1\)). In compression, we would expect these points to be re-located closer to the origin.
Among the options, the fourth graph shows an x-axis that ranges from -10 to 10 (which is the broadest) but does not specifically indicate a compression. The other options provide a range of x-values either from -5 to 5 or -5 to 6.
To summarize:
- Since we need a graph where the x-axis limits are relatively smaller (indicating a horizontal compression effect), the best choice would be either the second or third options as they indicate a more reasonable range of values for a compressed horizontal function.
Based only on the visual compression expected, if I had to choose one, I would suggest going with the second graph's x-axis range from -5 to 6, as it provides the closest option to indicate that the transformation has occurred correctly.
Therefore, the response is: the second option.