To determine which graph represents the horizontally compressed function \( y = f(3x) \) derived from the parent function \( f(x) = x^2 \), it's important to understand how the horizontal compression affects the graph.
The parent function \( f(x) = x^2 \) is a standard parabola that opens upwards. When we apply a transformation such as \( f(kx) \), where \( k > 1 \), the graph will be compressed horizontally by a factor of \( \frac{1}{k} \). In this case, with \( k = 3 \), the graph compresses horizontally by a factor of \( \frac{1}{3} \).
This means that the x-coordinates of the points on the graph will be scaled down, making the parabola stretch more tightly around the y-axis. For example, the vertex of the original parabola \( (0, 0) \) remains the same, but the points that were at \( (1, 1) \) and \( (-1, 1) \) will now be closer to the y-axis at \( \left(\frac{1}{3}, 1\right) \) and \( \left(-\frac{1}{3}, 1\right) \).
Therefore, the correct graph should show the parabola opened upwards with a vertex at the origin, but it should be squeezed horizontally, meaning its width is reduced compared to the original \( f(x) = x^2 \).
Now, from the provided response options, selecting the one where the graph appears narrower can lead to the correct answer. Look for the parabola that has notable points closer to the y-axis compared to the standard form while ensuring it opens upwards.
If one of the graphs has a narrower width than the typical \( -5 \text{ to } 5 \) range for x-axis values, that would be the correct choice.
Without seeing the actual images, you should choose the graph reflecting this compressed nature of the parabola, likely one of the graphs with a narrower x-axis range.
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