To determine which graph represents the function \( y = f(3x) \), where \( f(x) = x^2 \), we need to understand the effect of the transformation.
The transformation from \( f(x) = x^2 \) to \( f(3x) \) compresses the graph horizontally by a factor of 3. This means that the x-coordinates of the points on the graph will be scaled by a factor of \( \frac{1}{3} \). Therefore, the parabola will appear "narrower" and the vertex will remain at the same y-coordinate (0), while the x-intercepts will also be reduced by a factor of 3.
For example, if the original function had points at (1, 1), (2, 4), and (-1, 1), the new function \( f(3x) \) will correspond to points at \( \left( \frac{1}{3}, 1\right) \), \( \left( \frac{2}{3}, 4\right) \), and \( \left(-\frac{1}{3}, 1\right) \).
Considering the options you provided, the key is in the x-axis range:
- The first graph ranges from -5 to 6.
- The second graph ranges from -10 to 10.
- The third graph and the fourth graph both range from -5 to 5.
Given that the transformation compresses the graph horizontally, we should look for the graph that shows a narrower parabola within a limited x-axis range.
Without the actual plotted points, you should choose the graph that has a narrower shape and fits within a limited x range after being compressed. Both the third and fourth options have the same range and description.
If all other factors such as vertex and y-range are similar and consistent, either of the two (third or fourth) could be correct, but typically one would refer to alterations in shape or placement to discern the correct one.
If you have further details regarding the characteristics of the upward parabola such as specific points it passes through, that would help in finalizing the answer. Otherwise, based on the given information, the third and fourth options look equivalent in description.