To determine which graph represents the function \( y = f(3x) \) when starting from the parent function \( f(x) = x^2 \), we must understand how the transformation affects the graph.
The parent function \( f(x) = x^2 \) is a parabola that opens upward with its vertex at the origin (0,0). The transformation \( f(3x) \) indicates a horizontal compression of the graph by a factor of 3. This means that the graph will be "squeezed" along the x-axis.
Key Changes:
- A horizontal compression by a factor of 3 will cause the x-coordinates of the points on the graph of the original function \( f(x) = x^2 \) to be reduced. For instance, a point that was at \( (1, 1) \) for the parent function will now be at \( \left(\frac{1}{3}, 1\right) \) for \( f(3x) \).
- The compressed graph will still open upward and maintain the same general parabolic shape but will fit within a smaller range on the x-axis.
Given the descriptions of the graphs:
- First Description: X-axis ranges from -5 to 6.
- Second Description: X-axis ranges from -10 to 10.
- Third Description: X-axis ranges from -5 to 5.
- Fourth Description: X-axis ranges from -5 to 5.
Since \( y = f(3x) \) would have its important points (such as the vertex and points like (1,1) compressed to \( \frac{1}{3} \)), we expect the x-values to be less than those of the original more widely spread values.
The graph that best represents \( y = f(3x) \) should fit well within a smaller x-range. The graph with an x-axis ranging from -5 to 5 will display such behavior. The first and second options are too broad.
Thus, either the third or fourth graph (both are described the same) would be correct.
Conclusion
The correct answer is either the third or the fourth option which both display the same characteristics. If you have to choose one based on the descriptions provided, select either of those as they fit the transformation correctly.