To understand how the graph of the function \( f(x) = x^2 \) is transformed into \( y = f(3x) \), we first need to clarify what the transformation means.
When we have a function of the form \( f(kx) \), where \( k > 1 \), it results in a horizontal compression of the graph. For \( f(3x) \), since \( k = 3 \), the graph of \( f(x) = x^2 \) will be compressed horizontally by a factor of \( 3 \).
To visualize this:
- The vertex of the parabola \( f(x) = x^2 \) is at the origin (0,0) and is symmetric about the y-axis.
- When compressed horizontally by a factor of \( 3 \), points on the graph will move closer to the y-axis. For example, the point (1,1) on the original graph will move to (1/3, 1) on the new graph; similarly, the point (2,4) will move to (2/3, 4).
Thus, the resulting graph \( y = f(3x) = (3x)^2 = 9x^2 \) will have the same basic 'U' shape as the original \( y = x^2 \), but it will be narrower.
If you're looking at multiple graphs to identify which one corresponds to \( y = f(3x) \), you should look for a parabola that is vertically aligned, has its vertex at the origin, and is narrower than the standard \( y = x^2 \) graph.
If you have specific graph options to choose from, the correct choice will be the graph that has the vertex at (0,0) and appears more narrow compared to the standard parabola \( y = x^2 \).