To understand the transformation of the parent function \( f(x) = x^2 \) when it is horizontally compressed to form \( y = f(3x) \), we need to analyze what this transformation entails.
The function \( f(3x) \) mathematically translates to \( (3x)^2 \), which simplifies to \( 9x^2 \). This indicates that we are compressing the graph of the parent function horizontally by a factor of 3.
Here are the key effects of this transformation:
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Horizontal Compression: The graph of \( f(x) = x^2 \) will get narrower. Each \( x \) value will yield the same \( y \) value as the original function, but it will happen at a smaller \( x \) value because we are effectively scaling the \( x \)-axis.
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Shape: The basic shape of the parabola (opening upwards) will remain the same. However, it will be narrower compared to the original parabola.
To visualize this:
- The vertex of the original parabola \( x^2 \) remains at \( (0,0) \).
- The points on the original graph which were at \( (1, 1) \) and \( (-1, 1) \) will now be at \( (1/3, 1) \) and \( (-1/3, 1) \) respectively after the transformation.
In conclusion, you would look for a graph that represents a narrower parabola, with the vertex still at the origin and points that have been compressed closer to the y-axis compared to the standard \( x^2 \) graph.