To visualize how the graph of the parent function \( f(x) = x^2 \) changes when it is horizontally compressed, we need to consider the transformation represented by \( y = f(3x) \).
- The parent function \( f(x) = x^2 \) is a parabola opening upwards with its vertex at the origin (0,0).
- The transformation \( f(3x) \) compresses the graph horizontally. Specifically, it compresses the graph by a factor of \( \frac{1}{3} \).
This means that all \( x \) values are effectively multiplied by \( \frac{1}{3} \). For example:
- The point \( (1, 1) \) on \( f(x) = x^2 \) will move to \( \left(\frac{1}{3}, 1\right) \) on \( f(3x) \).
- The point \( (-1, 1) \) will move to \( \left(-\frac{1}{3}, 1\right) \).
As a result of this transformation, the new graph of \( y = f(3x) \) appears "squeezed" horizontally compared to the original parabola.
To summarize, the graph of \( y = f(3x) \) will still be a parabola that opens upwards but will reach its values much quicker due to the horizontal compression by a factor of 3. The overall shape remains similar but is narrower. You would look for a graph that illustrates a parabola reaching its maximum more quickly in the horizontal direction.
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