The graph of $$ y=x^{2} $$ is shown on each grid in parts (a) and (b) below. (a) Use transformations to get the graph of $$ y=(2 x)^{2} $$. (a)

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To transform the graph of $$ y = x^2 $$ to the graph of $$ y = (2x)^2 $$, we can follow these steps:

1. **Identify the transformation**: The expression $$ (2x)^2 $$ can be rewritten as $$ 4x^2 $$. This indicates that we are scaling the input $$ x $$ by a factor of 2.

2. **Horizontal Scaling**: The transformation $$ y = (2x)^2 $$ means that for every $$ x $$ value, we are effectively compressing the graph horizontally by a factor of 2. This is because the input $$ x $$ is multiplied by 2, which means that the graph will reach the same $$ y $$ values at half the $$ x $$ values.

3. **Graphing the Transformation**:
   - The vertex of the original graph $$ y = x^2 $$ is at the point (0, 0).
   - For the transformed graph $$ y = (2x)^2 $$, we can find some key points:
     - When $$ x = 0 $$, $$ y = (2 \cdot 0)^2 = 0 $$ → point (0, 0)
     - When $$ x = 0.5 $$, $$ y = (2 \cdot 0.5)^2 = 1 $$ → point (0.5, 1)
     - When $$ x = 1 $$, $$ y = (2 \cdot 1)^2 = 4 $$ → point (1, 4)
     - When $$ x = -0.5 $$, $$ y = (2 \cdot -0.5)^2 = 1 $$ → point (-0.5, 1)
     - When $$ x = -1 $$, $$ y = (2 \cdot -1)^2 = 4 $$ → point (-1, 4)

4. **Conclusion**: The graph of $$ y = (2x)^2 $$ will be a vertically stretched version of the graph of $$ y = x^2 $$ that is also horizontally compressed. The shape remains a parabola, but it will be narrower than the original graph.

In summary, the transformation from $$ y = x^2 $$ to $$ y = (2x)^2 $$ involves a horizontal compression by a factor of 2.
To get the graph of $$ y = (2x)^2 $$ from $$ y = x^2 $$, compress the graph horizontally by a factor of 2.

The graph of \( y=x^{2} \) is shown on each grid in parts (a) and (b) below. (a) Use transformations to get the graph of \( y=(2 x)^{2} \). (a)

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