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)

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.