To determine the transformation from the solid-line graph (parent parabola) to the dotted-line graph, we need to understand the effects of horizontal compression and stretch on the graph of a parabola.
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A horizontal compression by a factor of \( k \) means that each point on the graph is moved closer to the y-axis, resulting in the parabola becoming "narrower." For a horizontal compression, the equation would be transformed to \( y = a(x/k)^2 \).
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A horizontal stretch by a factor of \( k \) means that each point on the graph is moved further from the y-axis, resulting in the parabola becoming "wider." For a horizontal stretch, the equation would be transformed to \( y = a(xk)^2 \).
- A horizontal compression with a factor of \( 2 \) means the parabola is half as wide as it was originally (i.e., \( k = 2\)).
- A horizontal compression with a factor of \( 0.5 \) means the parabola is twice as wide as it was originally (i.e., \( k = 0.5\)).
- A horizontal stretch with a factor of \( 2 \) means the parabola is wider by a factor of two (i.e., \( k = 2 \)).
- A horizontal stretch with a factor of \( 0.5 \) means the parabola is narrower than the original (i.e., \( k = 0.5 \)).
To deduce the transformation properly, observe where the vertex and points of the given parabolas lie. If the dotted-line graph is narrower than the solid line, it indicates a horizontal compression. If it's wider, it indicates a horizontal stretch.
If you can visually confirm that the dotted parabola is narrower (indicating it's a horizontal compression), then you would look for \( k = 2 \), since this indicates the graph is closer to the y-axis.
Therefore, given the options, the best description of the transformation from the solid-line graph to the dotted-line graph is:
horizontal compression with a factor k = 2.