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Horizontal Stretches & Compressions Quick Check
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Question
Use the image to answer the question.

Two parabolas with a common vertex are drawn on a coordinate plane. The x-axis ranges from negative 3 to 3 in one-unit increments and the y-axis ranges from negative 4 to 2 in one-unit increments.

Which of the following best describes the transformation from the solid-line graph to the dotted-line graph and correctly states the factor k?

(1 point)
Responses

horizontal compression with a factor k = 0.5
horizontal compression with a factor k = 0.5

horizontal stretch with a factor k = 2
horizontal stretch with a factor k = 2

horizontal stretch with a factor k = 0.5
horizontal stretch with a factor k = 0.5

horizontal compression with a factor k = 2
horizontal compression with a factor k = 2

1 answer

To determine the correct transformation from the solid-line graph to the dotted-line graph based on the options provided, we need to analyze how the graphs of parabolas are affected by horizontal stretches and compressions.

  1. Horizontal Compression: If the graph is compressed horizontally, it means that for a function \( f(x) \), the new function after compression would look like \( f(kx) \) where \( k > 1 \). This results in the graph being "squeezed" towards the y-axis.

  2. Horizontal Stretch: If the graph is stretched horizontally, it means that the new function would look like \( f(kx) \) where \( 0 < k < 1 \). This results in the graph being "stretched" away from the y-axis.

  • In the options provided:
    • A horizontal compression with a factor of \( k = 0.5 \) means that the graph is being compressed to half its width.
    • A horizontal compression with a factor of \( k = 2 \) means that the graph is being compressed to twice its width.
    • A horizontal stretch with a factor of \( k = 0.5 \) means it's actually compressed, as this is less than 1 (a factor less than 1 indicates compression).
    • A horizontal stretch with a factor of \( k = 2 \) correctly indicates the graph is being stretched.

Without the visual, based on the transformation description:

  • If the solid-line graph has a wider spread than the dotted-line graph, it indicates a compression; if it's narrower, it indicates a stretch.

Based on the possible answers, please analyze how wide the dotted-line graph is compared to the solid-line graph:

  • If the dotted-line graph is narrower compared to the solid-line graph, then the relation indicates a horizontal compression with \( k > 1 \).
  • Conversely, if the dotted-line graph is wider, then it indicates a horizontal stretch with \( 0 < k < 1 \).

Without the image, I can't definitively tell which transformation it is, but you can use the above analysis to check the stretching or compressing nature of the graphs accordingly.

If you conclude the graph went from wider to narrower (indicating compression), look for options with horizontal compression. If it’s from narrower to wider, pick the horizontal stretch options.

Let me know if you have any questions or further details!