Use the image to answer the question.

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 = 2
horizontal compression with a factor k = 2

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

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

horizontal stretch with a factor k = 2

Answer 1

To determine the correct transformation from the solid-line graph to the dotted-line graph, we need to analyze the nature of the change.

If you are observing a horizontal transformation of a function \( f(x) \) to produce a new function \( f(kx) \), the factor \( k \) influences how the width of the parabola changes:

• If \( k > 1 \), it compresses the graph horizontally.
• If \( 0 < k < 1 \), it stretches the graph horizontally.

Since you mentioned that both graphs share a common vertex, the transformation is likely one of compression or stretching rather than a shift.

Given your options regarding the factor \( k \):

• A horizontal compression is associated with \( k > 1 \). If \( k = 2 \), it would imply that every x-value is halved, compressing the graph.
• A horizontal stretch is associated with \( 0 < k < 1\). If \( k = 0.5\), the x-values are doubled, which would stretch the graph horizontally.

Without the visual information, I cannot directly observe the graphs to find the transformation, but based on your list of options, if the dotted line looks narrower than the solid line, it indicates a horizontal compression.

Thus, based on the choices provided, if the dotted line is narrower than the solid line, the correct response would be:

horizontal compression with a factor k = 2.