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f(x)=3x^2-6 to f(x)=3(0.5x)^2-6

According to the graph, what is the value of k? Does the transformation show a stretch or a compression?

(1 point)
Responses

0.5; stretch
0.5; stretch

0.5; compression
0.5; compression

0.5x
; compression
0.5 x ; compression

0.5x
; stretch

3 answers

To analyze the transformations from the function \( f(x) = 3x^2 - 6 \) to \( f(x) = 3(0.5x)^2 - 6 \), we can look at the change in the \( x \) value.

  1. Identify the transformation: In the second function, the substitution \( 0.5x \) in place of \( x \) represents a horizontal transformation. Specifically, it is a horizontal compression by a factor of \( 0.5 \). This means that the graph of the function will be squeezed horizontally because every input \( x \) is now halved, bringing points closer to the y-axis.

  2. Value of \( k \): The coefficient \( 0.5 \) is the value of \( k \) in this scenario.

Based on this analysis, the answer would be:

0.5; compression.

Which of the following best describes the transformation from the solid-line graph to the dotted-line graph and correctly states the factor k?
the solid line points:(-1,0) , (0,-3) , (1,0)
dotted line points: (-2,0) , (0,-3) , (2,0)
(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

To analyze the transformation from the solid-line graph to the dotted-line graph based on the given points, let's look at each point:

  • The points for the solid line are:

    • (-1, 0)
    • (0, -3)
    • (1, 0)

  • The points for the dotted line are:

    • (-2, 0)
    • (0, -3)
    • (2, 0)

Analyzing the Transformation:

  1. x-coordinates:

    • The x-coordinates of the solid line are -1, 0, and 1.
    • The corresponding x-coordinates on the dotted line are -2, 0, and 2.

  2. Determining the Transformation:

    • Each x-coordinate in the dotted line is double that of the corresponding x-coordinate in the solid line:
      • (-1 becomes -2)
      • (0 remains 0)
      • (1 becomes 2)
    • This indicates a horizontal stretch by a factor of \( k = 2 \).

Conclusion:

The transformation from the solid-line graph to the dotted-line graph is a horizontal stretch with a factor k = 2.

Thus, the correct response is:

horizontal stretch with a factor k = 2.