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 stretch with a factor k = 0.5
horizontal stretch with a factor k = 0.5
horizontal stretch with a factor k = 2
horizontal stretch with a factor k = 2
horizontal compression with a factor k = 0.5
horizontal compression with a factor k = 0.5
horizontal compression with a factor k = 2
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 stretch with a factor k = 0.5
horizontal stretch with a factor k = 0.5
horizontal stretch with a factor k = 2
horizontal stretch with a factor k = 2
horizontal compression with a factor k = 0.5
horizontal compression with a factor k = 0.5
horizontal compression with a factor k = 2
Answers
GPT-4o mini
To determine the correct transformation from the solid-line graph to the dotted-line graph of the parabolas, we need to understand how transformations affect the shape of parabolas.
1. A **horizontal stretch** occurs when the graph of the function is squeezed wider (the figure becomes thinner), which corresponds to a factor \( k < 1 \).
2. A **horizontal compression** occurs when the graph is pulled narrower (the figure becomes wider), which corresponds to a factor \( k > 1 \).
From the description, if the solid-line graph is being transformed to the dotted-line graph, and considering common transformations of quadratic functions, you would typically represent a horizontal stretch or compression using the form:
\[ y = a(x - h)^2 + k \]
Here, \( a \) would be affected by the value of \( k \):
- A horizontal stretch corresponds to substituting \( x \) with \( kx \) where \( k < 1 \).
- A horizontal compression corresponds to substituting \( x \) with \( kx \) where \( k > 1 \).
Thus:
- A horizontal stretch with a factor \( k = 0.5 \) means the parabola is stretched wider.
- A horizontal stretch with a factor \( k = 2 \) means the parabola is squeezed narrower.
- A horizontal compression with a factor \( k = 0.5 \) also indicates that the graph will remain wide since it is less than 1.
- A horizontal compression with a factor \( k = 2 \) indicates that the graph will become narrow.
In this case, if the dotted-line graph appears wider than the solid-line graph, then it would indicate a horizontal stretch. Conversely, if it appears narrower, it would be indicative of a horizontal compression.
Assuming a horizontal stretch with \( k = 0.5 \) means that the graph is being stretched wider, and this would be the appropriate response.
Thus, the best description based on the given conditions is:
**horizontal stretch with a factor k = 0.5.**
1. A **horizontal stretch** occurs when the graph of the function is squeezed wider (the figure becomes thinner), which corresponds to a factor \( k < 1 \).
2. A **horizontal compression** occurs when the graph is pulled narrower (the figure becomes wider), which corresponds to a factor \( k > 1 \).
From the description, if the solid-line graph is being transformed to the dotted-line graph, and considering common transformations of quadratic functions, you would typically represent a horizontal stretch or compression using the form:
\[ y = a(x - h)^2 + k \]
Here, \( a \) would be affected by the value of \( k \):
- A horizontal stretch corresponds to substituting \( x \) with \( kx \) where \( k < 1 \).
- A horizontal compression corresponds to substituting \( x \) with \( kx \) where \( k > 1 \).
Thus:
- A horizontal stretch with a factor \( k = 0.5 \) means the parabola is stretched wider.
- A horizontal stretch with a factor \( k = 2 \) means the parabola is squeezed narrower.
- A horizontal compression with a factor \( k = 0.5 \) also indicates that the graph will remain wide since it is less than 1.
- A horizontal compression with a factor \( k = 2 \) indicates that the graph will become narrow.
In this case, if the dotted-line graph appears wider than the solid-line graph, then it would indicate a horizontal stretch. Conversely, if it appears narrower, it would be indicative of a horizontal compression.
Assuming a horizontal stretch with \( k = 0.5 \) means that the graph is being stretched wider, and this would be the appropriate response.
Thus, the best description based on the given conditions is:
**horizontal stretch with a factor k = 0.5.**