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 = 2
horizontal stretch with a factor k = 2
horizontal compression with a factor k = 2
horizontal compression 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 = 0.5
horizontal compression with a factor k = 0.5
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 = 2
horizontal stretch with a factor k = 2
horizontal compression with a factor k = 2
horizontal compression 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 = 0.5
horizontal compression with a factor k = 0.5
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 analyze how the shape of the parabola has changed in relation to the x-axis.
1. **Horizontal Stretch or Compression**:
- A horizontal stretch occurs when the parabola widens. This is when the x-values are scaled by a factor greater than 1, leading to the formula \( f(kx) \) where \( k > 1 \) results in a horizontal stretch.
- A horizontal compression occurs when the parabola narrows. This is represented by a factor \( k < 1 \) leading to a transformation of \( f(kx) \) which compresses the x-values.
2. **Determining the Factor**:
- If the distance between the vertices of the two parabolas is larger in the dotted-line graph compared to the solid-line graph, that indicates a horizontal stretch.
- Conversely, if the vertices are closer together in the dotted-line graph, it indicates a horizontal compression.
Since you specified a factor of 2 and a factor of 0.5, we should choose based on the transformation type:
- If \( k = 2 \), this corresponds to a horizontal stretch.
- If \( k = 0.5 \), this corresponds to a horizontal compression.
Without actually seeing the graphs, if, for example, the parabolas appear wider in the dotted-line graph compared to the solid-line graph, the appropriate response would be:
**Horizontal stretch with a factor k = 2**.
If they appear narrower, then it would be:
**Horizontal compression with a factor k = 0.5**.
Based on the parameters you've given, choose the option that matches your observation.
1. **Horizontal Stretch or Compression**:
- A horizontal stretch occurs when the parabola widens. This is when the x-values are scaled by a factor greater than 1, leading to the formula \( f(kx) \) where \( k > 1 \) results in a horizontal stretch.
- A horizontal compression occurs when the parabola narrows. This is represented by a factor \( k < 1 \) leading to a transformation of \( f(kx) \) which compresses the x-values.
2. **Determining the Factor**:
- If the distance between the vertices of the two parabolas is larger in the dotted-line graph compared to the solid-line graph, that indicates a horizontal stretch.
- Conversely, if the vertices are closer together in the dotted-line graph, it indicates a horizontal compression.
Since you specified a factor of 2 and a factor of 0.5, we should choose based on the transformation type:
- If \( k = 2 \), this corresponds to a horizontal stretch.
- If \( k = 0.5 \), this corresponds to a horizontal compression.
Without actually seeing the graphs, if, for example, the parabolas appear wider in the dotted-line graph compared to the solid-line graph, the appropriate response would be:
**Horizontal stretch with a factor k = 2**.
If they appear narrower, then it would be:
**Horizontal compression with a factor k = 0.5**.
Based on the parameters you've given, choose the option that matches your observation.