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
Answers
GPT-4o mini
To determine the transformation from the solid-line graph to the dotted-line graph of the parabolas, we'll analyze the effect of transformations on the equations of parabolas.
If we denote the original parabola by \( y = f(x) \) and the transformed parabola by \( y = f(kx) \), we can see the effects of different values of \( k \):
- If \( 0 < k < 1 \), the graph undergoes a horizontal compression.
- If \( k > 1 \), the graph undergoes a horizontal stretch.
From the options provided:
1. **Horizontal compression with a factor k = 0.5** implies the new parabola would be narrower than the original.
2. **Horizontal stretch with a factor k = 2** implies the new parabola would be wider than the original.
3. **Horizontal stretch with a factor k = 0.5** is incorrect by definition, as it cannot be less than 1 and still stretch.
4. **Horizontal compression with a factor k = 2** implies the new parabola would be wider than the original, which does not align with compression.
Without access to the graphs directly, you need to observe the parabolas:
- If the dotted-line parabola is narrower than the solid-line parabola, the correct transformation is a horizontal compression with \( k < 1 \).
- If the dotted-line parabola is wider than the solid-line parabola, the correct transformation is a horizontal stretch with \( k > 1 \).
Since we have a horizontal compression with a factor of \( k = 0.5 \) as one of the options that suggests the dotted-line graph is narrower, this would be the appropriate choice if observed that way.
**Final Answer: horizontal compression with a factor k = 0.5**.
If we denote the original parabola by \( y = f(x) \) and the transformed parabola by \( y = f(kx) \), we can see the effects of different values of \( k \):
- If \( 0 < k < 1 \), the graph undergoes a horizontal compression.
- If \( k > 1 \), the graph undergoes a horizontal stretch.
From the options provided:
1. **Horizontal compression with a factor k = 0.5** implies the new parabola would be narrower than the original.
2. **Horizontal stretch with a factor k = 2** implies the new parabola would be wider than the original.
3. **Horizontal stretch with a factor k = 0.5** is incorrect by definition, as it cannot be less than 1 and still stretch.
4. **Horizontal compression with a factor k = 2** implies the new parabola would be wider than the original, which does not align with compression.
Without access to the graphs directly, you need to observe the parabolas:
- If the dotted-line parabola is narrower than the solid-line parabola, the correct transformation is a horizontal compression with \( k < 1 \).
- If the dotted-line parabola is wider than the solid-line parabola, the correct transformation is a horizontal stretch with \( k > 1 \).
Since we have a horizontal compression with a factor of \( k = 0.5 \) as one of the options that suggests the dotted-line graph is narrower, this would be the appropriate choice if observed that way.
**Final Answer: horizontal compression with a factor k = 0.5**.