Question
Rachel drew the two graphs below.
Graph R
On a coordinate plane, a line goes through points (0, 0) and (2, 1).
Graph S
On a coordinate plane, a line goes through points (0, 0) and (1, 2).
Rachel says that Graph R has a greater constant of variation than Graph S. Which statement explains whether Rachel is correct?
Rachel is correct; Graph S is nonproportional.
Rachel is correct; Graph R goes farther along the x-axis.
Rachel is not correct; both graphs go through the origin and have the same constant of variation.
Rachel is not correct; Graph S is steeper than Graph R.
Graph R
On a coordinate plane, a line goes through points (0, 0) and (2, 1).
Graph S
On a coordinate plane, a line goes through points (0, 0) and (1, 2).
Rachel says that Graph R has a greater constant of variation than Graph S. Which statement explains whether Rachel is correct?
Rachel is correct; Graph S is nonproportional.
Rachel is correct; Graph R goes farther along the x-axis.
Rachel is not correct; both graphs go through the origin and have the same constant of variation.
Rachel is not correct; Graph S is steeper than Graph R.
Answers
GPT-4o mini
To determine whether Rachel is correct regarding the constants of variation for Graph R and Graph S, we first need to understand what the constant of variation means. The constant of variation (or slope) is calculated as the change in y divided by the change in x (rise/run).
**For Graph R:**
- Points: (0, 0) and (2, 1)
- Slope = (1 - 0) / (2 - 0) = 1 / 2 = 0.5
**For Graph S:**
- Points: (0, 0) and (1, 2)
- Slope = (2 - 0) / (1 - 0) = 2 / 1 = 2
Now, we compare the slopes (constants of variation):
- Graph R has a slope of 0.5.
- Graph S has a slope of 2.
Since 0.5 < 2, this means that Graph S has a greater constant of variation than Graph R.
Now, let's evaluate the statements:
1. **Rachel is correct; Graph S is nonproportional.** (Incorrect; both graphs are proportional since they go through the origin.)
2. **Rachel is correct; Graph R goes farther along the x-axis.** (Incorrect; while Graph R extends farther along the x-axis, that does not affect the constants of variation.)
3. **Rachel is not correct; both graphs go through the origin and have the same constant of variation.** (Incorrect; they do go through the origin, but they do not have the same constant of variation.)
4. **Rachel is not correct; Graph S is steeper than Graph R.** (Correct; Graph S has a greater slope, which means it is steeper.)
Thus, the correct statement is: **Rachel is not correct; Graph S is steeper than Graph R.**
**For Graph R:**
- Points: (0, 0) and (2, 1)
- Slope = (1 - 0) / (2 - 0) = 1 / 2 = 0.5
**For Graph S:**
- Points: (0, 0) and (1, 2)
- Slope = (2 - 0) / (1 - 0) = 2 / 1 = 2
Now, we compare the slopes (constants of variation):
- Graph R has a slope of 0.5.
- Graph S has a slope of 2.
Since 0.5 < 2, this means that Graph S has a greater constant of variation than Graph R.
Now, let's evaluate the statements:
1. **Rachel is correct; Graph S is nonproportional.** (Incorrect; both graphs are proportional since they go through the origin.)
2. **Rachel is correct; Graph R goes farther along the x-axis.** (Incorrect; while Graph R extends farther along the x-axis, that does not affect the constants of variation.)
3. **Rachel is not correct; both graphs go through the origin and have the same constant of variation.** (Incorrect; they do go through the origin, but they do not have the same constant of variation.)
4. **Rachel is not correct; Graph S is steeper than Graph R.** (Correct; Graph S has a greater slope, which means it is steeper.)
Thus, the correct statement is: **Rachel is not correct; Graph S is steeper than Graph R.**