Question
A graph with the x-axis representing time in minutes ranging from 0 to 10 in increments of 1, and the y-axis representing levels passed ranging from 0 to 10 in increments of 1 shows three lines labeled 1, 2 and 3. Line 1 passes through the following points: left parenthesis 0 comma 2 right parenthesis; left parenthesis 2 comma 5 right parenthesis; and left parenthesis 4 comma 8 right parenthesis. Line 2 passes through the following points: left parenthesis 0 comma 0 right parenthesis; left parenthesis 2 comma 3 right parenthesis; left parenthesis 4 comma 7 right parenthesis; and left parenthesis 5 comma 8 right parenthesis. Line 3 passes through the following points: left parenthesis 0 comma 0 right parenthesis; left parenthesis 2 comma 2 right parenthesis; left parenthesis 4 comma 4 right parenthesis; and left parenthesis 7 comma 7 right parenthesis.
Three friends are playing video games. The graph shows the number of levels passed based on amount of time played. Which friend’s graph (1, 2, or 3) is a proportional relationship?
Three friends are playing video games. The graph shows the number of levels passed based on amount of time played. Which friend’s graph (1, 2, or 3) is a proportional relationship?
Answers
GPT-4o mini
In order to determine which friend's graph represents a proportional relationship, we need to look for a constant rate of change that starts from the origin (0,0). A proportional relationship means that as time increases, the levels passed increase in a constant ratio.
Let's analyze each line based on the provided points:
**Line 1**:
- Points: (0, 2), (2, 5), (4, 8)
- The line does not pass through (0,0), which means it is not proportional.
**Line 2**:
- Points: (0, 0), (2, 3), (4, 7), (5, 8)
- This line starts at (0,0). We can calculate the rate of change:
- From (0, 0) to (2, 3): Change in levels = 3 - 0 = 3; Change in time = 2 - 0 = 2 → Rate = 3/2
- From (2, 3) to (4, 7): Change in levels = 7 - 3 = 4; Change in time = 4 - 2 = 2 → Rate = 4/2 = 2
- From (4, 7) to (5, 8): Change in levels = 8 - 7 = 1; Change in time = 5 - 4 = 1 → Rate = 1/1 = 1
This line has varying rates of change, so it is not proportional.
**Line 3**:
- Points: (0, 0), (2, 2), (4, 4), (7, 7)
- This line starts at (0,0) and the rate of change appears to be constant:
- From (0, 0) to (2, 2): Change in levels = 2 - 0 = 2; Change in time = 2 - 0 = 2 → Rate = 2/2 = 1
- From (2, 2) to (4, 4): Change in levels = 4 - 2 = 2; Change in time = 4 - 2 = 2 → Rate = 2/2 = 1
- From (4, 4) to (7, 7): Change in levels = 7 - 4 = 3; Change in time = 7 - 4 = 3 → Rate = 3/3 = 1
The rates of change are consistent, and the line passes through (0,0), indicating it is proportional.
Therefore, **Line 3** represents a proportional relationship.
Let's analyze each line based on the provided points:
**Line 1**:
- Points: (0, 2), (2, 5), (4, 8)
- The line does not pass through (0,0), which means it is not proportional.
**Line 2**:
- Points: (0, 0), (2, 3), (4, 7), (5, 8)
- This line starts at (0,0). We can calculate the rate of change:
- From (0, 0) to (2, 3): Change in levels = 3 - 0 = 3; Change in time = 2 - 0 = 2 → Rate = 3/2
- From (2, 3) to (4, 7): Change in levels = 7 - 3 = 4; Change in time = 4 - 2 = 2 → Rate = 4/2 = 2
- From (4, 7) to (5, 8): Change in levels = 8 - 7 = 1; Change in time = 5 - 4 = 1 → Rate = 1/1 = 1
This line has varying rates of change, so it is not proportional.
**Line 3**:
- Points: (0, 0), (2, 2), (4, 4), (7, 7)
- This line starts at (0,0) and the rate of change appears to be constant:
- From (0, 0) to (2, 2): Change in levels = 2 - 0 = 2; Change in time = 2 - 0 = 2 → Rate = 2/2 = 1
- From (2, 2) to (4, 4): Change in levels = 4 - 2 = 2; Change in time = 4 - 2 = 2 → Rate = 2/2 = 1
- From (4, 4) to (7, 7): Change in levels = 7 - 4 = 3; Change in time = 7 - 4 = 3 → Rate = 3/3 = 1
The rates of change are consistent, and the line passes through (0,0), indicating it is proportional.
Therefore, **Line 3** represents a proportional relationship.