Question
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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?
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To determine which friend's graph represents a proportional relationship, we need to look for a graph where the ratio of levels passed to time played remains constant.
Let's examine each friend's graph.
Friend 1: The points on Friend 1's graph are (0, 2), (2, 5), and (4, 8).
Let's calculate the ratios of levels passed and time played for Friend 1:
- At 0 minutes, Friend 1 passed 2 levels.
- At 2 minutes, Friend 1 passed 5 levels.
- At 4 minutes, Friend 1 passed 8 levels.
The ratios are as follows:
- Between 0 and 2 minutes: 5/2 = 2.5
- Between 2 and 4 minutes: 8/5 = 1.6
The ratio is not constant for Friend 1, so their graph does not represent a proportional relationship.
Friend 2: The points on Friend 2's graph are (0, 0), (2, 3), (4, 7), and (5, 8).
Let's calculate the ratios of levels passed and time played for Friend 2:
- At 0 minutes, Friend 2 passed 0 levels.
- At 2 minutes, Friend 2 passed 3 levels.
- At 4 minutes, Friend 2 passed 7 levels.
- At 5 minutes, Friend 2 passed 8 levels.
The ratios are as follows:
- Between 0 and 2 minutes: 3/2 = 1.5
- Between 2 and 4 minutes: 7/2 = 3.5
- Between 4 and 5 minutes: 8/1 = 8
The ratio is not constant for Friend 2, so their graph does not represent a proportional relationship.
Friend 3: The points on Friend 3's graph are (0, 0), (2, 2), (4, 4), and (7, 7).
Let's calculate the ratios of levels passed and time played for Friend 3:
- At 0 minutes, Friend 3 passed 0 levels.
- At 2 minutes, Friend 3 passed 2 levels.
- At 4 minutes, Friend 3 passed 4 levels.
- At 7 minutes, Friend 3 passed 7 levels.
The ratios are as follows:
- Between 0 and 2 minutes: 2/2 = 1
- Between 2 and 4 minutes: 4/2 = 2
- Between 4 and 7 minutes: 7/3 = 2.33
The ratio is not constant for Friend 3, so their graph does not represent a proportional relationship.
None of the friends' graphs represent a proportional relationship, as the ratios of levels passed to time played are not constant for any of them.
To determine which friend's graph represents a proportional relationship, we need to understand what a proportional relationship means. In a proportional relationship, as one variable increases or decreases, the other variable also increases or decreases by a constant rate.
Looking at the graph, we can see that Friend 1's graph does not represent a proportional relationship because the rate of change is not constant. For example, from 0 to 2 minutes, the level passed increases by 3, but from 2 to 4 minutes, the level passed increases by 3 as well. This indicates a constant rate of change.
Friend 2's graph also does not represent a proportional relationship because the rate of change is not constant. From 0 to 2 minutes, the level passed increases by 3, and from 2 to 4 minutes, the level passed increases by 4. This indicates a changing rate of change.
However, Friend 3's graph does represent a proportional relationship. From 0 to 2 minutes, the level passed increases by 2, and from 2 to 4 minutes, the level passed also increases by 2. This indicates a constant rate of change.
Therefore, Friend 3's graph represents a proportional relationship.
To determine which friend's graph represents a proportional relationship, we need to look for a line that passes through the origin (0,0) and has a constant rate of change.
From the given information, line 2 passes through the origin (0,0) and has a constant rate of change. Thus, Friend 2's graph represents a proportional relationship.