In which of the following graphs do you recognize a proportional relationship?(1 point)%0D%0AResponses%0D%0A%0D%0A%0D%0AImage with alt text: An illustration shows a coordinate plane with the horizontal axis and vertical axis labeled x and y, but the unit increments are not marked. The origin is marked as 0. Four points are plotted on the graph. The unmarked coordinates of the plotted points are left parenthesis 2 comma 1 right parenthesis, left parenthesis 4 comma 2 right parenthesis, left parenthesis 6 comma 3 right parenthesis, and left parenthesis 8 comma 4 right parenthesis. A solid arrow begins at the origin and passes through the four points and beyond.%0D%0A%0D%0A%0D%0AImage with alt text: An illustration shows a coordinate plane with the horizontal axis and vertical axis labeled x and y, but the unit increments are not marked. The origin is marked as 0. An inverted V-shaped curve with an arrow at the end is plotted on the graph. The unmarked coordinates of the inverted V-shaped curve are left parenthesis 0 comma 3 right parenthesis, left parenthesis 6 comma 9 right parenthesis, and left parenthesis 10 comma 4 right parenthesis.%0D%0A%0D%0A%0D%0AImage with alt text: An illustration shows a coordinate plane with the horizontal axis and vertical axis labeled x and y, but the unit increments are not marked. The origin is marked as 0. An upward diagonal line is plotted on the graph. The unmarked coordinates of the upward line are left parenthesis 0 comma 3 right parenthesis, and left parenthesis 8 comma 10 right parenthesis. The line has an arrow at the end.
The graph that represents a proportional relationship is the third graph with the upward diagonal line.
To determine which of the graphs represents a proportional relationship, we need to understand what a proportional relationship is. In a proportional relationship, the ratio between the two variables remains constant. This means that for every increase or decrease in one variable, the other variable will also increase or decrease by the same ratio.
Now let's analyze each graph to see if they exhibit a proportional relationship:
1. Image with the arrow passing through four points: This graph represents a proportional relationship. As we move along the x-axis from 2 to 8 (increasing by a factor of 4), the corresponding points on the y-axis move from 1 to 4 (also increasing by a factor of 4). Thus, there is a constant ratio of 1:4 between the two variables, indicating a proportional relationship.
2. Image with an inverted V-shaped curve: This graph does not represent a proportional relationship. The points on the graph do not maintain a constant ratio. For example, the change from (0,3) to (6,9) on the x-axis increases by a factor of 6, but the corresponding change on the y-axis increases by a factor of 3. Therefore, the ratio is not constant, and it does not represent a proportional relationship.
3. Image with an upward diagonal line: This graph represents a proportional relationship. As we move along the x-axis from 0 to 8 (increasing by a factor of 8), the corresponding points on the y-axis move from 3 to 10 (increasing by a factor of 7). Thus, there is a constant ratio of 7:8 between the two variables, indicating a proportional relationship.
Therefore, the graphs that represent a proportional relationship are the first and third graphs.