Question

select the graphs that show a proportional relationship between x and y.

Explanation:

Step1: Recall Proportional Relationship

A proportional relationship between \(x\) and \(y\) is represented by \(y = kx\) (where \(k\) is a constant), and its graph is a straight line passing through the origin \((0,0)\) with a constant slope (constant rate of change).

Step2: Analyze First Graph

The first graph has a straight line starting at \((0,0)\) and has a consistent slope (for example, when \(x = 2\), \(y = 2\); \(x = 4\), \(y = 4\)? Wait, no, looking at the grid, when \(x = 2\), \(y = 2\)? Wait, no, the first graph: when \(x = 1\), \(y = 1\)? Wait, the first graph's line passes through the origin, and has a constant slope (since it's a straight line through the origin), so it represents a proportional relationship.

Step3: Analyze Second Graph

The second graph: the line starts at the origin, but let's check the slope. For example, when \(x = 10\), \(y\) is around 2.5? Wait, no, the second graph's line: let's check if it's a straight line through the origin with constant slope. Wait, the second graph's line is straight and passes through the origin? Wait, the second graph's line starts at \((0,0)\) and is a straight line, so it also represents a proportional relationship? Wait, no, wait the first graph: when \(x = 2\), \(y = 2\) (if the grid is 1 unit per square). Wait, the first graph: when \(x = 1\), \(y = 1\); \(x = 2\), \(y = 2\); so slope \(k = 1\). The second graph: when \(x = 10\), \(y\) is 2.5? Wait, no, maybe I misread. Wait the first graph: the line goes from (0,0) to (8,10)? Wait no, the first graph's y-axis: 0,2,4,6,8,10. x-axis: 0,2,4,6,8,10. So when x=8, y=10? Wait no, the first graph's line: at x=2, y=2? No, at x=2, y=2? Wait, no, the first graph: the line passes through (0,0) and (8,10)? Wait, no, the first graph's line: when x=1, y=1? Wait, maybe the first graph has a slope of 1 (since it's a straight line through the origin), and the second graph: let's check the slope. For the second graph, when x=10, y is 2.5? Wait, no, the second graph's line: let's see, the second graph's line is straight and passes through the origin, so it's a proportional relationship. Wait, but wait, the problem is to select the graphs that show a proportional relationship. So both? Wait, but maybe I made a mistake. Wait, a proportional relationship graph is a straight line through the origin (since \(y = kx\) implies when \(x = 0\), \(y = 0\)) and has a constant slope (since it's linear). So both graphs are straight lines through the origin, so they represent proportional relationships. Wait, but the first graph: when x=2, y=2 (if the grid is 1 unit per square), so slope 1. The second graph: when x=10, y=2.5? Wait, no, the second graph's line: let's check the coordinates. Wait, the first graph: the blue line (first graph) is steeper, the second is less steep, but both are straight lines through the origin, so both are proportional. Wait, but maybe the second graph is not? Wait, no, a proportional relationship is \(y = kx\), which is a linear function with \(b = 0\) (y-intercept 0) and constant slope \(k\). So any straight line through the origin is a proportional relationship. So both graphs (the left one and the right one) show proportional relationships? Wait, but the first graph is checked, the second is also checked? Wait, the original problem's image: the first graph (left) has a blue check, the second (right) has a light blue check. So both are proportional. Wait, but let's confirm: a proportional relationship graph must be a straight line passing through the origin (0,0) because the equation is \(y = kx\), so when \(x = 0\), \(y = 0\), and it's a linear function with constant slope. So both graphs are straight lines through the origin, so they represent proportional relationships.

Answer:

Both the left graph (first graph) and the right graph (second graph) show a proportional relationship between \(x\) and \(y\) (since they are straight lines passing through the origin, indicating \(y = kx\) with constant \(k\)).