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 a straight line that passes through the origin \((0,0)\) and has a constant ratio \(\frac{y}{x}=k\) (constant of proportionality).

Step2: Analyze First Graph

The first graph has a line starting at \((0,0)\) and going up with a constant slope. For example, when \( x = 2 \), \( y = 3 \) (since from \( (0,0) \) to \( (2,3) \), the slope is \(\frac{3}{2}\)). Checking other points: at \( x = 4 \), \( y = 6 \) (\(\frac{6}{4}=\frac{3}{2}\)), at \( x = 6 \), \( y = 9 \) (\(\frac{9}{6}=\frac{3}{2}\)), at \( x = 7 \), \( y = 10.5 \)? Wait, no, looking at the grid, the first graph's line at \( x = 7 \) (since the arrow is at \( x = 7 \)? Wait, no, the x-axis goes to 10, and the first graph's line ends at \( x = 7 \) (since the arrow is at \( x = 7 \), \( y = 10.5 \)? Wait, no, maybe I misread. Wait, the first graph: when \( x = 1 \), \( y = 1.5 \)? Wait, no, let's check the grid. The first graph: from (0,0), when x=2, y=3 (since y-axis: 0,2,4,6,8,10; x-axis: 0,2,4,6,8,10. So each grid is 1 unit. So first graph: at x=2, y=3? Wait, no, the first graph's line: when x=1, y=1.5? No, maybe the first graph: when x=2, y=3 (since from (0,0) to (2,3), then (4,6), (6,9), (7, 10.5)? Wait, but the key is it passes through (0,0) and has constant slope.

Step3: Analyze Second Graph

The second graph: line starts at (0,0). Let's check points. At x=2, y=2? Wait, no, x=2, y=2? Wait, no, the second graph: when x=2, y=2? Wait, no, looking at the grid, the second graph's line: at x=2, y=2? Wait, no, x=4, y=4? Wait, no, x=2, y=2? Wait, no, let's see: the second graph's line: when x=2, y=2? Wait, no, the second graph's line: at x=2, y=2? Wait, no, x=4, y=4? Wait, no, the second graph: from (0,0), the slope is 1? Wait, no, when x=2, y=2? Wait, no, the second graph: at x=2, y=2? Wait, no, x=4, y=4? Wait, no, the second graph's line: when x=2, y=2? Wait, no, let's check the coordinates. Wait, the second graph: when x=2, y=2? No, x=2, y=2? Wait, no, the second graph's line: at x=2, y=2? Wait, no, x=4, y=4? Wait, no, the second graph: from (0,0), the line goes to (10,9)? Wait, no, the second graph's arrow is at x=10, y=9? Wait, no, the second graph: when x=10, y=9? Wait, no, maybe I made a mistake. Wait, the key is both graphs pass through (0,0) and have constant slope. Wait, but the first graph: when x=2, y=3 (slope 3/2), second graph: when x=2, y=2 (slope 1)? Wait, no, looking at the grid again. Wait, the first graph: the line is steeper. The second graph: line is less steep. But both pass through (0,0). Wait, but the problem is to select the graphs that show proportional relationship. A proportional relationship graph is a straight line through the origin. So both graphs are straight lines through (0,0), so both show proportional relationships? Wait, but maybe I misread the first graph. Wait, the first graph: does it pass through (0,0)? Yes. The second graph: passes through (0,0). So both are proportional. Wait, but let's check the slopes. First graph: slope = rise/run. From (0,0) to (2,3): slope 3/2. From (2,3) to (4,6): slope 3/2. So constant. Second graph: from (0,0) to (2,2): slope 1. From (2,2) to (4,4): slope 1. So constant. So both graphs are proportional. Wait, but the original problem: the first graph has a checkmark, the second too? Wait, the user's image shows two graphs, both with checkmarks? Wait, no, the user's image: first graph (left) has a checkmark, second (right) too? Wait, the problem is to select the graphs that show proportional relationship. Since both are straight lines through the origin, both are proportional. Wait, but maybe I made a mistake. Wait, the first graph: when x=1, y=1.5 (slope 1.5), second graph: slope 1. But both pass through (0,0), so both are proportional.

Answer:

Both the Left Graph and the Right Graph (the two given graphs) show a proportional relationship between \( x \) and \( y \) because they are straight lines passing through the origin \((0,0)\) with constant slopes (constant \(\frac{y}{x}\) ratio).