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 of the form \(y = kx\) (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 Left Graph

The left graph is a straight line passing through \((0,0)\). Let's check the slope. For example, when \(x = 10\), \(y = 2\), so the slope \(k=\frac{y}{x}=\frac{2}{10}=0.2\) (constant). So it follows \(y = 0.2x\), a proportional relationship.

Step3: Analyze Right Graph

The right graph is a straight line passing through \((0,0)\). Let's check the slope. When \(x = 2\), \(y = 6\) (wait, no, let's check coordinates. Wait, at \(x = 1\), maybe? Wait, looking at the grid, when \(x = 2\), \(y = 6\)? Wait, no, the line goes from \((0,0)\) to \((3,10)\)? Wait, no, the right graph: when \(x = 1\), \(y = 2.5\)? Wait, no, let's see the grid. Wait, the right graph: the line passes through \((0,0)\) and has a constant slope. Let's take two points. At \(x = 2\), \(y = 6\)? Wait, no, the grid lines: each square is 1 unit. Wait, the right graph: when \(x = 1\), \(y = 2.5\)? No, maybe better to check if it's a straight line through origin. Both left and right graphs are straight lines through \((0,0)\), so both show proportional relationships? Wait, wait, the left graph: when \(x = 10\), \(y = 2\); slope \(2/10 = 0.2\). The right graph: when \(x = 4\), \(y = 10\)? Wait, no, the arrow is at \(x = 3\) or \(x = 4\)? Wait, the right graph's line: from \((0,0)\) to, say, \(x = 2\), \(y = 6\)? Wait, no, the y-axis is 10 at top, x-axis 10 at right. Wait, maybe the right graph: when \(x = 1\), \(y = 2.5\)? No, perhaps I misread. Wait, the key is: proportional relationship graph is a straight line through the origin. Both graphs are straight lines through \((0,0)\), so both show proportional relationships? Wait, but let's check the left graph: it's a straight line from (0,0) to (10,2), so slope 2/10 = 0.2, constant. The right graph: straight line from (0,0) to (4,10)? Wait, no, the arrow is at x=3 or x=4? Wait, the right graph's line: when x=2, y=6? So slope 6/2=3, constant. So both are straight lines through origin, so both show proportional relationships. Wait, but maybe the left graph is a straight line, right graph too. So the graphs that show proportional relationship are both the left graph and the right graph? Wait, the question is "Select the graphs that show a proportional relationship between x and y." So both graphs (left and right) are straight lines passing through the origin, hence proportional.

Answer:

Both the left - hand graph and the right - hand graph (the graph on the Left and the graph on the Right) show a proportional relationship between \(x\) and \(y\) since their graphs are straight lines passing through the origin \((0,0)\) (a key characteristic of a proportional relationship \(y = kx\) where \(k\) is the constant of proportionality).