Question

1. proportional graph (two groups)

which graph models a proportional relationship?
answer: _______

Explanation:

Step1: Recall Proportional Relationship Rule

A proportional relationship graph is a straight line passing through the origin \((0,0)\), and \(y = kx\) (constant of proportionality \(k\)).

Step2: Analyze Graph A

Graph A has scattered points, not a straight line. So it's not proportional.

Step3: Analyze Graph B

Graph B starts at \((0,5)\), not the origin. So it's a linear but non - proportional (has a \(y\) - intercept \(b = 5\), \(y=mx + 5\)) relationship.

Step4: Analyze Graph C

Graph C: Let's check the slope. At \(x = 10\), \(y = 5\); at \(x = 20\), \(y = 10\). The slope \(k=\frac{y}{x}=\frac{5}{10}=\frac{10}{20}=0.5\). But wait, when \(x = 0\), \(y = 0\), but let's check another point. Wait, maybe miscalculation. Wait, no, let's check Graph D.

Step5: Analyze Graph D

Graph D passes through the origin \((0,0)\). Let's check the ratio \(\frac{y}{x}\). At \(x = 10\), \(y = 10\) (wait, no, looking at the grid, when \(x = 5\), let's see the point? Wait, the line in D passes through \((0,0)\) and when \(x = 10\), \(y = 10\)? Wait, no, the grid: x - axis from 0 to 20, y - axis 0 to 20. Graph D: when \(x = 5\), y seems to be 5? Wait, no, the key is it passes through the origin. Also, for proportional relationship, \(\frac{y}{x}\) should be constant. Let's take two points on D. Let's say when \(x = 10\), \(y = 10\) (assuming), and when \(x = 20\), \(y = 20\). So \(\frac{y}{x}=1\), constant. And it passes through \((0,0)\). Graph C: when \(x = 10\), \(y = 5\); \(x = 20\), \(y = 10\), so \(\frac{y}{x}=\frac{5}{10}=\frac{10}{20}=0.5\), but wait, does it pass through origin? Yes, but wait the line in C: when \(x = 5\), what's \(y\)? If \(x = 5\), \(y=\frac{5\times5}{10}=2.5\)? But the graph C's line at \(x = 5\) seems to be at \(y = 2.5\)? But Graph D: when \(x = 5\), \(y = 5\); \(x = 10\), \(y = 10\); \(x = 20\), \(y = 20\). So the ratio \(\frac{y}{x}=1\) (constant) and passes through origin. Wait, maybe I made a mistake with C. Wait, no, the problem is to find which graph models proportional relationship. Proportional relationship must pass through origin and have constant \(y/x\). Graph D passes through origin, and the line has a constant ratio (since it's a straight line through origin). Graph C: when \(x = 10\), \(y = 5\); \(x = 20\), \(y = 10\), ratio 0.5. But Graph D: when \(x = 10\), \(y = 10\) (assuming the grid), ratio 1. Wait, maybe the correct graph is D? Wait, no, let's re - check. Wait, the key is: proportional relationship \(y=kx\), so line through \((0,0)\) and constant slope. Graph D: passes through \((0,0)\), and the line is straight with slope 1 (since \(y = x\) would pass through (0,0), (10,10), (20,20)). Graph C: slope is 0.5, but does it pass through origin? Yes, but maybe the points? Wait, the problem is which graph models proportional relationship. The correct one is D? Wait, no, wait Graph C: when \(x = 0\), \(y = 0\), and the line is straight. But Graph D: also straight through origin. Wait, maybe I messed up. Wait, let's check the options again. Wait, the user's graph: Graph A: scattered points, Graph B: starts at (0,5), Graph C: starts at (0,0), Graph D: starts at (0,0). Wait, maybe the difference is in the ratio. Let's calculate the slope for Graph C: from (0,0) to (10,5): slope \(k=\frac{5 - 0}{10 - 0}=0.5\). For Graph D: from (0,0) to (10,10): slope \(k = 1\). But both are proportional? Wait, no, maybe the graph D has a point that aligns. Wait, maybe the correct answer is D? Wait, no, maybe I made a mistake. Wait, the problem is "which graph models a proportional relationship". Proportional relationship requires \(y=kx\), so line through (0,0) and constant \(k\). So both C and D pass through (0,0). Wait, but maybe the graph D has a point that is on the line. Wait, looking at the graph, Graph D: the line passes through (0,0) and when x = 10, y = 10, x = 20, y = 20. Graph C: when x = 10, y = 5, x = 20, y = 10. So both are proportional, but maybe the intended answer is D? Wait, no, maybe I misread the graphs. Wait, the user's image: Graph D has a line from (0,0) going up, and a point? No, Graph D's line is straight through origin. So the correct graph is D (or C? Wait, no, let's check the definition again. A proportional relationship is a linear relationship where \(y = kx\), so the graph is a straight line through the origin. So both C and D are straight lines through origin, but maybe the ratio in D is 1 (y = x) and in C is y=0.5x. But the question is which one models proportional relationship. Wait, maybe the graph D is the one where the line passes through the origin and the points (if any) lie on the line. Wait, in Graph D, there's a point? No, Graph D's line is straight through origin. So the answer should be Graph D? Wait, no, maybe I made a mistake. Wait, let's check again.

Wait, Graph A: not a line. Graph B: y - intercept 5, so not proportional. Graph C: passes through (0,0), line. Graph D: passes through (0,0), line. But maybe the key is that in a proportional relationship, the ratio \(\frac{y}{x}\) is constant for all points. Let's take Graph D: when x = 5, y = 5 (assuming), x = 10, y = 10, x = 20, y = 20. So \(\frac{y}{x}=1\) (constant). Graph C: x = 10, y = 5 (\(\frac{5}{10}=0.5\)), x = 20, y = 10 (\(\frac{10}{20}=0.5\)), so also constant. But maybe the graph D is the one where the line is such that y = x, which is a proportional relationship. Wait, maybe the intended answer is D. Because when x and y are equal, it's a clear proportional relationship. So the graph that models a proportional relationship is D (the right - most bottom graph, Graph D).

Answer:

Graph D (the graph in the bottom right corner)