In Grade 6 you graphed ratio tables and saw the points line up toward $(0, 0)$. That picture is the test for proportional relationships, and it has two parts.
Key idea
A graph shows a proportional relationship when it is a straight line (or points on one) and it passes through the origin, $(0, 0)$.
Why it works
A proportional relationship is $y = kx$. Put in $x = 0$: $y = k \cdot 0 = 0$, so $(0, 0)$ is always on the graph. And each step of 1 to the right adds the same amount, $k$, to $y$, so the graph never bends. A line like $y = 2x + 3$ is straight, but it starts at $(0, 3)$, and the extra 3 breaks the "same ratio" rule.
Worked examples
Worked example: Points through the origin
A graph shows the points $(0, 0)$, $(2, 3)$, $(4, 6)$ and $(6, 9)$ on one straight line. Is it proportional?
- The points line up, and the line passes through $(0, 0)$.
- Check the ratios: $\frac{3}{2} = \frac{6}{4} = \frac{9}{6} = 1.5$. They match, so $y = 1.5x$.
Answer: Yes, proportional, with $k = 1.5$
Worked example: Straight, but misses the origin
A straight line passes through $(0, 2)$ and $(3, 8)$. Is it proportional?
- At $x = 0$ the line is at $y = 2$, not $0$.
- So it is not proportional. Check the ratio at $(3, 8)$: $\frac{8}{3}$. At a third point, $(6, 14)$, it is $\frac{14}{6} = \frac{7}{3}$. The ratios differ.
Answer: No
Worked example: Through the origin, but curved
A square's area $A$ is graphed against its side $s$: the points $(0, 0)$, $(1, 1)$, $(2, 4)$, $(3, 9)$. Is area proportional to side length?
- The graph starts at the origin, but it curves upward.
- $\frac{1}{1} = 1$, $\frac{4}{2} = 2$, $\frac{9}{3} = 3$: the ratio grows, so it is not proportional.
Answer: No
Common mistake
Accepting any straight line. A proportional graph must pass through $(0, 0)$, because zero of one quantity goes with zero of the other. Check by looking where the line crosses the $y$-axis: if it is not at $0$, the relationship is not proportional.