A graph represents a function with direct variation if it is a straight line that passes through the origin (0, 0). Those are the only two visual requirements. If a graph is curved, or if a straight line crosses the y-axis at any point other than zero, it does not show direct variation.
What Direct Variation Looks Like on a Graph
Direct variation follows the equation y = kx, where k is a constant number. This is really just the slope-intercept form y = mx + b with b set to zero. Because b is zero, the line has no vertical shift, which is why it must pass through the origin. The value of k determines how steep the line is and whether it angles upward or downward. A positive k means the line rises from left to right; a negative k means it falls.
So when you’re looking at multiple graphs and need to pick the one showing direct variation, check two things in order:
- Is it a straight line? Curves, parabolas, and hyperbolas are not direct variation.
- Does it pass through (0, 0)? A straight line that crosses the y-axis above or below zero is a linear function, but it is not direct variation.
Only a graph that satisfies both conditions represents direct variation.
Why the Origin Matters
The origin requirement comes from the underlying math. In direct variation, the ratio y/x always equals the same constant, k. If x is 0, then y must also be 0 for that ratio to hold. If a line crossed the y-axis at, say, 3, then when x = 0, y would equal 3 instead of 0. That breaks the proportional relationship. The two quantities would no longer scale by the same factor, and y/x would change from point to point rather than staying constant.
This is the key difference between direct variation and a general linear function. Every direct variation is linear, but not every linear function is a direct variation. The line y = 2x + 5 is perfectly linear, yet it is not direct variation because of that +5 shifting it away from the origin.
How to Verify With Points on the Graph
If the graph provides specific coordinate points, you can double-check by dividing y by x at each point. Pick any two or three points along the line and calculate y/x for each one. If every ratio gives you the same number, the graph shows direct variation and that number is your constant of variation, k.
For example, if a line passes through (2, 6) and (5, 15), dividing gives you 6/2 = 3 and 15/5 = 3. The ratio is constant at 3, confirming direct variation with k = 3. If instead the points were (2, 7) and (5, 15), the ratios would be 3.5 and 3, which don’t match, so the relationship is not a direct variation.
This ratio test is especially useful when a graph passes close to the origin but not exactly through it. Eyeballing can be tricky, so the numbers give you a definitive answer.
Graphs That Look Similar but Aren’t Direct Variation
A few common graph types can fool you if you’re scanning quickly:
- A line through the origin that curves: Some functions like y = x² also pass through (0, 0), but they produce a curve, not a straight line. Direct variation requires both conditions, not just one.
- A straight line that doesn’t touch the origin: A line with a y-intercept of 1 or -2 is a regular linear function (y = mx + b where b ≠ 0). It may look almost proportional, but the constant shift disqualifies it.
- An inverse variation curve: Inverse variation (y = k/x) produces a hyperbola where y gets smaller as x gets larger. The shape curves and never passes through the origin, making it visually distinct from direct variation once you know what to look for.
The Constant of Variation as Slope
The constant k in y = kx is identical to the slope of the line. You can read it directly from the graph using rise over run, or calculate it by picking any point on the line (other than the origin) and dividing the y-coordinate by the x-coordinate. Both methods give you the same value.
If the line passes through (4, 12), for instance, k = 12/4 = 3, and the equation is y = 3x. This means for every 1 unit increase in x, y increases by 3. The relationship scales perfectly: double x and y doubles, triple x and y triples. That proportional scaling is what makes direct variation useful for modeling real-world relationships like speed and distance, or price and quantity.