Yes, every direct variation passes through the origin. This is not a coincidence or a special case. It is built into the equation itself. If y = kx, then plugging in x = 0 always gives y = 0, no matter what the constant k equals. The point (0, 0) is always on the graph.
Why the Origin Is Guaranteed
Direct variation means one variable is a constant multiple of another. The equation is y = kx, where k is called the constant of variation. Notice there’s no added or subtracted number at the end. Compare this to the general slope-intercept form, y = mx + b. In direct variation, the y-intercept (b) is always zero. That zero y-intercept is what forces the line through the origin.
The algebra is straightforward. Set x to 0 in y = kx, and you get y = k(0) = 0. This holds regardless of whether k is positive, negative, large, or small. As long as the relationship is truly a direct variation, the graph passes through (0, 0).
What the Graph Looks Like
A direct variation graph is a straight line with a single intercept at (0, 0). The slope of that line equals k, the constant of variation. If k is positive, the line rises from left to right. If k is negative, it falls. Either way, the line cuts through the origin and nowhere else on the axes.
This is one of the quickest ways to identify direct variation visually. If a line on a graph passes through the origin and is straight, it could represent a direct variation. If it crosses the y-axis at any point other than zero, it is not direct variation.
How to Verify Direct Variation in a Table
When you’re given a table of x and y values, you can check for direct variation by dividing y by x for every data pair. If the ratio y/x is the same constant for every row, and the table includes (or would include) the point (0, 0), you have direct variation. That constant ratio is k.
For example, if your table shows (2, 6), (4, 12), and (5, 15), dividing y by x gives 3 every time. The relationship is y = 3x, and when x = 0, y = 0. If even one row produces a different ratio, the relationship is not a direct variation.
Direct Variation vs. Partial Variation
A common source of confusion is the difference between direct and partial variation. Direct variation follows y = kx, where one variable is purely a constant multiple of the other. The graph goes through the origin, and the table always contains the value (0, 0).
Partial variation follows y = kx + b, where b is some nonzero number. Examples include equations like y = 2x + 1 or y = -3x + 100. The graph of a partial variation does not pass through the origin. It crosses the y-axis at b instead. So if you see a linear relationship with a nonzero y-intercept, you’re looking at partial variation, not direct variation.
This distinction matters in practice. A cell phone plan that charges a flat monthly fee plus a per-minute rate is partial variation: even with zero minutes used, you still owe the base fee. A job paying strictly by the hour with no base salary is direct variation: zero hours means zero pay.
Real-World Examples
Direct variation shows up whenever two quantities scale together from zero. If a car uses 5 liters of gasoline to drive 40 kilometers, the fuel consumption varies directly with distance. At zero kilometers driven, zero fuel is used, putting the starting point right at the origin.
Other everyday examples:
- Hourly wages. If you earn $15 per hour, your total pay is y = 15x. Work zero hours, earn zero dollars.
- Buying by weight. The total cost of strawberries is directly proportional to the number of pounds you purchase. Buy nothing, pay nothing.
- Distance at constant speed. The distance you travel varies directly with time when your speed stays the same. At time zero, distance is zero.
In each case, the relationship starts at the origin because there is no fixed starting value. The output is purely a multiple of the input.
What If k Is Zero?
The constant of variation k must be a nonzero value. If k were zero, the equation y = 0x would just give y = 0 for every input, which is a flat horizontal line along the x-axis. That technically passes through the origin, but it doesn’t describe a meaningful proportional relationship between two variables. By definition, k is nonzero, so the line always has some tilt to it.