A linear function shows direct variation when it has the form y = kx, where k is a nonzero constant. The key requirement is that the y-intercept equals zero, meaning there is no added or subtracted constant in the equation. If you’re looking at a set of answer choices, the one with no “+ b” or “- b” term is the direct variation.
What Makes Direct Variation Different
Every linear function can be written in slope-intercept form: y = mx + b. Direct variation is a special case where b (the y-intercept) is exactly zero. That simplifies the equation to y = kx, where k is called the constant of variation or constant of proportionality. It plays the same role as the slope.
So if you’re comparing functions like y = 3x + 2, y = -5x, and y = x – 7, only y = -5x is a direct variation. The other two have a nonzero constant term that disqualifies them. A linear function with a nonzero b value is sometimes called a partial variation, since y depends partly on x and partly on a fixed amount.
The Graph Always Passes Through the Origin
On a graph, direct variation is easy to spot: the line passes straight through (0, 0). That’s because when x = 0, y must also equal 0 (since y = k times 0 = 0, regardless of what k is). If a line crosses the y-axis anywhere other than the origin, it is not a direct variation.
The slope of that line equals k, the constant of variation. A steeper line means a larger constant. If k is positive, the line rises from left to right. If k is negative, it falls.
How to Check a Table of Values
When you’re given a table instead of an equation, divide every y value by its corresponding x value. If y/x produces the same number for every row in the table, the relationship is a direct variation, and that repeated ratio is your constant k.
For example, if a table shows (2, 6), (4, 12), and (5, 15), dividing y by x gives 3 each time. The function is y = 3x, a direct variation with k = 3. If even one pair produces a different ratio, it’s not direct variation.
Finding k From a Single Point
If you already know a relationship is a direct variation, you only need one pair of x and y values to find k. Rearrange y = kx to k = y/x. Suppose you know that 40 boxes require 160 iron blocks, and the relationship is directly proportional. Then k = 160 ÷ 40 = 4, and the function is y = 4x. You can now predict y for any value of x.
Real-World Examples
Direct variation shows up whenever doubling the input exactly doubles the output, with no base amount involved. A few common cases:
- Hourly wages: If you earn $15 per hour, your pay equals 15 times the number of hours worked. Zero hours means zero pay, and the relationship is y = 15x.
- Unit pricing: Strawberries at $2.50 per pound cost exactly $2.50 times however many pounds you buy. Five pounds costs $12.50; two pounds costs $5.00.
- Distance at constant speed: Traveling at 60 miles per hour, your distance equals 60 times the number of hours driven.
In every case, the pattern is the same: no starting fee, no flat charge, no initial value. The output is purely the constant times the input. The moment a situation includes a base cost (like a $5 delivery fee plus a per-item charge), it becomes a partial variation with a nonzero b, and the graph no longer passes through the origin.
Quick Checklist for Multiple-Choice Problems
When a problem asks you to identify which function shows direct variation, run through three checks. First, the equation should be in the form y = kx with no constant added or subtracted. Second, if a graph is shown, the line must pass through (0, 0). Third, if a table is given, every y/x ratio should be identical. Any function that fails one of these tests is a standard linear function, not a direct variation.