Which Graph Represents A Direct Variation

The Unmistakable Graphical Signature of Direct Variation

Imagine you are planning a road trip. The total distance you travel is directly proportional to your speed and the time you spend driving. If you drive twice as long at a constant speed, you cover twice the distance. This intuitive relationship is a classic example of direct variation. In mathematics, when we say y varies directly as x, we mean that as x increases, y increases at a constant rate, and crucially, if x is zero, y must also be zero. This fundamental constraint gives the relationship its most powerful visual identifier on a coordinate plane. The graph of a direct variation is always a straight line that passes directly through the origin, the point (0,0). Understanding this graphical signature is key to distinguishing direct variation from other linear and nonlinear relationships.

What Exactly is Direct Variation?

At its core, a direct variation is a specific type of proportional relationship between two variables. It is defined by an equation in the form: y = kx Here, k is a non-zero constant called the constant of variation or the constant of proportionality. This constant k represents the fixed ratio between y and x (k = y/x for any non-zero x). The equation is deliberately simple: it contains no added constants or other terms. This algebraic form has immediate and important graphical consequences. Since there is no y-intercept term (often denoted as b in the slope-intercept form y = mx + b), the line must intersect the y-axis at zero. When x = 0, substituting into y = kx yields y = k(0) = 0. Therefore, the point (0,0) is always a solution and must lie on the graph.

The Graphical Hallmark: A Line Through the Origin

When you plot the points that satisfy y = kx on a Cartesian plane, they will always align to form a straight line. The steepness of this line is determined by the value of k.

• If k is positive, the line slopes upward from left to right.
• If k is negative, the line slopes downward from left to right.
• The magnitude of k (its absolute value) is the slope of the line. A larger |k| means a steeper line.

The non-negotiable feature is that this line must pass through the origin (0,0). This is the single most reliable visual test. You can think of it logically: if there is no "starting amount" or fixed fee (the b in y=mx+b), then when the input x is zero, the output y has to be zero. A line that crosses the y-axis above or below the origin represents a linear relationship, but not a direct variation.

How to Identify Direct