Scale Factor in a Proportional Relationship

In a proportional relationship, two quantities always have the same ratio between them. That ratio is the constant of proportionality, k, and the relationship is y = kx. Because k never changes, multiplying x by any factor forces y to be multiplied by that exact same factor. Triple the notebooks, triple the cost, because the cost per notebook stays fixed. This is why scale factor problems are really just ratio problems: find k once, then apply it wherever you need it, or find the ratio between two amounts of x and apply that same ratio to y.

This constant ratio is also what makes the graph a straight line through the origin. Every point on the graph satisfies y = kx, and when x is 0, y must be 0 too, since k times 0 is 0. There is no leftover amount, no starting value. Compare that to y = f + rx, a line with a flat fee f. That graph is still straight, and y still increases steadily with x, but when x is 0, y equals f, not 0. Doubling x no longer doubles y because the fixed fee refuses to double along with it. Linear only requires a constant rate of change; proportional requires that constant rate to pass through the origin with nothing added.

This idea connects directly to unit rates and derived units elsewhere in this unit. The constant k in y = kx is the unit rate, dollars per notebook, miles per hour, grams per milliliter. If you flip which variable is x and which is y, the new constant of proportionality is the reciprocal of the old one, since the two quantities' roles in the ratio swap.