Intercepts (x- and y-intercepts)

Intercepts are the points where a curve meets the coordinate axes. Because every point on the x-axis has a y-coordinate of zero, and every point on the y-axis has an x-coordinate of zero, finding an intercept is always the same two-step move: set the other variable to zero and solve.

To find the y-intercept, substitute x = 0 into the equation. To find the x-intercept, substitute y = 0 and solve for x. Take y = 2x − 6. Setting x = 0 gives y = −6, so the y-intercept is (0, −6). Setting y = 0 gives 0 = 2x − 6, so x = 3 and the x-intercept is (3, 0). The same method works for any equation, not just lines: for the parabola y = x² − 4, the y-intercept is (0, −4) and the x-intercepts are (2, 0) and (−2, 0).

Intercepts are shortcuts as much as they are features. A line with two distinct intercepts is fully determined by them, which makes them the fastest way to sketch it. The x-intercepts of a function are also its zeros, or roots — the solutions to f(x) = 0 — so factoring a quadratic hands you its x-intercepts directly. A function can have at most one y-intercept (otherwise it would fail the vertical line test), but it may have many x-intercepts, or none at all: y = x² + 1 never crosses the x-axis.

Intercepts appear throughout standardized test math. The GRE quantitative section asks you to read intercepts off graphs of lines and parabolas, and both the SAT and ACT use the slope-intercept form y = mx + b, where b is the y-intercept, as the backbone of their linear equation questions.