Xintercept

An x-intercept, or x-intercept point, is a point where a graph in the Cartesian plane meets the x-axis (the line y = 0). For a real-valued function y = f(x), the x-intercepts are the solutions to the equation f(x) = 0, and the intercept points have coordinates (x0, 0). A graph may have zero, one, or many x-intercepts depending on the function or curve.

Examples help illustrate: the graph of y = x^2 − 4 intersects the x-axis at (-2, 0) and (2, 0). The graph of y = x^3 intersects at (0, 0). If a root occurs with even multiplicity, the graph touches the x-axis and turns around rather than crossing; with odd multiplicity, it crosses the axis.

For curves that are not explicit functions y = f(x) but are given implicitly by F(x, y) = 0, x-intercepts are found by solving F(x, 0) = 0, yielding points (x, 0) where the curve meets the x-axis. A general approach to locating intercepts is to set y to zero and solve for x, using algebraic methods such as factoring, the quadratic formula, or numerical techniques when necessary.

Relation to other intercepts: the x-intercept is complementary to the y-intercept, which is the point where the graph crosses the y-axis (x = 0). In applications, x-intercepts often correspond to zeros of a function, times when a quantity becomes zero, or threshold points in models.