intercepts

Intercepts are the points where a graph crosses an axis in a coordinate plane. In two-variable relationships, the x-intercept is the point where the graph meets the x-axis (y = 0), and the y-intercept is where it meets the y-axis (x = 0). Intercepts provide a quick way to understand a graph’s placement and scale.

For a straight line given by ax + by = c, the x-intercept is (c/a, 0) and the y-intercept is (0, c/b), provided a and b are nonzero. A common form is the intercept form x/a + y/b = 1, which has intercepts at (a, 0) and (0, b). Such forms are useful for graphing lines by plotting the two axis-crossing points.

For a function y = f(x), the y-intercept is f(0). The x-intercepts, also called real zeros or roots, are the solutions to f(x) = 0. For example, the line 3x + 2y = 12 has x-intercept (4, 0) and y-intercept (0, 6). The quadratic f(x) = x^2 − 4 has x-intercepts at x = −2 and x = 2, and its y-intercept is f(0) = −4.

In multivariate contexts, intercepts are found similarly by setting one variable to zero and solving for the other. In statistics, the intercept (often denoted as the constant term) is the predicted value when all predictors are zero and is a parameter of the model that requires careful interpretation.