Y-intercept (coordinate geometry)

Definition and notation

In the familiar slope–intercept form of a linear equation, y = m x + b, the constant term b represents the y-intercept. Graphically this corresponds to the coordinate (0, b). The same idea applies to other functions: the y-intercept is the value of the function when x = 0, provided the function is defined there.

How to find a y-intercept

There are several routine ways to determine the y-intercept depending on the information available:

• From slope–intercept form y = m x + b: read off b directly.
• From a general equation: set x = 0 and solve for y; the result is the y-intercept (0, y).
• From two points on a line: compute slope m = (y2 − y1)/(x2 − x1), then use b = y1 − m x1 to get the intercept.

For example, the line y = 2x + 3 has y-intercept (0, 3).

Special cases and common points

Horizontal lines have the form y = c and intersect the y-axis at (0, c). Vertical lines, with form x = k, do not have a y-intercept unless k = 0; the line x = 0 coincides with the y-axis and intersects it at every point. If the y-intercept is negative the point lies below the origin; if it is zero the line passes through the origin.

Uses and interpretations

In applied contexts the y-intercept often represents an initial value or baseline when the independent variable is zero — for instance, a starting concentration, a fixed fee, or an offset in a measurement. In algebra and calculus the y-intercept is useful for quick sketching, solving systems of linear equations by graphing, and understanding translations of graphs under vertical shifts.

Notable remarks

The y-intercept is one of two simple intercepts commonly used to describe a line; the other is the x-intercept, where the graph meets the horizontal axis. Together these intercepts can be used to write a line in intercept form and to gain geometric insight into a function's behavior near the axes.