y-intercept

Graph y=ƒ(x) with the x-axis as the horizontal axis and the y-axis as the vertical axis. The y-intercept of ƒ(x) is indicated by the red dot at (x=0, y=1).

In analytic geometry, using the common convention that the horizontal axis represents a variable x and the vertical axis represents a variable y, a y-intercept is a point where the graph of a function or relation intersects with the y-axis of the coordinate system. As such, these points satisfy x = 0.

Using equations

If the curve in question is given as $y= f(x),$ the y-coordinate of the y-intercept is found by calculating $f(0).$ Functions which are undefined at x = 0 have no y-intercept.

If the function is linear and is expressed in slope-intercept form as $f(x)=a+bx,$ the constant term $a$ is the y-coordinate of the y-intercept.

Multiple y-intercepts

Some 2-dimensional mathematical relationships such as circles, ellipses, and hyperbolas can have more than one y-intercept. Because functions associate x values to no more than one y value as part of their definition, they can have at most one y-intercept.

x-intercepts

Main article: x-intercept

Analogously, an x-intercept is a point where the graph of a function or relation intersects with the x-axis. As such, these points satisfy y=0. The zeros, or roots, of such a function or relation are the x-coordinates of these x-intercepts.

Unlike y-intercepts, functions of the form y = f(x) may contain multiple x-intercepts. The x-intercepts of functions, if any exist, are often more difficult to locate than the y-intercept, as finding the y intercept involves simply evaluating the function at x=0.

In higher dimensions

The notion may be extended for 3-dimensional space and higher dimensions, as well as for other coordinate axes, possibly with other names. For example, one may speak of the I-intercept of the current-voltage characteristic of, say, a diode. (In electrical engineering, I is the symbol used for electric current.)

Application to Electric Circuits

In the specific case of electrical circuits, the y-intercept of the graph of terminal potential difference versus current through the circuit is equal to the electromotive force (e.m.f) of the cell/battery. The general equation to calculate terminal potential difference is $V=-rI+E.$ Compare this to the slope-intercept form f a linear function, the relationship is clear:

$y=mx+b$

$V=-rI+E$