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.^[1] As such, these points satisfy x = 0.

Using equations

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

If the function is linear and is expressed in slope-intercept form as ${\displaystyle f(x)=a+bx,}$ the constant term $a$ is the y-coordinate of the y-intercept.^[2]

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

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.^[3]

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 ${\displaystyle V=-rI+E.}$ Compare this to the slope-intercept form f a linear function, the relationship is clear:^[4]

$y=mx+b$

${\displaystyle V=-rI+E}$

See also

• Regression intercept

References

1. Weisstein, Eric W. "y-Intercept". MathWorld--A Wolfram Web Resource. Retrieved 2010-09-22.
2. Stapel, Elizabeth. "x- and y-Intercepts." Purplemath. Available from http://www.purplemath.com/modules/intrcept.htm.
3. Weisstein, Eric W. "Root". MathWorld--A Wolfram Web Resource. Retrieved 2010-09-22.
4. http://physicsnet.co.uk/a-level-physics-as-a2/current-electricity/electromotive-force-and-internal-resistance/