In analytic geometry, using the common convention that the horizontal axis represents a variable x {\displaystyle x} and the vertical axis represents a variable y {\displaystyle y} , a y {\displaystyle y} -intercept or vertical intercept is a point where the graph of a function or relation intersects the y {\displaystyle y} -axis of the coordinate system. As such, these points satisfy x = 0 {\displaystyle x=0} .
Using equations If the curve in question is given as y = f ( x ) , {\displaystyle y=f(x),} the y {\displaystyle y} -coordinate of the y {\displaystyle y} -intercept is found by calculating f ( 0 ) {\displaystyle f(0)} . Functions which are undefined at x = 0 {\displaystyle x=0} have no y {\displaystyle y} -intercept. If the function is linear and is expressed in slope-intercept form as f ( x ) = a + b x {\displaystyle f(x)=a+bx} , the constant term a {\displaystyle a} is the y {\displaystyle y} -coordinate of the y {\displaystyle y} -intercept.
Multiple y {\displaystyle y} -intercepts Some 2-dimensional mathematical relationships such as circles, ellipses, and hyperbolas can have more than one y {\displaystyle y} -intercept. Because functions associate x {\displaystyle x} -values to no more than one y {\displaystyle y} -value as part of their definition, they can have at most one y {\displaystyle y} -intercept.
x {\displaystyle x} -intercepts
Analogously, an x {\displaystyle x} -intercept is a point where the graph of a function or relation intersects with the x {\displaystyle x} -axis. As such, these points satisfy y = 0 {\displaystyle y=0} . The zeros, or roots, of such a function or relation are the x {\displaystyle x} -coordinates of these x {\displaystyle x} -intercepts. Functions of the form y = f ( x ) {\displaystyle y=f(x)} have at most one y {\displaystyle y} -intercept, but may contain multiple x {\displaystyle x} -intercepts. The x {\displaystyle x} -intercepts of functions, if any exist, are often more difficult to locate than the y {\displaystyle y} -intercept, as finding the y {\displaystyle y} -intercept involves simply evaluating the function at x = 0 {\displaystyle 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 {\displaystyle I} -intercept of the current–voltage characteristic of, say, a diode. (In electrical engineering, I {\displaystyle I} is the symbol used for electric current.)
See also Regression intercept
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