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In analytic geometry, using the common convention that the horizontal axis represents a variable ${\displaystyle x}$ and the vertical axis represents a variable ${\displaystyle y}$, a ${\displaystyle y}$-intercept or vertical intercept is a point where the graph of a function or relation intersects the ${\displaystyle y}$-axis of the coordinate system. ^[1] As such, these points satisfy ${\displaystyle x=0}$.

Using equations

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

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

Multiple ${\displaystyle y}$-intercepts

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

${\displaystyle x}$-intercepts

Analogously, an ${\displaystyle x}$-intercept is a point where the graph of a function or relation intersects with the ${\displaystyle x}$-axis. As such, these points satisfy ${\displaystyle y=0}$. The zeros, or roots, of such a function or relation are the ${\displaystyle x}$-coordinates of these ${\displaystyle x}$-intercepts. ^[3]

Functions of the form ${\displaystyle y=f(x)}$ have at most one ${\displaystyle y}$-intercept, but may contain multiple ${\displaystyle x}$-intercepts. The ${\displaystyle x}$-intercepts of functions, if any exist, are often more difficult to locate than the ${\displaystyle y}$-intercept, as finding the ${\displaystyle y}$-intercept involves simply evaluating the function at ${\displaystyle x=0}$.

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 ${\displaystyle I}$-intercept of the current–voltage characteristic of, say, a diode. (In electrical engineering, ${\displaystyle I}$ is the symbol used for electric current.)

See also

• Regression intercept

References