y-intercept

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Image:Y-intercept.svg, 300px, 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 or vertical intercept is a point where the
graph of a function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

or relation intersects the ''y''-axis of the
coordinate system In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...
. As such, these points satisfy ''x'' = 0.

# Using equations

If the curve in question is given as $y= f\left(x\right),$ the ''y''-coordinate of the ''y''-intercept is found by calculating $f\left(0\right).$ Functions which are undefined at ''x'' = 0 have no ''y''-intercept. If the function is
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and is expressed in slope-intercept form as $f\left(x\right)=a+bx,$ the constant term $a$ is the ''y''-coordinate of the ''y''-intercept.

# Multiple y-intercepts

Some 2-dimensional mathematical relationships such as
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s,
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s, and
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s can have more than one ''y''-intercept. Because
functions Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

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
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of, say, a
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. (In
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, ''I'' is the symbol used for
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.)