Y-intercept

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

In analytic geometry, using the common convention that the horizontal axis represents a variable <math>x</math> and the vertical axis represents a variable <math>y</math>, a <math>y</math>-intercept or vertical intercept is a point where the graph of a function or relation intersects the <math>y</math>-axis of the coordinate system.<ref> {{#invoke:citation/CS1|citation |CitationClass=web }} </ref> As such, these points satisfy <math>x = 0</math>.

Using equationsEdit

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

If the function is linear and is expressed in slope-intercept form as <math>f(x) = a + bx</math>, the constant term <math>a</math> is the <math>y</math>-coordinate of the <math>y</math>-intercept.<ref>Stapel, Elizabeth. "x- and y-Intercepts." Purplemath. Available from

   http://www.purplemath.com/modules/intrcept.htm.</ref>

Multiple <math>y</math>-interceptsEdit

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

<math>x</math>-interceptsEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Analogously, an <math>x</math>-intercept is a point where the graph of a function or relation intersects with the <math>x</math>-axis. As such, these points satisfy <math>y = 0</math>. The zeros, or roots, of such a function or relation are the <math>x</math>-coordinates of these <math>x</math>-intercepts.<ref> {{#invoke:citation/CS1|citation |CitationClass=web }} </ref>

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

In higher dimensionsEdit

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 <math>I</math>-intercept of the current–voltage characteristic of, say, a diode. (In electrical engineering, <math>I</math> is the symbol used for electric current.)

ReferencesEdit