User:Lfahlberg/sb y-intercept

In two-dimensional coordinate geometry, a y-intercept of a function or relation is the y-coordinate of a point at which the graph intersects the y-axis. Because the y-axis is the set of points for which x=0, one finds y-intercept(s) by substituting x=0 into the function or relation and solving for y.

y-intercept of a line in the plane
The y-intercept of a line in the plane is the y-coordinate of the point at which the line crosses the y-axis. . It may also refer to the point (and not just the y-coordinate of this point) at which the line crosses the y-axis. To find the y-intercept of a line, substitute x=0 into the equation of the line. The resulting value for y is the y-intercept. The y-intercept of the line $$y(x)=m x+b$$  is  $$y(0)=b$$  or the point  $$(0,b)$$.
 * If the line is given as: $$y(x)=mx+b$$  or just  $$y=mx+b$$  where m, b are real numbers, it follows that for x=0:  $$y(0)=m \cdot 0+b \,\, \Rightarrow \,\, y(0)=b \,.$$
 * Example: Given the linear function y=3x-2. Here m=3 and b=–2. So the y-intercept is b=–2 or the point (0,–2).

The y-intercept of the line $$Ax+By=C$$  is  $$y(0)=\frac{C}{B}$$  or the point  $$(0,\frac{C}{B})$$.
 * If the line is given in standard form: $$Ax+By=C$$  where A, B and C are real numbers with B&ne;0, it follows that for x=0:  $$A \cdot 0 +B \cdot y=C \,\, \Rightarrow \,\, y=\frac{C}{B} \,.$$


 * Every non-vertical line has exactly one y-intercept.


 * Two lines with the same slope, but different y-intercepts are parallel non-intersecting lines.

Analogously, an x-intercept of a function or relation is the x-coordinate of a point at which the graph intersects the x-axis. These values are also called roots or zeros of the function since the value of the function at an x-intercept is y=0.

y-intercept of a function
By definition, a function assigns each value in its domain to exactly one output value. This means a function can have at most one y-intercept.
 * If x=0 is in the domain of the function, the function will have exactly one y-intercept.
 * If x=0 is not in the domain of the function, the function will have no y-intercept and the function does not cross the y-axis.

y-intercepts of a relation
Some 2-dimensional mathematical relationships such as circles, ellipses, and hyperbolas can have more than one y-intercept.

Examples

 * 1) The y-intercept of the function y=4 is the point (0,4). (This is a constant function whose graph is a horizontal line passing through the point (0,4).)
 * 2) The y-intercept of the linear function y=3x–2 is the point (0,–2). (This is a line in slope-intercept form y=mx+b with b= –2)
 * 3) The y-intercept of the function  30x+2y=120 is the point (0,60). (This is a line with slope m=–15 passing through the point on the y-axis (0,60).)
 * 4) The y-intercept of the polynomial y=anxn+an-1xn-1+...+a2x&sup2;+a1x+a0 is a0; that is, the y-intercept is the constant term.
 * 5) The function y=1/x has no y-intercept because the rational function 1/x is not defined for x=0. That is, x=0 is not in the domain of this function.
 * 6) The function y=log(x) has no y-intercept because the logarithmic function log(x) is not defined for x=0. That is, x=0 is not in the domain of this function.
 * 7) The y-intercept of the function y=x&sup2;–4x+3/(x+2) is the point (0,1.5).
 * 8) The y-intercepts of the relation (x–2)&sup2;+(y-1)&sup2;=8 are the points (0,3) and (0,-1). The graph is a circle that crosses the y-axis twice.