User:Lfahlberg/sb y-intercept

The y-intercept of the line y=mx+b is the blue point (0,b). (The line here is generic. See examples with concrete values below.)

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.

• 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:  $y(0)=m\cdot 0+b\,\,\Rightarrow \,\,y(0)=b\,.$

The y-intercept of the line  $y(x)=mx+b$  is  $y(0)=b$  or the point  $(0,b)$ .

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).

• If the line is given in standard form:  $Ax+By=C$  where A, B and C are real numbers with B≠0, it follows that for x=0:  $A\cdot 0+B\cdot y=C\,\,\Rightarrow \,\,y={\frac {C}{B}}\,.$

The y-intercept of the line  $Ax+By=C$  is  $y(0)={\frac {C}{B}}$  or the point  $(0,{\frac {C}{B}})$ .

• 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=a_nxⁿ+a_n-1x^n-1+...+a₂x²+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=^x²–4x+3/_(x+2) is the point (0,1.5).
8. The y-intercepts of the relation (x–2)²+(y-1)²=8 are the points (0,3) and (0,-1). The graph is a circle that crosses the y-axis twice.

1. The y-intercept of a constant function.	2. The y-intercept of a linear function in slope-intercept form.	3. The y-intercept of a linear function in standard form.	4. The y-intercept of a polynomial is the constant term.
5. The rational function y=1/x does not cross the y-axis.	6. The logarithmic function y=log(x) does not cross the y-axis. (Here base=10.)	7. The y-intercept of a rational function is the point where the numerator is 0.	8. This circle relation has two y-intercepts.

See also

• Coordinate geometry
• Function, Relation
• Graph of a function
• Root of a function
• Linear function

External links

• "Intercept of a Line". Math Open Reference. 2009. Retrieved December 2013. {{cite web}}: Check date values in: |accessdate= (help) (in English) Interactive
• "Line in Plane". Planet Math. Retrieved January 2014. {{cite web}}: Check date values in: |accessdate= (help) (in English)