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.^[1]^[2]^[3]^[4]^[5] 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.^[6]^[7]^[8]. 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.^[9]

Two lines with the same slope, but different y-intercepts are parallel non-intersecting lines.^[10]

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.^[11]^[12]

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.^[13]

Examples

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).)
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)
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).)
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.^[14]
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.^[15]
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.^[16]
The y-intercept of the function y=^x²–4x+3/_(x+2) is the point (0,1.5).
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.

References

^ Zike, Dinah; Sloan, Leon L.; Willard, Teri (2005). Pre-Algebra, Student Edition (1 ed.). Glencoe/McGraw-Hill School Pub Co. p. 381. ISBN 978-0078651083. (in English)
^ Dawkins, Paul (2007). "College Algebra". Lamar University. p. 156. Retrieved January 2014. {{cite web}}: Check date values in: |accessdate= (help) (in English)
^ Tanton, James (2005). Encyclopedia of Mathematics. Facts on File, New York. p. 284. ISBN 0-8160-5124-0. (in English)
^ Weisstein, Eric W. "y-intercept". From MathWorld--A Wolfram Web Resource. Retrieved January 2014. {{cite web}}: Check date values in: |accessdate= (help) (in English)
^ Staple, E. (2013). "x- and y- intercepts". Purple Math. Retrieved January 2014. {{cite web}}: Check date values in: |accessdate= (help) (in English)
^ Marks, Daniel; Cuevas, Gilbert J. (2005). Algebra 1, Student Edition (1 ed.). Glencoe/McGraw-Hill School Pub Co. p. 220. ISBN 978-0078651137. (in English)
^ Beecher, Judith A.; Penna, Judith A.; Bittinger, Marvin L. (2007). Algebra and Trigonometry. Addison Wesley. p. 60. ISBN 978-0321466204. (in English)
^ Carter, John A.; Cuevas, Gilbert J.; Holliday, Berchie; Marks, Daniel; McClure, Melissa S. (2005). Advanced Mathematical Concepts - Pre-calculus with Applications, Student Edition (1 ed.). Glencoe/McGraw-Hill School Pub Co. p. 20. ISBN 978-0078682278. (in English)
^ "Intercept of a Line". Math Open Reference. 2009. Retrieved December 2013. {{cite web}}: Check date values in: |accessdate= (help) (in English) Interactive
^ Carter, John A.; Cuevas, Gilbert J.; Holliday, Berchie; Marks, Daniel; McClure, Melissa S. (2005). Advanced Mathematical Concepts - Pre-calculus with Applications, Student Edition (1 ed.). Glencoe/McGraw-Hill School Pub Co. p. 68. ISBN 978-0078682278. (in English)
^ Clapham, C.; Nicholson, J. (2009). "Oxford Concise Dictionary of Mathematics, Root" (PDF). Addison-Wesley. p. 695. Retrieved January 2013. {{cite web}}: Check date values in: |accessdate= (help) (in English)
^ Tanton, James (2005). Encyclopedia of Mathematics. Facts on File, New York. p. 451. ISBN 0-8160-5124-0. (in English)
^ Staple, E. (2013). "Functions versus Relationships". Purple Math. Retrieved January 2014. {{cite web}}: Check date values in: |accessdate= (help) (in English)
^ Jones, James (1997). "Polynomial Functions of Higher Degree". Retrieved January 2013. {{cite web}}: Check date values in: |accessdate= (help) (in English)
^ Clapham, C.; Nicholson, J. (2009). "Oxford Concise Dictionary of Mathematics, Rational function" (PDF). Addison-Wesley. p. 664. Retrieved January 2013. {{cite web}}: Check date values in: |accessdate= (help) (in English)
^ Clapham, C.; Nicholson, J. (2009). "Oxford Concise Dictionary of Mathematics, Logarithmic function" (PDF). Addison-Wesley. p. 487. Retrieved January 2013. {{cite web}}: Check date values in: |accessdate= (help) (in English)

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)

[PreAlg-1] Zike, Dinah; Sloan, Leon L.; Willard, Teri (2005). Pre-Algebra, Student Edition (1 ed.). Glencoe/McGraw-Hill School Pub Co. p. 381. ISBN 978-0078651083. (in English)

[DawkinsAlg-2] Dawkins, Paul (2007). "College Algebra". Lamar University. p. 156. Retrieved January 2014. {{cite web}}: Check date values in: |accessdate= (help) (in English)

[3] Tanton, James (2005). Encyclopedia of Mathematics. Facts on File, New York. p. 284. ISBN 0-8160-5124-0. (in English)

[4] Weisstein, Eric W. "y-intercept". From MathWorld--A Wolfram Web Resource. Retrieved January 2014. {{cite web}}: Check date values in: |accessdate= (help) (in English)

[5] Staple, E. (2013). "x- and y- intercepts". Purple Math. Retrieved January 2014. {{cite web}}: Check date values in: |accessdate= (help) (in English)

[Alg1-6] Marks, Daniel; Cuevas, Gilbert J. (2005). Algebra 1, Student Edition (1 ed.). Glencoe/McGraw-Hill School Pub Co. p. 220. ISBN 978-0078651137. (in English)

[Alg2B-7] Beecher, Judith A.; Penna, Judith A.; Bittinger, Marvin L. (2007). Algebra and Trigonometry. Addison Wesley. p. 60. ISBN 978-0321466204. (in English)

[8] Carter, John A.; Cuevas, Gilbert J.; Holliday, Berchie; Marks, Daniel; McClure, Melissa S. (2005). Advanced Mathematical Concepts - Pre-calculus with Applications, Student Edition (1 ed.). Glencoe/McGraw-Hill School Pub Co. p. 20. ISBN 978-0078682278. (in English)

[9] "Intercept of a Line". Math Open Reference. 2009. Retrieved December 2013. {{cite web}}: Check date values in: |accessdate= (help) (in English) Interactive

[10] Carter, John A.; Cuevas, Gilbert J.; Holliday, Berchie; Marks, Daniel; McClure, Melissa S. (2005). Advanced Mathematical Concepts - Pre-calculus with Applications, Student Edition (1 ed.). Glencoe/McGraw-Hill School Pub Co. p. 68. ISBN 978-0078682278. (in English)

[11] Clapham, C.; Nicholson, J. (2009). "Oxford Concise Dictionary of Mathematics, Root" (PDF). Addison-Wesley. p. 695. Retrieved January 2013. {{cite web}}: Check date values in: |accessdate= (help) (in English)

[12] Tanton, James (2005). Encyclopedia of Mathematics. Facts on File, New York. p. 451. ISBN 0-8160-5124-0. (in English)

[13] Staple, E. (2013). "Functions versus Relationships". Purple Math. Retrieved January 2014. {{cite web}}: Check date values in: |accessdate= (help) (in English)

[14] Jones, James (1997). "Polynomial Functions of Higher Degree". Retrieved January 2013. {{cite web}}: Check date values in: |accessdate= (help) (in English)

[15] Clapham, C.; Nicholson, J. (2009). "Oxford Concise Dictionary of Mathematics, Rational function" (PDF). Addison-Wesley. p. 664. Retrieved January 2013. {{cite web}}: Check date values in: |accessdate= (help) (in English)

[16] Clapham, C.; Nicholson, J. (2009). "Oxford Concise Dictionary of Mathematics, Logarithmic function" (PDF). Addison-Wesley. p. 487. Retrieved January 2013. {{cite web}}: Check date values in: |accessdate= (help) (in English)

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