User:Lfahlberg/sb y-intercept

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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.[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:    or just    where m, b are real numbers, it follows that for x=0:  
The y-intercept of the line    is    or the point   .
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:    where A, B and C are real numbers with B≠0, it follows that for x=0:  
The y-intercept of the line    is    or the point   .
  • 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

  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+...+a2x²+a1x+a0 is a0; that is, the y-intercept is the constant term.[14]
  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.[15]
  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.[16]
  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

References

  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)
  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)
  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)
  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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