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In [[calculus]] and related areas of mathematics, a '''linear function''' from the real numbers to the real numbers is a function whose graph is a line in the plane.<ref>Stewart 20132012, p. 23</ref>
== Properties of Linear Functions ==
* A linear function is a [[polynomial]] function of first degree with one independent variable ''х'', i.e.
*: <math>y(x)=ax+b</math> (In the USA, the coefficient letter ''a'' is usually replaced by ''m''.)
**To use the function or graph the line, the coefficient letters ''a'' and ''b'' must be given as actual real numbers. For example: <math>y(x)=2x-1</math>. Here ''a''=2 and ''b''=-1.
**In the function, ''x'' is the independent variable and ''y'' is the dependent variable.
** The [[Domain_of_a_function|___domain]] or set of allowed values for ''x'' of a linear function is <math>\mathbb{R}</math> (all real numbers). This means that any real number can be substituted for ''x''.
**The set of all points: (''x'',''y''(''x'')) is the line.
**Since two points determine a line, it is enough to substitute two different values for ''x'' in the linear function and determine
''y'' for each of these values (see videos below).
== Properties of linear functions ==
[[Image:wiki_linear_function.png|thumb|right|Graph of the linear function: ''y''(''x'')=-''x''+2''']]
A linear function is a [[polynomial]] function of first degree with one independent variable ''х'', i.e. <math>y(x)=ax+b</math>.<ref>Stewart 2012, page 242</ref> Here ''x'' is the independent variable and ''y'' is the dependent variable.
* Because the graph of a linear function is a line, a linear function has exactly one intersection point with the ''у''-axis. This point is (0,''b'').
* A nonconstant linear function has exactly one intersection point with the ''х''-axis. This point is ({{Fraction|-b|a}},0).
* From this, it follows that a nonconstant linear function has exactly one [[Zero_of_a_function|zero]] or root. That is, there is exactly one solution to the equation ''a''''x''+''b''=0. The zero is ''x''={{Fraction|-b|a}}.
The [[Domain_of_a_function|___domain]] or set of allowed values for ''x'' of a linear function is the entire set of real numbers, <math>\mathbb{R}</math>. This means that any real number can be substituted for ''x''.
*There are three standard forms for linear functions.
**[[#General Form|General form]]
**[[#Slope-Intercept Form|Slope-intercept form]]
**[[#Vector-Parametric Form|Vector-parametric form]]
The set of points (''x'',''y''(''x'')) is the line that is the graph of the function. Because two points determine a line, it is enough to substitute two different values for ''x'' in the linear function and determine ''y'' for each of these values. Because ''y''(''x'') is a function, the line cannot be vertical.
== General Form ==
Because the graph of a linear function is a nonvertical line, a linear function has exactly one intersection point with the ''у''-axis. This point is (0,''b'').
<math> Ax+By=C </math> where <math> A \ne 0 </math> and <math> B \ne 0 </math>.
A nonconstant linear function has exactly one intersection point with the ''х''-axis. This point is ({{Fraction|-b|a}},0). From this, it follows that a nonconstant linear function has exactly one [[Zero_of_a_function|zero]] or root. That is, there is exactly one solution to the equation ''a''''x''+''b''=0. The zero is ''x''={{Fraction|-b|a}}.
===Properties of the general form===
* The general form has 2 variables ''x'' and ''у'' and 3 coefficient letters A, B, and C.
* To use the function or graph the line, the coefficient letters ''A'', ''B'' and ''C'' must be given as actual real numbers: 3''x''-2''y''=1. Here ''A''=2, ''B''=-2 and ''C''=1.
* This form is not unique. If one multiplies ''A'', ''B'' and ''C'' by a factor ''k'', the coefficients change, but the line remains the same.
There are three standard forms for linear functions.
** Example: For ''k''=3, it follows that (e.g.) 3''x''-2''y''=1 and 9''x''-6''y''=3 are the same line.
*[[#General form|General form]]
**Example: For ''k''={{Fraction|-1|π}}, it follows that -3π''x''+2π''y''+π=0 and 3''x''-2''y''=1 are the same line.
*[[#Slope-intercept form|Slope-intercept form]]
*[[#Parametric form|Parametric form]]
== Slope-intercept form ==
This general form is used mainly in geometry and in systems of two linear equations in two unknowns.
[[Image: wiki_linearna_funkcija_eks1.png|thumb||right]]
*The Example: 3''x''slope-2''y''=1intercept andform 6''x''-4''y''=2of are the samea linear function, i.e.is theiran graphequation isof the same line.form
:<math> y(x)=ax+b </math> or <math> y=ax+b </math>
** In the first equation the coefficients are: A=3, B=-2 and C=1.
where <math>{ a \ne 0} </math>. The slope-intercept form has 2 variables ''x'' and ''у'' and 2 coefficient letters ''а'' and ''b''.
** In the second equation the coefficients are: A=6, B=-4 and C=2.
** Notice that the second coefficients are all twice the first equations. This means the factor is ''k''=2.
** '''Further''', solving both of these equations for ''y'' gives the same slope-intercept form of this line.
*::''y''=1,5''x''-0,5
[[Image: wiki_linearna_funkcija_stand1.png|180px|right]]
The slope-intercept form is also called the ''explicit form'' because it defines ''y''(''x'') explicitly (directly) in terms of ''x''.
The slope-intercept form of a linear function is unique. That is, if the value of either or both of the coefficient letters ''a'' and ''b'' are changed, a different function is obtained.
== Slope-Intercept Form ==
The constant ''b'' is the so-called ''у''-intercept. It is the ''y''-value at which the line intersects the ''y''-axis.
<math> y(x)=ax+b </math> or <math> y=ax+b </math> where <math>{ a \ne 0} </math>.
The coefficient ''а'' is the [[slope]] of the line, which measures of the rate of change of the linear function. Since ''a'' is a constant, this rate of change is constant. Moving from any point on the line to the right by 1 (that is, increasing ''x'' by 1), the ''y''-value of the function changes by ''a''.
=== Properties of the Slope-Intercept Form ===
* The slope-intercept form is also called the ''explicit form'' because it defines ''y''(''x'') explicitly (directly) in terms of ''x''.
* The slope-intercept form has 2 variables ''x'' and ''у'' and 2 coefficient letters ''а'' and ''b''.
* To use the function or graph the line, the coefficient letters must be written as actual real numbers. For example: ''y''(''х'')=-2''$
*The slope-intercept form is unique. That is, if the value of either or both of the coefficient letters ''a'' and ''b'' are changes, th$
*Every linear function can be written uniquely in slope-intercept form.
* Intercepts (intersections of the line with the axes)
* The constant ''b'' is the so-called '''''у''-intercept'''. It is the ''y''-value at which the line intersects the ''y''-axis. This is$
* The number {{Fraction|-b|а}} is the [[http://en.wikipedia.org/wiki/Zero_of_a_function root]] or [[http://en.wikipedia.org/wiki/Zero_o$
*:: 0=''a''•''x''+''b''
* The number {{Fraction|-b|а}} is the [[http://en.wikipedia.org/wiki/Zero_of_a_function root]] or [[http://en.wikipedia.org/wiki/Zero_o$
*:: 0=''a''•''x''+''b''
*:: ''a''•''x''+''b''=0
*:: ''a''•''x'' = -''b''
*:: ''x''={{Fraction|-b|a}}
For example, the slope-intercept form <math> y(x)=-2x+4 </math> has ''a'' = -2 and ''b'' = 4. The point (0,''b'') = (0,4) is the intersection of the line and the ''у''-axis, the point {{Fraction|-b|а}},0) = ({{Fraction|-4|-2}},0) = (2,0) is the intersection of the line and the ''х''-axis, and ''а'' = -2 is the slope of the line. For every step to the right (''х'' increases by 1), the value of ''у'' changes by -2 (goes down).
This means that the point ({{Fraction|-b|a}},0) is both a point on the line and a point on the ''x''-axis. So it is ''the''
point where the line intersects the ''x''-axis.
== General form ==
* The coefficient ''а'' is the so-called '''[[slope]]''' of the line and is a measure of the rate of change of the linear function. Since ''a'' is a number (not a variable), this rate of change is constant. Moving from any point on the line to the right by 1 (that is, increasing ''x'' by 1), the ''y''-value of the function changes by ''a''. See [[slope]].
[[Image: wiki_linearna_funkcija_stand1.png|180px|right]]
The general form for a linear function is an equation of the form
; Example: <math> y(x)=-2x+4 </math> where ''a'' = -2 and ''b'' = 4
: <math> Ax+By=C </math>
* (0,''b'') = (0,4) is the intersection of the line and the ''у''-axis
where <math> A \ne 0 </math>.
* {{Fraction|-b|а}},0) = ({{Fraction|-4|-2}},0) = (2,0) is the intersection of the line and the ''х''-axis and
* ''а'' = -2 is the slope of the line. For every step to the right (''х'' increases by 1), the value of ''у'' changes by -2 (goes down).
The general form has 2 variables ''x'' and ''у'' and 3 coefficient letters A, B, and C.
[[Image: wiki_linearna_funkcija_eks1.png|thumb||right]]
This form is not unique. If one multiplies ''A'', ''B'' and ''C'' by a constant factor ''k'', the coefficients change, but the line remains the same. For example, 3''x''-2''y''=1 and 9''x''-6''y''=3 are general forms of the same line.
This general form is used mainly in geometry and in systems of two linear equations in two unknowns.
==Vector-Parametric Form==
==Parametric form==
[[Image: wiki_linearna_funkcija_par1.png|300px|right]]
The [[parametric form]] of a line consists of two equations:
Parametric form:
:x(t) = {b_1}+{a_1}t
*y(t) = {b_2}+{a_2}t
where <math> a_1 \ne 0 </math>.
The parametric form has one [[parameter]] ''t'', two variables ''x'' and ''у'', and four coefficients а<sub>1</sub>, а<sub>2</sub>, b<sub>1</sub>, and b<sub>2</sub>. The coefficients are not unique, but they are related.
=== Properties of the vector-parametric form ===
*The Vector-parametricline formpasses hasthrough 1the [[parameter]] ''t'', 2points variables ''x'' and ''у'', and 4 coefficients а(b<sub>1</sub>, аb<sub>2</sub>,) and (b<sub>1</sub>+a<sub>1</sub>, and b<sub>2</sub>+a<sub>2</sub>).
* The coefficients are not unique, but they are related.
The vector-parametric form is used in engineering (it is simple to model the path from one point to another point with ''t''=time). Engineers tend to use parametric notation and the letter ''t'' for the parameter; mathematicians use vector notation and the letter λ.
* The line passes through the points (b<sub>1</sub>,b<sub>2</sub>) and (b<sub>1</sub>+a<sub>1</sub>,b<sub>2</sub>+a<sub>2</sub>).
* The vector-parametric form is used in engineering (it is simple to model the path from one point to another point with ''t''=time).
This form of a linear function can easily be extended to lines in higher dimensions, which is not true of the other forms.
* Engineers tend to use parametric notation and the letter ''t'' for the parameter; mathematicians use vector notation and the letter λ.
* This form of a linear function can easily be extended to lines in higher dimensions, which is not true of the other forms.
* This form of a linear function can easily be extended to lines in higher dimensions, which is not true of the other forms.
; Example: <math>{{X}} = ({-1},{1}) + t({2},{3})</math>
* The slope-intercept form of this line is: ''y''(''x'')=1,5''x''+2,5 (solve the first parametric equation for ''t'' and substitute in$
* One general form of this line is: -3''x''+2''y''=5.
[[Image: wiki_linearna_funkcija_par1.png|300px|right]]
== References ==
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