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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 2013, p. 23</ref>
{{underconstruction}}
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.
== Properties of Linear Functions ==
* A linear function is a [[polynomial]] function of first degree with one independent variable <em>''х</em>'', i.e.
*: <math>y(x)=ax+b</math> (In the USA, the coefficient letter <em>''a</em>'' is usually replaced by <em>''m</em>''.)
**To use the function or graph the line, the coefficient letters <em>''a</em>'' and <em>''b</em>'' must be given as actual real numbers. For example: <math>y(x)=2x-1</math>. Here <em>''a</em>''=2 and <em>''b</em>''=-1.
**In the function, <em>''x</em>'' is the independent variable and <em>''y</em>'' is the dependent variable.</li>
** The [[Domain_of_a_function|___domain]] or set of allowed values for <em>''x</em>'' of a linear function is <math>\Remathbb{R}</math> (all real numbers). This means that any real number can be substituted for <em>''x</em>''. (Of course, the value of <em>y</em> depends on the substituted value for x.)</li>
**The set of all points: (<em>''x</em>'',<em>''y</em>''(<em>''x</em>'')) is the line.</li>
**Since two points determine a line, it is enough to substitute two different values for <em>''x</em>'' in the linear function and determine <em>
''y</em>'' for each of these values (see videos below).
[[Image:wiki_linear_function.png|thumb|right|Graph of the linear function: ''y''(''x'')=-''x''+2''']]
* 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'').
[[Image: wiki_linear_function.png|right|160px Graph of the linear function: <em>y</em>(<em>x</em>)=-<em>x</em>+2</strong>]]
* 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}}.
*There are three standard forms for linear functions.
**[[#General Form|General form]]
**[[#Slope-Intercept Form|Slope-intercept form]]
**[[#Vector-Parametric Form|Vector-parametric form]]
* Because the graph of a linear function is a line, a linear function has exactly one intersection point with the <em>у</em>-axis. This point is (0,<em>b</em>).</li>
* A nonconstant linear function has exactly one intersection point with the <em>х</em>-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 <em>a</em><em>x</em>+<em>b</em>=0. <br /> The zero is <em>x</em>={{Fraction|-b|a}}.
*There are three standard forms for linear functions.
**[[#General Form|General form]]</li>
**[[#Slope-Intercept Form|Slope-intercept form]]</li>
**[[#Vector-Parametric Form|Vector-parametric form]]</li>
== General Form ==
<math> Ax+By=C </math> <big>where</big> <math> A \ne 0 </math> and <math> B \ne 0 </math>.
===Properties of the general form===
* The general form has 2 variables <em>''x</em>'' and <em>''у</em>'' and 3 coefficient letters A, B, and C.
* To use the function or graph the line, the coefficient letters <em>''A</em>'', <em>''B</em>'' and <em>''C</em>'' must be given as actual real numbers: 3<em>''x</em>''-2<em>''y</em>''=1. Here <em>''A</em>''=2, <em>''B</em>''=-2 and <em>''C</em>''=1.
* This form is not unique. If one multiplies <em>''A</em>'', <em>''B</em>'' and <em>''C</em>'' by a factor <em>''k</em>'', the coefficients change, but the line remains the same.
<ul>
<li>Example: For <em>k</em>=3, it follows that (e.g.) 3<em>x</em>-2<em>y</em>=1 and 9<em>x</em>-6<em>y</em>=3 are the same line. </li>
<li>Example: For <em>k</em>={{Fraction|-1|π}}, it follows that -3π<em>x</em>+2π<em>y</em>+π=0 and 3<em>x</em>-2<em>y</em>=1 are the same line. </li>
</ul>
</li>
<li>This general form is used mainly in geometry and in systems of two linear equations in two unknowns.</li>
<li>The general form of a line is a [[linear equation]]; the opposite is not necessarily true. </li>
</ul>
** Example: For ''k''=3, it follows that (e.g.) 3''x''-2''y''=1 and 9''x''-6''y''=3 are the same line.
----
**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.
<table border="0" cellspacing="5" width="850" >
<tr><td valign="top">Example: 3<em>x</em>-2<em>y</em>=1 and 6<em>x</em>-4<em>y</em>=2 are the same linear function, i.e. their graph is the same line.
<ul>
<li>In the first equation the coefficients are: A=3, B=-2 and C=1.</li>
<li>In the second equation the coefficients are: A=6, B=-4 and C=2.</li>
<li>Notice that the second coefficients are all twice the first equations. This means the factor is <em>k</em>=2.</li>
<li><strong>Further</strong>, solving both of these equations for <em>y</em> gives the same slope-intercept form of this line.<br />
<big><em>y</em>=1,5<em>x</em>-0,5</big></li>
</ul>
This general form is used mainly in geometry and in systems of two linear equations in two unknowns.
</td><td>
<p align="center">[[Image: wiki_linearna_funkcija_stand1.png|180px ]]</p></td></tr></table>
* Example: 3''x''-2''y''=1 and 6''x''-4''y''=2 are the same linear function, i.e. their graph is the same line.
** In the first equation the coefficients are: A=3, B=-2 and C=1.
** 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]]
<h1>Slope-Intercept Form</h1>
<div style="margin-left:15px">
<p><math> y(x)=ax+b </math> <big>or</big> <math> y=ax+b </math> <big>where</big> <math>{ a \ne 0} </math>.</p>
</div>
<p style=" font-size:1.2em; font-weight:bold">Properties of the= Slope-Intercept Form</p> ==
<ul>
<li>The slope-intercept form is also called the <em>explicit form</em> because it defines <em>y</em>(<em>x</em>) explicitly (directly) in terms of <em>x</em>.</li>
<li>The slope-intercept form has 2 variables <em>x</em> and <em>у</em> and 2 coefficient letters <em>а</em> and <em>b</em>. </li>
<li>To use the function or graph the line, the coefficient letters must be written as actual real numbers. For example: <em>y</em>(<em>х</em>)=-2<em>х</em>+4 </li>
<li>The slope-intercept form is unique. That is, if the value of either or both of the coefficient letters <em>a</em> and <em>b</em> are changes, the result is a different line!</li>
<li>Every linear function can be written uniquely in slope-intercept form.</li>
<li>Intercepts (intersections of the line with the axes)
<ul>
<li>The constant <em>b</em> is the so-called <strong><em>у</em>-intercept</strong>. It is the <em>y</em>-value at which the line intersects the <em>y</em>-axis. This is because the <em>y</em>-axis is the line where <em>x</em>=0. Substituting <em>x</em>=0 into the linear function <em>y</em>(<em>x</em>)=<em>a</em><em>x</em>+<em>b</em>, it follows that: <em>y</em>(0)=<em>a</em>•0+<em>b</em>=<em>b</em>. This means that the point (0,b) is both a point on the line and a point on the <em>y</em>-axis. So it is <em>the</em> point where the line intersects the <em>y</em>-axis.</li>
<li>The number {{Fraction|-b|а}} is the [[http://en.wikipedia.org/wiki/Zero_of_a_function root]] or [[http://en.wikipedia.org/wiki/Zero_of_a_function zero]] of the function. It is the <em>x</em>-value at which the line intersects the <em>x</em>-axis. This is because the <em>x</em>-axis is the line where <em>y</em>=0. Substituting <em>y</em>=0 into the linear function and solving (backwards) for <em>x</em>, it follows that: <br />
0=<em>a</em>•<em>x</em>+<em>b</em> <br />
<em>a</em>•<em>x</em>+<em>b</em>=0 <br />
<em>a</em>•<em>x</em> = -<em>b</em> <br />
<em>x</em>={{Fraction|-b|a}} <br />
This means that the point ({{Fraction|-b|a}},0) is both a point on the line and a point on the <em>x</em>-axis. So it is <em>the</em> point where the line intersects the <em>x</em>-axis. </li>
</ul></li></ul></li>
<li>The coefficient <em>а</em> is the so-called <strong>[[slope]]</strong> of the line and is a measure of the rate of change of the linear function. Since <em>a</em> 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 <em>x</em> by 1), the <em>y</em>-value of the function changes by <em>a</em>. See [[slope]].</li>
</ul>
<math> y(x)=ax+b </math> or <math> y=ax+b </math> where <math>{ a \ne 0} </math>.
----
<div style="margin-left:15px">
<table border="0" cellspacing="5" width="850">
<tr><td valign="top">
Example: <math> y(x)=-2x+4 </math> where <br />
<ul>
<li><big><em>a</em> = -2</big> and <big><em>b</em> = 4</big> <br /> </li>
<li><big>(0,<em>b</em>) = (0,4)</big> is the intersection of the line and the <em>у</em>-axis <br /> </li>
<li><big>(</big>{{Fraction|-b|а}}<big>,0) = (</big>{{Fraction|-4|-2}}<big>,0) = (2,0)</big> is the intersection of the line and the <em>х</em>-axis and <br /> </li>
<li><big><em>а</em> = -2</big> is the slope of the line. For every step to the right (<em>х</em> increases by 1), the value of <em>у</em> changes by -2 (goes down).</li>
</ul></td>
<td><div align="center">[[Image: wiki_linearna_funkcija_eks1.png|180px]]</div></td></tr></table>
=== 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}}
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''
<h1>Vector-Parametric Form</h1>
point where the line intersects the ''x''-axis.
<div style="margin-left:15px">
<p>Parametri: <math>\left\{ {\begin{array}{*{20}{l}} {x(t) = {b_1}+{a_1}t }\\ {y(t) = {b_2}+{a_2}t } \end{array}} \right.</math> or Vector: <math>{{X}} = ({b_1},{b_2}) + t({a_1},{a_2})</math> <big>where</big> <math>{ a_1 \ne 0} </math> and <math>{ a_2 \ne 0} </math>.</p>
</div>
* 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]].
<p style=" font-size:1.2em; font-weight:bold">Properties of the vector-parametric form</p>
<ul>
<li>Vector-parametric form has 1 [[parameter]] <em>t</em>, 2 variables <em>x</em> and <em>у</em>, and 4 coefficients а<sub>1</sub>, а<sub>2</sub>, b<sub>1</sub>, and b<sub>2</sub>. </li>
<li>The coefficients are not unique, but they are related. </li>
<li>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>). </li>
<li>The vector-parametric form is used in engineering (it is simple to model the path from one point to another point with <em>t</em>=time).</li>
<li>Engineers tend to use parametric notation and the letter <em>t</em> for the parameter; mathematicians use vector notation and the letter λ.</li>
<li>This form of a linear function can easily be extended to lines in higher dimensions, which is not true of the other forms. </li>
</ul>
; Example: <math> y(x)=-2x+4 </math> where ''a'' = -2 and ''b'' = 4
----
* (0,''b'') = (0,4) is the intersection of the line and the ''у''-axis
<div style="margin-left:15px">
* {{Fraction|-b|а}},0) = ({{Fraction|-4|-2}},0) = (2,0) is the intersection of the line and the ''х''-axis and
<table border="0" cellspacing="5" width="850px" >
* ''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).
<tr><td valign="top">
Example: <math>{{X}} = ({-1},{1}) + t({2},{3})</math>
<ul>
<li>Here: <big><em>a</em><sub>1</sub> = 2</big> and <big><em>a</em><sub>2</sub> = 3</big> and <big><em>b</em><sub>1</sub> = -1</big> and <big><em>b</em><sub>2</sub> = 1</big> </li>
<li>The line passes through the points> <big>(<em>b</em><sub>1</sub>, <em>b</em><sub>2</sub>)=(-1,1)</big> and <big>(b<sub>1</sub>+a<sub>1</sub>,b<sub>2</sub>+a<sub>2</sub>)=(1,4)</big> <br /> These are the points where <em>t</em>=0 and <em>t</em>=1 (<em>t</em> is not visible on the graph!). </li>
<li>The parametric form of this line is: <math>\left\{ {\begin{array}{*{20}{l}} {x(t) = {-1}+{2}t }\\ {y(t) = {1}+{3}t } \end{array}} \right.</math> <br /> </li>
<li>The slope-intercept form of this line is: <em>y</em>(<em>x</em>)=1,5<em>x</em>+2,5 (solve the first parametric equation for <em>t</em> and substitute into the second).</li>
<li>One general form of this line is: -3<em>x</em>+2<em>y</em>=5.</li>
</ul>
</td>
<td><p>[[Image: wiki_linearna_funkcija_par1.png|300px ]]</p></td></tr></table>
</div>
[[Image: wiki_linearna_funkcija_eks1.png|thumb||right]]
==Vector-Parametric Form==
Parametric form:
:x(t) = {b_1}+{a_1}t
*y(t) = {b_2}+{a_2}t
where <math> a_1 \ne 0 </math>.
=== Properties of the vector-parametric form ===
* Vector-parametric form has 1 [[parameter]] ''t'', 2 variables ''x'' and ''у'', and 4 coefficients а<sub>1</sub>, а<sub>2</sub>, b<sub>1</sub>, and b<sub>2</sub>.
<h1>References</h1>
* The coefficients are not unique, but they are related.
* 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).
* 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>
* Here: ''a''<sub>1</sub> = 2 and ''a''<sub>2</sub> = 3 and ''b''<sub>1</sub> = -1 and ''b''<sub>2</sub> = 1
* The line passes through the points> (''b''<sub>1</sub>, ''b''<sub>2</sub>)=(-1,1) and (b<sub>1</sub>+a<sub>1</sub>,b<sub>2</sub>+a<s$
* The parametric form of this line is:
*:x(t) = {-1}+{2}t
*:y(t) = {1}+{3}t
* 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 ==
* James Stewart (2012), ''Calculus: Early Transcendentals'', edition 7E, Brooks/Cole. ISBN 978-0-538-49790-9
== External links ==
* http://www.columbia.edu/itc/sipa/math/linear.html
* http://www.math.okstate.edu/~noell/ebsm/linear.html
* http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf#page=55
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[[Category: Mathematics]]
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