Linear function (calculus): Difference between revisions

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<p>Definition: In mainstream mathematics, a <strong>linear function</strong> is a [[Function_(mathematics)|function]] whose graph is a slanted line in the plane.</p>

<p>&nbsp;&nbsp;&nbsp;<small>In advanced mathematics, a [[linear mapping]] is sometimes referred to as a linear function, but this is not the mainstream definition. A linear function as defined here is not a linear mapping.</small></p>

<p>It should be noted that a horizontal line is usually not considered to be linear function, but is just called a <em>constant function</em> and a vertical line is not a function since it does not pass the [[Vertical_line_test|vertical line test]].</p>

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.

<h1>== Properties of Linear Functions</h1> ==

* A linear function is a [[polynomial]] function of first degree with one independent variable <em>х</em>, i.e.

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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;*: <math>y(x)=ax+b</math> &nbsp;&nbsp;(In the USA, the coefficient letter <em>a</em> is usually replaced by <em>m</em>.)

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<li>**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.</li>

<ul>

**In the function, <liem>x</em>A linear function is a [[polynomial]] function of first degree with onethe independent variable and <em>хy</em>, i.e.is the dependent variable.<br/li>

** <li>The [[Domain_of_a_function|___domain]] or set of allowed values for <em>x</em> of a linear function is &nbsp;<math>\Re</math>&nbsp; (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>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <math>y(x)=ax+b</math> &nbsp;&nbsp;(In the USA, the coefficient letter <em>a</em> is usually replaced by <em>m</em>.)

<li>**The set of all points: (<em>x</em>,<em>y</em>(<em>x</em>)) is the line.</li>

<ul>

<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).</li>

<li>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.</li>

<li>In the function, <em>x</em> is the independent variable and <em>y</em> is the dependent variable.</li>

<li>The [[Domain_of_a_function|___domain]] or set of allowed values for <em>x</em> of a linear function is &nbsp;<math>\Re</math>&nbsp; (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>

<li>The set of all points: (<em>x</em>,<em>y</em>(<em>x</em>)) is the line.</li>

<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).</li>

</ul>

</li>

</ul></td>

<td align="center" style="border: 1px solid #444444;">[[Image: wiki_linear_function.png|160px]]</td></tr>

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<strong>Graph of the linear function: <em>y</em>(<em>x</em>)=-<em>x</em>+2</strong>

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<ul>

<li>Because the graph of a linear function is a slanted line:

<ul>

<li>A linear function has exactly one intersection point with the <em>у</em>-axis. This point is (0,<em>b</em>).</li>

<li>A linear function has exactly one intersection point with the <em>х</em>-axis. This point is ({{Fraction|-b|a}},0). </li>

<li>From this, it follows that a 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}}.</li>

</ul>

</li>

<li>There are three standard forms for linear functions.

<ul>

<li>[[#General Form|General form]]</li>

<li>[[#Slope-Intercept Form|Slope-intercept form]]</li>

<li>[[#Vector-Parametric Form|Vector-parametric form]]</li>

</ul>

</li>

<li>Quite often the term <em>[[linear equation]]</em> is used interchangably with <em>linear function</em>. While a linear function in general form is indeed a linear equation, the opposite is definitely not true.</li>

</ul>

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<strong>[[Image: wiki_linear_function.png|right|160px Graph of the linear function: <em>y</em>(<em>x</em>)=-<em>x</em>+2</strong>]]

<h1>General Form</h1>

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<p><math> Ax+By=C </math> &nbsp;&nbsp;&nbsp; <big>where</big> &nbsp;&nbsp; <math> A \ne 0 </math> &nbsp;and&nbsp; <math> B \ne 0 </math>.</p>

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* Because the graph of a <li>Alinear 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>

<p style=" font-size:1.2em; font-weight:bold">Properties of the general form</p>

* A <li>Anonconstant linear function has exactly one intersection point with the <em>х</em>-axis. This point is ({{Fraction|-b|a}},0). </li>

<ul>

* <li>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}}.</li>

<li>The general form has 2 variables <em>x</em> and <em>у</em> and 3 coefficient letters A, B, and C. </li>

<li>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. </li>

<li>*There are three standard forms for linear functions.

<li>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.

<li>**[[#General Form|General form]]</li>

<li>**[[#Slope-Intercept Form|Slope-intercept form]]</li>

<li>**[[#Vector-Parametric Form|Vector-parametric form]]</li>

<h1>== General Form</h1> ==

<p><math> Ax+By=C </math> &nbsp;&nbsp;&nbsp; <big>where</big> &nbsp;&nbsp; <math> A \ne 0 </math> &nbsp;and&nbsp; <math> B \ne 0 </math>.</p>

===Properties of the general form===

* <li>The general form has 2 variables <em>x</em> and <em>у</em> and 3 coefficient letters A, B, and C. </li>

* <li>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. </li>

* <li>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>

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<h1>Video - How to sketch the graph of <em>A</em><em>x</em>+<em>B</em><em>y</em>=<em>C</em></h1>

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<p>[[Image: wiki_linear_standard_xy_en.ogv ]]</p></td></tr></table>

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<h1>Video - How to sketch the graph of <em>y</em>(<em>x</em>)=<em>a</em><em>x</em>+<em>b</em></h1>

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<p>[[Image: wiki_linear_explicit_zoom_out.ogv ]]</p></td></tr></table>

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