Linear function (calculus): Difference between revisions

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{{db-a10|article=Linear function}}

<p>Definition: AIn mainstream mathematics, a <strong>linear function</strong> is a [[Function_(mathematics)|function]] whose graph is a slanted¹ 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>

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<ul>

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

&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>.)

<ul>

<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>

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<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, weit getfollows 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>

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<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 weone multiplymultiplies <em>A</em>, <em>B</em> and <em>C</em> by a factor <em>k</em>, wethe willcoefficients havechange, but the same line remains the same.

<ul>

<li>Example: withFor <em>k</em>=3, we haveit 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: withFor <em>k</em>={{Fraction|-1|&pi;}}, weit havefollows that -3&pi;<em>x</em>+2&pi;<em>y</em>+&pi;=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>

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<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 we change the value of either or both of the coefficient letters <em>a</em> and <em>b</em> are changes, wethe result getis 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. and if we substituteSubstituting <em>x</em>=0 into the linear function <em>y</em>(<em>x</em>)=<em>a</em><em>x</em>+<em>b</em>, weit getfollows 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. and if we substituteSubstituting <em>y</em>=0 into the linear function and solvesolving (backwards) for <em>x</em>, weit follows getthat: <br />

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0=<em>a</em>•<em>x</em>+<em>b</em> <br />

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <em>a</em>•<em>x</em>+<em>b</em>=0 <br />

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <em>a</em>•<em>x</em> = -<em>b</em> <br />

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <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. ForMoving everyfrom increaseany inpoint 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>

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Example: <math>{{X}} = ({-1},{1}) + t({2},{3})</math> &nbsp;

<ul>

<li>We haveHere: <big><em>a</em>₁ = 2</big> &nbsp;and&nbsp; <big><em>a</em>₂ = 3</big> &nbsp;and&nbsp; <big><em>b</em>₁ = -1</big> &nbsp;and&nbsp; <big><em>b</em>₂ = 1</big> &nbsp;</li>

<li>The line passes through the points> <big>(<em>b</em>₁, <em>b</em>₂)=(-1,1)</big> and <big>(b₁+a₁,b₂+a₂)=(1,4)</big>&nbsp; <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 />&nbsp; </li>

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</div>

¹ Slanted meaning neither vertical nor horizontal.

<h1>References</h1>

* 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

* http://www.math.okstate.edu/~noell/ebsm/linear.html

* http://www.columbia.edu/itc/sipa/math/linear.html