Linear function (calculus): Difference between revisions

Revision as of 17:18, 19 June 2013

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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 2012, p.  23</ref>

== Properties of linear functions ==

[[Image:wiki_linear_function.png|thumb|right|Graph of the linear function: {{math|1=''y''(''x'') =- −''x'' + 2'''}}]]<!-- people, find an SVG image please instead of this abomination -->

A linear function is a [[polynomial]] function of first degree with one independent variable ''х''{{mvar|x}}, i.e. <math>y(x)=ax+b</math>.<ref>Stewart 2012, page p. 24</ref> Here ''{{mvar|x''}} is the independent variable and ''{{mvar|y''}} is the dependent variable.

The [[Domain_of_a_function|___domain]] or set of allowed values for ''{{mvar|x''}} of a linear function is the entire set of [[real numbers,number]]s <math>\mathbb{{math|'''R'''}}</math>. This means that any real number can be substituted for ''{{mvar|x''}}.

The set of points {{math|(''x'', ''y''(''x''))}} is the line that is the [[graph of a function|graph]] of the function. Because two points determine a line, it is enough to substitute two different values for ''{{mvar|x''}} in the linear function and determine ''{{mvar|y''}} for each of these values. Because {{math|''y''(''x'')}} is a function, the line cannot be vertical.

Because the graph of a linear function is a nonvertical line, a linear function has exactly one intersection point with the ''у''{{mvar|y}}-axis. This point is {{math|(0, ''b'')}}.

A nonconstant linear function has exactly one intersection point with the ''х''{{mvar|x}}-axis. This point is {{math|({{Fractionsfrac|-−''b''|''a''}},0 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 {{math|1=''a''''xax'' + ''b'' = 0}}. The zero is {{math|1=''x'' =}}&nbsp;{{Fractionsfrac|-−''b''|''a''}}.

There are three standard forms for linear functions.

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== Slope-intercept form ==

[[Image: wiki_linearna_funkcija_eks1.png|thumb||right]]<!-- are PNG and a translit from Ukrainian necessary? -->

The slope-intercept form of a linear function is an equation of the form

:<math> y(x)=ax+b </math> or <math> y=ax+b </math>

where <math>{ a \ne 0} </math>. The slope-intercept form has 2 variables ''{{mvar|x''}} and ''у''{{mvar|y}} and 2 coefficientcoefficients letters ''а''{{mvar|a}} and ''{{mvar|b''}}.

The slope-intercept form is also called the ''explicit form'' because it defines {{math|''y''(''x'')}} explicitly (directly) in terms of ''{{mvar|x''}}.

The slope-intercept form of a linear function is unique. That is, if the value of either or both of the coefficient letters ''{{mvar|a''}} and ''{{mvar|b''}} are changed, a different function is obtained.

The constant ''{{mvar|b''}} is the so-called ''у''{{mvar|y}}-intercept. It is the ''{{mvar|y''}}-value at which the line intersects the ''{{mvar|y''}}-axis.

The coefficient ''а''{{mvar|a}} is the [[slope]] of the line, which measures of the rate of change of the linear function. Since ''{{mvar|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 ''{{mvar|x''}} by 1), the ''{{mvar|y''}}-value of the function changes by ''{{mvar|a''}}.

For example, the slope-intercept form <math> y(x)=-2x+4 </math> has {{math|1=''a'' = -2−2}} and {{math|1=''b'' = 4}}. The point {{math|1=(0, ''b'') = (0,4 4)}} is the intersection of the line and the ''у''{{mvar|y}}-axis, the point {{Fractionmath|-1=({{sfrac|−''b''|а''a''}},0 0)}} = ({{Fractionsfrac|-4−4|-2−2}},0 0) = (2,0 0)}} is the intersection of the line and the ''х''{{mvar|x}}-axis, and {{math|1=''аa'' = -2−2}} is the slope of the line. For every step to the right (''х''{{mvar|x}} increases by 1), the value of ''у''{{mvar|y}} changes by -2−2 (goes down).

== General form ==

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where <math> A \ne 0 </math>.

The general form has 2 variables ''{{mvar|x''}} and ''у''{{mvar|y}} and 3 coefficient letterscoefficients {{mvar|A}}, {{mvar|B}}, and {{mvar|C}}.

This form is not unique. If one multiplies ''{{mvar|A''}}, ''{{mvar|B''}} and ''{{mvar|C''}} by a constant factor ''{{mvar|k''}}, the coefficients change, but the line remains the same. For example, {{math|1=3''x''- − 2''y'' = 1}} and {{math|1=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.

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The [[parametric form]] of a line consists of two equations:

:<math>x(t) = {b_1}+{a_1}t </math>

*:<math>y(t) = {b_2}+{a_2}t </math>

where <math> a_1 \ne 0 </math>.

The parametric form has one [[parameter]] ''{{mvar|t''}}, two variables ''{{mvar|x''}} and ''у''{{mvar|y}}, and four coefficients а{{math|''a''₁}}, а{{math|''a''₂}}, {{math|''b''₁}}, and {{math|''b''₂}}. The coefficients are not unique, but they are related.

The line passes through the points {{math|(''b''₁, ''b''₂)}} and {{math|(''b''₁ + ''a''₁, ''b''₂ + ''a''₂)}}.

The vector-parametric form is used in engineering (it is simple to model the path from one point to another point with ''{{mvar|t''}}=time). Engineers tend to use parametric notation and the letter ''{{mvar|t''}} for the parameter; mathematicians use vector notation and the letter &lambda;.

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: {{math|1=''a''₁ = 2}} and {{math|1=''a''₂ = 3}} and {{math|1=''b''₁ = -1−1}} and {{math|1=''b''₂ = 1}}

* The line passes through the points> {{math|1=(''b''₁,  ''b''₂) = (-1−1,1 1)}} and {{math|(''b''₁ + ''a''₁, ''b''₂ + ''a''<s$sub>2</sub>)}}

* The parametric form of this line is:

*:<math>x(t) = {-1}+{2}t </math>

*:<math>y(t) = {1}+{3}t </math>

* The slope-intercept form of this line is: {{math|1=''y''(''x'') = 1,.5''x'' + 2,.5 }} (solve the first parametric equation for ''{{mvar|t''}} and substitute in$

* One general form of this line is: -3{{math|1=−3''x'' + 2''y'' = 5}}.

== Notes ==

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* http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

{{Polynomials}}

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[[Category: MathematicsCalculus]]

[[Category:Polynomials]]