Linear function (calculus): Difference between revisions

Revision as of 10:16, 22 June 2013

Revision as of 12:05, 28 June 2013

Line 1:

{{confuse|linear map}}

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 (geometry)|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]] in which the variable {{mvar|x}} has degree at most one, i.e. <math>f(x)=ax+b</math>.<ref>Stewart 2012, p. 24</ref> Here {{mvar|x}} is the variable. The [[graph of a function|graph]] of a linear function, that is, the set of all points whose coordinates have the form ({{mvar|x}}, {{mvarmath|''f''(''x'')}}), is a line, which is why this type of [[Function (mathematics)|function]] is called ''linear''. Some authors, for various reasons, also require that the coefficient of the variable (the {{mvar|a}} in {{mvar|ax + b}}) should not be zero.<ref>{{harvnb|Swokowski|1983|loc=pgp. 34}} is but one of many well known references that could be cited.</ref> This requirement can also be expressed by saying that the degree of the polynomial defining the function is exactly one, or by saying that the line which is the graph of a linear function is a ''slanted'' line (neither vertical nor horizontal). This requirement will not be imposed in this article, thus constant functions, <math>f(x) = b</math>, will be considered to be linear functions (their graphs are horizontal lines).

The [[Domain_of_a_function|___domain]] or set of allowed values for {{mvar|x}} of a linear function is the entire set of [[real number]]s {{math|'''R'''}}, or whatever [[field (mathematics)|field]] that is in use. This means that any (real) number can be substituted for {{mvar|x}}.

The [[graph of a function|graph]] of a linear function is a [[line (geometry)|line]] on the [[Cartesian plain]] (if over real numbers). Because two different points determine a line, it is enough to substitute two different values for {{mvar|x}} in the linear function and determine {{mvarmath|''f''(''x'')}} for each of these values. This will give the coordinates of two different points that lie on the line. Because {{mvar|f}} is a function, this line will not be vertical. If the value of either or both of the coefficient letters {{mvar|a}} and {{mvar|b}} are changed, a different line is obtained.

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

A the graph of a nonconstant linear function has exactly one intersection point with the {{mvar|x}}-axis. This point is {{math|({{sfrac|−''b''|''a''}}, 0)}}. From this, it follows that a nonconstant linear function has exactly one [[Zero_of_a_functionzero of a function|zero]] or root. That is, there is exactly one solution to the equation {{math|1=''ax'' + ''b'' = 0}}. The zero is {{math|1=''x'' =}}&nbsp;{{sfrac|−''b''|''a''}}.

The points on a line have coordinates which can also be thought of as the solutions of [[linear equation]]s in two variables (the equation of the line). These solution sets define functions which are linear functions. This connection between linear equations and linear functions provides the most common way to produce linear functions.

== Linear function and linear equation ==

There are many standard forms of linear equations, but only three will be examined to see how they are related to linear functions:

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

*[[#Slope-intercept form|Slope-intercept form]]

InThe [[equation]] {{math|1=''y'' = ''ax'' + ''b''}} is referred to as the slope-intercept form of a [[linear equation]]. In this form, the variable is {{mvar|x}}, and {{mvar|y}}, is the value of the function. It also has two coefficients, {{mvar|a}} and {{mvar|b}}. In this instance, the fact that the values of {{mvar|y}} depend on the values of {{mvar|x}} is an expression of the functional relationship between them. To be very explicit, the linear equation is expressing the equality of values of the dependent variable {{mvar|y}} with the functional values of the linear function <math>f(x) = ax + b</math>, in other words <math>y = f(x)</math> for this particular linear function {{mvar|f}}.

*[[#General form|General form]]

*[[#Parametric form|Parametric form]]

If the linear function {{mvar|f}} is given, the linear equation of the graph of this function is obtained by ''defining'' the variable {{mvar|y}} to be the functional value {{mvarmath|''f''(''x'')}}, that is, setting <math>y = f(x) = ax + b</math> and suppressing the functional notation in the middle. Starting with a linear equation, one can create linear functions, but this is a more subtle operation and must be done with care. Why this is so is not immediately apparent when the linear equation has the slope-intercept form, so this discussion will be postponed. For the moment observe that if the linear equation has the slope-intercept form, then the expression that the dependent variable {{mvar|y}} is equal to is the linear function whose graph is the line satisfying the linear equation.

== Slope-intercept form ==

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

The constant {{mvar|b}} is the so-called {{mvar|y}}-intercept. It is the {{mvar|y}}-value at which the line intersects the {{mvar|y}}-axis. The coefficient {{mvar|a}} is the [[slope]] of the line. This measures of the rate of change of the linear function associated with the line. Since {{mvar|a}} is a constant, this rate of change is constant. Moving from any point on the line to the right by one unit (that is, increasing {{mvar|x}} by 1), the {{mvar|y}}-value of the point's coordinate changes by {{mvar|a}}. This is expressed functionally by the statement that <math>f(x+1) = f(x) +a</math> when <math>f(x) = ax + b</math>.

The slope-intercept form of a linear equation is an equation in two variables of the form

:<math> y=ax+b </math>.

In the slope-intercept form, the variable is {{mvar|x}}, and {{mvar|y}}, is the value of the function. It also has two coefficients, {{mvar|a}} and {{mvar|b}}. In this instance, the fact that the values of {{mvar|y}} depend on the values of {{mvar|x}} is an expression of the functional relationship between them. To be very explicit, the linear equation is expressing the equality of values of the dependent variable {{mvar|y}} with the functional values of the linear function <math>f(x) = ax + b</math>, in other words <math>y = f(x)</math> for this particular linear function {{mvar|f}}.

If the linear function {{mvar|f}} is given, the linear equation of the graph of this function is obtained by ''defining'' the variable {{mvar|y}} to be the functional value {{mvar|f(x)}}, that is, setting <math>y = f(x) = ax + b</math> and suppressing the functional notation in the middle. Starting with a linear equation, one can create linear functions, but this is a more subtle operation and must be done with care. Why this is so is not immediately apparent when the linear equation has the slope-intercept form, so this discussion will be postponed. For the moment observe that if the linear equation has the slope-intercept form, then the expression that the dependent variable {{mvar|y}} is equal to is the linear function whose graph is the line satisfying the linear equation.

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

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

The coefficient {{mvar|a}} is the [[slope]] of the line. This measures of the rate of change of the linear function associated with the line. Since {{mvar|a}} is a constant, this rate of change is constant. Moving from any point on the line to the right by one unit (that is, increasing {{mvar|x}} by 1), the {{mvar|y}}-value of the point's coordinate changes by {{mvar|a}}. This is expressed functionally by the statement that <math>f(x+1) = f(x) +a</math> when <math>f(x) = ax + b</math>.

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

== General form ==

[[Image: wiki_linearna_funkcija_stand1.png|180px|right]]

If the linear equation in the general form

The general form for the equation of a line is:

: <math> Ax+By=C. </math>

Whenhas <math> B \ne 0 </math>, thisthen equationit may be solved for the variable {{mvar|y}} and thus used to define a linear function (namely, <math> y = - \tfrac{A}{B} x + \tfrac{C}{B} = f(x)</math>). While all lines have equations in the general form, only the non-vertical lines have equations which can give rise to linear functions.

The general form has 2 variables {{mvar|x}} and {{mvar|y}} and 3 coefficients {{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. The linear function obtained from this form is unique since it depends only on the coordinates of the points on the line. For example, {{math|1=3''x'' − 2''y'' = 1}} and {{math|1=9''x'' − 6''y'' = 3}} are general forms of the equation of the same line which is associated with the linear function <math> f(x) = \tfrac{3}{2} x - \tfrac{1}{2} </math>.

This general form is used mainly in geometry and in systems of two linear equations in two unknowns.

==Parametric form==

[[Image: wiki_linearna_funkcija_par1.png|300px|right]]

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 {{mvar|y}}, and four coefficients {{math|''a''₁}}, {{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}} and {{math|1=''b''₂ = 1}}

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

* 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: {{math|1=−3''x'' + 2''y'' = 5}}.

== Notes ==