Linear function (calculus): Difference between revisions

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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.

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

*[[#General form|General form]]

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

*[[#General form|General form]]

*[[#Parametric form|Parametric form]]

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

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

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

The slope-intercept form has twothe variables {{mvar|x}}, the ''independent'' variable, and {{mvar|y}}, andthe ''dependent'' variable. It also has two coefficients, {{mvar|a}} and {{mvar|b}}. TheIn slope-interceptthis forminstance, the fact that the values of {{mvar|y}} depend on the values of {{mvar|x}} is alsoan calledexpression of the ''functional relationship between them. To be very explicit, form''the becauselinear itequation definesis expressing the equality of values of the dependent variable {{mathmvar|''y''(''x'')}} explicitlywith the functional values of the linear function <math>f(directlyx) = ax + b</math>, in termsother ofwords <math>y = f(x)</math> for this particular linear function {{mvar|xf}}.

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 functionequation is unique. That is, if the value of either or both of the coefficient letters {{mvar|a}} and {{mvar|b}} are changed, a different functionline 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,. whichThis 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 functionpoint'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(x)=-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 ==