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== Properties ==
A linear function is a [[polynomial function]] in which the [[variable (mathematics)|variable]] {{mvar|x}} has degree at most one, which means it is of the form<ref>Stewart 2012, p. 24</ref>
:{{math|1=''f''(''x'') = ''ax'' + ''b''|size=120%}}.
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 {{math|(''x'', ''f''(''x''))}}, is a [[line (geometry)|line]] on the [[Cartesian plane]] (if over real numbers). That 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 {{math|''ax'' + ''b''}}) should not be zero.<ref>{{harvnb|Swokowski|1983|loc=p. 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|1=''f''(''x'') = ''b''}}, will be considered to be linear functions (their graphs are horizontal lines).
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