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==Slope==
[[File:Slope picture.svg|thumb|right|128px|The slope of a line is the ratio <math>\tfrac{\Delta y}{\Delta x}</math> between a change in {{mvar|x}}, denoted {{math|Δ''x''}}, and the corresponding change in {{mvar|y}}, denoted {{math|Δ''y''}}]]
The [[slope (mathematics)|slope]] of a nonvertical line is a number that measures how steeply the line is slanted (rise-over-run). The [[first derivative]] of a linear function, inIf the sense of calculus,line is exactly this slope of the graph of the function.linear Forfunction {{math|1=''f''(''x'') = ''ax'' + ''b''}}, this slope and derivative is given by the constant {{mvar|a}}. Linear functions can be characterized as the only real-valued functions that are defined on the entire real line and have a [[constant function|constant]] derivative.
InThe calculus,slope measures the derivative of a general function measures itsconstant rate of change. Becauseof a linear function {{<math|1=''>f''(''x'')</math> =per ''ax''unit +change in ''bx''}} has a constant rate of change {{mvar|a}}, it has the property that: whenever the input {{mvar|x}} is increased by one unit, the output changes by {{mvar|a}} units.: If<math>f(x{+}1)=f(x)+a</math>, and more generally <math>f(x{{mvar|a}+}\Delta isx)=f(x)+a\Delta positive,x</math> thisfor willany causenumber the<math>\Delta valuex</math>. ofIf the functionslope tois increasepositive, while if {{mvar|<math>a}} is> negative it0</math>, will causethen the valuefunction to<math>f(x)</math> decrease.is More generally,increasing; if the<math>a input< increases by some other amount0</math>, {{mvar|c}},then the<math>f(x)</math> outputis changes by {{math|''ca''}}.decreasing
The fundamental idea ofIn [[differential calculus|calculus]], isthe thatderivative anyof [[differentiablea general function|smooth]] measures its rate of change. A linear function <math>f(x)=ax+b</math> (nothas necessarilya linear)constant canrate beof closelychange [[linearequal approximation|approximated]]to byits a unique linear function nearslope {{mvar|a}}, givenso pointits <math>x=c</math>. The [[derivative]] <math>f'(a)</math> is the slope of this linearconstant function, and the approximation is: <math>f(x) \approx f,'(c)(x{-}c)+f(c)=a</math>. for <math>x\approx c</math>.
The fundamental idea of differential calculus is that any [[differentiable function|smooth]] function <math>f(x)</math> (not necessarily linear) can be closely [[linear approximation|approximated]] near a given point <math>x=c</math> by a unique linear function. The [[derivative]] <math>f\,'(c)</math> is the slope of this linear function, and the approximation is: <math>f(x) \approx f\,'(c)(x{-}c)+f(c)</math> for <math>x\approx c</math>. The graph of the linear approximation is the [[tangent line]] of the graph <math>y=f(x)</math> at the point <math>(c,f(c))</math>. The derivative slope <math>f\,'(c)</math> generally varies with the point ''c''. Linear functions can be characterized as the only real functions whose derivative is constant: if <math>f\,'(x)=a</math> for all ''x'', then <math>f(x)=ax+b</math> for <math>b=f(0)</math>.
==Slope-intercept, point-slope, and two-point forms==
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