Linear function (calculus): Difference between revisions

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Revision as of 19:54, 29 June 2018

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where we denote <math>a=-\tfrac{A}{B}</math> and <math>b=\tfrac{C}{B}</math>. That is, one may consider ''y'' as a dependent variable (output) obtained from the independent variable (input) ''x'' via a linear function: <math>y = f(x) = ax+b</math>. In the ''xy''-coordinate plane, the possible values of <math>(x,y)</math> form a line, the graph of the function <math>f(x)</math>. If <math>B=0</math> in the original equation, the resulting line <math>x=\tfrac{C}{A}</math> is vertical, and cannot be written as <math>y=f(x)</math>.

The features of the graph <math>y = f(x) = ax+b</math> can be interpreted in terms of the variables ''x'' and ''y''. The ''y''-intercept is the intialinitial value <math>y=f(0)=b</math> at <math>x=0</math>. The slope ''a'' measures the rate of change of the output ''y'' per unit change in the input ''x''. In the graph, moving one unit to the right (increasing ''x'' by 1) moves the ''y''-value up by ''a'': that is, <math>f(x{+}1) = f(x) + a</math>. Negative slope ''a'' indicates a decrease in ''y'' for each increase in ''x''.

For example, the linear function <math>y = -2x + 4</math> has slope <math>a=-2</math>, ''y''-intercept point <math>(0,b)=(0,4)</math>, and ''x''-intercept point <math>(2,0)</math>.