the slope goes by several names
• average rate of change
• rate of change
• deltaY over deltaX
• Δy over Δx
• rise over run
• gradient
• constant of proportionality
• constant of variation
however, is the same cat wearing different costumes.
[tex]\begin{array}{|c|c|ll} \cline{1-2} x&y\\ \cline{1-2} 7&11\\ 8&13\\ 9&15\\ 10&17\\ \cline{1-2} \end{array} \begin{array}{llll} \\[-2.8em] \leftarrow \textit{let's use this point}\\ \leftarrow \textit{and this one} \end{array}\qquad \qquad (\stackrel{x_1}{7}~,~\stackrel{y_1}{11})\qquad (\stackrel{x_2}{8}~,~\stackrel{y_2}{13})[/tex]
[tex]\stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{13}-\stackrel{y1}{11}}}{\underset{run} {\underset{x_2}{8}-\underset{x_1}{7}}}\implies \cfrac{2}{1}\implies 2 \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{11}=\stackrel{m}{2}(x-\stackrel{x_1}{7}) \\\\\\ y-11=2x-14\implies \stackrel{\textit{constant of variation}}{y = \stackrel{\stackrel{m = k}{\downarrow ~\hfill }~\hfill }{2 x-3}}[/tex]
