Question

How does the rate of change vary from point to point?

Answer 1

$\bf \begin{array}{|cc|ll} \cline{1-2} x&y\\ \cline{1-2} 40&32\\ 28&16\\ 16&12\\ \cline{1-2} \end{array}~\hfill \stackrel{\textit{average rate of change}}{slope} \\\\[-0.35em] ~\dotfill\\\\ (\stackrel{x_1}{40}~,~\stackrel{y_1}{32})\qquad (\stackrel{x_2}{28}~,~\stackrel{y_2}{16}) \\\\\\ slope = m\implies \cfrac{\stackrel{rise}{ y_2- y_1}}{\stackrel{run}{ x_2- x_1}}\implies \cfrac{16-32}{28-40}\implies \cfrac{-16}{-12}\implies \boxed{\cfrac{4}{3}} \\\\[-0.35em] ~\dotfill$

$\bf (\stackrel{x_1}{28}~,~\stackrel{y_1}{16})\qquad (\stackrel{x_2}{16}~,~\stackrel{y_2}{12}) \\\\\\ slope = m\implies \cfrac{\stackrel{rise}{ y_2- y_1}}{\stackrel{run}{ x_2- x_1}}\implies \cfrac{12-16}{16-28}\implies \cfrac{-4}{-12}\implies \boxed{\cfrac{1}{3}}$