Question

The function f(x) = 20,000(0.8)^x, represented in the table, shows the value of a car (in dollars) in a certain state where x is the number of years after 2015. Find the average rate of change from 2015 to 2019 and interpret its meaning in terms of the context.

Answer 1

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

however, is the same cat wearing different costumes.

to get the slope of any straight line, we simply need two points off of it, let's use those two in the picture below.

`(\stackrel{x_1}{2015}~,~\stackrel{y_1}{20000})\qquad (\stackrel{x_2}{2019}~,~\stackrel{y_2}{8192}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{8192}-\stackrel{y1}{20000}}}{\underset{\textit{\large run}} {\underset{x_2}{2019}-\underset{x_1}{2015}}} \implies \cfrac{ \stackrel{ value~\$ }{-11808 }}{ \underset{ year }{4} } \implies -2952 ~~ \cfrac{\$}{year} \\\\\\ \textit{average depreciation of -2952 per year from 2015 to 2019}`