# Linear Interpolation and Extrapolation

## Key Questions

• Linear extrapolation is used to answer certain questions like:

(insert name here) and (insert name here) start a (insert business thingy here) business and start with $2,000 per year. They get more customers each year. For each they they continue to work, they gain$2,000 more than the last year, and then they take away the cost of materials, $500. The equation is: $2 , 000 X - 500 = Y$Pretend that there is a graph that lasts 20 years. Now, figure out how much money they will have after 30 years. By the way, that's: $2 , 000 \left(30\right) - 500 = Y$$60 , 000 - 500 = Y$$59 , 500 = Y$• Let's say you have two points: $\left(x , y\right)$co-ordinates. An equation of the line will be of the form $y = m \cdot x + b$where $m =$the slope and $b =$the so-called $y -$intercept. Example : Let's take $\left(- 6 , 0\right)$and $\left(4 , 5\right)$graph{0.5x+3 [-9.61, 12.89, -2.795, 8.455]} Then first we determine the slope $m$Difference in $y = \Delta y = 5 - 0 = 5$Difference in $x = \Delta x = 4 - \left(- 6\right) = 10$To find the slope we divide $\frac{\Delta y}{\Delta x} = \frac{5}{10} = \frac{1}{2}$We fill this in in one of the points to get $b$$y = m x + b \to 0 = \frac{1}{2} \cdot \left(- 6\right) + b \to b = + 3$So the equation goes $y = \frac{1}{2} \cdot x + 3$And you can fill in any $x$to get the $y$Check (allways check!) with the other point $\left(4 , 5\right)$: $\frac{1}{2} \cdot 4 + 3 = 5\$ is OK