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# Exponential vs. linear models: table

CCSS.Math:

## Video transcript

the table represents the cost of buying a small piece of land in a remote village since the year 1990 which kind of function best models this relationship and I'm using this is an example from the Khan Academy exercises and we're really trying to pick between whether a linear model or linear function models this relationship or an exponential model or exponential function will model this relationship so like always pause this video and see if you can figure it out on your own alright so now let's think about this together so as the time goes by on this the time variable right over here we see that we keep incrementing it by to go from 0 to 2 2 to 4 4 to 6 so on and so forth it keeps going up by 2 so if this is a linear relationship given that our change in time is constant our change in cost should increase by a constant amount does it have to be this constant but has to be a constant amount if we're dealing with an exponential relationship we would multiply by the same amount for a constant change in time let's see what's going on here let's just first look at the difference between these numbers to go from 30 to 36 point 9 you would have to add 6.9 now to go from thirty six point nine to forty four point one what do you have to add you have to add seven point two and now to go from 44 point one to 51 point one you would have to add seven now to go to 51 point one to fifty seven point nine you are adding six point eight six point eight and then finally going from 57.9 to sixty five point one let's see this is almost eight seven point one this is what seven point two we're adding plus seven point two so you might say anyway we're not adding the exact same amount every time but remember this is intended to be data that you might actually get from real from a real-world situation and the data that you get from a real-world situation will never be exactly a linear model or exactly an exponential model but every time we add two years it does look like we're getting pretty close to adding \$7,000 in cost 6.9 is pretty close to seven that's pretty close to seven that is seven is pretty close to seven that's pretty close to seven so this is looking like a linear model to me you could test whether it's an exponential model you see well what factor am i multiplying each time but that doesn't seem to be as this doesn't seem to be growing exponentially it doesn't seem like we're multiplying by the same factor every time it seems like we're multiplying by a slightly lower factor as we get to higher cost so the linear model seems to be a pretty good thing if I see every time I increase by two years I'm increasing cost by six point nine or seven point two or seven it's pretty close to seven so it's not exactly the cost but the model predicts it pretty well if you were to plot these on a on a on a on a coordinate plane and try to connect the dots you could it would look pretty close to a line or you could draw a line that gets pretty close to those dots