a Since we have a decrease, the multiplier will be less than 1. The decimal form of 20 % is 0.2, and if we subtract this from 1 we can write the multiplier.
1-0.2=0.8
b The decrease in cost is 20 % per year. Therefore, if we multiply the cost of the item by the multiplier three times, we can obtain the cost three years from now. This results in using a positive exponent of 3.
1200* 0.8* 0.8 * 0.8 =1200(0.8)^3≈ 614
The cost in three years is about $614.
c To obtain the cost n years in the future, we have to multiply the cost by the multiplier n times which gives us a positive exponent. Conversely, to obtain the cost n year in the past, we have to use a negative exponent. With this information, we can calculate the cost two years ago.
1200(0.8)^(-2) = 1875
The cost two years ago was $1875.
d We are asked to write the equation to model the situation. In Part C we noticed that if we want to know what the price will be after n years, we should multiply the cost by the multiplier n times.
1200(0.8)^n
We also found that if we want to find to know what the price was n years ago, we should put - n instead of n in the exponent.
1200(0.8)^(- n)
In general, we should use the following exponential function.
f(x)=1200(0.8)^x
When x is positive, the expression above will tell us the price x years from now. However, for negative x this expression tells us what the price was in the past.
