Begin by writing an expression that models a single year. Then, consider how that expression could change to account for the other time frames using exponents.
$125
Practice makes perfect
To find the budget from 2 years ago, let's create a model of the budget using some concrete examples.
Creating a Model Expression
Recall that the budget started as $ 500. The budget then doubles each year.
0 Year(s)
1 Year(s)
2 Year(s)
3 Year(s)
Total
500
2* 500
2* 2* 500
2* 2* 2* 500
Using Exponents
2^0* 500
2^1* 500
2^2* 500
2^3* 500
Compare the number of years with the exponents in the final expressions. Can you find a pattern? In each of the final expressions, there is an exponent whose value matches the number of years that have passed. Therefore, if y years have passed, we would get the following expression.
2^y* 500
Using this expression, we can find the budget for any number of years! Let's use this expression to find the budget for two years ago.
The Budget 2 Years Ago
To use our expression, we first need to know what value to use. We have established that when y=0, we have the initial budget. When the value of y is positive, then we have the budget y years in the future. Therefore, to go backwards in time, we need to use a negative value. Thus, the number we need to use is - 2.
