# Modeling with Functions

## Key Questions

• If a value changes based on its relationship to some other value, then a function can (often) be used to model the change in that primary value.

Some examples might help.

Example 1
Suppose you purchase a $1000 Guaranteed Investment Certificate with an annual (compounded) return rate of 3% and you want to know the value of that GIC (after having held it for some number of years). We can model the value of your GIC using a function: $v \left(t\right) = 1000 \times {\left(1 + 0.03\right)}^{t}$(where $t$is the number of years you have owned the GIC). Even if you put in some value for $t$, the function is not really a monetary value (you can't buy anything with the number that comes out of the function, but it reflects or " models " that value. Example 2 If you drop an object the speed at which that object is falling (ignoring air resistance and assuming it doesn't hit something else) changes with the distance that the object has already fallen. Specifically, if the distance is measured in feet and the speed in feet per second, then we could model the speed of our falling object dependent upon the distance it has fallen by a function: $s \left(d\right) = 2 \times d$$2 \times d$is just a number, but it is a number we can think of as being related to a specific speed. In that sense, $s \left(d\right)$is a function that models speed depending upon distance ($d\$).