Creating and Using Exponential Functions in Context
An exponential function models a quantity that grows or shrinks by the same percent every period. It has the form y = a·bx, where a is the starting value and b is the growth factor.
A quantity that adds the same amount every period grows in a straight line. A quantity that multiplies by the same factor every period runs away from it, because each gain itself earns more gain. That is the whole difference between linear and exponential change, and it is why populations, interest, and radioactive decay all need a different kind of function than a line.
The function itself is y = a times b to the x. The letter a is the starting amount, the value when x is 0. The letter b is the growth factor, what you multiply by after one full period. For a quantity growing by r percent per period, b = 1 + r/100. For a quantity shrinking by p percent per period, b = 1 - p/100. Once you have a and b, every other value comes from plugging x into the exponent — no need to multiply year by year by hand.
This topic sits right next to linear functions and right before more complex growth models like compound interest with multiple periods per year. The same a times b to the x skeleton handles quarterly or monthly compounding — you just shrink the growth factor and multiply the number of periods, swapping x for a count of quarters or months instead of years. Recognizing which situations call for a changing-by-percent model versus a changing-by-fixed-amount model is the real skill being tested.
Key terms
- exponential function
- A function of the form y = a times b to the x, where the output changes by a constant percent each step.
- growth factor
- The number b you multiply by each period, equal to 1 + r/100 for growth or 1 - p/100 for decay.
- initial value
- The starting amount a, the value of the function when x equals 0.
- compounding period
- The fixed interval, such as a year or a quarter, after which the growth factor is applied once.
Watch it explained
- How it worksHow do the starting amount and the growth factor drop into the two slots of a times b to the x?
- Worked exampleFind the value after 10 years of 4,000 dollars invested at 6 percent growth per year.
- Worked exampleFind the function for a 200 gram sample that loses 15 percent of its mass every hour.
- Common mistakeWhy isn't a 20 percent rise followed by a 20 percent fall a return to where you started?
- Going furtherWhat changes inside the function when the growth is compounded quarterly instead of once a year?
Check yourself
End of topic.