Exponential Decay
In short: Exponential decay is what happens when a quantity falls by one fixed percent per period, so each period removes less than the one before.
Algebra 1Functions and GraphsF-LE.A.1c
What is Exponential Decay?
A new truck sheds about 15% of its value every year, and each year’s loss is smaller in dollars because it comes out of a smaller value; this steady fading is exponential decay.
The model is \(A = P(1 – r)^t\), with starting amount \(P\), decay rate \(r\) as a decimal and decay factor \(1 – r\), which lies strictly between 0 and 1. The curve drops steeply at first, then levels off toward the x-axis without ever touching zero.
How to spot it
- A quantity drops by a percent each period: depreciation, cooling, fading or medicine leaving the body.
- The wording includes half-life, loses 8% per year or decreases by a factor of.
- A table shows outputs multiplied by a number less than 1 between equal steps.
- A graph plunges early on and then levels off near the horizontal axis, typical of exponential decay.
The method
- Turn the percent loss into a decimal and subtract it from 1, so 15% becomes the decay factor 0.85.
- Name the starting amount P and the count of periods t.
- Place the numbers in \(A = P(1 – r)^t\) to model the exponential decay.
- Work out the power first, then multiply, and round only at the very end.
Exponential Decay: solved example
Problem
A pickup truck purchased at $24,000 depreciates 15% every year. Find its value at the four-year mark, with cents included.
Solution
The decay factor is \(1 – 0.15 = 0.85\). Use \(A = 24{,}000(0.85)^4\).
\(0.85^4 = 0.52200625\), so \(A = 24{,}000 \times 0.52200625 = 12{,}528.15\). At the four-year mark the truck is worth $12,528.15.
Try it
Question 1
A 500 mg dose of medicine halves every 4 hours. How much remains after 12 hours?
Show the answer
12 hours is 3 half-lives: \(500\left(\frac{1}{2}\right)^3 = 62.5\) mg.
Question 2
What is the decay factor for an 8% decrease per period?
Show the answer
\(1 – 0.08 = 0.92\).
Exponential Decay vs. Exponential Growth
Same family of function, opposite direction. The factor tells you which: above 1 the amount climbs, below 1 it falls. A common slip is using 1.15 for a 15% drop; subtract the rate from 1 for a drop and add it to 1 for a rise.
Keep going
Lessons
Worksheets and quizzes
Related words
- Exponential Function
- Exponential Growth
- Percent Decrease
- Geometric Sequence
- Compound Interest
- Discount
Whole dictionary: A–Z · Algebra 1 · Functions and Graphs