Exponential decay
Model exponential decay with y=ab^x where 0<b<1: identify the decay factor, calculate half-life, and graph decreasing exponential curves that approach but never reach zero.
Key Concepts
Exponential decay models a quantity that always decreases by the same percent over a period of time. It uses the function $f(x) = ab^x$, where 'a' is the starting amount and the decay factor 'b' is between 0 and 1 ($0 < b < 1$). This can also be written as $f(x) = ab^{ x}$ where $b 1$.
A car worth 25000 dollars depreciates by 15% each year. The model is $y = 25000(0.85)^t$. After 5 years, its value is $25000(0.85)^5 \approx 11092.63$ dollars. A pendulum's swing starts at 2 meters. Each swing is 90% of the previous one. The model is $y = 2(0.9)^s$. The 4th swing will be $2(0.9)^4 = 1.3122$ meters.
Imagine your phone battery draining. It starts full ('a') but loses a fraction ('b') of its remaining power each hour. The drop is steepest at the start, then slows as it gets closer to zero.
Common Questions
What distinguishes exponential decay from exponential growth?
In exponential decay the base b satisfies 0<b<1, so each multiplication by b reduces the quantity. In exponential growth b>1 and the quantity increases. Both have the form y=ab^x; the base determines whether the curve rises or falls as x increases.
How do you find the decay factor from a real-world decay problem?
If a quantity decreases by r percent per period, the decay factor is b=1-r expressed as a decimal. For example, a substance losing 20% per year has b=0.80. Substitute into y=ab^x where a is the initial amount and x is the number of periods elapsed.
What does the horizontal asymptote of an exponential decay graph represent?
The graph of y=ab^x approaches y=0 but never touches it, because multiplying any positive number by a fraction repeatedly reduces it but never reaches zero. This models real situations like radioactive decay where a substance diminishes but never fully disappears in finite time.