Algebra
Exponential functions
Exponential decay
20 practice questions
2 video lessons
Theory + worked examples
Theory
Exponential decay decreases by a constant percent each period:
\[y=a(1-r)^t,\]
where \(a\) is the initial amount and \(r\) the decay rate. The base \(0<1-r<1\).
Each period multiplies by \(1-r\) β the amount approaches zero.
Decay curves downward toward zero.
The decay model.
The decay model:
\[y=a(1-r)^t\]
The base \(1-r\) is between 0 and 1 for decay.
How to model decay
- Identify the initial amount \(a\).
- Convert the decay rate to a decimal \(r\).
- Write \(y=a(1-r)^t\).
- Substitute \(t\) to predict.
Example 1 β Build a model
A \(\$20{,}000\) car loses \(15\%\) of its value yearly. Write the model.
Solution
Use \(y=a(1-r)^t\).
| \(y\) | \(=\) | \(20000(0.85)^t\) |
Example 2 β Evaluate
Find the value after \(2\) years for \(y=20000(0.85)^t\).
Solution
Substitute \(t=2\).
| \(y\) | \(=\) | \(20000(0.85)^2\) |
| \(=\) | \(\$14{,}450\) |
Example 3 β Half-life idea
A sample halves each hour from \(80\) g. Write the model.
Solution
Halving means base \(0.5\).
| \(y\) | \(=\) | \(80(0.5)^t\) |
Example 4 β Decay rate to base
A \(15\%\) decay rate gives what base?
Solution
Base is \(1-r\).
| \(1-0.15\) | \(=\) | \(0.85\) |
Common pitfalls
The base is \(1-r\), so \(15\%\) decay gives \(0.85\).
Decay approaches zero but never reaches it.
Multiply, don't subtract, each period.
Frequently asked questions
What is exponential decay?
Decrease by a constant percent each period.
What is the decay model?
\(y=a(1-r)^t\).
What base gives \(20\%\) decay?
\(0.80\).
Does decay reach zero?
No β it approaches zero as an asymptote.
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