Algebra
Exponential functions
Exponential growth
20 practice questions
2 video lessons
Theory + worked examples
Theory
Exponential growth increases by a constant percent each period:
\[y=a(1+r)^t,\]
where \(a\) is the initial amount and \(r\) the growth rate. The base \(1+r>1\).
Each period multiplies by \(1+r\) β the amount grows faster and faster.
Growth curves upward at an increasing rate.
The growth model.
The growth model:
\[y=a(1+r)^t\]
Convert the percent to a decimal for \(r\).
How to model growth
- Identify the initial amount \(a\).
- Convert the growth rate to a decimal \(r\).
- Write \(y=a(1+r)^t\).
- Substitute \(t\) to predict.
Example 1 β Build a model
\(\$500\) grows \(4\%\) per year. Write the model.
Solution
Use \(y=a(1+r)^t\).
| \(y\) | \(=\) | \(500(1.04)^t\) |
Example 2 β Evaluate
Find the value after \(3\) years for \(y=500(1.04)^t\).
Solution
Substitute \(t=3\).
| \(y\) | \(=\) | \(500(1.04)^3\) |
| \(\approx\) | \(\$562.43\) |
Example 3 β Doubling
A colony doubles each hour from \(100\). Write the model.
Solution
Doubling means base \(2\).
| \(y\) | \(=\) | \(100(2)^t\) |
Example 4 β Growth rate to base
A \(7\%\) growth rate gives what base?
Solution
Base is \(1+r\).
| \(1+0.07\) | \(=\) | \(1.07\) |
Common pitfalls
The base is \(1+r\), so \(5\%\) growth gives \(1.05\).
Convert the percent to a decimal.
Multiply, don't add, each period.
Frequently asked questions
What is exponential growth?
Increase by a constant percent each period.
What is the growth model?
\(y=a(1+r)^t\).
What base gives \(5\%\) growth?
\(1.05\).
How is growth different from a linear increase?
Growth multiplies each period; linear adds a constant.
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