Exponential growth and decay - Higher
Money invested in a bank can generate two different types of interest. This arises when interest on an investment is calculated and added and then this interest payment also earns interest. occurs when interest is added to the balance at the end of a time period and interest is then calculated on the total of these at the end of the next time period.
If ÂŁ10,000 is invested into an account offering 2% compound interest each year then the balance grows as shown in the table.
| Year | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Balance | 1,000.00 | 1,020.00 | 1,040.40 | 1,061.21 | 1,082.43 | 1,104.08 | 1,126.16 | 1,148.69 | 1,171.66 | 1,195.09 | 1,218.99 |
| Balance | |
|---|---|
| 0 | 1,000.00 |
| 1 | 1,020.00 |
| 2 | 1,040.40 |
| 3 | 1,061.21 |
| 4 | 1,082.43 |
| 5 | 1,104.08 |
| 6 | 1,126.16 |
| 7 | 1,148.69 |
| 8 | 1,171.66 |
| 9 | 1,195.09 |
| 10 | 1,218.99 |
This is an example of an exponential graph in the form \(y = k^x\).
\(y = k^x\) graphs increase in value.
Solving problems with exponential growth
The graph shows how bacteria \((y)\) grows over time \((t)\). The equation of the graph is:
\(y = 20g^t\)
From the graph, when \(t = 2\) \(y = 2,000\).
Substituting these values into \(y = 20g^t\):
\(2,000 = 20g^2\)
Dividing by 20 gives:
\(100 = g^2\)
Square rooting gives:
\(10 = g\)
So \(y = 20 \times 10^t\).