Exponential growth and decay - Higher

Money invested in a bank can generate two different types of interest. Compound 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 £10,000 is invested into an account offering 2% compound interest each year then the balance grows as shown in the table.

Year012345678910
Balance1,000.001,020.001,040.401,061.211,082.431,104.081,126.161,148.691,171.661,195.091,218.99
Balance of compound interest graph

This is an example of an exponential graph in the form \(y = k^x\).

Graphs in the form of \(y = k^x\) increase in value.

Exponential graph showing y = k to the power of x

Solving problems with exponential growth

The graph shows how bacteria \((y)\) grows over time \((t)\). The equation of the graph is:

\[y = 20g^t\]

Exponential graph of 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 \(y = 20 \times 10^t\).

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