Exponential Growth & Decay

Exponential Growth & Decay

Exponential growth & decay

What is exponential growth?

• When a quantity grows exponentially it is increasing from an original amount by a percentage each year for $n$ years

  • Some questions use a different timescale, such as each day, or each minute

• Real-life examples of exponential growth include:

  • Population increases

  • Bacterial growth

  • The number of people infected by a virus

What is exponential decay?

• When a quantity exponentially decays it is decreasing from an original amount by a percentage each year for $n$ years

  • Some questions use a different timescale, such as each day, or each minute

• Real-life examples of exponential decay include:

  • The temperature of hot water cooling down

  • The value of a car decreasing over time

  • Radioactive decay (the mass of a radioactive substance over time)

How can I model a scenario as exponential growth or decay?

• Scenarios which exponentially grow or decay can be modelled with an equation

• A useful format for this equation is

  • $B=A\times k^n$ where:

    • $A$ is the starting (initial) amount

    • $B$ is the new amount

    • $k$ is the appropriate multiplier or scale factor for the growth or decay in the time period

      • E.g. $k=0.8$ for a 20% decay, $k=1.2$ for a 20% growth

    • $n$ is the number of time periods

  • Note if $k>1$ then it is exponential growth

    • If $0<k<1$ then it is exponential decay

    • $k$ cannot be negative

How do I use the exponential growth & decay equation?

• You may need to rearrange the equation $B=A\times k^n$

  • To find $A$ giving $A=\frac{B}{k^n}$

  • To find $k$ giving $k^n=\frac{B}{A}$ so $k=\sqrt[n]{\frac{B}{A}}$

  • To find $n$, using trial and improvement

    • Test different whole-number values for $n$ until both sides of the equation balance

How does exponential growth and decay relate to exponential graphs?

• Plotting the exponential model $B=A\times k^n$ on a graph where:

  • $n$ is on the x-axis

  • and $B$ is on the y-axis

  • gives the shape of an exponential graph

    • often written as $y=a{k}^x$