Exponential Growth & Decay (Cambridge (CIE) IGCSE International Maths: Extended): Revision Note

Exponential Growth & Decay

The ideas of compound interest and depreciation can be applied to other (non-money) situations, such as increasing or decreasing populations.

What is exponential growth?

• When a quantity grows exponentially it is increasing from an original amount by a scale factor (or percentage) each year for $x$ 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 scale factor (or percentage) each year for $x$ 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 do I use models for exponential growth or decay?

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

• A useful format for this equation is

  • $y=a^x$

    • E.g. $y=4^x$ or $y=0.8^x$

  • If $a>1$ then it is exponential growth

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

  • $a$ cannot be negative

• $y=3^x$ could model the number of flies, $y$, in a population of flies, against the number of days, $x$

  • The initial number of flies, when $x=0$, is $3^0=1$

  • After 2 days, the number would be $3^2=9$

  • After 4 days, the number would be $3^4=81$

  • From the values above, and the fact that $a>1$, this is exponential growth

• $y=0.5^x$ could model the radius of a water droplet, $y\mathrm{cm}$, as it evaporates where $x$ is measured in minutes

  • The initial radius of the droplet, when $x=0$, is $0.5^0=1\mathrm{cm}$

  • After 2 minutes, the radius would be $0.5^2=0.25\mathrm{cm}$

  • After 4 minutes, the radius would be $0.5^4=0.0625\mathrm{cm}$

  • From the values above, and the fact that $0<a<1$, this is exponential decay

How are exponential models related to exponential graphs?

• Plotting an exponential model, $y=a^x$, on a set of axes gives an exponential graph with the following shape

• The $\mathit{y}$-intercept is always 1

  • This is because the $y$-intercept is where $x=0$

  • $y=a^0=1$

• There is an asymptote at $y=0$ (the curve gets closer and closer to the x-axis without crossing it)

  • This is because there are no solutions to $0=a^x$

Are there any other exponential models I may encounter?

• You may see exponential models in the form $y=b{a}^x$

  • The coefficient, $b$, scales (stretches) the graph vertically

  • The $y$-intercept will be $\left(0,b\right)$

    • This is because when $x=0$, $y=b{a}^0=b\left(1\right)=b$

  • $b$ is the initial (starting) amount of the quantity being modelled