Modelling Exponential Growth and Decay

A LevelAQAEdexcelOCR

Modelling Exponential Functions and the Natural Logarithm

It is important to know how to use $e^{x}$ and $ln(x)$ in real life. This means you will need to be able to sketch graphs of $e^{ax+b}+c$ and $ln(ax+b)$ , and interpret word-based problems about growth and decay.

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Exponential Function Graphs

You need to know how to sketch $y=e^{ax+b}+c$ . The best way to do this is step by step.

Step 1: Find where the graph intercepts the axes.

y-axis is at $x=0$ , so y-intercept is at $y=e^{b}+c$

x-axis is at $y=0$ , so x-intercept is at $e^{ax+b}+c=0$ , which is an equation we know how to solve. Note that if $c>0$ the graph does not cross the x-axis.

Step 2: Find the asymptotes by looking at the behaviour of the graph as $xrightarrowpminfty$ .

As $xrightarrowinfty$ , $y=e^{ax+b}+crightarrowinfty$ if $a>0$
and $y=e^{ax+b}+crightarrow c$ if $a<0$

As $xrightarrow -infty$ , $y=e^{ax+b}+crightarrow c$ if $a>0$
and $y=e^{ax+b}+crightarrowinfty$ if $a<0$

Step 3: Mark all of this information on a graph and use it to plot the graph.

Natural Logarithm Graphs

You need to know how to sketch $y=ln(ax+b)$ . The best way to do this is step by step.

Step 1: Find where the graph intercepts the axes.

y-axis is at $x=0$ , so y-intercept is at $y=ln(b)$

x-axis is at $y=0$ , so x-intercept is at $ax+b=1$ , which is
$x=dfrac{1-b}{a}$

Step 2: Find the asymptotes by looking at the behaviour of the graph as $xrightarrowpminfty$ .

If $a>0$ , as $xrightarrowinfty$ , $y=ln(ax+b)rightarrowinfty$ and $x$ can decrease until $ax+b=0$ , so there is an asymptote at $x=dfrac{-b}{a}$ where
$yrightarrow -infty$

If $a<0$ , as $xrightarrow -infty$ , $y=ln(ax+b)rightarrow infty$ and $x$ can increase until $ax+b=0$ , so there is an asymptote at $x=dfrac{-b}{a}$ where
$yrightarrow -infty$

Step 3: Mark all of this information on a graph and use it to plot the graph.

Modelling Exponential Growth and Decay

Most real world problems about exponential functions involve either something increasing (growth) or decreasing (decay). The maths in these problems is not new, but it is not trivial to interpret what the questions are asking.

Example: A radioactive particle decays according to $R=Ae^{-0.05t}$ , where $R$ is radioactivity and $t$ is time. It starts with a radioactivity of $500$ .

i) Find $A$

ii) Find the half-life of the particle.

[6 marks]

i) At $t=0$ , $R=500$

$500=Ae^{-0.05times0}$

$500=Ae^{0}$

$500=Atimes1$

$A=500$

ii) The half-life is when the radioactivity has halved, so at $R=dfrac{500}{2}=250$

$250=500e^{-0.05t}$

$dfrac{250}{500}=e^{-0.05t}$

$e^{-0.05t}=dfrac{1}{2}$

$-0.05t=lnleft(dfrac{1}{2}right)$

$begin{aligned}t&=dfrac{-lnleft(dfrac{1}{2}right)}{0.05}[1.2em]&=13.9end{aligned}$

Example 1: Sketching an Exponential Graph

Sketch $y=e^{3x+2}-11$ , marking any asymptotes and intersections with the axes.

[3 marks]

Example 2: Sketching a Logarithmic Graph

Sketch $y=ln(6x+5)$ , marking any asymptotes and intersections with the axes.

[3 marks]

Modelling Exponential Growth and Decay Example Questions

Question 1: Plot the graph $y=e^{0.3x-0.8}-1$ , labelling where the graph crosses the axes and any asymptotes.

[3 marks]

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Question 2: Plot the graph $y=ln(22x+35)$ , labelling where the graph crosses the axes and any asymptotes.

[3 marks]

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Question 3: The population of a small city grows according to $P=10000e^{0.1(t-2008)}$ where $P$ is the population and $t$ is the current year.

a) In which year was the city’s population first measured, and what was the population in this year?

b) What is the city’s population in $2021$ ?

c) How many years will it take for the city’s population to reach $100000$ ?

[7 marks]

a) From the equation, we can see that the population was first measured in $2008$ .

The population in $2008$ was:

$begin{aligned}P&=10000e^{0.1(2008-2008)}[1.2em]&=10000e^{0.1times0}[1.2em]&=10000e^{0}[1.2em]&=10000times1[1.2em]&=10000end{aligned}$

b) $t=2021$

$begin{aligned}P&=10000e^{0.1(2021-2008)}[1.2em]&=10000e^{0.1times13}[1.2em]&=10000e^{1.3}[1.2em]&=36700end{aligned}$

c) $P = 100000$

$begin{aligned} 100000 &=10000e^{0.1(t-2008)} [1.2em] dfrac{100000}{10000}&=e^{0.1(t-2008)} [1.2em] 10 &=e^{0.1(t-2008)} [1.2em] 0.1(t-2008)&=ln(10) [1.2em] t-2008 & =10ln(10) [1.2em] t &=2008+10ln(10) [1.2em] &=2031end{aligned}$

The city’s population will reach $100000$ in $2031$ .

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Question 4: A colony of bacteria grow very quickly according to $y=Ae^{kt}$ where $y$ is the population and $t$ is time in hours. At time $0$ there is only $1$ bacterium. At time $10$ there are $1000$ bacteria.

i) Find $A$ and $t$ .

ii) The container in which the bacteria are in can only support $72000$ of them. How long until it is at capacity?

[5 marks]

i) At $t=0$ , $y=1$

$Ae^{ktimes0}=1$

$Ae^{0}=1$

$Atimes1=1$

$A=1$

At $t=10$ , $y=1000$

$e^{10k}=1000$

$10k=ln(1000)$

$begin{aligned}k&=dfrac{ln(1000)}{10}&=0.691end{aligned}$

ii) $y=e^{0.691t}$

$72000=e^{0.691t}$

$0.691t=ln(72000)$

$begin{aligned}t&=dfrac{ln(72000)}{0.691}&=16.2text{ hours}end{aligned}$

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Additional Resources

Exam Tips Cheat Sheet

A Level

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Formula Booklet

A Level

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Specification Points Covered

F7 – Understand and use exponential growth and decay; use in modelling (examples may include the use of $e$ in continuous compound interest, radioactive decay, drug concentration decay, exponential growth as a model for population growth); consideration of limitations and refinements of exponential models

Modelling Exponential Growth and Decay Worksheet and Example Questions

Logarithms

Exam Questions Mark Scheme

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Try it yourself

Modelling Exponential Growth and Decay

Modelling Exponential Functions and the Natural Logarithm

A Level Maths Predicted Papers 2025

A-A* A Level Maths Practice Papers

Exponential Function Graphs

Natural Logarithm Graphs

Modelling Exponential Growth and Decay

Example 1: Sketching an Exponential Graph

Example 2: Sketching a Logarithmic Graph

Modelling Exponential Growth and Decay Example Questions