Shape of y=a^x for a1 and for 0<a<1?

Growth (increasing) and decay (decreasing)

What is special about e?

The gradient of y=e^x equals e^x everywhere.

When is an exponential model suitable?

When the rate of change is proportional to the current value.

In y=Ae^{kt}, what do A and k represent?

A is the initial value; k is the rate constant (growth if k0, decay if k<0).

Name a limitation of a pure exponential growth model.

Real growth is limited by resources such as food and space.

Exponential functions and e^xEdexcel A-Level Maths: Flashcards

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Where does $y=a^x$ cross the $y$-axis?

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Where does $y=a^x$ cross the $y$ -axis?: $(0,1)$ , for every $a>0$
What is the asymptote of $y=a^x$ ?: $y=0$ (the $x$ -axis)
Shape of $y=a^x$ for $a>1$ and for $0<a<1$ ?: Growth (increasing) and decay (decreasing)
How is $y=(\frac1a)^x$ related to $y=a^x$ ?: Reflection in the $y$ -axis
What is special about $e$ ?: The gradient of $y=e^x$ equals $e^x$ everywhere.
$\frac{d}{dx}(e^{kx})$ ?: $ke^{kx}$
$\frac{d}{dx}(5e^{-2x})$ ?: $-10e^{-2x}$
When is an exponential model suitable?: When the rate of change is proportional to the current value.
In $y=Ae^{kt}$ , what do $A$ and $k$ represent?: $A$ is the initial value; $k$ is the rate constant (growth if $k>0$ , decay if $k<0$ ).
Asymptote of $y=e^{ax+b}+c$ ?: $y=c$
$y$ -intercept of $y=e^{2x-1}+3$ ?: $\frac1e+3$
Why does $y=e^{2x-1}+3$ not cross the $x$ -axis?: $e^{2x-1}>0$ , so $y>3$ .
Name a limitation of a pure exponential growth model.: Real growth is limited by resources such as food and space.

Exam questions on Exponential functions and e^x

A function is given by $f(x)=a^x$ , where $a>0$ , and its graph passes through the point $(2,\,25)$ .
Solve $f(x)=\dfrac{1}{125}$ .2 marks
Consider the curve $y=e^{2x-1}+3$ .
Explain why the curve never crosses the $x$ -axis.2 marks
The mass $M$ grams of a radioactive sample $t$ days after it is first measured is modelled by $M=80e^{-0.05t}$ .
Find $\dfrac{dM}{dt}$ and hence show that the rate of change of $M$ is proportional to $M$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).