Exponential functions and e^xEdexcel A-Level Maths: Revision notes

An exponential function has the variable in the power, $f(x)=a^x$ with $a>0$ and $a\ne1$. Its graph:

• passes through $(0,1)$, because $a^0=1$;
• lies above the $x$-axis for all $x$ (since $a^x>0$), with the $x$-axis as a horizontal asymptote;
• for $a>1$ it increases (growth), rising steeply as $x$ increases; for $0<a<1$ it decreases (decay). The graph of $y=\left(\frac1a\right)^x=a^{-x}$ is the reflection of $y=a^x$ in the $y$-axis. Example: if $a^2=25$ and $a>0$ then $a=5$, $f(-1)=\frac15$ and $f(x)=\frac1{125}$ gives $x=-3$.

The number $e\approx2.718$ is the special base for which the gradient of $y=a^x$ at $(0,1)$ is exactly $1$. The function $y=e^x$ has the remarkable property that its gradient at every point equals its $y$-value: $\frac{d}{dx}(e^x)=e^x$. Its graph passes through $(0,1)$, lies above the $x$-axis, and increases at an increasing rate. For any other base the gradient is $a^x$ multiplied by a constant, which is $1$ only for $a=e$.

For a constant $k$, $\frac{d}{dx}\left(e^{kx}\right)=ke^{kx}.$ The gradient is $k$ times the function. If $k>0$ the function grows; if $k<0$ it decays. Examples: $\frac{d}{dx}(e^{3x})=3e^{3x}$ and $\frac{d}{dx}(5e^{-2x})=-10e^{-2x}$. For $M=80e^{-0.05t}$, $\frac{dM}{dt}=-0.05M$.

Since $\frac{d}{dx}(Ae^{kx})=k\times Ae^{kx}$, the rate of change is proportional to the current value $y$. Many real situations behave like this: population growth (more individuals produce more offspring), radioactive decay (each nucleus decays with a fixed probability), compound interest and cooling. So $y=Ae^{kt}$, with $A$ the initial value and $k$ the rate constant, is a suitable model. Limits: unlimited growth is unrealistic (food, space) and a continuous model cannot show whole items.

Start from $y=e^x$ and transform it:

• $e^{ax}$: a stretch parallel to the $x$-axis, scale factor $\frac1a$;
• $e^{ax+b}$: a horizontal translation (written $e^{a(x+b/a)}$);
• $+c$: translation $c$ units up, so the horizontal asymptote is $y=c$. The $y$-intercept is $e^{b}+c$ and the range is $y>c$ (for $a$ positive or negative). Example: $y=e^{2x-1}+3$ has asymptote $y=3$, $y$-intercept $\frac1e+3$ and never meets the $x$-axis because $y>3$.