Exponential Functions (Edexcel IGCSE Further Pure Maths): Revision Note

Exponential Functions & Graphs

What is an exponential function?

• An exponential function is of the form $\mathrm{f}\left(x\right)=a^x,a>0$

• Its domain is the set of all real numbers

• Its range is the set of all positive real numbers

• An important exponential function is $\mathrm{f}\left(x\right)={\mathrm{e}}^x$

  • e is the mathematical constant $2.718281\dots$

  • See the following note on "$\mathrm{e}$" for more details

What are the key features of exponential graphs?

• The graphs have a $\mathit{y}$-intercept at $\left(0,1\right)$

  • Because  $a^0=1$

• The graph will always pass through the point $\left(1,a\right)$

  • Because  $a^1=a$

• The graphs do not intersect the $\mathit{x}$-axis

  • The graphs have a horizontal asymptote at the x-axis $y=0$

• The graphs do not have any minimum or maximum points

• When $\mathit{a}\mathbf{>}\mathbf{1}$

  • For $\mathit{x}\mathbf{<}\mathbf{0}$ the higher value of $a$ is the “lower” graph

  • Where $\mathit{x}\mathbf{>}\mathbf{0}$ the higher value of $a$ is the “higher” graph
    

    

• When $\mathbf{0}\mathbf{<}\mathit{a}\mathbf{<}\mathbf{1}$

  • Where $\mathit{x}\mathbf{<}\mathbf{0}$ the higher value of $a$ is the “lower” graph

  • Where $\mathit{x}\mathbf{>}\mathbf{0}$ the higher value of $a$ is the “higher” graph
    

    

• When $\mathit{a}\mathbf{=}\mathbf{1}$

  • The graph is a horizontal line through $y=1$

  • Because  $1^x=1$  for all values of $x$

"e"

What is e, the exponential constant?

• $\mathrm{e}$  is one of the most important mathematical constants

  • $\mathrm{e}=2.718281...$

  • $\mathrm{e}$ is an irrational number

  • $\mathrm{f}\left(x\right)={\mathrm{e}}^x$  is often referred to as 'the exponential function'

• As with other exponential graphs, $y={\mathrm{e}}^x$

  • passes through $\left(0,1\right)$

  • has the $x$-axis as an asymptote

  

Why is e so important?

• $y={\mathrm{e}}^x$ has the particular property that  $\frac{\mathrm{d}y}{\mathrm{d}x}={\mathrm{e}}^x$

  • i.e. if  $\mathrm{f}\left(x\right)={\mathrm{e}}^x$, then ${\mathrm{f}}^{\text{'}}\left(x\right)$ is also equal to ${\mathrm{e}}^x$

• This means that for every real number $x$, the gradient of $y={\mathrm{e}}^x$ is also equal to ${\mathrm{e}}^x$

The negative exponential graph

• $y={\mathrm{e}}^{-x}$ is a reflection in the $y$-axis of $y={\mathrm{e}}^x$

  • Note by laws of indices that  ${\mathrm{e}}^{-x}={\left({\mathrm{e}}^{-1}\right)}^x={\left(\frac{1}{\mathrm{e}}\right)}^x$

• They are of the form $y=\mathrm{f}\left(x\right)$ and $y=\mathrm{f}\left(-x\right)$