Exponential Functions (Edexcel IGCSE Further Maths)

Exponential Functions & Graphs

Spec point — 4A

What is an exponential function?

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

• Its domain is the set of all real numbers

• Its range is the set of all positive real numbers

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

  • e is the mathematical constant $2.718281\dots$
  • See the following note on "$e$" for more details

What are the key features of exponential graphs?

• The graphs have a $y$-intercept at $( 0 , 1 )$

  • Because $a^{0}=1$

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

  • Because $a^{1}=a$

• The graphs do not intersect the$x$-axis

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

• The graphs do not have any minimum or maximum points

• When$a>1$

  • For $x<0$ the higher value of $a$ is the “lower” graph
  • Where $x>0$ the higher value of $a$ is the “higher” graph

• When$0<a<1$

  • Where $x<0$ the higher value of $a$ is the “lower” graph
  • Where $x>0$ the higher value of $a$ is the “higher” graph

• When $a=1$

  • The graph is a horizontal line through $y=1$
  • Because $1^{x}=1$ for all values of $x$

Worked Example On the same set of axes, sketch the graphs of $y=3^{x}$ and $y=4^{x}$.

Both graphs will have the 'typical' $y=a^{x}$ exponential shape for the $a>1$case

$y=4^{x}$ will be the 'lower' graph for $x<0$,and the 'higher' graph for $x>0$

Both graphs go through $( 0 , 1 )$ and have an asymptote at the $x$-axis

"e"

What is e, the exponential constant?

• $e$ is one of the most important mathematical constants

  • $e=2.718281...$
  • $e$ is an irrational number
  • $\mathrm{f} ( x ) = \mathrm{e}^{x}$ is often referred to as 'the exponential function'

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

  • passes through $( 0 , 1 )$
  • has the $x$-axis as an asymptote

Why is e so important?

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

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

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

The negative exponential graph

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

  • Note by laws of indices that e ^ - x = ( e ^ - 1 ) ^ x = ( 1 / e ) ^ x

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

Worked Example On the same set of axes, sketch the graphs of $y=e^{x}$, $y=e^{2x}$ and $y=e^{-2x}$.

By laws of indices, $\mathrm{e}^{2 x} = ( \mathrm{e}^{2} )^{x}$

$e^{2}>e$, so $y=e^{2x}$ will be the 'lower' graph for $x<0$,and the 'higher' graph for $x>0$

$y=e^{-2x}$ is the reflection of $y=e^{2x}$in the $y$-axis

All three graphs go through $( 0 , 1 )$ and have an asymptote at the $x$-axis