Exponential Functions (Edexcel IGCSE Further Pure Maths)

Revision Note

Paul

Written by: Paul

Reviewed by: Dan Finlay

Exponential Functions & Graphs

What is an exponential function?

  • An exponential function is of the form space straight f left parenthesis x right parenthesis equals a to the power of x comma space a greater than 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 space straight f left parenthesis x right parenthesis equals straight e to the power of x

    • e is the mathematical constant 2.718281 horizontal ellipsis

    • See the following note on "straight e" for more details

What are the key features of exponential graphs?

  • The graphs have a bold italic y-intercept at open parentheses 0 comma space 1 close parentheses

    • Because  a to the power of 0 equals 1

  • The graph will always pass through the point open parentheses 1 comma space a close parentheses

    • Because  a to the power of 1 equals a

  • The graphs do not intersect the bold italic x-axis

    • The graphs have a horizontal asymptote at the x-axis space y equals 0

  • The graphs do not have any minimum or maximum points

Exponential Functions fig1
  • When bold italic a bold greater than bold 1

    • For bold italic x bold less than bold 0 the higher value of a is the “lower” graph

    • Where bold italic x bold greater than bold 0 the higher value of a is the “higher” graph

      Exponential Functions fig3, A Level & AS Maths: Pure revision notes
  • When bold 0 bold less than bold italic a bold less than bold 1

    • Where bold italic x bold less than bold 0 the higher value of a is the “lower” graph

    • Where bold italic x bold greater than bold 0 the higher value of a is the “higher” graph

      6-1-1-notes-fig4
  • When bold italic a bold equals bold 1

    • The graph is a horizontal line through y equals 1

    • Because  1 to the power of x equals 1  for all values of x

Worked Example

On the same set of axes, sketch the graphs of  y equals 3 to the power of x  and  y equals 4 to the power of x.

Both graphs will have the 'typical' y equals a to the power of x exponential shape for the a greater than 1case

y equals 4 to the power of x will be the 'lower' graph for x less than 0,and the 'higher' graph for x greater than 0

Both graphs go through open parentheses 0 comma space 1 close parentheses and have an asymptote at the x-axis

jAyO5R1a_picture1

"e"

What is e, the exponential constant?

  • straight e  is one of the most important mathematical constants

    • straight e equals 2.718281...

    • straight e is an irrational number

    • straight f open parentheses x close parentheses equals straight e to the power of x  is often referred to as 'the exponential function'

  • As with other exponential graphs, y equals straight e to the power of x

    • passes through open parentheses 0 comma space 1 close parentheses

    • has the x-axis as an asymptote

    e Notes fig1, A Level & AS Maths: Pure revision notes

Why is e so important?

  • y equals straight e to the power of x has the particular property that  fraction numerator straight d y over denominator straight d x end fraction equals straight e to the power of x

    • i.e. if  straight f open parentheses x close parentheses equals straight e to the power of x, then straight f to the power of apostrophe open parentheses x close parentheses is also equal to straight e to the power of x

  • This means that for every real number x, the gradient of y equals straight e to the power of x is also equal to straight e to the power of x

e Notes fig2, A Level & AS Maths: Pure revision notes
e Notes fig3, A Level & AS Maths: Pure revision notes

The negative exponential graph

  • y equals straight e to the power of negative x end exponent is a reflection in the y-axis of y equals straight e to the power of x

    • Note by laws of indices that  straight e to the power of negative x end exponent equals open parentheses straight e to the power of negative 1 end exponent close parentheses to the power of x equals open parentheses 1 over straight e close parentheses to the power of x

  • They are of the form y equals straight f open parentheses x close parentheses and y equals straight f open parentheses negative x close parentheses

    e Notes fig4, A Level & AS Maths: Pure revision notes

Worked Example

On the same set of axes, sketch the graphs of  y equals straight e to the power of xy equals straight e to the power of 2 x end exponent  and  y equals straight e to the power of negative 2 x end exponent.

By laws of indices, straight e to the power of 2 x end exponent equals open parentheses straight e squared close parentheses to the power of x

straight e squared greater than straight e,  so  y equals straight e to the power of italic 2 x end exponent will be the 'lower' graph for x less than 0,and the 'higher' graph for x greater than 0

y italic equals straight e to the power of negative 2 x end exponent is the reflection of  y italic equals straight e to the power of 2 x end exponentin the y-axis

All three graphs go through open parentheses 0 comma space 1 close parentheses and have an asymptote at the x-axis

Aj8jgIxw_picture2

Last updated:

You've read 0 of your 5 free revision notes this week

Sign up now. It’s free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

Did this page help you?

Paul

Author: Paul

Expertise: Maths Content Creator (Previous)

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.

Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.