Exponential Functions (Cambridge O Level Additional Maths)

Revision Note

Test yourself
Amber

Author

Amber

Last updated

Did this video help you?

Exponential Functions

What is an exponential function?

  • An exponential functions in a function where the variable is the power
  • They are of the form y = ax with a > 0 

What is an exponential graph?

  • All graphs of the form y = ax will pass through (0, 1) because a0 = 1
  • The x-axis is an asymptoteExponential curves

Exponential graphs when 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
  • a > 1 is exponential growth 

The graph y=3^x increases faster then the graph y=2^x

What about when a ≤ 1?

  • You may like to think about why a = 1 is not considered... If a = 1, y = 1x = 1 for all values of x
  • 0 < a < 1 represents exponential decay
    • Where x < 0 the higher value of a is the “higher” graph
    • Where x > 0 the higher value of a is the “lower” graph

    The graphs y=a^x decrease if a is between 0 and 1

Worked example

Exponential Functions Example fig1, A Level & AS Maths: Pure revision notes

Did this video help you?

"e"

What is e, the exponential function?

  • The exponential function is y = ex
    • e is an irrational number
    • e ≈ 2.718
  • As with other exponential graphs y = ex
    • passes through (0, 1)
    • has the x-axis as an asymptote

    The graph of y=e^x in comparison to y=3^x and y=2^x

What is the big deal with e?

  • y = ex has the particular property

begin mathsize 26px style fraction numerator straight d y over denominator straight d x end fraction italic equals straight e to the power of x end style

  • i.e. for every real number x, the gradient of y = ex is also equal to ex (see Differentiating e^x and lnx)

The graph of y=e^x and some of its tangents

Some values of y=e^x and the gradients at the corresponding points

The negative exponential graph

  • y = e-x is a reflection in the y-axis of y = ex

    y=^x and y=e^-x are reflections in the y-axis

 

What is exponential growth and decay?

Exponential growth and decay

 

  • y = Aekx (k > 0) is exponential growth
  • y = Ae-kx (k > 0) is exponential decay
  • A is the initial value
  • k is a (usually positive) constant
  • A negative sign is used in the equation making clear whether it is growth or decay

Worked example

e Example fig1, A Level & AS Maths: Pure revision notese Example fig2, A Level & AS Maths: Pure revision notes

You've read 0 of your 10 free revision notes

Unlock more, it's free!

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

the (exam) results speak for themselves:

Did this page help you?

Amber

Author: Amber

Expertise: Maths

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.