"e" (Cambridge (CIE) AS Maths) : Revision Note

"e" Exponential Function

What is e, the exponential function?

• The exponential function is y = e^x

  • e is an irrational number

  • e ≈ 2.718

• As with other exponential graphs y = e^x

  • passes through (0, 1)

  • has the x-axis as an asymptote

  

What is the big deal with e?

• y = e^x has the particular property

• ie for every real number x, the gradient of y = e^x is also equal to e^x (see Derivatives of Exponential Functions)

The negative exponential graph

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

• They are of the form y = f(x) and y = f(-x) (see Transformations of Functions - Reflections)

  

What is exponential growth and decay?

• y = Ae^kx (k > 0) is exponential growth

• y = Ae^-kx (k > 0) is exponential decay

• A is the initial value

• k is a (usually positive) constant

• “-“ is used in the equation making clear whether it is growth or decay