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