toPi 2e

7.01 + 7.02 - Exponential Functions and Euler's Number

Properties1
tagsnote

1 min read

Exponential Functions and Euler’s Number

Exponential Functions

𝑦=𝑎𝑥

Exponential functions come in the aforementioned form. This type of function must have a to be greater than 0.

Tip

Only vertical translations will change the asmyptote.

Important

Exponential functions typically have the following special properties (presuming it is in the form 𝑎𝑥).

  • The domain of 𝑎𝑥 is (−∞,∞) or all real x.
  • The range of 𝑎𝑥 is (0,∞) or 𝑦>0.
  • An asymptote is present. It sits at 𝑦=0 (the x-axis).

In regards to its special behaviours…

  • If 𝑎>1 and as 𝑥→∞, 𝑎𝑥→∞
  • If 0<𝑎<1 and as 𝑥→∞, 𝑎𝑥→0+.
  • These behaviours are flipped (e.g. 0<𝑎<1,𝑥→−∞,𝑎𝑥→−∞) when 𝑥→−∞.
    If 𝑦=𝑘𝑎𝑥, all y-values of 𝑎𝑥 are dilated (vertical dilation). The scale factor of 𝑦=𝑘𝑎𝑥 is 𝑘, as seen in 8.04 + 8.05 - Dilations.

Euler’s Number

Euler’s number, called 𝑒, is a special number in which its derivative is the same as its original function when placed into the form 𝑦=𝑎𝑥, where 𝑎=𝑒.

ddx(𝑒𝑥)=𝑒𝑥

𝑒 is also used as the base of the natural log function ln.

Graph View

Backlinks