Properties of Further Graphs (DP IB Applications & Interpretation (AI)) : Revision Note

A logarithmic function is of the form 

Remember the natural logarithmic function 

• This is the inverse of 

  • and 

• The graphs will always pass through the point (1, a)

• The graphs do not have a y-intercept

  • The graphs have a vertical asymptote at the y-axis: 

• The graphs have one root at 

  • This can be found using your GDC

• The graphs do not have any minimum or maximum points

• The value of b determines whether the graph is increasing or decreasing

  • If b is positive then the graph is increasing

  • If b is negative then the graph is decreasing

A logistic function is of the form 

• L, C & k are positive constants

Its domain is the set of all real values

Its range is the set of real positive values less than L

The y-intercept is at the point 

There are no roots

There is a horizontal asymptote at y = L

• This is called the carrying capacity

  • This is the upper limit of the function

  • For example: it could represent the limit of a population size

There is a horizontal asymptote at y = 0

The graph is always increasing