IBMaths: AI HLDPRevision Notes2. Functions2.6 Further Modelling with Functions2.6.1 Properties of Further Graphs

Properties of Further Graphs (DP IB Maths: AI HL)

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Logarithmic Functions & Graphs

What are the key features of logarithmic graphs?

  • A logarithmic function is of the form space f left parenthesis x right parenthesis equals a plus b space ln space x comma space x greater than 0
  • Remember the natural logarithmic function ln x identical to log subscript straight e open parentheses x close parentheses
    • This is the inverse of space f left parenthesis x right parenthesis equals straight e to the power of x
      • ln open parentheses straight e to the power of x close parentheses equals x and straight e to the power of ln x end exponent equals x
    • 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 open parentheses straight e to the power of negative a over b end exponent comma space 0 close parentheses
      • 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

2-6-1-ib-ai-hl-natural-logarithm-graph

Logistic Functions & Graphs

What are the key features of logistic graphs?

  • A logistic function is of the form space f open parentheses x close parentheses equals fraction numerator L over denominator 1 plus C straight e to the power of negative k x end exponent end fraction
    • 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 open parentheses 0 comma space fraction numerator L over denominator 1 plus C end fraction close parentheses
  • 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

2-6-1-ib-ai-hl-logistic-graphs

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    Dan

    Author: Dan

    Expertise: Maths

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.