IB Maths AI 2.7 Definitions

This page contains our IB Maths AI definitions for 2.7. By learning each one of these definitions, you will fully cover the content for IB Maths AI 'Advanced modelling functions'.

All Topic 2 Sub-topics 2.1: Forms Of Lines 2.2: Introduction To Functions 2.3: Introduction To Modelling Functions 2.4: Modelling Applications (HL)2.5: Further functions (HL)2.6: Graph transformations (HL)2.7: Advanced modelling functions (HL)2.8: Logarithmic graphs

amplitude

The distance from the principal axis to a maximum or minimum of a sinusoidal graph, given by '

\left|a\right|

asymptote

A line that a graph approaches but does not reach; for '

f(x)=a+b\ln x

' there is a vertical asymptote at '

x=0

', and for a logistic model there is a horizontal asymptote at '

y=L

half-life

The time taken for a quantity to reduce to half of its original value; for '

f(t)=Ae^{-kt}

' with '

k\gt0

', it is '

t=\frac{\ln2}{k}

logarithmic

Based on logarithms, so equal distances on an axis represent equal ratios (multiplicative changes) rather than equal differences.

Logistic

A model of the form '

f(x)=\frac{L}{1+Ce^{-kx}}

' with '

L,C,k\gt0

', used when growth begins quickly but then slows towards a limiting value '

L

period

The horizontal length of one complete cycle of a trigonometric graph; for '

y=\sin x

' and '

y=\cos x

' it is '

2\pi

', and for '

y=\tan x

' it is '

\pi

phase

The angle that determines the horizontal shift of a sinusoidal function; in

a\cos\left(\omega t+\alpha\right)

it is

\alpha

piecewise

Defined by different algebraic rules on different parts of the domain, for example using one rule when

x\geq0

and another when

x\lt0

principal

The original amount borrowed (or the remaining loan balance) before interest is added; part of each payment reduces this amount.

Sinusoidal

A type of model used for periodic behaviour, where the same pattern repeats regularly, often written as '

f(x)=a\sin\left(b(x-c)\right)+d

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