IB Maths AI Topic 2 Definitions

This page contains our IB Maths AI definitions for topic 2. By learning each one of these definitions, you will fully cover the content for IB Maths AI '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

asymptotes

Lines that a rational function’s graph approaches; can be vertical where the denominator is

0

(and the numerator is not

0

there), horizontal depending on degrees, or oblique when the numerator’s degree is one more than the denominator’s.

axis of symmetry

The vertical line through the vertex of a quadratic, with equation

x=-\frac{b}{2a}

for

f(x)=ax^{2}+bx+c

composite functions

Functions formed by using the output of one function as the input of another, written $\left(f\left(g(x)\right)\right)=(fg)(x)$ , where the order matters.

composite transformation

A transformation formed by applying more than one transformation to the same base graph, for example combining translations, reflections and stretches in a single function rule.

compression

A stretch with scale factor between $0$ and $1$ , which makes the graph closer to an axis; for example $y=pf(x)$ with $0\lt p\lt1$ compresses vertically and $y=f(qx)$ with $q\gt1$ compresses horizontally by scale factor $\frac{1}{q}$ .

domain

The set of input values for which a model is intended to be used, chosen using the context rather than only the algebraic definition.

exponential model

A model of the form

y=ab^x

; taking logarithms gives

\log y=\log a+x\log b

, which is linear in

x

extrapolation

Using a regression line to predict a value that lies outside the range of the observed data, which is generally less reliable.

function

A rule that takes each input and gives exactly one output; used to model relationships between quantities, such as velocity as a function of time.

function notation

A way to name the output of a function for a given input, for example writing

f(x)

to mean the output of the function

f

when the input is

x

gradient

The coefficient

a

y=ax+b

, giving the predicted change in

y

when

x

increases by

1

gradient-intercept

A form of a straight-line equation written as

y=mx+c

, where

m

is the gradient and

c

is the

y

-intercept.

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}

horizontal asymptote

A horizontal line

y=c

that an exponential graph approaches as

x

increases or decreases, for models such as

f(x)=ka^{x}+c

f(x)=ke^{rx}+c

horizontal stretch

A transformation that multiplies the $x$ -coordinate by a scale factor $k$ while leaving the $y$ -coordinate unchanged, using $\begin{pmatrix}k&&0\\0&&1\end{pmatrix}$

horizontal translation

A shift of $y=f(x)$ by $a$ units in the $x$ -direction, given by $y=f(x-a)$ .

identity function

A function that returns the input unchanged, given by

I(x)=x

; composing a function with its inverse gives this function.

intercept

The value where a straight line crosses the vertical axis; in the linearized forms it equals

\log a

Intercepts

The points where a line crosses the coordinate axes; the

y

-intercept occurs when

x=0

and the

x

-intercept occurs when

y=0

interpolation

Estimating a value at an unknown point using known values at nearby points; in nearest neighbour interpolation, the estimate is taken from the closest site.

inverse function

A function that reverses the effect of another function, so that if

f

maps

x

y

then

f^{-1}

maps

y

back to

x

, with

\left(f\left(f^{-1}(x)\right)\right)=x

and

\left(f^{-1}\left(f(x)\right)\right)=x

when both are defined.

linear scale

An axis scale with equal spacing for equal differences in value.

linearization

Transforming a curved relationship into a straight-line form by taking logarithms, so that a best-fit straight line can be used to estimate model parameters.

log-log graph

A graph with both axes on logarithmic scales; used to test for power relationships by plotting

$\log y$ against

$\log x$ .

logarithmic

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

logarithmic scale

An axis scale that increases by equal ratios; for example, values like

1

10

100

, and

1000

are equally spaced because each is

10

times the previous value.

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

order

The size of a matrix written as

m \times n

, where

m

is the number of rows and

n

is the number of columns.

parallel

Checks consistency by comparing results from two different versions designed to measure the same thing.

parameters

Fixed constants in a chosen model that determine its specific form and are found from data, known points, or conditions.

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

perpendicular

Meeting at a right angle, so the angle between the two lines is

90^{\circ}

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

point-gradient

A form of a straight-line equation using a point

(x_1,y_1)

and gradient

m

y-y_1=m\left(x-x_1\right)

power model

A model of the form

y=ax^n

; taking logarithms gives

\log y=\log a+n\log x

, which is linear in

\log x

principal

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

quadratic function

A function of the form

f(x)=ax^2+bx+c

whose graph is a parabola, with

y

-intercept

(0,c)

and axis of symmetry

x=\frac{-b}{2a}

quadratic model

A function of the form

f(x)=ax^{2}+bx+c

with

a \neq 0

, whose graph is a parabola and is suitable when there is a turning point.

range

The set of possible outputs a function can produce.

reasonableness

The extent to which a model’s outputs and predictions make sense in context, including whether values are realistic over the intended domain.

reflection

A transformation that flips points in a line (such as an axis), reversing orientation while preserving distances.

roots

Values of

w

that satisfy an equation of the form

w^n=z

; for

z=r\operatorname{cis}\theta

, the

n

th roots are

w_k=\sqrt[n]{r}\operatorname{cis}\left(\frac{\theta+2k\pi}{n}\right)

for

k=0,1,\dots,n-1

scale

The multiplicative factor that determines how much a graph is stretched or compressed, such as $p$ in $y=pf(x)$ for vertical scaling or $\frac{1}{q}$ in $y=f(qx)$ for horizontal scaling.

semi-log graph

A graph with one axis on a logarithmic scale and the other on a linear scale; used to test for exponential relationships by plotting

$\log y$ against $x$ .

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

translation

A movement of a graph that shifts every point the same distance without changing the shape of the graph.

variables

Quantities that can change in a situation and are represented by symbols so relationships can be modelled mathematically.

vertex

A point in a graph.

vertical

Describing changes parallel to the

y

-axis, such as a stretch by factor

|a|

or a shift up or down by

d

vertical asymptote

A vertical line that a graph approaches but never reaches; for inverse variation models with

n \lt 0

f(x)=ax^{n}

, the line

x=0

(the

y

-axis) is a vertical asymptote.

vertical stretch

A transformation that multiplies the $y$ -coordinate by a scale factor $k$ while leaving the $x$ -coordinate unchanged, using $\begin{pmatrix}1&&0\\0&&k\end{pmatrix}$

vertical translation

A shift of $y=f(x)$ by $b$ units in the $y$ -direction, given by $y=f(x)+b$ .

x-intercept

The point where a line crosses the

x

-axis, found by setting

y=0

and solving for

x

, giving a point of the form

(x,0)

y-intercept

The point where a line crosses the $y$ -axis, found by setting $x=0$ in the equation of the line.

Next Up

You have completed the topic 2 definitions for IB Maths AI - continue with related resources below or explore the full IB Maths AI course from the IBO.

Explored IB Maths AI Definitions?

Get stuck into one of our other subjects!

Other subjects' Definitions

Biology

Chemistry

Physics

Maths AA

Join 85,000 students, across 130+ countries, in 500+ IB schools. That's half of the IB science graduates worldwide.

Start a 7d free trial