IB Maths AA Topic 2 Definitions

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

All Topic 2 Sub-topics 2.1: Forms Of Lines 2.2: Functions 2.3: Solving Functions 2.4: Graph Transformations (HL)2.5: Advanced functions (HL)2.6: Inequalities (HL)2.7: Additional transformations

accuracy

The stated precision of an approximate solution, such as to

3

decimal places or to

2

significant figures.

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.

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 $\frac{1}{q}$ .

degree

The highest power of

x

with a non-zero coefficient in a polynomial.

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.

equations

Statements that two expressions are equal, often written as

f(x)=g(x)

f(x)=0

, and solved by finding the input values that make the statement true.

f(x)

A function rule that assigns each input

x

a single output value, written as

f(x)

factor

A polynomial expression that divides another polynomial exactly, leaving remainder

0

factor theorem

The statement that

f(a)=0

if and only if

(x-a)

is a factor of

f(x)

factorising

Rewriting an expression as a product of simpler factors so that solving an equation can be reduced to setting each factor equal to zero.

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

g(x)

A function used in comparisons such as

g(x)\geq f(x)

, where solutions are the

x

-values for which the graph

y=g(x)

lies on or above

y=f(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.

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.

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

intersect

A calculator feature or graphical method that finds the approximate

x

-value where

y=f(x)

and

y=g(x)

meet, corresponding to a solution of

f(x)=g(x)

interval notation

A way to write solution sets as intervals of

x

, using brackets to show whether endpoints are included (for

\leq

\geq

) or excluded (for

\lt

\gt

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.

modulus equations

Equations involving modulus that are typically solved by splitting into cases, such as

|A|=k

giving

A=k

A=-k

parallel

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

perpendicular

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

90^{\circ}

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)

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}

range

The set of possible outputs a function can produce.

rational functions

Functions that can be written as a quotient of two polynomials, for example

f(x)=\frac{ax+b}{cx^2+dx+e}

, with a domain excluding values that make the denominator

0

reciprocal

The multiplicative inverse of a non-zero number, so multiplying a value by its reciprocal gives

1

(for example, the reciprocal of

a

\frac{1}{a}

reflection

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

remainder theorem

When a polynomial

f(x)

is divided by

(x-a)

, the remainder is

f(a)

root

A value

a

such that

f(a)=0

for a given function or polynomial; corresponds to an

x

-intercept of the graph.

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.

technology

Digital tools such as a graphic display calculator (GDC) or a spreadsheet used to carry out calculations, solve equations, and model financial situations efficiently.

translation

A movement of a graph that shifts every point the same distance without changing the shape of the 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 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 = |f(x)|

Formed by keeping the parts of

y=f(x)

that are on or above the

x

-axis unchanged and reflecting any parts below the

x

-axis in the

x

-axis, so all outputs are non-negative.

y = f(|x|)

Formed by taking the part of

y=f(x)

for

x\geq0

and reflecting it in the

y

-axis to create the part for

x\lt0

, giving a graph symmetric about the

y

-axis.

y = f(ax + b)

A horizontal transformation determined by the expression inside the function: the factor

a

produces a horizontal stretch or compression and the constant

b

produces a horizontal shift, often clarified by rewriting

ax+b=a\left(x+\frac{b}{a}\right)

y-intercept

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

y-variable

The dependent variable giving the output of a function; on a graph it is the vertical coordinate of each point.

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