IB Math AA SL Revision Notes

Formulas, conditions, key connections and common exam traps across the five AA SL topics. Use it alongside the formula booklet and regular question practice.

Written by IB Math tutor Abdulhamid

IB Math Master · Updated November 2026

For the current AA SL syllabus, for exams up to November 2028.

Get the printable PDF

How to use these notes

Each topic is split into short sections. Every section keeps its formulas, conditions and diagrams together.

Connection (blue dot): links between formulas that the formula booklet does not state.
Exam note (gold diamond): short reminders and common traps.
Video: a link to my explanation of the topic.

Topic 1 Number and Algebra

Exponents and Logarithms

Video: Logarithms (opens YouTube in a new tab)

Exponent laws

Name	Formula	Condition
Same base	$a^{m} \times a^{n} = a^{m + n}$ $\frac{a^{m}}{a^{n}} = a^{m - n}$	$a > 0$ , $m, n \in ℚ$
Powers of products	${(a^{m})}^{n} = a^{m n}$ $(a b)^{n} = a^{n} b^{n}$ ${(\frac{a}{b})}^{n} = \frac{a^{n}}{b^{n}}$	$a, b > 0$
Zero and negative	$a^{0} = 1$ $a^{- n} = \frac{1}{a^{n}}$	$a \neq 0$
Fractional	$a^{\frac{1}{n}} = \sqrt[n]{a}$ $a^{\frac{m}{n}} = {(\sqrt[n]{a})}^{m} = \sqrt[n]{a^{m}}$	$a > 0$
Standard form	$a \times 10^{k}$	$1 \leq a < 10$ , $k \in ℤ$

Connection: $\sqrt{a} = a^{\frac{1}{2}}$ and $\frac{1}{\sqrt{a}}$ $= a^{- \frac{1}{2}}$
Connection: Standard form products: multiply the $a$ parts, add the powers of 10, then rescale to $1 \leq a < 10$ .

Logarithms

Name	Formula	Condition
Definition of a log	$a^{x} = b ⟺ x = \log_{a} b$	$a > 0$ , $b > 0$ , $a \neq 1$
Inverses	$\log_{a} a^{x} = x = a^{\log_{a} x}$	$x > 0$ on the right
Base $e$	$a^{x} = e^{x \ln a}$	$a > 0$
Product and quotient	$\log_{a} (x y) = \log_{a} x + \log_{a} y$ $\log_{a} \frac{x}{y} = \log_{a} x - \log_{a} y$	$x, y > 0$
Power	$\log_{a} (x^{m}) = m \log_{a} x$	$x > 0$
Change of base	$\log_{b} a = \frac{\log_{c} a}{\log_{c} b}$	$a, b, c > 0$ , $b, c \neq 1$
Base 10 and base $e$	$\log x = \log_{10} x$ $\ln x = \log_{e} x$	$e \approx 2.718$

Connection: $\log_{a} 1 = 0$ and $\log_{a} a = 1$
Connection: $e^{\ln x} = x$ and $\ln (e^{x}) = x$
Connection: $\log_{a} \frac{1}{x} = - \log_{a} x$
Connection: $\log_{b} a = \frac{1}{\log_{a} b}$
Connection: $\log_{a^{n}} x$ $= \frac{1}{n} \log_{a} x$
Connection: $a^{x}$ $= b$ $\Rightarrow x$ $= \log_{a} b$ $= \frac{\ln b}{\ln a}$
Exam note: No law exists for $\log (x + y)$ , $(\log x)^{2}$ or $(\log x) (\log y)$ . Leave them as they are.
Exam note: $\log_{a} f (x)$ exists only when $f (x) > 0$ . Reject any answer that makes $f (x) \leq 0$ .

Back to topics

Sequences and Series

Arithmetic and geometric sequences

Item	Arithmetic common difference $d$	Geometric common ratio $r$
$n$ th term	$u_{n} = u_{1} + (n - 1) d$	$u_{n} = u_{1} r^{n - 1}$
Sum of $n$ terms	$\begin{aligned} S_{n} & = \frac{n}{2} (2 u_{1} + (n - 1) d) \\ = \frac{n}{2} (u_{1} + u_{n}) \end{aligned}$	$S_{n}$ $= \frac{u_{1} (r^{n} - 1)}{r - 1}$ $= \frac{u_{1} (1 - r^{n})}{1 - r}$ , $r \neq 1$
Sum to infinity		$S_{\infty} = \frac{u_{1}}{1 - r}$ , only when $\| r \| < 1$
Finding $d$ or $r$	$u_{2} - u_{1} = u_{3} - u_{2} = \dots = d$	$\frac{u_{2}}{u_{1}} = \frac{u_{3}}{u_{2}} = \dots = r$

Connection: For any sequence:
$S_{1} = u_{1}$
$S_{2} = u_{1} + u_{2}$
$S_{3} = u_{1} + u_{2} + u_{3}$
$u_{n} = S_{n} - S_{n - 1}$ for $n \geq 2$
Exam note: $n$ counts terms, so $n \in ℤ^{+}$ .

Sigma notation

Name	Formula	Condition
Sigma notation	$\sum_{i = 1}^{n} u_{i} = u_{1} + u_{2} + \dots + u_{n}$	$i$ runs through whole numbers

Connection: Sums split: $\sum (a_{i} + b_{i})$ $= \sum a_{i} + \sum b_{i}$ . Constants come out: $\sum c a_{i} = c \sum a_{i}$ .
Exam note: Check the first and last values of $i$ .

Money: interest and depreciation

Name	Formula	Condition
Compound interest	$F V = P V \times {(1 + \frac{r}{100 k})}^{k n}$ $F V$ = future value, $P V$ = present value, $n$ = number of years, $k$ = number of compounding periods per year, $r %$ = nominal annual rate of interest	geometric, ratio $1 + \frac{r}{100 k}$ per period
Depreciation	$V_{n} = V_{0} \times {(1 - \frac{r}{100})}^{n}$ $V_{0}$ = starting value, $V_{n}$ = value after $n$ years, $r %$ = annual rate of depreciation	geometric, ratio $1 - \frac{r}{100}$ per year

Back to topics

The Binomial Theorem

Video: Binomial expansion (opens YouTube in a new tab)

The binomial theorem

Name	Formula	Condition
Factorial	$n! = n (n - 1) (n - 2) \dots 2 \times 1$	$0! = 1$
Combinations	$^{n} C_{r} = \frac{n!}{r! (n - r)!}$
Binomial theorem	$\begin{aligned} (a + b)^{n} & = \sum_{r = 0}^{n}^{n} C_{r} a^{n - r} b^{r} \\ = a^{n} +^{n} C_{1} a^{n - 1} b + \dots +^{n} C_{r} a^{n - r} b^{r} + \dots + b^{n} \end{aligned}$	$n \in ℕ$
General term	$^{n} C_{r} a^{n - r} b^{r}$	the term in $b^{r}$ , $r = 0, 1, \dots, n$
Pascal’s triangle	$n = 0$ 1 $n = 1$ 11 $n = 2$ 121 $n = 3$ 1331 $n = 4$ 14641 $n = 5$ 15101051	row $n$ gives $^{n} C_{0},^{n} C_{1}, \dots,^{n} C_{n}$

Connection: $n! = n (n - 1) (n - 2)!$ for $n \geq 2$
Connection: $^{n} C_{0} =^{n} C_{n} = 1$ , $^{n} C_{1} = n$ and $^{n} C_{r} =^{n} C_{n - r}$
Connection: $(a + b)^{n}$ has $n + 1$ terms. In each term, the powers of $a$ and $b$ add up to $n$ .
Exam note: A coefficient is the number only. A term includes the power of $x$ .
Exam note: In $(a - b)^{n}$ , use $- b$ as the second term. The signs alternate.
Exam note: Bracket the whole term: $(2 x)^{r} = 2^{r} x^{r}$ , not $2 x^{r}$ .

Back to topics

Proof

Deductive proof

Name	Formula	Condition
Identity	The symbol $\equiv$ means true for every value. In an equation, $=$ holds for some values only.
LHS to RHS	Start from one side. Change it one step at a time until it becomes the other side.	LHS $= \dots =$ RHS

Connection: Even numbers: $2 m$ . Odd numbers: $2 m \pm 1$ . Consecutive integers: $n$ , $n + 1$ . Here $m, n \in ℤ$ .
Exam note: LHS to RHS: work on one side only. Do not rewrite the whole identity on every line.

Back to topics

Topic 2 Functions

Lines and Quadratics

Video: Graphing quadratics (opens YouTube in a new tab)

Straight lines

Name	Formula	Condition
Straight line	$y = m x + c$ $a x + b y + d = 0$ $y - y_{1} = m (x - x_{1})$
Gradient	$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$	$x_{1} \neq x_{2}$
Parallel lines	$m_{1} = m_{2}$
Perpendicular lines	$m_{1} \times m_{2} = - 1$ $m_{2} = - \frac{1}{m_{1}}$	$m_{1} \neq 0$

The three forms of a quadratic

Form	Equation	What it shows
Standard	$f (x) = a x^{2} + b x + c$	$y$ -intercept $(0, c)$ axis of symmetry $x = - \frac{b}{2 a}$ vertex $(- \frac{b}{2 a}, f (- \frac{b}{2 a}))$
Factor	$f (x) = a (x - p) (x - q)$	$x$ -intercepts $(p, 0)$ , $(q, 0)$ axis of symmetry $x = \frac{p + q}{2}$
Vertex	$f (x) = a (x - h)^{2} + k$	vertex $(h, k)$ axis of symmetry $x = h$

Connection: $a > 0$ : minimum at the vertex. $a < 0$ : maximum at the vertex.

f (x)

= x^{2} - 6 x + 5

= (x - 1) (x - 5)

= (x - 3)^{2} - 4

Roots and the discriminant

Name	Formula	Condition
Quadratic formula	$a x^{2} + b x + c = 0 \Rightarrow x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$	$a \neq 0$
Discriminant	$Δ = b^{2} - 4 a c$
$Δ > 0$	two distinct real roots: the graph crosses the $x$ -axis twice
$Δ = 0$	two equal real roots: the graph touches the $x$ -axis
$Δ < 0$	no real roots: the graph misses the $x$ -axis
Always positive	$f (x) > 0$ for all $x ⟺ a > 0$ and $Δ < 0$
Always negative	$f (x) < 0$ for all $x ⟺ a < 0$ and $Δ < 0$

Exam note: “Real roots” means $Δ \geq 0$ . “Two distinct real roots” means $Δ > 0$ .

Back to topics

Functions, Composites and Inverses

Video: Inverse functions (opens YouTube in a new tab)

Functions and composites

Name	Formula	Condition
Function	Each $x$ in the domain gives exactly one $y$
Domain, range	Domain: the allowed inputs. Range: the outputs.
Composite	$(f \circ g) (x) = f (g (x))$	apply $g$ first

Connection: In general $f \circ g \neq g \circ f$ .

Inverse functions

Name	Formula	Condition
Inverse	$y = f (x)$ : swap $x$ and $y$ , then make $y$ the subject. $f$ must be one-to-one.
Identity	$(f \circ f^{- 1}) (x) = (f^{- 1} \circ f) (x) = x$

Connection: The graph of $f^{- 1}$ is the reflection of $f$ in $y = x$ .
Connection: Domain of $f^{- 1}$ = range of $f$
Range of $f^{- 1}$ = domain of $f$
Exam note: $f^{- 1} (x)$ is not $\frac{1}{f (x)}$ .

(1, 2)

f

and

(2, 1)

f^{- 1}

Back to topics

Graphs and Transformations

Video: Transformations of graphs: translations, stretches and reflections (opens YouTube in a new tab)

Transformations of $y = f (x)$

New graph	Transformation	Point $(x, y)$ moves to
$y = f (x) + b$	translation $b$ units up	$(x, y + b)$
$y = f (x) - b$	translation $b$ units down	$(x, y - b)$
$y = f (x - a)$	translation $a$ units to the right	$(x + a, y)$
$y = f (x + a)$	translation $a$ units to the left	$(x - a, y)$
$y = - f (x)$	reflection in the $x$ -axis	$(x, - y)$
$y = f (- x)$	reflection in the $y$ -axis	$(- x, y)$
$y = p f (x)$	vertical stretch, scale factor $p$	$(x, p y)$
$y = f (q x)$	horizontal stretch, scale factor $\frac{1}{q}$	$(\frac{x}{q}, y)$

Exam note: In this table $a, b, p, q > 0$ . So $f (x) + b$ always moves up and $f (x) - b$ always moves down.
Connection: Inside the bracket: changes $x$ and works the opposite way to how it looks.
Connection: Outside the bracket: changes $y$ and works the way it looks.
Connection: $y = p f (x) + b$ : stretch first, then translate.
Exam note: Describe each transformation fully: type, direction and scale factor or distance.

Dashed

y = x^{2}

. Left:

y = (x - 2)^{2}

moves right 2. Right:

y = (2 x)^{2}

divides each

x

by 2.

Sketching

Exam note: Label key features on a sketch: intercepts, turning points, asymptotes and domain end points.
Exam note: Paper 2: find intersections, zeros and turning points with the GDC.

Back to topics

Rational Functions

Video: Rational functions: linear over linear (opens YouTube in a new tab)

Types and their asymptotes

Type	$f (x)$	Asymptotes	Example
Reciprocal	$\frac{1}{x}$	vertical $x = 0$ horizontal $y = 0$ self-inverse: $f^{- 1} (x) = \frac{1}{x}$
Linear over linear	$\frac{a x + b}{c x + d}$	vertical: where the denominator $c x + d = 0$ horizontal $y = \frac{a}{c}$	$y = \frac{2 x + 1}{x - 1}$

Type

f (x)

Asymptotes

Example

Reciprocal

\frac{1}{x}

vertical $x = 0$

horizontal $y = 0$

self-inverse: $f^{- 1} (x) = \frac{1}{x}$

Linear over linear

\frac{a x + b}{c x + d}

vertical: where the denominator $c x + d = 0$

horizontal $y = \frac{a}{c}$

y = \frac{2 x + 1}{x - 1}

Finding asymptotes and intercepts

Connection: Vertical asymptote: the $x$ that makes the denominator 0 (numerator $\neq 0$ ).
Connection: $x$ -intercept: put $y = 0$ . $y$ -intercept: put $x = 0$ .
Connection: Horizontal asymptote, compare degrees:
top < bottom: $y = 0$
top = bottom: $y =$ ratio of leading coefficients
Exam note: Write asymptotes as equations.

Back to topics

Exponential and Logarithmic Functions

The two graphs

Name	Formula	Condition
Exponential	$f (x) = a^{x}$ $f (x) = e^{x}$ $a > 0, a \neq 1$ domain $ℝ$ , range $y > 0$ asymptote $y = 0$ , through $(0, 1)$
Logarithmic	$f (x) = \log_{a} x$ $f (x) = \ln x$ domain $x > 0$ , range $ℝ$ asymptote $x = 0$ , through $(1, 0)$

Connection: $a^{x}$ and $\log_{a} x$ are inverses: reflections in $y = x$ . The same holds for $e^{x}$ and $\ln x$ .

Changing the graphs

Connection: $a > 1$ : increasing. $0 < a < 1$ : decreasing.
Connection: $a^{- x}$ $= {(\frac{1}{a})}^{x}$ : the reflection of $a^{x}$ in the $y$ -axis
Connection: $y = A e^{k x} + c$ : asymptote $y = c$ . $k > 0$ : growth. $k < 0$ : decay.
Connection: $\log_{a} g (x)$ needs $g (x) > 0$ . Its vertical asymptote is where $g (x) = 0$ .
Exam note: Translations move the asymptote too.
Exam note: In $y = A e^{k t} + c$ , the value at $t = 0$ is $A + c$ , not $A$ .

Back to topics

Solving Equations and Inequalities

Video: Solving quadratic inequalities (opens YouTube in a new tab)

Using graphs

Name	Formula	Condition
Graphical solution	Solutions of $f (x) = g (x)$ are the $x$ -coordinates where the graphs meet

Connection: Number of solutions = number of intersection points

Quadratic inequalities

Name	Formula	Condition
Roots $α < β$ , $a > 0$	$a x^{2} + b x + c > 0 ⟺ x < α$ or $x > β$ $a x^{2} + b x + c < 0 ⟺ α < x < β$

Name

Formula

Condition

Roots

α < β

a > 0

$a x^{2} + b x + c > 0 ⟺ x < α$ or $x > β$

$a x^{2} + b x + c < 0 ⟺ α < x < β$

Exam note: Sketch the graph, then read off where it is above or below the $x$ -axis.
Exam note: Multiplying or dividing by a negative number reverses the inequality.

Back to topics

Topic 3 Geometry and Trigonometry

3D Geometry and Triangles

Points in 3D

Name	Formula	Condition
Distance	$d = \sqrt{(x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2} + (z_{1} - z_{2})^{2}}$
Midpoint	$(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2}, \frac{z_{1} + z_{2}}{2})$

Solids

Name	Formula	Condition
Right pyramid	$V = \frac{1}{3} A h,$ $A$ = area of base, $h$ = height
Right cone	$V = \frac{1}{3} π r^{2} h$ curved surface $A = π r l$ total surface $= π r l + π r^{2}$ $l^{2} = r^{2} + h^{2}$ , $l$ = slant height
Sphere	$V = \frac{4}{3} π r^{3}$ $A = 4 π r^{2}$
Hemisphere	$V = \frac{2}{3} π r^{3}$ curved surface $2 π r^{2}$

Right-angled triangles

Name	Formula	Condition
Ratios	$\sin θ = \frac{opp}{hyp}$ $\cos θ = \frac{adj}{hyp}$ $\tan θ = \frac{opp}{adj}$
Pythagoras	$a^{2} + b^{2} = c^{2},$ $c$ = hypotenuse

Opposite and adjacent are named from

θ

Angles of elevation and depression

Exam note: Angles of elevation and depression are measured from the horizontal.
Connection: The angle of elevation of $B$ from $A$ equals the angle of depression of $A$ from $B$ .

Both angles are measured from the horizontal.

Any triangle

Name	Formula	Condition
Sine rule	$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
Cosine rule	$c^{2} = a^{2} + b^{2} - 2 a b \cos C$ $\cos C = \frac{a^{2} + b^{2} - c^{2}}{2 a b}$
Area	$A = \frac{1}{2} a b \sin C$

Side

a

is opposite angle

A

, and so on.

Use the sine rule when you know	Use the cosine rule when you know
two angles and one side two sides and an angle not between them (this can be the ambiguous case)	two sides and the angle between them all three sides

Exam note: Set the GDC to degrees or radians to match the question.

The ambiguous case of the sine rule

Exam note: It happens only when you know two sides and an angle not between them, and you use the sine rule to find an angle.
Exam note: $\sin θ = k$ gives two angles: $θ$ and $180^{\circ} - θ$ . Check both. Keep $180^{\circ} - θ$ only if the angles of the triangle still add up to $180^{\circ}$ with every angle positive.

Ambiguous case only: two triangles fit the same

A

c

and

a

Back to topics

Radians and the Unit Circle

Radians, arcs and sectors

Name	Formula	Condition
Radians	$π radians = 180^{\circ}$
Arc length	$l = r θ$	$θ$ in radians
Sector area	$A = \frac{1}{2} r^{2} θ$	$θ$ in radians

Exam note: Arc length and sector area need $θ$ in radians.

The unit circle and exact values

Name	Formula	Condition
Unit circle	$\cos θ = x$ -coordinate $\sin θ = y$ -coordinate
Tan	$\tan θ = \frac{\sin θ}{\cos θ}, \cos θ \neq 0$

$θ$	$0$	$\frac{π}{6}$	$\frac{π}{4}$	$\frac{π}{3}$	$\frac{π}{2}$
$\sin θ$	$0$	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	$1$
$\cos θ$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	$0$
$\tan θ$	$0$	$\frac{\sqrt{3}}{3}$	$1$	$\sqrt{3}$	undefined

Exam note: Exact values of multiples: find the reference angle, then the sign from the quadrant.

The full unit circle is on the next page.

The unit circle

Each point is

(\cos θ, \sin θ)

. Angles are measured anticlockwise from the positive

x

-axis.

\tan θ = \frac{y}{x}

Example: $θ$ $= 30^{\circ}$ $= \frac{π}{6}$

The point on the unit circle at $30^{\circ}$ is $(\frac{\sqrt{3}}{2}, \frac{1}{2})$ .

$\cos 30^{\circ} = x$ -coordinate $= \frac{\sqrt{3}}{2}$

$\sin 30^{\circ} = y$ -coordinate $= \frac{1}{2}$

$\tan 30^{\circ}$ $= \frac{y}{x}$ $= \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}$ $= \frac{1}{\sqrt{3}}$ $= \frac{\sqrt{3}}{3}$

Back to topics

Trigonometric Identities

Video: Double angle identities for sine and cosine (opens YouTube in a new tab)

Pythagorean and double angle identities

Name	Formula	Condition
Pythagorean	$\cos^{2} θ + \sin^{2} θ = 1$
Double angle	$\sin 2 θ = 2 \sin θ \cos θ$ $\cos 2 θ = \cos^{2} θ - \sin^{2} θ = 2 \cos^{2} θ - 1 = 1 - 2 \sin^{2} θ$

Connection: $\cos^{2} θ$ $= \frac{1 + \cos 2 θ}{2}$ and $\sin^{2} θ$ $= \frac{1 - \cos 2 θ}{2}$
Exam note: $\sin 2 θ \neq 2 \sin θ$ .
Exam note: Pick the $\cos 2 θ$ form that matches the rest of the equation: all sin or all cos.

Back to topics

Trigonometric Functions

The three graphs

Name	Formula	Condition
$\sin x$ , $\cos x$	period $2 π$ range $- 1 \leq y \leq 1$ sin (blue). cos (dark).
$\tan x$	period $π$ asymptotes $x = \frac{π}{2} + k π$ , $k \in ℤ$

Transformed sine and cosine

Name	Formula	Condition
Functions	$f (x) = a \sin (b (x - c)) + d$ $f (x) = a \cos (b (x - c)) + d$	$b > 0$

Feature	From the function	From the graph
Amplitude	$\| a \|$	$\frac{max - min}{2}$
Period	radians: $\frac{2 π}{b}$ degrees: $\frac{360^{\circ}}{b}$	from a maximum to the next maximum, or from a minimum to the next minimum then $b = \frac{2 π}{period}$ or $b$ $= \frac{360^{\circ}}{period}$
Principal axis	$y = d$	$y = \frac{max + min}{2}$
Maximum, minimum	$d + \| a \|, d - \| a \|$	the highest and lowest $y$ -values
Horizontal shift	$c$ to the right ( $c < 0$ : to the left)	sine with $a > 0$ : the $x$ where the graph crosses $y = d$ going up cosine with $a > 0$ : the $x$ of a maximum point

Connection: Example: amplitude 1.5, period $\frac{2 π}{2} = π$ , principal axis $y = 2$ , shift $\frac{π}{4}$ to the right, maximum $2 + 1.5 = 3.5$ , minimum $2 - 1.5 = 0.5$ .

f (x)

= 1.5 \sin (2 (x - \frac{π}{4})) + 2

a = 1.5

b = 2

c = \frac{π}{4}

d = 2

Back to topics

Topic 4 Statistics and Probability

Data and Statistics

Mean and spread

Name	Formula	Condition
Mean	$\bar{x} = \frac{\sum f_{i} x_{i}}{n}$	$n = \sum f_{i}$
Variance	$σ^{2} = \frac{\sum f_{i} (x_{i} - μ)^{2}}{n} = \frac{\sum f_{i} x_{i}^{2}}{n} - μ^{2}$
Standard deviation	$σ = \sqrt{\frac{\sum f_{i} (x_{i} - μ)^{2}}{n}}$
Grouped data	use mid-interval values as $x_{i}$	gives an estimate

Change to every value	Mean	Standard deviation $σ$	Variance $σ^{2}$	IQR
Add $k$ to every value	$mean + k$	unchanged	unchanged	unchanged
Subtract $k$ from every value	$mean - k$	unchanged	unchanged	unchanged
Multiply every value by $k$	$mean \times k$	$σ \times \| k \|$	$σ^{2} \times k^{2}$	$I Q R \times \| k \|$
Divide every value by $k$ , $k \neq 0$	$mean \div k$	$σ \div \| k \|$	$σ^{2} \div k^{2}$	$I Q R \div \| k \|$

Connection: Adding or subtracting $k$ moves every value by the same amount, so the spread does not change.
Connection: Example: 2, 4, 6 has mean 4. Add 10 to every value: 12, 14, 16, mean 14, same spread. Multiply every value by 3: 6, 12, 18, mean 12, and $σ$ is 3 times larger.
Exam note: Use $σ_{x}$ from the GDC, not $s_{x}$ .

Quartiles, outliers and box plots

Name	Formula	Condition
Interquartile range	$I Q R = Q_{3} - Q_{1}$
Outlier	below $Q_{1} - 1.5 \times I Q R$ or above $Q_{3} + 1.5 \times I Q R$
Cumulative frequency	read across at $\frac{n}{4}$ for $Q_{1}$ , $\frac{n}{2}$ for the median, $\frac{3 n}{4}$ for $Q_{3}$	percentile $p$ : at $p %$ of $n$

Data: 1, 3, 4, 6, 7, 8, 10, 11, 13, 14, 16. Read each value from the scale.

Sampling and data

Name	Formula	Condition
Population, sample	The population is the whole group being studied. A sample is part of it.
Simple random sample	Every possible sample of that size is equally likely to be chosen.
Bias, reliability	A biased sample does not represent the population. Check that the source of the data is reliable.
Frequency distribution	A table of values or classes with their frequencies. A histogram shows it as bars with no gaps.
Mode, modal class	The value with the highest frequency. For grouped data: the class with the highest frequency.

Connection: Sampling methods: simple random, systematic, stratified, quota, convenience
Exam note: Discrete data is counted. Continuous data is measured.

Back to topics

Correlation and Regression

Video: Regression line (opens YouTube in a new tab)

Correlation

Name	Formula	Condition
Pearson’s $r$	$- 1 \leq r \leq 1$	from the GDC

Connection: $r$ close to $\pm 1$ : strong linear correlation. $r$ close to 0: weak or no linear correlation. The sign of $r$ gives the direction.
Connection: Scatter diagram: plot each pair $(x, y)$ as a point. A line of best fit drawn by eye passes through the mean point $(\bar{x}, \bar{y})$ .
Exam note: Correlation does not mean causation.

Regression lines

Name	Formula	Condition
$y$ on $x$	$y = a x + b$ : predicts $y$ from $x$
$x$ on $y$	$x = c y + d$ : predicts $x$ from $y$

Connection: For each increase of 1 in $x$ , $y$ changes by $a$ .
When $x = 0$ , $y = b$ .
Connection: Both lines pass through $(\bar{x}, \bar{y})$ .
Exam note: The $y$ on $x$ line predicts $y$ from $x$ only. Using it to predict $x$ from $y$ is not reliable: use the $x$ on $y$ line.
Exam note: Interpolation is reliable when the correlation is strong ( $r$ is close to $\pm 1$ ). Extrapolation is not reliable.

Both lines pass through

(\bar{x}, \bar{y})

Back to topics

Probability

Video: Basic theoretical probability (opens YouTube in a new tab)

Basic rules

Name	Formula	Condition
Probability of $A$	$P (A)$ $= \frac{n (A)}{n (U)}$ , equally likely outcomes
Complementary events	$P (A) + P (A') = 1$
Combined events	$P (A \cup B) = P (A) + P (B) - P (A \cap B)$
Mutually exclusive	$P (A \cap B) = 0$ so $P (A \cup B)$ $= P (A) + P (B)$
Expected number	$n p$ for $n$ trials with probability $p$

P (A) + P (B)

counts

A \cap B

twice, so subtract

P (A \cap B)

once.

Conditional probability and independence

Name	Formula	Condition
Conditional	$P (A ∣ B) = \frac{P (A \cap B)}{P (B)}$
Independent	$P (A \cap B) = P (A) P (B)$

Connection: Independent $⟺ P (A ∣ B)$ $= P (A)$ $= P (A ∣ B')$
Connection: $P (A \cap B)$ $= P (A ∣ B) P (B)$ $= P (B ∣ A) P (A)$
Exam note: “Given that” means conditional: divide by the probability of the given event.

Tree diagrams

Connection: Tree diagram: multiply along branches and add the end results.
Exam note: Without replacement, the second-stage probabilities change.

P (B ∣ A)

= \frac{0.01 \times 0.95}{0.01 \times 0.95 + 0.99 \times 0.08}

Back to topics

Discrete Distributions

Video: Binomial distribution (opens YouTube in a new tab)

Discrete random variables

Name	Formula	Condition
Probability distribution	$\sum P (X = x) = 1$	$0 \leq P (X = x) \leq 1$
Expected value	$E (X) = μ = \sum x P (X = x)$

Connection: Fair game: $E (gain) = 0$

Binomial distribution

Name	Formula	Condition
Binomial	$X \sim B (n, p) \Rightarrow P (X = x) =^{n} C_{x} p^{x} (1 - p)^{n - x}$	$x = 0, 1, \dots, n$
Mean and variance	$E (X) = n p$ $Var (X) = n p (1 - p)$

Connection: Conditions: fixed $n$ , two outcomes (success and failure), constant $p$ , independent trials
Connection: $P (X \geq k)$ $= 1 - P (X \leq k - 1)$
$P (X > k)$ $= 1 - P (X \leq k)$
Exam note: GDC: binomial pdf for $P (X = x)$ and binomial cdf for $P (X \leq x)$ .
Exam note: “At least”, “more than” and “fewer than”: rewrite with $\leq$ before using the cdf.

X \sim B (6, 0.4)

P (X \geq 3)

= 1 - P (X \leq 2)

Back to topics

The Normal Distribution

The normal curve

Name	Formula	Condition
Normal	$X \sim N (μ, σ^{2})$	$σ^{2} =$ variance
Properties	symmetric about $μ$ , mean = median = mode, total area 1

Connection: Normal probabilities such as $P (a < X < b)$ : use the normal cdf on the GDC with the lower bound, the upper bound, $μ$ and $σ$ .
Exam note: $N (μ, σ^{2})$ gives the variance. Take the square root for $σ$ .

The empirical rule

Blue numbers: the example

X \sim N (50, 10^{2})

, so

μ = 50

and

σ = 10

. Each percentage is the area to 2 decimal places.

Using symmetry: an example

Connection: $X \sim N (50, 10^{2})$
Connection: $P (X < 50) = 0.5$
Connection: $P (X < 40)$ $= P (X > 60)$ , because 40 and 60 are both 10 from the mean.
Connection: $P (X = 45) = 0$ , so $P (X < 45)$ $= P (X \leq 45)$ .

X \sim N (50, 10^{2})

: the two gold areas are equal.

Standardising and the inverse normal

Name	Formula	Condition
Standardised variable	$z = \frac{x - μ}{σ}$	$Z \sim N (0, 1)$

Connection: $μ$ or $σ$ unknown, with a given probability such as $P (X < x) = p$ :
1. Standardise with the $z$ formula: $P (X < x)$ $= P (Z < \frac{x - μ}{σ})$ $= p$ , where $Z \sim N (0, 1)$ .
2. Use the inverse normal on $N (0, 1)$ , with mean 0 and standard deviation 1, to find the $z$ -value with $P (Z < z) = p$ .
3. Set $\frac{x - μ}{σ} = z$ and solve for the unknown.
Connection: Example: $X \sim N (μ, 4^{2})$ and $P (X < 30) = 0.9$ . Inverse normal: $z = 1.2816$ . So $\frac{30 - μ}{4} = 1.2816$ , giving $μ = 24.87$ (2 d.p.).
Connection: Both $μ$ and $σ$ unknown: two probabilities are given. Do steps 1 to 3 for each one. This gives two equations in $μ$ and $σ$ . Solve them together.
Exam note: Inverse normal: check which tail the GDC uses.

Back to topics

Topic 5 Calculus

Limits and the Derivative

Limits

Name	Formula	Condition
Limit	$\lim_{x \to a} f (x)$ : the value $f (x)$ approaches as $x \to a$

The derivative

Name	Formula	Condition
Derivative	$y = f (x) \Rightarrow \frac{d y}{d x} = f' (x)$ the gradient function and the rate of change of $y$ with respect to $x$
Second derivative	$f ″ (x) = \frac{d^{2} y}{d x^{2}}$

h \to 0

the chord turns into the tangent.

Back to topics

Differentiation Rules

Standard derivatives

$f (x)$	$f' (x)$
$x^{n}$	$n x^{n - 1}$
$\sin x$	$\cos x$
$\cos x$	$- \sin x$

$f (x)$	$f' (x)$
$e^{x}$	$e^{x}$
$\ln x$	$\frac{1}{x}$

Exam note: Trig derivatives need $x$ in radians.

Chain, product and quotient rules

Name	Formula	Condition
Chain rule	$y = g (u)$ , where $u = f (x) \Rightarrow \frac{d y}{d x} = \frac{d y}{d u} \times \frac{d u}{d x}$
Product rule	$y = u v \Rightarrow \frac{d y}{d x} = u \frac{d v}{d x} + v \frac{d u}{d x}$
Quotient rule	$y = \frac{u}{v} \Rightarrow \frac{d y}{d x} = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v^{2}}$

Connection: $\frac{d}{d x} f (a x + b)$ $= a f' (a x + b)$
Connection: $\frac{d}{d x} e^{g (x)}$ $= g' (x) e^{g (x)}$ and $\frac{d}{d x} \ln g (x)$ $= \frac{g' (x)}{g (x)}$
Connection: $\frac{d}{d x} [g (x)]^{n}$ $= n [g (x)]^{n - 1} g' (x)$
Exam note: Simplify first: split fractions and use log laws before differentiating.

Back to topics

Using Derivatives

Shape of a graph

If	Then
$f' (x) > 0$	$f$ is increasing
$f' (x) < 0$	$f$ is decreasing
$f' (a) = 0$	stationary point at $x = a$
$f' (a) = 0$ and $f ″ (a) < 0$	local maximum at $x = a$
$f' (a) = 0$ and $f ″ (a) > 0$	local minimum at $x = a$
$f' (a) = 0$ and $f ″ (a) = 0$	the test fails: check the sign of $f'$ on each side of $a$
$f' (a) = 0$ , $f ″ (a) = 0$ and $f ″$ changes sign at $a$	horizontal (stationary) point of inflection at $x = a$
$f ″ (x) > 0$	concave up
$f ″ (x) < 0$	concave down
$f ″ (a) = 0$ and $f ″$ changes sign at $a$	point of inflection at $x = a$

Connection: Second derivative test: first solve $f' (x) = 0$ to find the stationary point $x = a$ . Then test the sign of $f ″ (a)$ .
Connection: $f$ turns where $f' = 0$ and $f'$ changes sign. $f$ has an inflection where $f'$ turns.
Connection: Horizontal (stationary) inflection: the tangent is horizontal, $f' = 0$ there. Non-stationary inflection: $f' \neq 0$ there.
Exam note: $f ″ (a) = 0$ alone does not prove an inflection. Check the sign change.

f (x) = x^{3} - 3 x

. Step 1:

f' (x) = 3 x^{2} - 3 = 0

gives the turning points

a = - 1

and

a = 1

.
Step 2: test each one with

f ″ (x) = 6 x

f

has an inflection where

f'

turns.

f (x) = x^{3} - 3 x^{2} + 4

f ″ (x) = 6 x - 6

: concave down for

x < 1

, concave up for

x > 1

, inflection at

(1, 2)

f (x) = x^{3}

: a horizontal point of inflection at the origin. The tangent there is horizontal.

Sketching the first and second derivatives

Graph of $f$	Graph of $f'$	Graph of $f ″$
increasing	above the $x$ -axis ( $f' > 0$ )
decreasing	below the $x$ -axis ( $f' < 0$ )
max or min	crosses the $x$ -axis ( $f' = 0$ )
point of inflection	max or min	crosses the $x$ -axis ( $f ″ = 0$ )
concave up	increasing	above the $x$ -axis ( $f ″ > 0$ )
concave down	decreasing	below the $x$ -axis ( $f ″ < 0$ )

Connection: $f'$ touches the $x$ -axis without crossing: a horizontal point of inflection of $f$ .
Exam note: Mark the $x$ -values of the key points of $f$ first. Then sketch $f'$ through them.

f (x) = x^{3} - 3 x^{2} + 2

Max at

x = 0

and min at

x = 2

f'

crosses the axis.
Inflection at

x = 1

f'

has its min and

f ″

crosses the axis.

Tangents and normals

Connection: Tangent at $x = a$ : find the point $(a, f (a))$ and the gradient $m = f' (a)$ . Put them into the equation of a straight line.
Connection: Normal at $x = a$ : the same point, with gradient $- \frac{1}{m}$ , because the normal is perpendicular to the tangent. If $m = 0$ , the normal is $x = a$ .

Optimisation

Connection: Write the quantity in one variable, set the derivative to 0, then confirm max or min.
Exam note: Check the end points of the domain too.

Back to topics

Integration

Video: Integration by substitution (opens YouTube in a new tab)

Standard integrals

\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C, n \neq - 1

\int \frac{1}{x} d x = \ln | x | + C

\int \sin x d x = - \cos x + C

\int \cos x d x = \sin x + C

\int e^{x} d x = e^{x} + C

Exam note: Add $+ C$ to every indefinite integral. Keep the modulus in $\ln | x |$ .
Connection: A boundary condition finds $C$ .

Recognising the form

Name	Formula	Condition
Linear inside	$\int f (a x + b) d x = \frac{1}{a} F (a x + b) + C$	$F' = f$
Top is the derivative of the bottom	$\int \frac{g' (x)}{g (x)} d x = \ln \| g (x) \| + C$
Derivative times exponential	$\int g' (x) e^{g (x)} d x = e^{g (x)} + C$

Connection: $\sin^{2} x$ , $\cos^{2} x$ : use the double angle formula (Topic 3).

Substitution and definite integrals

Name	Formula	Condition
Substitution	$\int f (g (x)) g' (x) d x$ : let $u = g (x)$ , $d u = g' (x) d x$ , then find $\int f (u) d u$
Definite integral	$\int_{a}^{b} f (x) d x = F (b) - F (a)$	$F' = f$

Connection: Example: $\int 2 x {(x^{2} + 1)}^{3} d x$ . Let $u = x^{2} + 1$ , so $d u = 2 x d x$ . This gives $\int u^{3} d u$ $= \frac{u^{4}}{4} + C$ $= \frac{{(x^{2} + 1)}^{4}}{4} + C$ .
Exam note: Substitution in a definite integral: change the limits to $u$ -values too.

Back to topics

Areas and Kinematics

Areas

Name	Formula	Condition
With the $x$ -axis	$A = \int_{a}^{b} \| y \| d x$ $A = \int_{a}^{b} y d x$ when $y > 0$
Between curves	$A = \int_{a}^{b} (f (x) - g (x)) d x, f (x) \geq g (x)$

Connection: Below the $x$ -axis, the integral is negative. Split at the roots or integrate $| y |$ .
Exam note: Area between curves: find where they meet for the limits.

y = x (x - 2) (x - 4)

0 \leq x \leq 4

area

= 4 + 4 = 8

integral

= 4 + (- 4) = 0

Kinematics

Name	Formula	Condition
Velocity, acceleration	$v = \frac{d s}{d t}$ $a = \frac{d v}{d t} = \frac{d^{2} s}{d t^{2}}$
Going back	$s = \int v d t$ $v = \int a d t$ use the starting values to find $C$
Displacement from $t_{1}$ to $t_{2}$	$\int_{t_{1}}^{t_{2}} v (t) d t$
Distance travelled from $t_{1}$ to $t_{2}$	$\int_{t_{1}}^{t_{2}} \| v (t) \| d t$
Speed	$\| v \|$

v = 3 (t - 1) (t - 3)

0 \leq t \leq 4

displacement

= 4 + (- 4) + 4 = 4

distance

= 4 + 4 + 4 = 12

Connection: $v > 0$ : moving in the positive direction. $v < 0$ : moving in the negative direction.
Connection: At rest: $v = 0$ . Changes direction: $v$ changes sign.
Connection: Speeding up: $v$ and $a$ have the same sign. Slowing down: opposite signs.
Connection: Maximum speed from a $v$ – $t$ graph: speed is $| v |$ , so look for the point furthest from the $t$ -axis. It can be above or below the axis.

Motion along a straight line:

s

is the displacement from 0.

v = (t - 1) (t - 7)

0 \leq t \leq 7

Furthest from the

t

-axis:

v = - 9

t = 4

, not

v = 7

t = 0

.
Maximum speed

= 9

Back to topics

Cover of the AA SL Revision Notes PDF, updated November 2026

Prefer a printable copy?

Receive the AA SL Revision Notes as a PDF, ready to print or keep offline. Free to share, unchanged.

24 pages · 5 topics · 24 sections

Need help applying these ideas?

These notes show the formulas and their conditions. One-to-one tutoring looks at your own work, finds what is limiting your grade, chooses the right practice, and gives feedback on your reasoning.

Explore one-to-one tutoring

Prefer a fixed programme? See the revision groups starting January 2027

Related pages and updates

Looking for what the course covers? View the AA SL syllabus
Found an error? Email contact@ibmathmaster.com. Corrections appear in the next update.
Last updated: November 2026.

How to use these notes

Topic 1 Number and Algebra

1Exponents and Logarithms

Exponent laws

Logarithms

2Sequences and Series

Arithmetic and geometric sequences

Sigma notation

Money: interest and depreciation

3The Binomial Theorem

The binomial theorem

4Proof

Deductive proof

Topic 2 Functions

1Lines and Quadratics

Straight lines

The three forms of a quadratic

Roots and the discriminant

2Functions, Composites and Inverses

Functions and composites

Inverse functions

3Graphs and Transformations

Transformations of y=f(x)

Sketching

4Rational Functions

Types and their asymptotes

Finding asymptotes and intercepts

5Exponential and Logarithmic Functions

The two graphs

Changing the graphs

6Solving Equations and Inequalities

Using graphs

Quadratic inequalities

Topic 3 Geometry and Trigonometry

13D Geometry and Triangles

Points in 3D

Solids

Right-angled triangles

Angles of elevation and depression

Any triangle

The ambiguous case of the sine rule

2Radians and the Unit Circle

Radians, arcs and sectors

The unit circle and exact values

The unit circle

3Trigonometric Identities

Pythagorean and double angle identities

4Trigonometric Functions

The three graphs

Transformed sine and cosine

Topic 4 Statistics and Probability

1Data and Statistics

Mean and spread

Quartiles, outliers and box plots

Sampling and data

2Correlation and Regression

Correlation

Regression lines

3Probability

Basic rules

Conditional probability and independence

Tree diagrams

4Discrete Distributions

Discrete random variables

Binomial distribution

5The Normal Distribution

The normal curve

The empirical rule

Using symmetry: an example

Standardising and the inverse normal

Topic 5 Calculus

1Limits and the Derivative

Limits

The derivative

2Differentiation Rules

Standard derivatives

Chain, product and quotient rules

3Using Derivatives

Shape of a graph

Sketching the first and second derivatives

Exponents and Logarithms

Sequences and Series

The Binomial Theorem

Proof

Lines and Quadratics

Functions, Composites and Inverses

Graphs and Transformations

Transformations of $y = f (x)$

Rational Functions

Exponential and Logarithmic Functions

Solving Equations and Inequalities

3D Geometry and Triangles

Radians and the Unit Circle

Trigonometric Identities

Trigonometric Functions

Data and Statistics

Correlation and Regression

Probability

Discrete Distributions

The Normal Distribution

Limits and the Derivative

Differentiation Rules

Using Derivatives

Integration

Areas and Kinematics