IB Math AA HL Revision Notes

Formulas, conditions, key connections and common exam traps across the five AA HL 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 HL syllabus, for exams up to November 2028.

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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

Counting and the Binomial Theorem

Counting

Name	Formula	Condition
Factorial	$n! = n (n - 1) (n - 2) \dots 2 \times 1$	$0! = 1$
Permutations	$^{n} P_{r} = \frac{n!}{(n - r)!}$	order matters, $0 \leq r \leq n$
Combinations	$^{n} C_{r} = \frac{n!}{r! (n - r)!}$	order does not matter

Connection: $n! = n (n - 1) (n - 2)!$ for $n \geq 2$
Connection: $^{n} C_{0} =^{n} C_{n} = 1$ , $^{n} C_{1} = n$ , $^{n} C_{r} =^{n} C_{n - r}$ and $^{n} P_{n} = n!$

The binomial theorem

Name	Formula	Condition
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: $(a + b)^{n}$ 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 $2 x^{r}$ .

Extended binomial

Name	Formula	Condition
Extended binomial	$(a + b)^{n} = a^{n} {(1 + \frac{b}{a})}^{n} = a^{n} (1 + n (\frac{b}{a}) + \frac{n (n - 1)}{2!} {(\frac{b}{a})}^{2} + \dots)$	$n \in ℚ$ , $\| \frac{b}{a} \| < 1$

Connection: $(1 - x)^{- 1} = 1 + x + x^{2} + \dots$ is the geometric series with $r = x$ , valid for $| x | < 1$ .

Back to topics

Partial Fractions

Video: Partial fractions (opens YouTube in a new tab)

Two linear factors

Name	Formula	Condition
Two linear factors	$\frac{p x + q}{(a x + b) (c x + d)} \equiv \frac{A}{a x + b} + \frac{B}{c x + d}$	the two factors are different

Connection: Clear the denominators: $p x + q \equiv A (c x + d) + B (a x + b)$ . This holds for every $x$ , so substitute the $x$ that makes each factor zero.
Connection: In Topic 5: $\int \frac{A}{c x + d} d x$ $= \frac{A}{c} \ln | c x + d | + C$
Exam note: Factorise the denominator fully first.
Exam note: The numerator must have lower degree than the denominator. If not, divide first.
Exam note: Watch the signs: $\frac{1}{a - x}$ $= - \frac{1}{x - a}$ .

Back to topics

Systems of Linear Equations

Up to three equations in three unknowns

Eliminate one unknown at a time.

Result	What you see after eliminating
Unique solution	Each unknown gets one value.
Infinitely many solutions	One equation becomes $0 = 0$ . It gives no new information. Let one unknown be any number $λ$ , then write the other unknowns in terms of $λ$ . If two equations become $0 = 0$ , let two unknowns be $λ$ and $μ$ .
No solution	One equation becomes $0 = c$ with $c \neq 0$ , for example $0 = 5$ . This is impossible, so the system has no solution.

Result

What you see after eliminating

Unique solution

Each unknown gets one value.

Infinitely many solutions

One equation becomes $0 = 0$ . It gives no new information. Let one unknown be any number $λ$ , then write the other unknowns in terms of $λ$ .

If two equations become $0 = 0$ , let two unknowns be $λ$ and $μ$ .

No solution

One equation becomes $0 = c$ with $c \neq 0$ , for example $0 = 5$ . This is impossible, so the system has no solution.

Back to topics

Complex Numbers

Cartesian form

Name	Formula	Condition
Cartesian form	$z = a + b i$ $R e (z) = a$ $I m (z) = b$ $i^{2} = - 1$	$a, b \in ℝ$
Conjugate and modulus	$z^{*} = a - b i$ $\| z \| = r = \sqrt{a^{2} + b^{2}}$
Argument	$\arg z = θ$ $\tan θ = \frac{b}{a}$ $a = r \cos θ$ $b = r \sin θ$ check the quadrant	Argand diagram

Connection: $z z^{*} = | z |^{2}$
$z + z^{*} = 2 R e (z)$
$z - z^{*} = 2 i I m (z)$
Connection: $| z^{*} |$ $= | z |$ and $\arg z^{*} = - \arg z$ . Add $2 π$ if this is outside the required range.
Connection: $i^{1}$ $= i, i^{2}$ $= - 1, i^{3}$ $= - i, i^{4}$ $= 1$
Exam note: $a + b i = c + d i ⟺ a = c$ and $b = d$ , for $a, b, c, d \in ℝ$ . This gives two equations.
Exam note: Give $\arg z$ in the range the question states, often $- π < θ \leq π$ or $0 \leq θ < 2 π$ .

Arguments on the axes

$z$ lies on the	$z$	$\arg z$	All its arguments
positive real axis	$a$	$0$	$0 + 2 k π$
negative real axis	$- a$	$π$	$π + 2 k π$
positive imaginary axis	$b i$	$\frac{π}{2}$	$\frac{π}{2} + 2 k π$
negative imaginary axis	$- b i$	$- \frac{π}{2}$	$- \frac{π}{2} + 2 k π$

Here $a > 0$ , $b > 0$ and $k \in ℤ$ . The $\arg z$ column uses $- π < θ \leq π$ .

Exam note: Adding a multiple of $2 π$ gives the same point. Give the argument in the range the question asks.
Exam note: Do not use $\tan θ = \frac{b}{a}$ here. On the imaginary axis $a = 0$ , and on the real axis it gives $0$ for both $3$ and $- 3$ . Sketch $z$ , then read the angle.
Exam note: $z = 0$ has no argument.

Measure from the positive real axis.
Anticlockwise is positive. Clockwise is negative.

Polar and Euler forms

Name	Formula	Condition
Polar and Euler forms	$z = r (\cos θ + i \sin θ) = r e^{i θ} = r cis θ$
Product	$z_{1} z_{2} = r_{1} r_{2} cis (θ_{1} + θ_{2})$
Quotient	$\frac{z_{1}}{z_{2}} = \frac{r_{1}}{r_{2}} cis (θ_{1} - θ_{2})$	$z_{2} \neq 0$

Connection: $| z_{1} z_{2} |$ $= | z_{1} | | z_{2} |$ and $\arg (z_{1} z_{2})$ $= \arg z_{1} + \arg z_{2}$
$| \frac{z_{1}}{z_{2}} |$ $= \frac{| z_{1} |}{| z_{2} |}$ and $\arg (\frac{z_{1}}{z_{2}})$ $= \arg z_{1} - \arg z_{2}$
Adjust arguments by $2 π$ if needed.
Connection: $z_{1} + z_{2}$ : vector addition (the diagonal of the parallelogram).
$z^{*}$ is the reflection of $z$ in the real axis.
Connection: Multiplying by $i$ : rotate $90^{\circ}$ anticlockwise about 0.
Connection: Multiplying by $r cis θ$ : enlarge by $r$ and rotate by $θ$ .
Connection: $| z_{1} - z_{2} |$ is the distance between the points $z_{1}$ and $z_{2}$ .
Connection: $e^{i π} = - 1$ and $| e^{i θ} |$ $= 1$

Reflection and

90^{\circ}

turn

z_{2}

= (r cis θ) z_{1}

| z_{2} |

= r | z_{1} |

Powers and roots

Name	Formula	Condition
De Moivre	$\begin{aligned} {[r (\cos θ + i \sin θ)]}^{n} & = r^{n} (\cos n θ + i \sin n θ) \\ = r^{n} e^{i n θ} = r^{n} cis n θ \end{aligned}$	$n \in ℤ$ . For fractional powers, use the $n$ th roots formula.
$n$ th roots	$z^{n} = r cis θ \Rightarrow z = r^{\frac{1}{n}} cis (\frac{θ + 2 π k}{n})$	$n \in ℤ^{+}$ , $r > 0$ $k = 0, 1, \dots, n - 1$

Connection: If $| z | = 1$ : $z^{n} + z^{- n} = 2 \cos n θ$
$z^{n} - z^{- n} = 2 i \sin n θ$
Connection: The roots of $z^{n} = w$ , $w \neq 0$ , have equal modulus, and their arguments are $\frac{2 π}{n}$ apart. For $n \geq 3$ they form a regular $n$ -gon centred at 0.
Connection: The $n$ roots of $z^{n} = 1$ (the $n$ th roots of unity) add to 0, for $n \geq 2$ .
Exam note: Conjugate roots: if the coefficients are real and $z$ is a root, then $z^{*}$ is a root.
Exam note: $n$ th roots: list all $n$ of them. Start at $k = 0$ , then put $k = 1, 2, \dots, n - 1$ . Keep every argument in the required range.

Example with

n = 6

: a regular hexagon

Back to topics

Proof

Video: Proof by induction (opens YouTube in a new tab)

Deductive proof, contradiction and counterexample

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
Contradiction	Assume the statement is false. Reach something impossible. So the statement is true.
Counterexample	One case where the statement fails shows it is not always true.

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.
Exam note: A counterexample is one specific value, with the check shown.

Proof by induction

Name	Formula	Condition
Induction	Show the statement is true for $n = 1$ (or for the first value in the question, such as $n = 0$ or $n = 2$ ). Assume it is true for $n = k$ . Prove it is true for $n = k + 1$ . This step always uses the assumption for $n = k$ . Conclude: true for $n = 1$ , and true for $n = k$ implies true for $n = k + 1$ , so true for all $n \in ℤ^{+}$ by mathematical induction.	$n \in ℤ^{+}$ , or $n$ from the first value

Exam note: Write “Assume true for $n = k$ ”, not “for all $k$ ”.
Exam note: In the $n = k + 1$ step, show the line where you use the assumption for $n = k$ .
Exam note: Write the full concluding sentence. Mark schemes usually give the final mark only when the earlier steps are there.

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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 $Δ > 0$ .

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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	$(f \circ f^{- 1}) (x) = (f^{- 1} \circ f) (x) = x$
Self-inverse	$f^{- 1} (x) = f (x)$ , so $f (f (x)) = x$ . Symmetric in $y = x$ .

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 $\frac{1}{f (x)}$ .
Exam note: Not one-to-one? Restrict the domain first, usually to one side of a turning point.

(1, 2)

f

and

(2, 1)

f^{- 1}

Odd and even functions

Name	Formula	Condition
Odd	$f (- x) = - f (x)$ Half-turn symmetry about the origin
Even	$f (- x) = f (x)$ Symmetric in the $y$ -axis

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 $\frac{1}{q}$	$(\frac{x}{q}, y)$
$y = f (a x + b)$	translation $b$ units to the left, then horizontal stretch, scale factor $\frac{1}{a}$	$(\frac{x - b}{a}, y)$

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: Papers 2 and 3: find intersections, zeros and turning points with the GDC.

Back to topics

Special Function Graphs

Video: Modulus transformations: $| f (x) |$ and $f (| x |)$ (opens YouTube in a new tab)

Each graph is drawn from $f (x) = x^{2} - 4 x + 3$ , shown dashed. Gold lines are asymptotes.

$y = | f (x) |$

On $y = f (x)$	On $y = \| f (x) \|$
Part on or above the $x$ -axis, $f (x) \geq 0$	stays the same
Part below the $x$ -axis, $f (x) < 0$	reflected in the $x$ -axis
Graph	on or above the $x$ -axis: $y \geq 0$

$y = f (| x |)$

On $y = f (x)$	On $y = f (\| x \|)$
Part for $x \geq 0$	stays the same
Part for $x < 0$	removed. It is replaced by the reflection of the $x \geq 0$ part in the $y$ -axis.
Symmetry	symmetric in the $y$ -axis: an even function

$y = \frac{1}{f (x)}$

On $y = f (x)$	On $y = \frac{1}{f (x)}$
Zero at $x = a$ , $f (a) = 0$	vertical asymptote $x = a$
Vertical asymptote $x = a$	$\frac{1}{f (x)} \to 0$ as $x \to a$ hole at $(a, 0)$ : still undefined at $x = a$
$f (x) \to \pm \infty$ as $x \to \pm \infty$	horizontal asymptote $y = 0$
Horizontal asymptote $y = k$ , $k \neq 0$	horizontal asymptote $y = \frac{1}{k}$
Positive or negative	same sign
Points where $f (x) = 1$ or $f (x) = - 1$	stay fixed
Increasing	decreasing
Decreasing	increasing
Maximum point, not on the $x$ -axis	minimum point, same $x$
Minimum point, not on the $x$ -axis	maximum point, same $x$

$y = [f (x)]^{2}$

On $y = f (x)$	On $y = [f (x)]^{2}$
Zero at $x = a$	touches the $x$ -axis at $x = a$
Negative values	become positive, so $y \geq 0$
Values with $\| f (x) \| > 1$	move away from the $x$ -axis
Values with $\| f (x) \| < 1$	move closer to the $x$ -axis
Points where $f (x) = 0$ or $f (x) = 1$	stay fixed
Points where $f (x) = - 1$	move to $y = 1$

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 $y = \frac{a}{c}$	$y = \frac{2 x + 1}{x - 1}$
Linear over quadratic	$\frac{a x + b}{c x^{2} + d x + e}$	vertical: where the denominator $c x^{2} + d x + e = 0$ horizontal $y = 0$	$y = \frac{x + 1}{x^{2} - 4}$
Quadratic over linear	$\frac{a x^{2} + b x + c}{d x + e}$	vertical: where the denominator $d x + e = 0$ oblique $y = m x + k$	$y = \frac{x^{2} + 1}{x - 1}$

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
Connection: Oblique asymptote when the top degree is one more: divide to get $f (x) = m x + k + \frac{R}{d x + e}$ , so $y = m x + k$ .
Exam note: Write asymptotes as equations.
Exam note: A graph can cross a horizontal or oblique asymptote. It never crosses a vertical one.

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
$f (x) > g (x)$	The $x$ -values where the graph of $f$ is above $g$

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.

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Polynomials

Factor and remainder theorems

Name	Formula	Condition
Polynomial	$P (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0}$	$a_{n} \neq 0$ , degree $n$
Factor theorem	$P (k) = 0 ⟺ (x - k)$ is a factor
Remainder theorem	$P (x) \div (x - k)$ leaves remainder $P (k)$

Connection: Divisor $(a x - b)$ : remainder $P (\frac{b}{a})$

Sum and product of roots

Name	Formula	Condition
Sum of roots	$\frac{- a_{n - 1}}{a_{n}}$	for $\sum_{r = 0}^{n} a_{r} x^{r}$ $= 0$
Product of roots	$\frac{(- 1)^{n} a_{0}}{a_{n}}$

Connection: Quadratic $a x^{2} + b x + c = 0$ : $α + β = - \frac{b}{a}$ and $α β = \frac{c}{a}$
Exam note: Write the equation as … = 0 before using the sum and product formulas.
Exam note: The sum and product include repeated and complex roots.

Shape of the graph

Name	Formula	Condition
Single root	The graph crosses the $x$ -axis.
Double root	$(x - a)^{2}$ is a factor, so $x = a$ is a root twice. The graph touches the $x$ -axis at $x = a$ and turns there.

Connection: Degree $n$ : at most $n$ real roots and at most $n - 1$ turning points
Connection: For large $| x |$ , $P (x)$ behaves like $a_{n} x^{n}$ .

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

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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

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 $= \frac{1}{2}$

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

Symmetries from the unit circle

sin	cos	tan
$\sin (π - θ) = \sin θ$	$\cos (π - θ) = - \cos θ$	$\tan (π - θ) = - \tan θ$
$\sin (- θ) = - \sin θ$	$\cos (- θ) = \cos θ$	$\tan (- θ) = - \tan θ$
$\sin (π + θ) = - \sin θ$	$\cos (π + θ) = - \cos θ$	$\tan (π + θ) = \tan θ$
$\sin (\frac{π}{2} - θ) = \cos θ$	$\cos (\frac{π}{2} - θ) = \sin θ$

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

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Trigonometric Identities

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

Reciprocal ratios and Pythagorean identities

Name	Formula	Condition
Reciprocal ratios	$\sec θ = \frac{1}{\cos θ}$ $cosec θ = \frac{1}{\sin θ}$ $\cot θ = \frac{1}{\tan θ} = \frac{\cos θ}{\sin θ}$
Pythagorean	$\cos^{2} θ + \sin^{2} θ = 1$ $1 + \tan^{2} θ = \sec^{2} θ$ $1 + \cot^{2} θ = {cosec}^{2} θ$

Connection: Divide $\cos^{2} θ + \sin^{2} θ = 1$ by $\cos^{2} θ$ to get $1 + \tan^{2} θ = \sec^{2} θ$ , or by $\sin^{2} θ$ to get $1 + \cot^{2} θ$ $= {cosec}^{2} θ$ .

Compound and double angles

Name	Formula	Condition
Compound angle	$\sin (A \pm B) = \sin A \cos B \pm \cos A \sin B$ $\cos (A \pm B) = \cos A \cos B \mp \sin A \sin B$ $\tan (A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$
Double angle	$\sin 2 θ = 2 \sin θ \cos θ$ $\cos 2 θ = \cos^{2} θ - \sin^{2} θ = 2 \cos^{2} θ - 1 = 1 - 2 \sin^{2} θ$ $\tan 2 θ = \frac{2 \tan θ}{1 - \tan^{2} θ}$

Connection: Put $B = A$ in the compound angle formulas to get the double angle formulas.
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.

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Trigonometric Functions

The three graphs

Name	Formula	Condition
$\sin x$ , $\cos x$	period $2 π$ range $- 1 \leq y \leq 1$ sin is odd (blue). cos is even (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

Inverse trig functions

Name	Formula	Condition
$\arcsin x$	domain $- 1 \leq x \leq 1$ range $- \frac{π}{2} \leq y \leq \frac{π}{2}$
$\arccos x$	domain $- 1 \leq x \leq 1$ range $0 \leq y \leq π$
$\arctan x$	domain $ℝ$ range $- \frac{π}{2} < y < \frac{π}{2}$

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Vectors

Video: Vector product (opens YouTube in a new tab)

Vector basics

Name	Formula	Condition
Components	$𝐯 = v_{1} 𝐢 + v_{2} 𝐣 + v_{3} 𝐤 = (\begin{array}{c} v_{1} \\ v_{2} \\ v_{3} \end{array})$
Magnitude	$\| 𝐯 \| = \sqrt{v_{1}^{2} + v_{2}^{2} + v_{3}^{2}}$
Unit vector	$\frac{𝐯}{\| 𝐯 \|}$
Displacement	$\vec{A B} = 𝐛 - 𝐚$

Connection: Parallel: $𝐰 = k 𝐯$ . Two vectors are parallel when one is a multiple of the other.

Scalar product

Name	Formula	Condition
Scalar product	$𝐯 \cdot 𝐰 = v_{1} w_{1} + v_{2} w_{2} + v_{3} w_{3} = \| 𝐯 \| \| 𝐰 \| \cos θ$ $θ$ = angle between $𝐯$ and $𝐰$
Angle between vectors	$\cos θ = \frac{v_{1} w_{1} + v_{2} w_{2} + v_{3} w_{3}}{\| 𝐯 \| \| 𝐰 \|}$

Connection: Perpendicular: $𝐯 \cdot 𝐰 = 0$
Connection: $𝐯 \cdot 𝐯$ $= | 𝐯 |^{2}$ and $𝐰 \cdot 𝐯$ $= 𝐯 \cdot 𝐰$
Exam note: $𝐯 \cdot 𝐰$ is a number.
Exam note: For the angle at a vertex, both vectors must start at that vertex.

θ

is between the two vectors, both starting at the same point.

Vector product

Name	Formula	Condition
Vector product	$𝐯 \times 𝐰 = (\begin{array}{c} v_{2} w_{3} - v_{3} w_{2} \\ v_{3} w_{1} - v_{1} w_{3} \\ v_{1} w_{2} - v_{2} w_{1} \end{array})$ $\| 𝐯 \times 𝐰 \| = \| 𝐯 \| \| 𝐰 \| \sin θ$
Area of parallelogram	$A = \| 𝐯 \times 𝐰 \|,$ $𝐯$ , $𝐰$ adjacent sides

Connection: $𝐯 \times 𝐰$ is perpendicular to both $𝐯$ and $𝐰$ .
Connection: $𝐰 \times 𝐯$ $= - (𝐯 \times 𝐰)$
Connection: Parallel: $𝐯 \times 𝐰$ $= 𝟎$
Connection: Area of triangle $= \frac{1}{2} | 𝐯 \times 𝐰 |$
Exam note: $𝐯 \times 𝐰$ is a vector.

Parallelogram area

| 𝐯 \times 𝐰 |

. Triangle: half of it.

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Lines and Planes

Lines

Name	Formula	Condition
Vector form	$𝐫 = 𝐚 + λ 𝐛$ $(\begin{array}{c} x \\ y \\ z \end{array}) = (\begin{array}{c} x_{0} \\ y_{0} \\ z_{0} \end{array}) + λ (\begin{array}{c} l \\ m \\ n \end{array})$	$𝐚$ $= (\begin{array}{c} x_{0} \\ y_{0} \\ z_{0} \end{array})$ : a point on the line $𝐛$ $= (\begin{array}{c} l \\ m \\ n \end{array})$ : direction vector
Parametric form	$x = x_{0} + λ l, y = y_{0} + λ m, z = z_{0} + λ n$
Cartesian form	$\frac{x - x_{0}}{l} = \frac{y - y_{0}}{m} = \frac{z - z_{0}}{n}$
Parallel	Directions are parallel. Coincident if they also share a point.
Intersecting	One solution for $λ$ and $μ$ in all three components
Skew	Not parallel and no common point (3D only): one line passes over the other

Connection: Kinematics $𝐫$ $= 𝐚 + t 𝐯$ : $𝐚$ = initial position, $𝐯$ = velocity, speed = $| 𝐯 |$
Exam note: Intersecting lines: solve two components, then check the third.

Planes

Name	Formula	Condition
Vector form	$𝐫 = 𝐚 + λ 𝐛 + μ 𝐜$ $(\begin{array}{c} x \\ y \\ z \end{array}) = (\begin{array}{c} x_{0} \\ y_{0} \\ z_{0} \end{array}) + λ (\begin{array}{c} b_{1} \\ b_{2} \\ b_{3} \end{array}) + μ (\begin{array}{c} c_{1} \\ c_{2} \\ c_{3} \end{array})$	$𝐚$ : a point on the plane $𝐛$ , $𝐜$ : two directions in the plane, not parallel
Normal form	$𝐫 \cdot 𝐧 = 𝐚 \cdot 𝐧$ $(\begin{array}{c} x \\ y \\ z \end{array}) \cdot (\begin{array}{c} n_{1} \\ n_{2} \\ n_{3} \end{array}) = (\begin{array}{c} x_{0} \\ y_{0} \\ z_{0} \end{array}) \cdot (\begin{array}{c} n_{1} \\ n_{2} \\ n_{3} \end{array})$	$𝐧$ $= (\begin{array}{c} n_{1} \\ n_{2} \\ n_{3} \end{array})$ : the normal vector
Cartesian form	$a x + b y + c z = d$	$𝐧$ $= (\begin{array}{c} a \\ b \\ c \end{array})$

Connection: Normal to $𝐫$ $= 𝐚 + λ 𝐛 + μ 𝐜$ : $𝐧$ $= 𝐛 \times 𝐜$
Connection: Line meets plane: substitute the line into the plane and solve for $λ$ .
No solution: line parallel to plane
True for all $λ$ : line lies in the plane
Connection: Two non-parallel planes meet in a line with direction $𝐧_{1} \times 𝐧_{2}$ .
Connection: Three planes: solve as a system (Topic 1).

Angles

Name	Formula	Condition
Two lines	$\cos θ = \frac{\| 𝐛_{1} \cdot 𝐛_{2} \|}{\| 𝐛_{1} \| \| 𝐛_{2} \|},$ directions
Line and plane	$\sin θ = \frac{\| 𝐛 \cdot 𝐧 \|}{\| 𝐛 \| \| 𝐧 \|},$ direction and normal
Two planes	$\cos θ = \frac{\| 𝐧_{1} \cdot 𝐧_{2} \|}{\| 𝐧_{1} \| \| 𝐧_{2} \|},$ normals

Exam note: Each formula gives the acute angle $θ$ : the modulus makes the right side positive.
Exam note: Two lines or two planes: without the modulus you may get the obtuse angle. The acute angle is then $180^{\circ} - θ$ . Line and plane: always use $| 𝐛 \cdot 𝐧 |$ .
Exam note: Line and plane: use sin, because $𝐧$ is at $90^{\circ}$ to the plane.

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.

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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})

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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 and Bayes’ theorem

Name	Formula	Condition
Bayes’ theorem	$P (B ∣ A) = \frac{P (B) P (A ∣ B)}{P (B) P (A ∣ B) + P (B') P (A ∣ B')}$
Three events	$P (B_{i} ∣ A) = \frac{P (B_{i}) P (A ∣ B_{i})}{P (B_{1}) P (A ∣ B_{1}) + P (B_{2}) P (A ∣ B_{2}) + P (B_{3}) P (A ∣ B_{3})}$

Connection: Tree diagram: multiply along branches and add the end results.
Connection: Bayes is the tree read backwards: the branch you want over all branches that give $A$ .
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}

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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	$E (X) = μ = \sum x P (X = x)$
Variance	$Var (X) = \sum (x - μ)^{2} P (X = x) = \sum x^{2} P (X = x) - μ^{2}$ $Var (X) = E (X^{2}) - {[E (X)]}^{2}$

Connection: $E (a X + b)$ $= a E (X) + b$ and $Var (a X + b)$ $= a^{2} Var (X)$
Connection: Standard deviation $σ$ $= \sqrt{Var (X)}$
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

Continuous Random Variables

Probability density functions

Name	Formula	Condition
pdf conditions	$f (x) \geq 0$ and $\int_{- \infty}^{\infty} f (x) d x$ $= 1$
Probability	$P (a \leq X \leq b) = \int_{a}^{b} f (x) d x$

Connection: Outside the given domain, $f (x) = 0$ , so integrate over the domain only.
Exam note: Piecewise pdf: integrate each piece over its own interval.

f (x) = \frac{3}{32} x (4 - x)

0 \leq x \leq 4

: probability is area

Mean, variance, mode and median

Name	Formula	Condition
Expected value	$E (X) = μ = \int_{- \infty}^{\infty} x f (x) d x$
Variance	$Var (X) = E (X - μ)^{2} = E (X^{2}) - {[E (X)]}^{2}$ $Var (X) = \int_{- \infty}^{\infty} (x - μ)^{2} f (x) d x = \int_{- \infty}^{\infty} x^{2} f (x) d x - μ^{2}$
Mode	the $x$ where $f (x)$ is greatest
Median $m$	$\int_{- \infty}^{m} f (x) d x = \frac{1}{2}$

Connection: $E (a X + b)$ $= a E (X) + b$ and $Var (a X + b)$ $= a^{2} Var (X)$ , as for discrete $X$
Connection: $E (g (X))$ $= \int_{- \infty}^{\infty} g (x) f (x) d x$
Exam note: The mode can be at an end point of the domain, not only where $f' (x) = 0$ .
Exam note: Check the median lies inside the domain.

Back to topics

Topic 5 Calculus

Limits and First Principles

Video: Differentiating from first principles (opens YouTube in a new tab)

Limits and continuity

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

Connection: Continuous at $a$ : $\lim_{x \to a} f (x)$ $= f (a)$
Connection: Differentiable at $a$ means continuous at $a$ . The reverse is not necessarily true: $| x |$ is continuous at $x = 0$ but not differentiable there.
Connection: $\lim_{x \to 0} \frac{\sin x}{x}$ $= 1$ ( $x$ in radians)

First principles

Name	Formula	Condition
Derivative	$y = f (x) \Rightarrow \frac{d y}{d x} = f' (x) = \lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$ the gradient function and the rate of change of $y$ with respect to $x$
Higher derivatives	$f ″ (x) = \frac{d^{2} y}{d x^{2}}$ $f^{(n)} (x) = \frac{d^{n} y}{d x^{n}}$

Exam note: Expand $f (x + h)$ , cancel $h$ , then let $h \to 0$ .

h \to 0

the chord turns into the tangent.

L’Hôpital’s rule

Name	Formula	Condition
L’Hôpital’s rule	$\lim_{x \to a} \frac{f (x)}{g (x)} = \lim_{x \to a} \frac{f' (x)}{g' (x)}$	form $\frac{0}{0}$ or $\frac{\infty}{\infty}$ also as $x \to \infty$

Connection: Still $\frac{0}{0}$ or $\frac{\infty}{\infty}$ after one step: use l’Hôpital again.
Exam note: Check the form is $\frac{0}{0}$ or $\frac{\infty}{\infty}$ before using l’Hôpital.
Exam note: Differentiate the top and bottom separately. It is not the quotient rule.

Back to topics

Differentiation Rules

Standard derivatives

$f (x)$	$f' (x)$
$x^{n}$	$n x^{n - 1}$
$\sin x$	$\cos x$
$\cos x$	$- \sin x$
$\tan x$	$\sec^{2} x$
$e^{x}$	$e^{x}$
$\ln x$	$\frac{1}{x}$

$f (x)$	$f' (x)$
$\sec x$	$\sec x \tan x$
$cosec x$	$- cosec x \cot x$
$\cot x$	$- {cosec}^{2} x$
$a^{x}$	$a^{x} (\ln a)$
$\log_{a} x$	$\frac{1}{x \ln a}$
$\arcsin x$	$\frac{1}{\sqrt{1 - x^{2}}}$
$\arccos x$	$- \frac{1}{\sqrt{1 - x^{2}}}$
$\arctan x$	$\frac{1}{1 + x^{2}}$

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

Video: Implicit differentiation (opens YouTube in a new tab)

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 $x = a$ .

Implicit differentiation and related rates

Name	Formula	Condition
Implicit	$\frac{d}{d x} (y^{n}) = n y^{n - 1} \frac{d y}{d x}$
Related rates	$\frac{d y}{d t} = \frac{d y}{d x} \times \frac{d x}{d t}$

Connection: Implicit products: $\frac{d}{d x} (x^{2} y)$ $= 2 x y + x^{2} \frac{d y}{d x}$
Exam note: Related rates: a decreasing quantity has a negative rate.

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

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

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

\int \sec^{2} x d x = \tan x + C

\int \sec x \tan x d x = \sec x + C

\int {cosec}^{2} x d x = - \cot x + C

\int cosec x \cot x d x = - cosec x + C

\int \frac{1}{a^{2} + x^{2}} d x = \frac{1}{a} \arctan (\frac{x}{a}) + C

\int \frac{1}{\sqrt{a^{2} - x^{2}}} d x = \arcsin (\frac{x}{a}) + C, | x | < a

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).
Connection: Denominator with two distinct linear factors: partial fractions first (Topic 1).

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.

Integration by parts

Name	Formula	Condition
By parts	$\int u \frac{d v}{d x} d x = u v - \int v \frac{d u}{d x} d x$ or $\int u d v = u v - \int v d u$

Connection: By parts: choose $u$ that gets simpler when differentiated, such as $\ln x$ or $x^{n}$ . If $\ln x$ is in the integral, take $u = \ln x$ . For $\int \ln x d x$ , use $\frac{d v}{d x} = 1$ .
Connection: $\int e^{x} \sin x d x$ : parts twice, then the integral appears again. Solve for it.

Back to topics

Areas, Volumes 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 $y > 0$
With the $y$ -axis	$A = \int_{a}^{b} \| x \| d y$
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

Volumes of revolution

Name	Formula	Condition
About the $x$ -axis	$V = \int_{a}^{b} π y^{2} d x$
About the $y$ -axis	$V = \int_{a}^{b} π x^{2} d y$

Connection: Square $y$ first: $\int π y^{2} d x \neq π {(\int y d x)}^{2}$

Each slice is a disc of area

π y^{2}

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

Differential Equations

Euler’s method

Name	Formula	Condition
Euler’s method	$y_{n + 1} = y_{n} + h \times f (x_{n}, y_{n})$ $x_{n + 1} = x_{n} + h,$ $h$ = step length

Connection: A smaller $h$ gives a better approximation.
Exam note: Keep full accuracy at every step.

\frac{d y}{d x} = x + y

y (0) = 1

h = 0.5

Exact methods

Name	Formula	Condition
Variables separable	$\frac{d y}{d x} = f (x) g (y) \Rightarrow \int \frac{1}{g (y)} d y = \int f (x) d x$
Homogeneous	$\frac{d y}{d x} = f (\frac{y}{x})$ : put $y = v x$ , $\frac{d y}{d x}$ $= v + x \frac{d v}{d x}$	separable in $v$ and $x$
Integrating factor	$y' + P (x) y = Q (x)$ : $I = e^{\int P (x) d x}$ $\frac{d}{d x} (I y) = I Q \Rightarrow I y = \int I Q d x$	first write the equation in this form

Connection: An initial condition finds the constant.
Exam note: Add the constant before you divide or take exponentials.

Back to topics

Maclaurin Series

Standard series

Name	Formula	Condition
Maclaurin series	$f (x) = f (0) + x f' (0) + \frac{x^{2}}{2!} f ″ (0) + \dots$
$e^{x}$	$1 + x + \frac{x^{2}}{2!} + \dots$	all $x$
$\ln (1 + x)$	$x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \dots$	$- 1 < x \leq 1$
$\sin x$	$x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \dots$	all $x$
$\cos x$	$1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \dots$	all $x$
$\arctan x$	$x - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \dots$	$\| x \| \leq 1$

Building new series

Three separate methods. Each one starts from a standard series.

Name	Formula	Condition
Substitution	Replace $x$ in a standard series. Example: $e^{2 x}$ comes from the series for $e^{x}$ with $x$ replaced by $2 x$ .
Differentiation or integration	Differentiate or integrate a standard series term by term.
Products	Multiply two standard series and keep the terms up to the power asked.

Back to topics

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37 pages · 5 topics · 34 sections

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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

3Counting and the Binomial Theorem

Counting

The binomial theorem

Extended binomial

4Partial Fractions

Two linear factors

5Systems of Linear Equations

Up to three equations in three unknowns

6Complex Numbers

Cartesian form

Arguments on the axes

Polar and Euler forms

Powers and roots

7Proof

Deductive proof, contradiction and counterexample

Proof by induction

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

Odd and even functions

3Graphs and Transformations

Transformations of y=f(x)

Sketching

4Special Function Graphs

y=|f(x)|

y=f(|x|)

y=1f(x)

y=[f(x)]2

5Rational Functions

Types and their asymptotes

Finding asymptotes and intercepts

6Exponential and Logarithmic Functions

The two graphs

Changing the graphs

7Solving Equations and Inequalities

Using graphs

Quadratic inequalities

8Polynomials

Factor and remainder theorems

Sum and product of roots

Shape of the graph

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

Symmetries from the unit circle

3Trigonometric Identities

Reciprocal ratios and Pythagorean identities

Compound and double angles

4Trigonometric Functions

The three graphs

Transformed sine and cosine

Inverse trig functions

5Vectors

Vector basics

Scalar product

Vector product

6Lines and Planes

Exponents and Logarithms

Sequences and Series

Counting and the Binomial Theorem

Partial Fractions

Systems of Linear Equations

Complex Numbers

Proof

Lines and Quadratics

Functions, Composites and Inverses

Graphs and Transformations

Transformations of $y = f (x)$

Special Function Graphs

$y = | f (x) |$

$y = f (| x |)$

$y = \frac{1}{f (x)}$

$y = [f (x)]^{2}$

Rational Functions

Exponential and Logarithmic Functions

Solving Equations and Inequalities

Polynomials

3D Geometry and Triangles

Radians and the Unit Circle

Trigonometric Identities

Trigonometric Functions

Vectors

Lines and Planes

Data and Statistics

Correlation and Regression

Probability

Discrete Distributions

The Normal Distribution

Continuous Random Variables

Limits and First Principles

Differentiation Rules

Using Derivatives

Integration

Areas, Volumes and Kinematics

Differential Equations

Maclaurin Series