SLIBMathematics

Mathematics: Analysis and Approaches

The Standard Level pathway develops mathematical reasoning through functions, calculus, statistics, and proof.

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Course Overview

IB Mathematics: Analysis and Approaches SL is organized into 5 units and 51 focused topics with 737 Key Concepts sections, 792 questions, and 1020 flashcards.

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Begin with Topic 1.1: Scientific notation and operations with numbers.

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Course units

Each unit groups focused topics. Every topic follows the same clear rhythm: review Key Concepts, use the Question Bank, then strengthen recall.

Unit 1
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Number and Algebra
This topic builds the algebraic language used throughout AA SL. Sequences, exponents, logarithms, proof and the binomial theorem later reappear inside functions, probability and calculus.
9 topics
Unit 1 progress0%
Topics in this unit
Topic 1.1Not started
Scientific notation and operations with numbers
Scientific notation is a compact, reliable way to represent, compare and calculate with extremely large or small numbers. This guide builds from place value and conversions to arithmetic, significant figures, calculator use and IB-style multi-step problems.
105 min estimated
Topic 1.1 progress0%
Key Concepts0/8 sections complete Question Bank0/10 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Write every non-zero number in normalized scientific notation a × $10^{k}$ , where 1 ≤ |a| < 10 and k is an integer.
Convert accurately between ordinary decimal notation, scientific notation and calculator E-notation.
Multiply, divide and raise scientific-notation values to integer powers using exponent laws.
Add and subtract scientific-notation values by first expressing them with a common power of ten.
Round values to a stated number of significant figures and communicate trailing significant zeros clearly.
Use a scientific calculator efficiently while retaining guard digits until the final answer.
Estimate orders of magnitude and use them to check whether a calculated result is reasonable.
Present IB exam solutions with correct working, notation, accuracy and units.
Key formulas
Normalized scientific notation
$N=a\times10^k,\quad 1\le |a|<10,\quad k\in\mathbb Z$
Multiplication rule
$(a\times10^m)(b\times10^n)=(ab)\times10^{m+n}$
Division rule
$\frac{a\times10^m}{b\times10^n}=\frac{a}{b}\times10^{m-n},\quad b\ne0$
Power rule
$(a\times10^m)^p=a^p\times10^{mp},\quad p\in\mathbb Z,\quad a\ne0$
Addition with a common exponent
$a\times10^k+b\times10^k=(a+b)\times10^k$
Subtraction with a common exponent
$a\times10^k-b\times10^k=(a-b)\times10^k$
Calculator E-notation
$a\text{E}k=a\times10^k$
Topic 1.2Not started
Arithmetic sequences and series
Arithmetic sequences model quantities that change by the same additive amount each step. This guide develops common differences, nth terms, finite sums, sigma notation and IB-style applications from first principles.
110 min estimated
Topic 1.2 progress0%
Key Concepts0/8 sections complete Question Bank0/14 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Recognize an arithmetic sequence and calculate its common difference from consecutive or non-consecutive terms.
Use recurrence notation and the nth-term formula to generate and find terms.
Determine an unknown first term, common difference, term number or number of terms from given information.
Distinguish an arithmetic sequence from an arithmetic series and calculate the sum of a finite number of terms.
Represent and evaluate finite arithmetic sums using sigma notation.
Build and interpret arithmetic models for real-world situations with constant additive change.
Use calculator tables, summation tools and equation solvers efficiently while checking integer constraints.
Present IB exam solutions with clear formulas, substitutions, units and reasonableness checks.
Key formulas
Common difference
$d=u_{n+1}-u_n$
Recursive definition
$u_1=a,\qquad u_{n+1}=u_n+d$
Arithmetic nth term
$u_n=u_1+(n-1)d$
Common difference from two terms
$d=\frac{u_q-u_p}{q-p},\qquad p\ne q$
Term number
$n=1+\frac{u_n-u_1}{d},\qquad d\ne0$
Arithmetic series using first and last terms
$S_n=\frac n2(u_1+u_n)$
Arithmetic series using the common difference
$S_n=\frac n2\left[2u_1+(n-1)d\right]$
Arithmetic series in sigma notation
$S_n=\sum\limits_{r=1}^{n}\left[u_1+(r-1)d\right]$
Equivalent zero-based indexing
$S_n=\sum\limits_{r=0}^{n-1}(u_1+rd)$
Topic 1.3Not started
Geometric sequences and series
Geometric sequences model repeated multiplication. This guide develops common ratios, nth terms, finite and infinite sums, convergence, sigma notation and IB-style growth and decay applications.
120 min estimated
Topic 1.3 progress0%
Key Concepts0/9 sections complete Question Bank0/8 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Recognize a geometric sequence and calculate its common ratio, including fractional and negative ratios.
Use recurrence notation and the geometric nth-term formula to generate and find terms.
Determine an unknown first term, common ratio, term number or number of terms from given information.
Distinguish a geometric sequence from a geometric series and calculate finite sums.
Decide whether an infinite geometric series converges and calculate its sum when it exists.
Represent and evaluate finite and infinite geometric sums using sigma notation.
Build and interpret repeated-percentage growth and decay models in context.
Use calculator tables, logarithms and numerical solvers while respecting domain and integer constraints.
Present IB solutions with clear formulas, substitutions, units, accuracy and reasonableness checks.
Key formulas
Common ratio
$r=\frac{u_{n+1}}{u_n},\qquad u_n\ne0$
Recursive definition
$u_1=a,\qquad u_{n+1}=ru_n$
Geometric nth term
$u_n=u_1r^{n-1}$
Ratio from two terms
$r^{q-p}=\frac{u_q}{u_p}$
Term number for a positive ratio
$n=1+\frac{\ln(\frac{u_n}{u_1})}{\ln r},\qquad r>0,\ r\ne1,\quad u_1\ne0,\quad \frac{u_n}{u_1}>0$
Finite geometric sum
$S_n=\frac{u_1(1-r^n)}{1-r},\qquad r\ne1$
Equivalent finite-sum form
$S_n=\frac{u_1(r^n-1)}{r-1},\qquad r\ne1$
Special case $r=1$
$S_n=nu_1$
Infinite geometric sum
$S_\infty=\frac{u_1}{1-r},\qquad |r|<1$
Remainder after n terms
$R_n=S_\infty-S_n=\frac{u_1r^n}{1-r},\qquad |r|<1$
Finite geometric series in sigma notation
$S_n=\sum\limits_{k=0}^{n-1}u_1r^k$
Infinite geometric series in sigma notation
$\sum\limits_{k=0}^{\infty}u_1r^k=\frac{u_1}{1-r},\qquad |r|<1$
Repeated growth or decay
$Q_t=Q_0r^t$
Topic 1.4Not started
Financial applications
Financial mathematics models how money changes through simple interest, compound growth, depreciation, regular savings and loan repayments. Accurate solutions depend on matching the interest rate to the compounding or payment period.
140 min estimated
Topic 1.4 progress0%
Key Concepts0/9 sections complete Question Bank0/4 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Distinguish simple interest from compound interest and select the appropriate model.
Convert percentage rates to decimal multipliers for growth, appreciation, decay and depreciation.
Calculate future value, present value, interest rate or time using compound-interest models.
Match nominal annual rates, periodic rates, compounding frequency and elapsed time consistently.
Calculate and interpret effective annual interest rates.
Model regular savings deposits using finite geometric-series formulas.
Model amortizing loans, calculate periodic repayments and interpret outstanding balances.
Use GDC financial or TVM functions with a consistent cash-flow sign convention.
Solve financial word problems with appropriate units, accuracy, integer constraints and contextual interpretation.
Evaluate whether a financial model and its assumptions are reasonable in a real situation.
Key formulas
Simple interest earned
$I=Prt$
Simple-interest balance
$A=P(1+rt)$
Percentage multiplier
$\text{growth: }1+\frac{p}{100},\qquad \text{decay: }1-\frac{p}{100}$
Compound future value
$FV=PV(1+i)^n$
Periodic rate from two values
$i=\left(\frac{FV}{PV}\right)^{\frac{1}{n}}-1,\quad n>0,\ PV\ne0,\frac{\ FV}{PV}>0$
Number of periods
$\begin{gathered}n=\frac{\ln(FV/PV)}{\ln(1+i)}\\ PV\ne0,\quad FV/PV>0,\quad i>-1,\ i\ne0\end{gathered}$
Nominal-rate compound interest
$FV=PV\left(1+\frac{r}{m}\right)^{mt}$
Periodic rate and period count
$i=\frac{r}{m},\qquad n=mt$
Effective annual rate
$EAR=\left(1+\frac{r}{m}\right)^m-1$
Present value of one future amount
$PV=\frac{FV}{(1+i)^n}=FV(1+i)^{-n}$
Appreciation
$V_n=V_0(1+a)^n$
Reducing-balance depreciation
$V_n=V_0(1-d)^n$
Future value of an ordinary annuity
$FV=\begin{cases}R\frac{(1+i)^n-1}{i},&i\ne0\\ nR,&i=0\end{cases}$
Future value of an annuity due
$FV_{\mathrm{due}}=\begin{cases}R\frac{(1+i)^n-1}{i}(1+i),&i\ne0\\ nR,&i=0\end{cases}$
Deposit required for a target
$R=\begin{cases}FV\frac{i}{(1+i)^n-1},&i\ne0\\ FV/n,&i=0\end{cases}$
Present value of loan repayments
$PV=\begin{cases}R\frac{1-(1+i)^{-n}}{i},&i\ne0\\ nR,&i=0\end{cases}$
Periodic loan repayment
$R=\begin{cases}PV\frac{i}{1-(1+i)^{-n}},&i\ne0\\ PV/n,&i=0\end{cases}$
Outstanding balance after k payments
$B_k=\begin{cases}PV(1+i)^k-R\frac{(1+i)^k-1}{i},&i\ne0\\ PV-kR,&i=0\end{cases}$
Total interest on equal repayments
$I_{\mathrm{total}}=Rn-PV$
Topic 1.5Not started
Integer exponents and introduction to logarithms
Integer exponents describe repeated multiplication and reciprocal powers, while logarithms reverse exponentiation. This guide develops the exponent laws, logarithmic notation, exact and numerical equation solving, calculator methods, and IB exam reasoning.
125 min estimated
Topic 1.5 progress0%
Key Concepts0/9 sections complete Question Bank0/8 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Interpret positive, zero, and negative integer exponents and state the restrictions that make each expression defined.
Apply the product, quotient, and power laws accurately when simplifying algebraic expressions.
Rewrite answers with positive exponents when required and distinguish powers from ordinary multiplication.
Explain a logarithm as an exponent and convert fluently between exponential and logarithmic form.
Use the basic product, quotient, and power properties of logarithms with valid positive arguments.
Solve simple exponential equations exactly when a common base is available and numerically using logarithms otherwise.
Solve simple logarithmic equations while enforcing domain and base restrictions.
Use a scientific calculator or GDC efficiently and communicate IB solutions with appropriate working and accuracy.
Key formulas
Positive integer exponent
$a^n=\underbrace{a\cdot a\cdots a}_{n\text{ factors}},\quad n\in\mathbb N$
Zero exponent
$a^0=1,\quad a\ne0$
Negative integer exponent
$a^{-n}=\frac{1}{a^n},\quad a\ne0$
Product law
$a^m a^n=a^{m+n}$
Quotient law
$\frac{a^m}{a^n}=a^{m-n},\quad a\ne0$
Power of a power
$(a^m)^n=a^{mn}$
Power of a product
$(ab)^n=a^n b^n$
Power of a quotient
$\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n},\quad b\ne0$
Moving a negative power across a fraction bar
$\frac{a^{-m}}{b^{-n}}=\frac{b^n}{a^m},\quad a,b\ne0$
Exponential-logarithmic equivalence
$a^c=b\quad\Longleftrightarrow\quad\log_a b=c$
Common and natural logarithms
$\log x=\log_{10}x,\qquad\ln x=\log_e x$
Inverse identities
$\log_a(a^x)=x,\qquad a^{\log_a x}=x\quad(x>0)$
Product property
$\log_a(xy)=\log_a x+\log_a y$
Quotient property
$\log_a\left(\frac{x}{y}\right)=\log_a x-\log_a y$
Power property
$\log_a(x^k)=k\log_a x$
Useful exact values
$\log_a1=0,\qquad\log_a a=1$
Common-base principle
$a^{f(x)}=a^{g(x)}\Longrightarrow f(x)=g(x),\quad a>0,\ a\ne1$
Solving a basic exponential equation
$a^x=b\Longrightarrow x=\frac{\ln b}{\ln a},\quad a>0,\ a\ne1,\ b>0$
Single logarithm equation
$\log_a f(x)=c\Longleftrightarrow f(x)=a^c,\quad f(x)>0$
Equality of logarithms
$\log_a f(x)=\log_a g(x)\Longrightarrow f(x)=g(x),\quad f(x),g(x)>0$
Change-of-base formula
$\log_a b=\frac{\log b}{\log a}=\frac{\ln b}{\ln a}$
Topic 1.6Not started
Simple deductive proof
Deductive proof turns definitions and known facts into a logically certain conclusion. This guide develops direct proof, algebraic proof, parity, divisibility, consecutive-integer arguments, and the ability to distinguish proof from unsupported pattern spotting.
115 min estimated
Topic 1.6 progress0%
Key Concepts0/9 sections complete Question Bank0/15 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Explain deductive proof as a finite chain of justified statements that establishes a conclusion for every permitted case.
Identify hypotheses, conclusions, implications, universal statements, identities, and counterexamples.
Use mathematical definitions and previously established results explicitly rather than relying on numerical evidence.
Write a clear direct proof with a stated starting assumption and a conclusion that answers the original claim.
Prove algebraic identities by transforming one side into the other through valid reversible steps.
Represent even, odd, divisible, and consecutive integers algebraically and use those representations in proof.
Determine whether an argument is valid, detect hidden assumptions or circular reasoning, and explain why a proposed proof fails.
Communicate concise IB-style proofs using correct notation, complete sentences, and appropriate quantifiers.
Key formulas
Implication structure
$P\Rightarrow Q$
Universal statement
$\forall n\in S,\quad P(n)$
Even integer
$n=2k,\quad k\in\mathbb Z$
Odd integer
$n=2k+1,\quad k\in\mathbb Z$
Divisibility
$d\mid n\quad\Longleftrightarrow\quad n=dk\text{ for some }k\in\mathbb Z$
Consecutive integers
$n,\ n+1,\ n+2,\ldots\quad(n\in\mathbb Z)$
Direct-proof chain
$P\ \Longrightarrow\ R_1\ \Longrightarrow\ R_2\ \Longrightarrow\cdots\Longrightarrow\ Q$
Identity notation
$f(x)\equiv g(x)$
Difference of squares
$A^2-B^2=(A-B)(A+B)$
Square is non-negative
$u^2\ge0\quad\text{for every }u\in\mathbb R$
Even target form
$E=2K,\quad K\in\mathbb Z$
Odd target form
$O=2K+1,\quad K\in\mathbb Z$
Odd-square result
$n\text{ odd}\quad\Longrightarrow\quad n^2\equiv1\pmod8$
Divisibility target
$d\mid N\quad\Longleftrightarrow\quad N=dK\text{ for some }K\in\mathbb Z$
Linear-combination property
$d\mid a\ \land\ d\mid b\quad\Longrightarrow\quad d\mid(ra+sb),\quad r,s\in\mathbb Z$
Consecutive integers
$n,\ n+1,\ n+2$
Consecutive even integers
$2k,\ 2k+2,\ 2k+4$
Consecutive odd integers
$2k+1,\ 2k+3,\ 2k+5$
Counterexample test
$P(c)\text{ true and }Q(c)\text{ false}\quad\Longrightarrow\quad \forall x\,[P(x)\Rightarrow Q(x)]\text{ is false}$
Topic 1.7Not started
Rational exponents and logarithm laws
Connect roots with fractional powers, apply exponent and logarithm laws accurately, and solve exponential or logarithmic equations while respecting domain restrictions.
125 min estimated
Topic 1.7 progress0%
Key Concepts0/9 sections complete Question Bank0/32 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Interpret rational exponents and convert fluently between radical and exponent notation.
Apply exponent laws to rational powers and simplify exact expressions.
Use logarithm definitions, identities, and domain restrictions correctly.
Expand and condense logarithmic expressions using product, quotient, and power laws.
Use change of base and GDC methods to evaluate logarithms.
Solve exponential and logarithmic equations and reject invalid solutions.
Key formulas
Rational exponent definition
$a^{\frac{m}{n}}=\sqrt[n]{a^m}=(\sqrt[n]{a})^m$
Negative rational exponent
$a^{\frac{-m}{n}}=\frac{1}{a^{\frac{m}{n}}}$
Special root form
$a^{\frac{1}{n}}=\sqrt[n]{a}$
Radical–exponent conversion
$\sqrt[n]{x^m}=x^{\frac{m}{n}},\qquad x>0,\ m\in\mathbb Z,\ n\in\mathbb Z_{>0}$
Principal square root
$\sqrt{x^2}=|x|$
Odd root of a power
$\sqrt[2k+1]{x^{2k+1}}=x$
Product law
$a^r a^s=a^{r+s}$
Quotient law
$\frac{a^r}{a^s}=a^{r-s}$
Power of a power
$(a^r)^s=a^{rs}$
Power of a product
$(ab)^r=a^r b^r$
Power of a quotient
$\left(\frac ab\right)^r=\frac{a^r}{b^r}$
Negative exponent conversion
$x^{-r}=\frac{1}{x^r}$
Common-denominator strategy
$x^{\frac{p}{q}}x^{\frac{r}{s}}=x^{\frac{(ps+rq)}{(qs)}}$
Exact before decimal
$\sqrt[n]{a^m}=a^{\frac{m}{n}}$
Logarithm definition
$\log_a x=y\quad\Longleftrightarrow\quad a^y=x$
Logarithm of one
$\log_a1=0$
Logarithm of its base
$\log_a a=1$
Inverse relationships
$a^{\log_a x}=x,\qquad \log_a(a^x)=x$
Product law
$\log_a(xy)=\log_a x+\log_a y$
Quotient law
$\log_a\left(\frac{x}{y}\right)=\log_a x-\log_a y$
Power law
$\log_a(x^r)=r\log_a x$
Reciprocal result
$\log_a\left(\frac1x\right)=-\log_a x$
Change of base
$\log_a x=\frac{\log_b x}{\log_b a}$
Calculator forms
$\log_a x=\frac{\ln x}{\ln a}=\frac{\log x}{\log a}$
Condensing pattern
$r\log_a x-s\log_a y=\log_a\left(\frac{x^r}{y^s}\right)$
Same-base principle
$a^u=a^v\quad\Longrightarrow\quad u=v$
Logarithmic solution step
$a^{f(x)}=k\quad\Longrightarrow\quad f(x)=\frac{\ln k}{\ln a}$
Log equality principle
$\log_a u=\log_a v\quad\Longrightarrow\quad u=v$
Fractional exponent
$a^{\frac{m}{n}}=\sqrt[n]{a^m}$
Three log laws
$\log_a(xy)=\log_ax+\log_ay,\quad \log_a(\frac{x}{y})=\log_ax-\log_ay,\quad \log_a(x^r)=r\log_ax$
Change of base
$\log_a x=\frac{\ln x}{\ln a}$
Topic 1.8Not started
Infinite geometric series
Understand an endless geometric sum through its partial sums, test whether it converges, calculate its limiting value, and apply the model to algebraic and real-world problems.
110 min estimated
Topic 1.8 progress0%
Key Concepts0/9 sections complete Question Bank0/11 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Distinguish an infinite geometric sequence from an infinite geometric series.
Identify the first term and common ratio of a geometric series.
Determine convergence using the condition $|r|<1$ .
Derive and apply the sum-to-infinity formula.
Find unknown terms, first terms, or common ratios from series information.
Explain why a series diverges when the convergence condition fails.
Model recurring decimals, repeated doses, and bouncing-ball distances.
Use a GDC to investigate partial sums and solve unknown-ratio problems.
Key formulas
Infinite geometric series
$a+ar+ar^2+ar^3+\cdots$
Partial sum notation
$S_n=\sum\limits_{k=0}^{n-1}ar^k$
Common ratio
$r=\frac{u_{n+1}}{u_n}$
Geometric term
$u_n=ar^{n-1}$
Geometric convergence condition
$|r|<1\quad\Longleftrightarrow\quad -1<r<1$
Vanishing powers
$|r|<1\quad\Longrightarrow\quad r^n\to0$
Finite geometric sum
$S_n=\frac{a(1-r^n)}{1-r}$
Sum to infinity
$S_\infty=\frac{a}{1-r},\qquad |r|<1$
First term from the sum
$a=S_\infty(1-r)$
Ratio from first term and sum
$r=1-\frac{a}{S_\infty},\qquad S_\infty\ne0$
Term condition
$u_n=ar^{n-1}$
Remainder after n terms
$R_n=S_\infty-S_n=\frac{ar^n}{1-r}$
Approximation relation
$S_\infty=S_n+R_n$
Bouncing-ball distance
$D=h+\frac{2hq}{1-q}=h\left(\frac{1+q}{1-q}\right)$
Repeated fixed contribution
$A_\infty=\frac{d}{1-r},\qquad |r|<1$
Two-digit recurring decimal
$0.\overline{ab}=\frac{10a+b}{99},\qquad a,b\in\{0,1,\ldots,9\}$
Partial-sum calculator entry
$S_n=\frac{a(1-r^n)}{1-r}$
Solver interval
$-1<r<1$
Convergence and sum
$|r|<1\quad\Longrightarrow\quad S_\infty=\frac{a}{1-r}$
Term and remainder
$u_n=ar^{n-1},\qquad R_n=\frac{ar^n}{1-r}$
Topic 1.9Not started
Binomial theorem
Use Pascal’s triangle and combination notation to expand integer powers of binomials, identify individual terms and coefficients, and solve constant-term and parameter problems efficiently.
115 min estimated
Topic 1.9 progress0%
Key Concepts0/9 sections complete Question Bank0/43 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Explain the structure of the binomial theorem for non-negative integer powers.
Generate and interpret rows of Pascal’s triangle.
Calculate binomial coefficients using combination notation and a GDC.
Expand binomials containing variables, negative terms, and fractional coefficients.
Use the general term to find a specified term or coefficient without a full expansion.
Find constant terms by solving for the required variable exponent.
Use coefficient identities and substitutions to solve IB-style problems.
Key formulas
Binomial theorem
$(a+b)^n=\sum\limits_{r=0}^{n}\binom nr a^{n-r}b^r$
Number of formal binomial summands
$\text{formal summands before collection}=n+1$
Pascal recurrence
$\binom nr=\binom{n-1}{r-1}+\binom{n-1}{r}$
Coefficient symmetry
$\binom nr=\binom n{n-r}$
Row sum
$\sum\limits_{r=0}^{n}\binom nr=2^n$
Combination formula
$\binom nr=\frac{n!}{r!(n-r)!}$
Factorial
$n!=n(n-1)(n-2)\cdots2\cdot1,\qquad 0!=1$
Boundary values
$\binom n0=\binom nn=1$
Expanded form
$(a+b)^n=\binom n0a^n+\binom n1a^{n-1}b+\cdots+\binom nnb^n$
Exponent check
$(n-r)+r=n$
General term
$T_{r+1}=\binom nr a^{n-r}b^r$
pth-term index
$r=p-1$
Coefficient method
$\text{power of }x=k\quad\Longrightarrow\quad \text{solve for }r$
Quadratic coefficient
$\binom n2=\frac{n(n-1)}2$
Constant-term condition
$\text{net exponent of }x=0$
Admissible index
$r\in\{0,1,2,\ldots,n\}$
Sum of coefficients
$\sum\limits_{k=0}^{n}c_k=P(1)$
Alternating coefficient sum
$\sum\limits_{k=0}^{n}(-1)^kc_k=P(-1)$
GDC term calculation
$\binom nr a^{n-r}b^r$
Core theorem
$(a+b)^n=\sum\limits_{r=0}^{n}\binom nr a^{n-r}b^r$
Target term
$T_{r+1}=\binom nr a^{n-r}b^r$
Constant condition
$\text{net power of }x=0$
Unit 2
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Functions
Functions are one of the central threads of AA SL. Students learn to represent relationships, transform graphs, solve equations and connect symbolic structure with graphical behavior.
11 topics
Unit 2 progress0%
Topics in this unit
Topic 2.1Not started
Equations of Straight Lines
Build and interpret straight-line equations using gradients, points, intercepts and relationships between lines, then apply these ideas to intersections and real-world models.
135 min estimated
Topic 2.1 progress0%
Key ConceptsSubscription required Question BankSubscription required FlashcardsSubscription required
Learning objectives
Calculate and interpret the gradient of a line from a graph, two points or a real-world context.
Move confidently between gradient-intercept, point-slope and general forms of a straight-line equation.
Find the equation of a line from a point and gradient, from two points or from intercept information.
Use gradient conditions to identify and construct parallel and perpendicular lines.
Find intercepts, sketch straight lines and solve intersections algebraically and graphically.
Build and interpret linear models, including the units and practical meaning of the gradient and intercept.
Use a GDC to check graphs and intersections while still communicating sufficient mathematical working.
Recognize common special cases, including horizontal, vertical, coincident and parallel lines.
Key formulas
Gradient from two points
$m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1},\qquad x_2\ne x_1$
Gradient-intercept form
$y=mx+c$
Intercepts of $y=mx+c$
$y\text{-intercept}: (0,c),\qquad x\text{-intercept}: \left(-\frac{c}{m},0\right),\ m\ne0$
Point-slope form
$y-y_1=m(x-x_1)$
Two-point equation
$y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1),\qquad x_2\ne x_1$
Parallel-line condition
$m_1=m_2$
Perpendicular-line condition
$m_1m_2=-1\qquad\Longleftrightarrow\qquad m_2=-\frac{1}{m_1}$
Intercept method for $ax+by=d$
$x\text{-intercept}=\left(\frac{d}{a},0\right),\qquad y\text{-intercept}=\left(0,\frac{d}{b}\right)$
Generic linear model
$\text{output}=\text{rate}\times\text{input}+\text{initial value}$
Topic 2.2Not started
The Concept of a Function
Understand functions as input-output rules, use function notation, determine domain and range, and identify functions from mappings, tables, equations and graphs.
140 min estimated
Topic 2.2 progress0%
Key ConceptsSubscription required Question BankSubscription required FlashcardsSubscription required
Learning objectives
Explain what makes a relation a function and use the language of inputs, outputs and ordered pairs accurately.
Use function notation such as f(x), evaluate functions and solve for unknown inputs or parameters.
Identify independent and dependent variables in mathematical and real-world settings.
State and interpret the domain and range of a function using words, inequalities, intervals or sets.
Represent functions using mappings, tables, formulas, graphs and contextual descriptions.
Determine whether a relation is a function from ordered pairs, a mapping diagram, a table or a graph.
Apply the vertical line test correctly and explain why it works.
Use a GDC to evaluate, tabulate and graph functions while communicating appropriate IB working.
Key formulas
Function mapping
$f:A\to B,\qquad x\mapsto f(x)$
Output notation
$y=f(x)$
Evaluating a function
$f(a)=\left.f(x)\right|_{x=a}$
Domain notation
$\operatorname{Dom}(f)=\{x:\ f(x)\text{ is defined}\}$
Range notation
$\operatorname{Ran}(f)=\{f(x):\ x\in\operatorname{Dom}(f)\}$
Graph coordinates
$(x,y)=(x,f(x))$
Sideways parabola example
$x=y^2\quad\Longrightarrow\quad y=\pm\sqrt{x}$
Topic 2.3Not started
Graphs of Functions
Read, interpret and sketch function graphs; identify their domains, ranges, intercepts, extrema and intervals; and solve equations graphically with appropriate GDC methods.
145 min estimated
Topic 2.3 progress0%
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Learning objectives
Read coordinates and function values accurately from a graph and interpret them in context.
Determine domain and range from graphs, including restrictions caused by endpoints, gaps and asymptotes.
Find and interpret x-intercepts, y-intercepts, zeros and roots.
Identify local and absolute maximum and minimum points.
Describe increasing, decreasing, positive and negative intervals using correct notation.
Sketch graphs from equations, tables and important features.
Find approximate roots and graph intersections and explain what they mean.
Use a GDC efficiently while choosing a suitable window and reporting solutions to appropriate accuracy.
Key formulas
Point on a function graph
$(a,b)\text{ lies on }y=f(x)\iff b=f(a)$
Domain from a graph
$\operatorname{Dom}(f)=\{x\in\mathbb{R}:\text{the graph contains a point with x-coordinate }x\}$
Range from a graph
$\operatorname{Ran}(f)=\{y\in\mathbb{R}:y=f(x)\text{ for some }x\in\operatorname{Dom}(f)\}$
x-intercepts and roots
$f(x)=0\quad\Longleftrightarrow\quad (x,0)\text{ is an x-intercept}$
y-intercept
$x=0\quad\Longrightarrow\quad (0,f(0))$
Quadratic vertex form
$f(x)=a(x-h)^2+k\quad\Longrightarrow\quad\text{vertex }(h,k)$
Sign from graph position
$f(x)>0\iff\text{graph above x-axis},\qquad f(x)<0\iff\text{graph below x-axis}$
Useful quadratic form
$y=a(x-h)^2+k$
Useful reciprocal form
$y=\frac{a}{x-h}+k$
Root as an intercept
$f(r)=0\iff (r,0)\text{ lies on }y=f(x)$
Horizontal-level solution
$f(x)=k\iff y=f(x)\text{ intersects }y=k$
Intersection condition
$f(x)=g(x)\quad\Longleftrightarrow\quad f(x)-g(x)=0$
Point on graph
$(a,b)\text{ on }y=f(x)\iff f(a)=b$
Root or x-intercept
$f(x)=0$
y-intercept
$(0,f(0))$
Intersection
$f(x)=g(x)$
Sign
$f(x)>0\text{ above the x-axis},\qquad f(x)<0\text{ below the x-axis}$
Topic 2.4Not started
Key Features of Graphs
Identify, calculate and interpret the defining features of function graphs, including domains, ranges, intercepts, extrema, intervals, symmetry, end behavior, asymptotes and intersections.
150 min estimated
Topic 2.4 progress0%
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Learning objectives
Use a systematic feature scan to interpret a graph accurately and efficiently.
Determine domain and range from graphs and equations, including endpoint and asymptote restrictions.
Find x-intercepts, y-intercepts, zeros and roots algebraically and graphically.
Distinguish local extrema from global extrema on unrestricted and restricted domains.
State increasing, decreasing, positive and negative intervals using correct notation.
Find and use an axis of symmetry and describe the end behavior of common function families.
Identify vertical and horizontal asymptotes and explain their effect on domain, range and graph shape.
Find intersections exactly when possible and approximately with a GDC when necessary.
Interpret every important graph feature in its mathematical or real-world context.
Key formulas
Point on a graph
$(a,b)\text{ lies on }y=f(x)\iff f(a)=b$
Even-root condition
$\sqrt{u(x)}\text{ is real}\iff u(x)\geq0$
Denominator condition
$\frac{p(x)}{q(x)}\text{ is defined only when }q(x)\ne0$
Logarithm condition
$\log_a(u(x))\text{ is defined only when }u(x)>0$
Root condition
$f(r)=0\iff (r,0)\text{ is an x-intercept}$
y-intercept
$(0,f(0))$
Quadratic vertex x-coordinate
$x_{\mathrm v}=-\frac{b}{2a}\quad\text{for }f(x)=ax^2+bx+c$
Positive region
$f(x)>0\iff\text{the graph is above the x-axis}$
Negative region
$f(x)<0\iff\text{the graph is below the x-axis}$
Quadratic axis of symmetry
$x=-\frac{b}{2a}$
Even symmetry
$f(-x)=f(x)$
Odd symmetry
$f(-x)=-f(x)$
Positive even-degree end behavior
$a>0,\ n\text{ even}:\quad ax^n\to\infty\text{ as }x\to\pm\infty$
Positive odd-degree end behavior
$a>0,\ n\text{ odd}:\quad ax^n\to-\infty\text{ as }x\to-\infty,\quad ax^n\to\infty\text{ as }x\to\infty$
Transformed reciprocal asymptotes
$f(x)=\frac{a}{x-h}+k\quad\Longrightarrow\quad x=h\text{ and }y=k$
Horizontal asymptote statement
$\lim_{x\to\infty}f(x)=L\quad\Longrightarrow\quad y=L\text{ is a right-end horizontal asymptote}$
Intersection condition
$f(x)=g(x)\iff f(x)-g(x)=0$
Quadratic vertex input
$x_{\mathrm v}=-\frac{b}{2a}$
Root command target
$Y_1=f(x),\qquad\text{find }x\text{ where }Y_1=0$
Intersection command target
$Y_1=f(x),\quad Y_2=g(x),\qquad\text{find where }Y_1=Y_2$
x-intercepts and roots
$f(x)=0$
Quadratic axis
$x=-\frac{b}{2a}$
Intersection
$f(x)=g(x)$
Vertical asymptote of transformed reciprocal
$f(x)=\frac{a}{x-h}+k\quad\Longrightarrow\quad x=h$
Horizontal asymptote of transformed reciprocal
$f(x)=\frac{a}{x-h}+k\quad\Longrightarrow\quad y=k$
Topic 2.5Not started
Composite and Inverse Functions
Combine functions in the correct order, manage composite domains, construct and verify inverse functions, and connect inverse mappings with reflection in the line y equals x.
150 min estimated
Topic 2.5 progress0%
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Learning objectives
Explain composition as applying one function to the output of another and interpret both composite notations.
Evaluate composite functions numerically and derive composite formulas in the correct order.
Determine the domain of a composite by checking both the inner and outer functions.
Explain an inverse function as a rule that reverses a one-to-one mapping.
Find inverse formulas algebraically and state their appropriate domains and ranges.
Verify inverse pairs using both composite identities.
Determine whether a function is one-to-one using mappings, algebra or the horizontal line test.
Relate the domain and range of a function to those of its inverse.
Interpret inverse graphs as reflections in the line y equals x and use a GDC appropriately.
Key formulas
Composite notation
$(f\circ g)(x)=f(g(x))$
Reversed order
$(g\circ f)(x)=g(f(x))$
Substitution pattern
$f(u)\text{ with }u=g(x)\quad\Longrightarrow\quad f(g(x))$
Composite domain
$\operatorname{Dom}(f\circ g)=\{x\in\operatorname{Dom}(g):g(x)\in\operatorname{Dom}(f)\}$
Reversed mapping
$f(a)=b\iff f^{-1}(b)=a$
Inverse construction
$y=f(x)\quad\xrightarrow{\text{swap }x,y}\quad x=f(y)\quad\xrightarrow{\text{solve for }y}\quad y=f^{-1}(x)$
First inverse identity
$(f\circ f^{-1})(x)=f(f^{-1}(x))=x$
Second inverse identity
$(f^{-1}\circ f)(x)=f^{-1}(f(x))=x$
Algebraic one-to-one condition
$f(a)=f(b)\Longrightarrow a=b$
Inverse domain
$\operatorname{Dom}(f^{-1})=\operatorname{Ran}(f)$
Inverse range
$\operatorname{Ran}(f^{-1})=\operatorname{Dom}(f)$
Inverse graph reflection
$(a,b)\in f\iff(b,a)\in f^{-1}$
Fixed-point condition
$f(x)=x$
Numerical inverse value
$f^{-1}(k)=a\iff f(a)=k$
Composition
$(f\circ g)(x)=f(g(x))$
Inverse mapping
$f(a)=b\iff f^{-1}(b)=a$
Inverse identities
$f(f^{-1}(x))=x,\qquad f^{-1}(f(x))=x$
Domain and range swap
$\operatorname{Dom}(f^{-1})=\operatorname{Ran}(f),\qquad\operatorname{Ran}(f^{-1})=\operatorname{Dom}(f)$
Graph reflection
$(a,b)\mapsto(b,a)\text{ across }y=x$
Topic 2.6Not started
Quadratic Functions
Understand parabolas through standard, factorized and vertex forms; solve quadratic equations; analyze roots and extrema; and apply quadratic models to optimization problems.
165 min estimated
Topic 2.6 progress0%
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Learning objectives
Recognize a quadratic function and explain how its coefficients affect its parabola.
Convert between standard, factorized and vertex forms and select the form suited to a task.
Find intercepts, roots, the vertex, axis of symmetry and maximum or minimum value.
Complete the square accurately when the leading coefficient is one or another non-zero value.
Solve quadratic equations by factorization, the quadratic formula and graphing.
Use the discriminant to determine the number and type of real roots.
State the domain and range of a quadratic from its vertex and direction of opening.
Describe transformations of $y=x^2$ using vertex form.
Build and interpret quadratic models in optimization and motion contexts.
Use a GDC efficiently while retaining exact working and appropriate accuracy.
Key formulas
General quadratic form
$f(x)=ax^2+bx+c,\qquad a\ne0$
Standard form
$f(x)=ax^2+bx+c$
Factorized form
$f(x)=a(x-p)(x-q)$
Vertex form
$f(x)=a(x-h)^2+k$
y-intercept
$f(0)=c\quad\Longrightarrow\quad(0,c)$
Roots from factorized form
$a(x-p)(x-q)=0\quad\Longrightarrow\quad x=p\text{ or }x=q$
Vertex input
$x_{\mathrm v}=-\frac{b}{2a}$
Vertex
$\left(-\frac{b}{2a},\ f\left(-\frac{b}{2a}\right)\right)$
Axis from two roots
$x=\frac{p+q}{2}$
Square completion identity
$x^2+bx=\left(x+\frac{b}{2}\right)^2-\left(\frac{b}{2}\right)^2$
Standard-to-vertex conversion
$ax^2+bx+c=a\left(x+\frac{b}{2a}\right)^2+c-\frac{b^2}{4a}$
Zero-product rule
$AB=0\quad\Longrightarrow\quad A=0\text{ or }B=0$
Quadratic formula
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Discriminant
$\Delta=b^2-4ac$
Root classification
$\Delta>0:\ 2\text{ distinct real roots};\quad\Delta=0:\ 1\text{ repeated real root};\quad\Delta<0:\ 0\text{ real roots}$
Root as graph feature
$f(x)=0\iff(x,0)\text{ lies on }y=f(x)$
Intersection equation
$f(x)=g(x)$
Upward-opening range
$a>0:\quad\operatorname{Ran}(f)=[k,\infty)$
Downward-opening range
$a<0:\quad\operatorname{Ran}(f)=(-\infty,k]$
Transformation form
$f(x)=a(x-h)^2+k$
Optimization input
$x_{\mathrm{opt}}=-\frac{b}{2a}$
Standard form
$f(x)=ax^2+bx+c,\quad a\ne0$
Axis and vertex input
$x=-\frac{b}{2a}$
Completing the square
$x^2+bx=\left(x+\frac{b}{2}\right)^2-\left(\frac{b}{2}\right)^2$
Topic 2.7Not started
Quadratic Equations and Inequalities
Solve quadratic equations exactly and numerically, classify roots with the discriminant, interpret solutions as graph features, and solve quadratic inequalities using sign diagrams and graphs.
175 min estimated
Topic 2.7 progress0%
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Learning objectives
Rearrange a quadratic equation into the standard form ax squared plus bx plus c equals zero before choosing a solution method.
Solve factorable quadratic equations using the zero-product principle.
Solve quadratic equations by completing the square and by using the quadratic formula.
Use the discriminant to determine the number and nature of real roots.
Distinguish exact answers from decimal approximations and round only at the final step.
Interpret solutions as x-intercepts or intersections on graphs.
Solve quadratic inequalities using roots, sign diagrams, interval notation, and graph regions.
Find intersections of quadratic and linear functions algebraically and with a GDC.
Use calculator or GDC commands efficiently while recording sufficient mathematical working.
Check solutions by substitution, sign testing, or graphical verification and communicate them with correct notation.
Key formulas
Standard quadratic equation
$ax^2+bx+c=0,\quad a\ne0$
Zero-product principle
$PQ=0\quad\Longrightarrow\quad P=0\ \text{or}\ Q=0$
Roots from factored form
$a(x-p)(x-q)=0\quad\Longrightarrow\quad x=p\ \text{or}\ x=q$
Repeated factor
$a(x-p)^2=0\quad\Longrightarrow\quad x=p$
Completing-the-square identity
$x^2+bx=\left(x+\frac{b}{2}\right)^2-\left(\frac{b}{2}\right)^2$
Square-root step
$u^2=k,\ k>0\quad\Longrightarrow\quad u=\pm\sqrt{k}$
Quadratic formula
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Discriminant
$\Delta=b^2-4ac$
Two distinct real roots
$\Delta>0$
One repeated real root
$\Delta=0$
No real roots
$\Delta<0$
Exact-to-approximate reporting
$x=\frac{-b\pm\sqrt{\Delta}}{2a}\quad\Longrightarrow\quad x\approx x_1,\ x_2$
Graphical root condition
$f(x)=0\quad\Longleftrightarrow\quad (x,0)\ \text{lies on}\ y=f(x)$
Strict interval endpoints
$x<p\ \text{or}\ x>q\quad\Longleftrightarrow\quad (-\infty,p)\cup(q,\infty)$
Inclusive interval
$p\le x\le q\quad\Longleftrightarrow\quad [p,q]$
Union of outside intervals
$(-\infty,p)\cup(q,\infty)$
All real numbers except a point
$\mathbb{R}\setminus\{p\}=(-\infty,p)\cup(p,\infty)$
Empty solution set
$\varnothing$
Above the x-axis
$f(x)>0$
One graph above another
$f(x)>g(x)\quad\Longleftrightarrow\quad f(x)-g(x)>0$
Intersection equation
$f(x)=g(x)\quad\Longleftrightarrow\quad f(x)-g(x)=0$
Tangency condition
$\Delta=0$
GDC equation entry
$Y_1=ax^2+bx+c$
GDC intersection entry
$Y_1=f(x),\quad Y_2=g(x)$
Sum of roots check
$x_1+x_2=-\frac{b}{a}$
Product of roots check
$x_1x_2=\frac{c}{a}$
Root relationships
$x_1+x_2=-\frac{b}{a},\quad x_1x_2=\frac{c}{a}$
Interval notation reminder
$<,>:\ \text{exclude roots};\qquad \le,\ge:\ \text{include roots}$
Topic 2.8Not started
Reciprocal and Rational Functions
Understand reciprocal and rational functions through their domains, ranges, asymptotes, intercepts and transformations; solve rational equations and interpret graphs algebraically and with technology.
180 min estimated
Topic 2.8 progress0%
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Learning objectives
Explain reciprocal and rational functions and identify the restrictions created by their denominators.
Describe and sketch the basic reciprocal graph, including its domain, range, symmetry and asymptotes.
Analyze transformations of the form a divided by x minus h, plus k.
Find vertical and horizontal asymptotes, centres, excluded values and intercepts.
Analyze linear-over-linear functions of the form ax plus b divided by cx plus d.
Distinguish a vertical asymptote from a removable discontinuity or hole.
Sketch rational graphs using algebraic features and test values.
Solve rational equations while preserving restrictions and checking for extraneous solutions.
Find intersections between rational functions and other graphs algebraically and using a GDC.
Apply reciprocal models in context and communicate solutions with correct exact, approximate and interval notation.
Key formulas
General rational function
$f(x)=\frac{P(x)}{Q(x)},\quad Q(x)\ne0$
Reciprocal identity
$x\cdot\frac{1}{x}=1,\quad x\ne0$
Basic reciprocal function
$f(x)=\frac{1}{x},\quad x\ne0$
Odd symmetry
$f(-x)=-f(x)$
Self-inverse property
$f^{-1}(x)=f(x)=\frac{1}{x}$
Parent reciprocal domain and range
$D_f=\mathbb R\setminus\{0\},\quad R_f=\mathbb R\setminus\{0\}$
Transformed reciprocal domain and range
$D_f=\mathbb R\setminus\{h\},\quad R_f=\mathbb R\setminus\{k\}$
Transformed reciprocal asymptotes
$y=\frac{a}{x-h}+k\quad\Longrightarrow\quad x=h,\ y=k$
Centre
$(h,k)$
x-intercept condition
$\frac{P(x)}{Q(x)}=0\quad\Longleftrightarrow\quad P(x)=0,\ Q(x)\ne0$
y-intercept
$(0,f(0))$
Transformed reciprocal
$f(x)=\frac{a}{x-h}+k,\quad a\ne0$
Point from an offset
$x=h+t\quad\Longrightarrow\quad y=k+\frac{a}{t},\quad t\ne0$
Linear-over-linear function
$f(x)=\frac{ax+b}{cx+d},\quad c\ne0$
Vertical asymptote candidate
$x=-\frac{d}{c}$
Horizontal asymptote
$y=\frac{a}{c}$
Hyperbola condition
$ad-bc\ne0$
Linear-over-linear transformed form
$\frac{ax+b}{cx+d}=\frac{bc-ad}{c^2\left(x+\frac{d}{c}\right)}+\frac{a}{c}$
Centre from coefficients
$\left(-\frac{d}{c},\frac{a}{c}\right)$
Cancelled-factor model
$\frac{(x-r)P(x)}{(x-r)Q(x)}=\frac{P(x)}{Q(x)},\quad x\ne r$
Rational-equation workflow
$\text{restrictions}\ \longrightarrow\ \text{clear denominators}\ \longrightarrow\ \text{solve}\ \longrightarrow\ \text{check}$
Intersection condition
$f(x)=g(x)$
Intersection coordinates
$(x,f(x))=(x,g(x))$
Inverse variation
$y=\frac{k}{x},\quad k\ne0$
Average-cost model
$C(x)=\frac{F}{x}+v$
Fixed-distance time
$t=\frac{d}{v}$
GDC entry pattern
$Y_1=\frac{P(X)}{Q(X)}$
Numerical intersection
$Y_1=Y_2$
Parent reciprocal
$f(x)=\frac1x$
Transformed reciprocal
$f(x)=\frac{a}{x-h}+k$
Transformed asymptotes
$x=h,\quad y=k$
Linear-over-linear asymptotes
$x=-\frac dc,\quad y=\frac ac$
x-intercept condition
$P(x)=0,\quad Q(x)\ne0$
Topic 2.9Not started
Exponential and logarithmic functions
Exponential and logarithmic functions model repeated proportional change. Their inverse relationship connects growth and decay, graph transformations, equations and real-world predictions.
190 min estimated
Topic 2.9 progress0%
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Learning objectives
Describe exponential and logarithmic functions using their equations, graphs, domains, ranges, intercepts and asymptotes.
Model exponential growth and decay using discrete and continuous forms and interpret every parameter in context.
Convert between exponential and logarithmic form and explain why the two function families are inverses.
Apply the product, quotient and power laws of logarithms while respecting domain restrictions.
Solve exponential and logarithmic equations exactly where possible and numerically where appropriate.
Predict transformations of exponential and logarithmic graphs and identify the resulting asymptotes.
Use a GDC to graph models, find intersections and solve contextual growth and decay problems.
Communicate IB-style solutions with correct notation, appropriate accuracy and a contextual conclusion.
Key formulas
Basic exponential function
$f(x)=a^x,\qquad a>0,\ a\ne1$
Scaled exponential function
$f(x)=A a^x$
Natural exponential function
$f(x)=e^x,\qquad e\approx2.71828$
Basic exponential domain and range
$D=\mathbb R,\qquad R=(0,\infty)$
Basic exponential intercept
$a^0=1\quad\Longrightarrow\quad(0,1)$
Basic exponential asymptote
$y=0$
Discrete exponential model
$A=A_0b^t$
Growth condition
$b>1$
Decay condition
$0<b<1$
Percentage growth
$A=A_0(1+r)^n$
Percentage decay
$A=A_0(1-r)^n$
Compound interest
$A=P\left(1+\frac{r}{m}\right)^{mt}$
Continuous model
$A=A_0e^{kt}$
Discrete-continuous connection
$b=e^k,\qquad k=\ln b$
Doubling time
$T_2=\frac{\ln2}{k},\qquad k>0$
Half-life
$T_{\frac{1}{2}}=\frac{\ln2}{|k|},\qquad k<0$
Exponential-logarithmic equivalence
$\log_a x=y\quad\Longleftrightarrow\quad a^y=x$
Logarithm restrictions
$a>0,\qquad a\ne1,\qquad x>0$
Fundamental values
$\log_a1=0,\qquad\log_a a=1$
Natural inverse identities
$\ln(e^x)=x,\qquad e^{\ln x}=x\ (x>0)$
Basic logarithmic domain and range
$D=(0,\infty),\qquad R=\mathbb R$
Basic logarithmic intercept
$\log_a1=0\quad\Longrightarrow\quad(1,0)$
Basic logarithmic asymptote
$x=0$
Inverse pair
$f(x)=a^x,\qquad f^{-1}(x)=\log_a x$
Inverse compositions
$\log_a(a^x)=x,\qquad a^{\log_a x}=x\ (x>0)$
Reflected points
$(p,q)\longleftrightarrow(q,p)$
Product law
$\log_a(MN)=\log_aM+\log_aN$
Quotient law
$\log_a\left(\frac{M}{N}\right)=\log_aM-\log_aN$
Power law
$\log_a(M^p)=p\log_aM$
Argument conditions
$M>0,\qquad N>0$
Change of base
$\log_a x=\frac{\log_bx}{\log_ba}=\frac{\ln x}{\ln a}$
Common calculator form
$\log_a x=\frac{\log x}{\log a}$
Equal-base principle
$a^u=a^v\quad\Longrightarrow\quad u=v$
General exponential solution
$a^x=b\quad\Longrightarrow\quad x=\frac{\ln b}{\ln a}$
Natural exponential solution
$e^{kx}=c\quad\Longrightarrow\quad x=\frac{\ln c}{k}$
Single-log equation
$\log_a F(x)=c\quad\Longleftrightarrow\quad F(x)=a^c$
Equal-log principle
$\log_aM=\log_aN\quad\Longrightarrow\quad M=N$
Logarithmic domain
$M>0,\qquad N>0$
Transformed exponential
$y=Aa^{B(x-h)}+k$
Exponential asymptote
$y=k$
Transformed logarithm
$y=A\log_a(B(x-h))+k$
Logarithmic domain condition
$B(x-h)>0$
Time to a target
$A_0b^t=T\quad\Longrightarrow\quad t=\frac{\ln\left(\frac{T}{A_0}\right)}{\ln b}$
Continuous time to a target
$A_0e^{kt}=T\quad\Longrightarrow\quad t=\frac{\ln\left(\frac{T}{A_0}\right)}{k}$
Half-life model
$A=A_0\left(\frac{1}{2}\right)^{\frac{t}{h}}$
Intersection method
$Y_1=F(x),\qquad Y_2=G(x)$
Root method
$Y_1=F(x)-G(x)$
Approximation notation
$x=\frac{\ln7}{\ln3}-1\approx0.771$
Logarithmic validity check
$\text{every original argument}>0$
Topic 2.10Not started
Solving equations graphically and analytically
Equations can be solved exactly with algebra or approximately from roots and graph intersections. Successful problem solving means choosing an efficient method, respecting domains and checking every result.
200 min estimated
Topic 2.10 progress0%
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Learning objectives
Explain the meaning of analytical and graphical solutions and connect them to coordinates, roots and intersections.
Solve linear and quadratic equations using appropriate algebraic methods.
Solve exponential, logarithmic and rational equations while respecting domain restrictions.
Interpret roots, zeros and intersections from graphs, tables and GDC output.
Distinguish exact solutions from numerical approximations and use correct equality notation.
Choose an efficient solving method based on equation structure, required accuracy and available technology.
Use graphs to check analytical answers and detect missing or extraneous solutions.
Use GDC zero, intersection, table and numerical-solver tools with suitable windows and initial values.
Present IB-style solutions with visible setup, mathematical justification and appropriate accuracy.
Key formulas
Equation solution condition
$L(x)=R(x)$
Root form
$f(x)=0$
Equivalent intersection form
$F(x)=G(x)\quad\Longleftrightarrow\quad F(x)-G(x)=0$
Intersection equation
$Y_1=F(x),\qquad Y_2=G(x)$
Equivalent zero equation
$Y_1=F(x)-G(x)$
Root coordinate
$f(r)=0\quad\Longleftrightarrow\quad(r,0)\text{ lies on }y=f(x)$
Addition property
$A=B\quad\Longrightarrow\quad A+C=B+C$
Non-zero multiplication property
$A=B\quad\Longrightarrow\quad kA=kB,\qquad k\ne0$
Verification
$L(x_{\mathrm{candidate}})\stackrel{?}{=}R(x_{\mathrm{candidate}})$
General linear equation
$ax+b=cx+d$
Linear solution
$x=\frac{d-b}{a-c},\qquad a\ne c$
Parallel-line condition
$a=c,\qquad b\ne d$
Quadratic equation
$ax^2+bx+c=0,\qquad a\ne0$
Quadratic formula
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Discriminant
$\Delta=b^2-4ac$
Root classification
$\Delta>0:\ 2\text{ roots};\quad\Delta=0:\ 1\text{ repeated root};\quad\Delta<0:\ 0\text{ real roots}$
Root condition
$f(r)=0$
Crossing sign test
$f(a)f(b)<0$
Repeated quadratic factor
$f(x)=(x-r)^2g(x)$
Intersection condition
$F(x)=G(x)$
Intersection point
$\bigl(r,F(r)\bigr)=\bigl(r,G(r)\bigr)$
Restricted solution set
$\{x:F(x)=G(x)\}\cap D$
Equal-base principle
$a^u=a^v\quad\Longrightarrow\quad u=v$
Logarithmic solution
$a^x=b\quad\Longrightarrow\quad x=\frac{\ln b}{\ln a}$
Positive substitution
$u=a^x,\qquad u>0$
Logarithmic equivalence
$\log_aF(x)=c\quad\Longleftrightarrow\quad F(x)=a^c$
Equal-log principle
$\log_aM=\log_aN\quad\Longrightarrow\quad M=N$
Domain requirement
$M>0,\qquad N>0$
Rational restriction
$D(x)\ne0$
Zero of a rational expression
$\frac{N(x)}{D(x)}=0\quad\Longleftrightarrow\quad N(x)=0\text{ and }D(x)\ne0$
Cross multiplication condition
$\frac{A}{B}=\frac{C}{D}\quad\Longrightarrow\quad AD=BC,\qquad B\ne0,\ D\ne0$
Exact-to-approximate notation
$x=\frac{-1+\sqrt{33}}{4}\approx1.19$
Absolute checking error
$\left|L(x_{\mathrm{approx}})-R(x_{\mathrm{approx}})\right|$
Factor route
$P(x)=0\quad\longrightarrow\quad\text{factor if practical}$
Intersection route
$F(x)=G(x)\quad\longrightarrow\quad\text{graph }Y_1=F(x),\ Y_2=G(x)$
Logarithm route
$a^{g(x)}=b\quad\longrightarrow\quad g(x)=\frac{\ln b}{\ln a}$
Residual
$R(x)=F(x)-G(x)$
Exact check
$R(r)=0$
Approximate check
$|R(r)|\approx0$
One-graph zero setup
$Y_1=F(x)-G(x)$
Two-graph intersection setup
$Y_1=F(x),\qquad Y_2=G(x)$
Bracket condition
$f(a)f(b)<0$
Method-check structure
$\text{classify}\to\text{restrict}\to\text{solve}\to\text{check}\to\text{report}$
Approximation notation
$x\approx r\quad\text{when }r\text{ is rounded}$
Topic 2.11Not started
Transformations of graphs
Graph transformations predict new curves from familiar parent functions. Translations, reflections and scale changes move every point, domain boundary, asymptote and key feature in a systematic way.
200 min estimated
Topic 2.11 progress0%
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Learning objectives
Describe vertical and horizontal translations using function notation and point mappings.
Reflect graphs in the coordinate axes and distinguish $-f(x)$ from $f(-x)$ .
Apply vertical and horizontal stretches or compressions with the correct reciprocal horizontal scale factor.
Use the general form $g(x)=a f(b(x-h))+k$ to combine transformations reliably.
Map points, domains, ranges, intercepts, turning points, endpoints and asymptotes under transformations.
Identify transformations from equations, graphs and corresponding landmark points.
Sketch transformed graphs accurately from parent functions without relying on a large value table.
Use a GDC to compare parent and transformed graphs and verify predicted features.
Communicate transformation order, scale factors and graph features in IB-style notation.
Key formulas
Parent point condition
$v=f(u)$
General transformed function
$g(x)=a f\bigl(b(x-h)\bigr)+k$
General point mapping
$(u,v)\longmapsto\left(h+\frac{u}{b},\ k+av\right),\qquad b\ne0$
Vertical translation
$g(x)=f(x)+k$
Vertical point map
$(u,v)\longmapsto(u,v+k)$
Range translation
$R_g=\{v+k:v\in R_f\}$
Horizontal translation
$g(x)=f(x-h)$
Horizontal point map
$(u,v)\longmapsto(u+h,v)$
Domain translation
$D_g=\{u+h:u\in D_f\}$
Reflection in the $x$-axis
$g(x)=-f(x)$
$x$-axis point map
$(u,v)\longmapsto(u,-v)$
Reflection in the $y$-axis
$g(x)=f(-x)$
$y$-axis point map
$(u,v)\longmapsto(-u,v)$
Vertical scaling
$g(x)=af(x)$
Vertical scale point map
$(u,v)\longmapsto(u,av)$
Vertical stretch condition
$|a|>1$
Vertical compression condition
$0<|a|<1$
Horizontal scaling
$g(x)=f(bx),\qquad b\ne0$
Horizontal scale point map
$(u,v)\longmapsto\left(\frac{u}{b},v\right)$
Horizontal scale factor
$\frac{1}{|b|}$
Combined form
$g(x)=a f\bigl(b(x-h)\bigr)+k$
Combined point map
$(u,v)\longmapsto\left(h+\frac{u}{b},\ k+av\right)$
Inverse point recovery
$u=b(x-h),\qquad v=\frac{y-k}{a}$
Visual transformation
$g(x)=-2f(x-3)+1$
Visual point map
$(u,v)\longmapsto(u+3,1-2v)$
Transformed quadratic range
$R_g=(-\infty,1]$
Transformed domain
$D_g=\left\{h+\frac{u}{b}:u\in D_f\right\}$
Transformed range
$R_g=\{k+av:v\in R_f\}$
Horizontal asymptote mapping
$y=L\quad\longmapsto\quad y=k+aL$
Vertical asymptote mapping
$x=C\quad\longmapsto\quad x=h+\frac{C}{b}$
Transformed $y$-intercept
$g(0)=a f(-bh)+k$
Transformed zeros
$a f\bigl(b(x-h)\bigr)+k=0$
Inside factorization
$bx+c=b\left(x+\frac{c}{b}\right)$
Readable inside form
$b(x-h)$
Quadratic transformed form
$g(x)=a(x-h)^2+k$
Absolute-value transformed form
$g(x)=a|x-h|+k$
Reciprocal translated form
$g(x)=\frac{a}{x-h}+k$
Sketch workflow
$\text{parent}\to\text{asymptotes or boundaries}\to\text{landmarks}\to\text{intercepts}\to\text{shape}$
GDC parent entry
$Y_1=f(x)$
GDC image entry
$Y_2=a f\bigl(b(x-h)\bigr)+k$
Parameter checklist
$a:\text{ vertical};\quad b:\text{ reciprocal horizontal};\quad h:\text{ horizontal shift};\quad k:\text{ vertical shift}$
Reliable point check
$g\left(h+\frac{u}{b}\right)=k+a f(u)$
Unit 3
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Geometry and Trigonometry
This topic combines spatial reasoning, coordinate geometry and trigonometric functions. It moves from triangle methods to radians, identities, periodic functions and trigonometric equations.
8 topics
Unit 3 progress0%
Topics in this unit
Topic 3.1Not started
Three-dimensional geometry
Three-dimensional geometry connects spatial diagrams, right triangles, coordinates and mensuration. Students learn to expose hidden two-dimensional triangles inside solids and use them to calculate lengths, angles, surface areas and volumes.
190 min estimated
Topic 3.1 progress0%
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Learning objectives
Interpret and label points, lines, planes, faces, edges, vertices, diagonals and cross-sections in three-dimensional diagrams.
Distinguish intersecting, parallel, perpendicular and skew lines in space.
Apply Pythagoras in two stages to find face diagonals and space diagonals.
Calculate distance and midpoint between points in three-dimensional coordinates.
Find angles between a line and a plane by identifying the line's perpendicular projection on the plane.
Use right-triangle trigonometry and the cosine rule to determine angles and lengths in three-dimensional contexts.
Calculate surface area and volume for common and composite solids with consistent units.
Translate real-world descriptions into clear labelled diagrams and select an efficient calculation route.
Use a calculator or GDC with correct angle mode, full precision and appropriate rounding.
Present complete IB-style solutions with formulas, substitutions, units, diagrams and contextual conclusions.
Key formulas
Euler relation for a convex polyhedron
$V-E+F=2$
Perpendicular line-plane angle
$\theta=90^\circ$
Line-plane angle definition
$\theta=\angle(\text{line},\text{ perpendicular projection on plane})$
Pythagorean theorem
$c^2=a^2+b^2$
Rectangular base diagonal
$p=\sqrt{l^2+w^2}$
Cuboid space diagonal
$d=\sqrt{l^2+w^2+h^2}$
Distance in three dimensions
$AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$
Midpoint in three dimensions
$M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\right)$
Distance from the origin
$OP=\sqrt{x^2+y^2+z^2}$
Line-plane angle using tangent
$\tan\theta=\frac{\text{perpendicular height}}{\text{projection length}}$
Line-plane angle using sine
$\sin\theta=\frac{\text{perpendicular height}}{\text{line length}}$
Line-plane angle using cosine
$\cos\theta=\frac{\text{projection length}}{\text{line length}}$
Cosine rule for an included angle
$\cos C=\frac{a^2+b^2-c^2}{2ab}$
Square centre-to-corner distance
$p=\frac{s\sqrt2}{2}$
Cuboid surface area
$SA=2(lw+lh+wh)$
Closed cylinder surface area
$SA=2\pi r^2+2\pi rh$
Sphere surface area
$SA=4\pi r^2$
Closed cone surface area
$SA=\pi r^2+\pi rs$
Prism volume
$V=A_{\text{cross-section}}L$
Cuboid volume
$V=lwh$
Cylinder volume
$V=\pi r^2h$
Pyramid volume
$V=\frac13Ah$
Cone volume
$V=\frac13\pi r^2h$
Sphere volume
$V=\frac43\pi r^3$
Diagram-to-method chain
$\text{solid}\to\text{projection or cross-section}\to\text{2D triangle}\to\text{formula}$
Cubic metres to litres
$1\text{ m}^3=1000\text{ L}$
Maximum straight length in a cuboid
$d_{\max}=\sqrt{l^2+w^2+h^2}$
Inverse tangent entry
$\theta=\tan^{-1}\left(\frac{h}{p}\right)$
Inverse sine entry
$\theta=\sin^{-1}\left(\frac{h}{d}\right)$
Reliable 3D workflow
$\text{label}\to\text{project or cross-section}\to\text{calculate}\to\text{check}\to\text{interpret}$
Topic 3.2Not started
Trigonometry in triangles
Triangle trigonometry connects angles and side lengths in right and non-right triangles. Students learn to label diagrams accurately, choose an efficient rule, solve complete triangles and justify whether one, two or no configurations are possible.
205 min estimated
Topic 3.2 progress0%
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Learning objectives
Label right and non-right triangles using standard side-angle conventions.
Use sine, cosine and tangent ratios to find unknown sides and angles in right triangles.
Apply the Pythagorean theorem and recognize when it is the most efficient method.
Use the sine rule when a known side and its opposite angle form a matching pair.
Use the cosine rule with SSS or SAS information and rearrange it to find angles.
Calculate triangle area from two sides and their included angle.
Analyze the SSA ambiguous case and determine whether zero, one or two triangles exist.
Solve contextual triangle problems using clear diagrams, units and suitable accuracy.
Use a calculator or GDC in the correct angle mode without premature rounding.
Present complete IB-style reasoning and check that lengths and angles are geometrically valid.
Key formulas
Interior angle sum
$A+B+C=180^\circ$
Standard opposite-side convention
$a\leftrightarrow A,\quad b\leftrightarrow B,\quad c\leftrightarrow C$
Sine ratio
$\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$
Cosine ratio
$\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$
Tangent ratio
$\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$
Pythagorean theorem
$a^2+b^2=c^2$
Missing shorter side
$a=\sqrt{c^2-b^2}$
Complementary acute angles
$\alpha+\beta=90^\circ$
Sine rule
$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$
Reciprocal sine-rule form
$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$
Supplementary sine solution
$B_2=180^\circ-B_1$
SSA comparison height
$h=b\sin A$
Cosine rule for a side
$c^2=a^2+b^2-2ab\cos C$
Cosine rule for an angle
$\cos C=\frac{a^2+b^2-c^2}{2ab}$
Triangle area from two sides and included angle
$K=\frac{1}{2}ab\sin C$
Equivalent area forms
$K=\frac{1}{2}bc\sin A=\frac{1}{2}ca\sin B$
Method-selection guide
$\begin{aligned}\text{right triangle}&\Rightarrow\text{SOH-CAH-TOA or Pythagoras}\\\text{known opposite pair}&\Rightarrow\text{sine rule}\\\text{SAS or SSS}&\Rightarrow\text{cosine rule}\end{aligned}$
Triangle inequality
$a+b>c,\quad b+c>a,\quad c+a>b$
Right-triangle ratios
$\sin\theta=\frac{O}{H},\quad\cos\theta=\frac{A}{H},\quad\tan\theta=\frac{O}{A}$
Cosine rule
$c^2=a^2+b^2-2ab\cos C$
Triangle area
$K=\frac12 ab\sin C$
Topic 3.3Not started
Applications of trigonometry
Applied trigonometry turns real situations into mathematical triangles. Students learn to construct reliable diagrams, combine right-triangle ratios with the sine and cosine rules, solve navigation and spatial problems, and interpret answers with appropriate units and accuracy.
215 min estimated
Topic 3.3 progress0%
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Learning objectives
Translate written descriptions into clear, labelled two-dimensional and three-dimensional diagrams.
Solve heights and distances using angles of elevation and depression.
Interpret three-digit bearings and determine internal angles in navigation triangles.
Apply right-triangle trigonometry and Pythagoras in contextual problems.
Use the sine rule and cosine rule in surveying, navigation and measurement applications.
Calculate and interpret areas using two sides and an included angle.
Break multi-step problems into a sequence of linked triangles without premature rounding.
Expose useful right-triangle cross-sections in three-dimensional situations.
Use a calculator or GDC in the correct angle mode and report suitable precision.
Present complete IB-style solutions with assumptions, methods, units and contextual conclusions.
Key formulas
Triangle angle sum
$A+B+C=180^\circ$
Height from elevation angle
$h=d\tan\theta$
Horizontal distance from depression angle
$d=\frac{h}{\tan\theta}$
Two-position height relation
$x\tan\beta=(x+s)\tan\alpha$
Reverse bearing
$\theta_{\text{reverse}}=(\theta+180^\circ)\bmod 360^\circ$
Bearing from east and north components
$\theta=\tan^{-1}\left(\frac{E}{N}\right)$
Cosine rule for a route displacement
$d^2=p^2+q^2-2pq\cos C$
Sine rule for triangulation
$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$
Sine rule
$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$
Second possible sine-rule angle
$B_2=180^\circ-B_1$
Cosine rule for a side
$c^2=a^2+b^2-2ab\cos C$
Cosine rule for an angle
$C=\cos^{-1}\left(\frac{a^2+b^2-c^2}{2ab}\right)$
Triangle area
$K=\frac{1}{2}ab\sin C$
Equivalent area forms
$K=\frac12 bc\sin A=\frac12 ca\sin B$
Cuboid space diagonal
$d=\sqrt{l^2+w^2+h^2}$
Line-plane angle
$\tan\theta=\frac{\text{perpendicular height}}{\text{projection length}}$
Three-dimensional distance
$d=\sqrt{(\Delta x)^2+(\Delta y)^2+(\Delta z)^2}$
Route displacement bounds
$|p-q|\le d\le p+q$
Right-triangle ratios
$\sin\theta=\frac{O}{H},\quad\cos\theta=\frac{A}{H},\quad\tan\theta=\frac{O}{A}$
Pythagorean theorem
$a^2+b^2=c^2$
Cosine rule
$c^2=a^2+b^2-2ab\cos C$
Triangle area
$K=\frac12 ab\sin C$
Reverse bearing
$\theta_{\text{reverse}}=(\theta+180^\circ)\bmod360^\circ$
Topic 3.4Not started
Radian measure, arcs and sectors
Radian measure connects an angle directly to arc length. Students convert exactly between degrees and radians, solve arc and sector problems, and combine circular geometry in IB-style contexts.
210 min estimated
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Learning objectives
Explain one radian as the central angle subtended by an arc whose length equals the radius.
Convert angles exactly and approximately between degrees and radians.
Express standard and non-standard angles as exact multiples of pi.
Calculate arc length when the central angle is measured in radians.
Calculate the area of a sector when the central angle is measured in radians.
Rearrange circular-measure formulas to find an unknown radius or angle.
Calculate sector perimeters, chord lengths and circular-segment areas.
Solve composite problems involving sectors, triangles and annular regions.
Use radian mode and stored precision effectively on a calculator or GDC.
Present IB-style solutions with exact values, suitable units and reasonableness checks.
Key formulas
Radian definition
$\theta=\frac{s}{r}$
Full turn
$360^\circ=2\pi\text{ rad}$
Half turn
$180^\circ=\pi\text{ rad}$
Coterminal angles
$\theta_{\text{coterminal}}=\theta+2k\pi,\quad k\in\mathbb{Z}$
Degrees to radians
$\theta_{\mathrm{rad}}=\theta_{\mathrm{deg}}\frac{\pi}{180}$
Radians to degrees
$\theta_{\mathrm{deg}}=\theta_{\mathrm{rad}}\frac{180}{\pi}$
One-radian approximation
$1\text{ rad}=\frac{180^\circ}{\pi}\approx57.3^\circ$
Standard exact angles
$30^\circ=\frac{\pi}{6},\quad45^\circ=\frac{\pi}{4},\quad60^\circ=\frac{\pi}{3},\quad90^\circ=\frac{\pi}{2}$
Arc length in radians
$s=r\theta$
Arc length in degrees
$s=\frac{\theta^\circ}{360^\circ}(2\pi r)$
Major central angle
$\theta_{\text{major}}=2\pi-\theta_{\text{minor}}$
Sector area in radians
$K=\frac12r^2\theta$
Sector area in degrees
$K=\frac{\theta^\circ}{360^\circ}\pi r^2$
Major sector area
$K_{\text{major}}=\pi r^2-K_{\text{minor}}$
Radius from an arc
$r=\frac{s}{\theta}$
Angle from an arc
$\theta=\frac{s}{r}$
Radius from sector area
$r=\sqrt{\frac{2K}{\theta}}$
Angle from sector area
$\theta=\frac{2K}{r^2}$
Sector area from radius and arc
$K=\frac12rs$
Sector perimeter
$P=2r+r\theta$
Chord length
$c=2r\sin\left(\frac{\theta}{2}\right)$
Central triangle area
$K_{\triangle}=\frac12r^2\sin\theta$
Minor segment area
$K_{\text{segment}}=\frac12r^2(\theta-\sin\theta)$
Annular sector area
$K_{\text{annular}}=\frac12(R^2-r^2)\theta$
Annular-sector perimeter
$P=(R+r)\theta+2(R-r)$
Angular displacement after turns
$\theta=2\pi n$
Rolling distance
$s=r\theta=2\pi rn$
Angle conversions
$\theta_{\mathrm{rad}}=\theta_{\mathrm{deg}}\frac{\pi}{180},\quad\theta_{\mathrm{deg}}=\theta_{\mathrm{rad}}\frac{180}{\pi}$
Arc length
$s=r\theta$
Sector area
$K=\frac12r^2\theta=\frac12rs$
Topic 3.5Not started
Unit-circle definitions and exact values
The unit circle defines sine and cosine as coordinates, extends trigonometry through all four quadrants, and makes exact values and related angles systematic.
205 min estimated
Topic 3.5 progress0%
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Learning objectives
Define the unit circle and explain how an angle locates a point on it.
Interpret the coordinates of a unit-circle point as cosine and sine.
Define tangent as the ratio of sine to cosine where cosine is non-zero.
Determine the signs of sine, cosine and tangent in every quadrant.
Find and use reference angles in degrees and radians.
Derive exact trigonometric values from special right triangles.
Recall exact sine, cosine and tangent values for standard angles.
Evaluate related angles without a calculator using symmetry and periodicity.
Solve elementary exact-value equations over a stated interval.
Present IB-style reasoning with exact values, valid domains and clear quadrant checks.
Key formulas
Unit-circle equation
$x^2+y^2=1$
Coterminal angles
$\theta_{\mathrm{coterminal}}=\theta+2k\pi,\quad k\in\mathbb{Z}$
Unit-circle coordinates
$P=(\cos\theta,\sin\theta)$
Pythagorean identity
$\sin^2\theta+\cos^2\theta=1$
Coordinate definitions
$\cos\theta=x,\quad\sin\theta=y$
Tangent identity
$\tan\theta=\frac{\sin\theta}{\cos\theta}=\frac{y}{x},\quad\cos\theta\ne0$
Undefined tangent angles
$\theta=\frac{\pi}{2}+k\pi,\quad k\in\mathbb{Z}$
Quadrant I
$\sin\theta>0,\quad\cos\theta>0,\quad\tan\theta>0$
Quadrant II
$\sin\theta>0,\quad\cos\theta<0,\quad\tan\theta<0$
Quadrant III
$\sin\theta<0,\quad\cos\theta<0,\quad\tan\theta>0$
Quadrant IV
$\sin\theta<0,\quad\cos\theta>0,\quad\tan\theta<0$
QI reference angle
$\alpha=\theta$
QII reference angle
$\alpha=\pi-\theta$
QIII reference angle
$\alpha=\theta-\pi$
QIV reference angle
$\alpha=2\pi-\theta$
45-degree triangle ratio
$1:1:\sqrt2$
30-60-90 triangle ratio
$1:\sqrt3:2$
Rationalized tangent value
$\frac{1}{\sqrt3}=\frac{\sqrt3}{3}$
Values at zero
$\sin0=0,\quad\cos0=1,\quad\tan0=0$
Values at 30 degrees
$\sin\frac{\pi}{6}=\frac12,\quad\cos\frac{\pi}{6}=\frac{\sqrt3}{2},\quad\tan\frac{\pi}{6}=\frac{\sqrt3}{3}$
Values at 45 degrees
$\sin\frac{\pi}{4}=\frac{\sqrt2}{2},\quad\cos\frac{\pi}{4}=\frac{\sqrt2}{2},\quad\tan\frac{\pi}{4}=1$
Values at 60 degrees
$\sin\frac{\pi}{3}=\frac{\sqrt3}{2},\quad\cos\frac{\pi}{3}=\frac12,\quad\tan\frac{\pi}{3}=\sqrt3$
Values at 90 degrees
$\sin\frac{\pi}{2}=1,\quad\cos\frac{\pi}{2}=0,\quad\tan\frac{\pi}{2}\text{ is undefined}$
Degree-radian equivalence
$180^\circ=\pi\text{ rad}$
Degrees to radians
$\theta_{\mathrm{rad}}=\theta_{\mathrm{deg}}\frac{\pi}{180}$
Radians to degrees
$\theta_{\mathrm{deg}}=\theta_{\mathrm{rad}}\frac{180}{\pi}$
Coordinate recovery
$y=\pm\sqrt{1-x^2},\quad x=\pm\sqrt{1-y^2}$
Even and odd symmetry
$\cos(-\theta)=\cos\theta,\quad\sin(-\theta)=-\sin\theta,\quad\tan(-\theta)=-\tan\theta$
Sine and cosine periods
$\sin(\theta+2\pi)=\sin\theta,\quad\cos(\theta+2\pi)=\cos\theta$
Tangent period
$\tan(\theta+\pi)=\tan\theta$
Second-quadrant symmetry
$\sin(\pi-\alpha)=\sin\alpha,\quad\cos(\pi-\alpha)=-\cos\alpha$
Fourth-quadrant symmetry
$\sin(2\pi-\alpha)=-\sin\alpha,\quad\cos(2\pi-\alpha)=\cos\alpha$
Unit circle and coordinates
$x^2+y^2=1,\quad P=(\cos\theta,\sin\theta)$
Tangent
$\tan\theta=\frac{\sin\theta}{\cos\theta},\quad\cos\theta\ne0$
Standard sine magnitudes
$0,\ \frac12,\ \frac{\sqrt2}{2},\ \frac{\sqrt3}{2},\ 1$
Standard cosine magnitudes
$1,\ \frac{\sqrt3}{2},\ \frac{\sqrt2}{2},\ \frac12,\ 0$
Periods
$\sin(\theta+2\pi)=\sin\theta,\quad\cos(\theta+2\pi)=\cos\theta,\quad\tan(\theta+\pi)=\tan\theta$
Topic 3.6Not started
Trigonometric identities
Trigonometric identities connect sine, cosine and tangent throughout their valid domains. Students learn to simplify expressions, prove statements and select efficient identity forms for IB-style problems.
210 min estimated
Topic 3.6 progress0%
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Learning objectives
Distinguish an identity from an equation that is true only for particular angles.
Use quotient and reciprocal relationships while respecting their domains.
Derive and apply the Pythagorean identity and its rearranged forms.
Simplify trigonometric expressions by substitution, factoring and common denominators.
Prove simple identities by transforming one side into the other.
Choose an identity that matches the functions and powers in an expression.
Use the sine and cosine double-angle identities in exact calculations and simplification.
State restrictions caused by zero denominators and undefined trigonometric ratios.
Use a GDC to check numerical evidence without replacing an algebraic proof.
Present clear IB-style working with equivalent steps and justified restrictions.
Key formulas
Fundamental Pythagorean identity
$\sin^2\theta+\cos^2\theta\equiv1$
Example of an equation
$\sin\theta=\frac12$
Squared notation
$\sin^2\theta=(\sin\theta)^2,\quad\cos^2\theta=(\cos\theta)^2$
Tangent quotient identity
$\tan\theta=\frac{\sin\theta}{\cos\theta},\quad\cos\theta\ne0$
Reciprocal identities
$\sec\theta=\frac1{\cos\theta},\quad\csc\theta=\frac1{\sin\theta},\quad\cot\theta=\frac1{\tan\theta}$
Cotangent quotient relationship
$\cot\theta=\frac{\cos\theta}{\sin\theta},\quad\sin\theta\ne0$
Fundamental Pythagorean identity
$\sin^2\theta+\cos^2\theta=1$
Rearranged forms
$\sin^2\theta=1-\cos^2\theta,\quad\cos^2\theta=1-\sin^2\theta$
Tangent–secant identity
$1+\tan^2\theta=\sec^2\theta,\quad\cos\theta\ne0$
Cotangent–cosecant identity
$1+\cot^2\theta=\csc^2\theta,\quad\sin\theta\ne0$
Pythagorean replacements
$1-\cos^2x=\sin^2x,\quad1-\sin^2x=\cos^2x$
Reciprocal-family replacements
$\sec^2x-1=\tan^2x,\quad\csc^2x-1=\cot^2x$
Common-denominator pattern
$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd},\quad b\ne0,\ d\ne0$
Conjugate product
$(1+\cos x)(1-\cos x)=1-\cos^2x=\sin^2x$
Cosine double-angle choices
$\cos 2x=\cos^2x-\sin^2x=2\cos^2x-1=1-2\sin^2x$
Sine double angle
$\sin 2\theta=2\sin\theta\cos\theta$
Cosine double angle
$\cos 2\theta=\cos^2\theta-\sin^2\theta$
Alternative cosine forms
$\cos 2\theta=2\cos^2\theta-1=1-2\sin^2\theta$
Power-reduction forms
$\cos^2\theta=\frac{1+\cos 2\theta}{2},\quad\sin^2\theta=\frac{1-\cos 2\theta}{2}$
Cosine-zero angles
$\cos x=0\iff x=\frac{\pi}{2}+k\pi,\quad k\in\mathbb Z$
Sine-zero angles
$\sin x=0\iff x=k\pi,\quad k\in\mathbb Z$
Difference check
$D(x)=\mathrm{LHS}(x)-\mathrm{RHS}(x)$
Quotient identity
$\tan\theta=\frac{\sin\theta}{\cos\theta},\quad\cos\theta\ne0$
Reciprocal identities
$\sec\theta=\frac1{\cos\theta},\quad\csc\theta=\frac1{\sin\theta},\quad\cot\theta=\frac{\cos\theta}{\sin\theta}$
Pythagorean identity
$\sin^2\theta+\cos^2\theta=1$
Derived Pythagorean identities
$1+\tan^2\theta=\sec^2\theta,\quad1+\cot^2\theta=\csc^2\theta$
Cosine double-angle forms
$\cos 2\theta=\cos^2\theta-\sin^2\theta=2\cos^2\theta-1=1-2\sin^2\theta$
Topic 3.7Not started
Circular functions and their graphs
Circular functions turn rotation on the unit circle into periodic graphs. Students analyze sine, cosine and tangent, then transform and interpret these graphs in mathematical and real-world contexts.
215 min estimated
Topic 3.7 progress0%
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Learning objectives
Connect sine, cosine and tangent graphs to motion around the unit circle.
Sketch the parent sine, cosine and tangent functions from exact key points.
State the domain, range, period, intercepts and extrema of each parent function.
Identify the vertical asymptotes and repeating branches of the tangent function.
Find amplitude, midline, period and phase shift from a transformed sine or cosine equation.
Interpret vertical and horizontal transformations of trigonometric graphs.
Sketch transformed sine and cosine graphs using a five-point cycle.
Sketch transformed tangent graphs using their centre, period and asymptotes.
Build and interpret periodic models from contextual information.
Use a GDC effectively while presenting exact IB-style reasoning and notation.
Key formulas
Unit-circle coordinates
$P=(\cos\theta,\sin\theta)$
Tangent quotient
$\tan\theta=\frac{\sin\theta}{\cos\theta},\quad\cos\theta\ne0$
Parent periods
$\sin(x+2\pi)=\sin x,\quad\cos(x+2\pi)=\cos x,\quad\tan(x+\pi)=\tan x$
Quarter-cycle spacing
$\Delta x=\frac{T}{4}$
Sine domain and range
$\operatorname{Dom}(\sin)=\mathbb R,\quad\operatorname{Ran}(\sin)=[-1,1]$
Sine zeros
$x=k\pi,\quad k\in\mathbb Z$
Sine extrema
$\sin x=1\text{ at }x=\frac{\pi}{2}+2k\pi,\quad\sin x=-1\text{ at }x=\frac{3\pi}{2}+2k\pi$
Cosine domain and range
$\operatorname{Dom}(\cos)=\mathbb R,\quad\operatorname{Ran}(\cos)=[-1,1]$
Cosine zeros
$x=\frac{\pi}{2}+k\pi,\quad k\in\mathbb Z$
Cosine extrema
$\cos x=1\text{ at }x=2k\pi,\quad\cos x=-1\text{ at }x=\pi+2k\pi$
Tangent domain
$x\in\mathbb R,\quad x\ne\frac{\pi}{2}+k\pi,\quad k\in\mathbb Z$
Tangent range
$\operatorname{Ran}(\tan)=\mathbb R$
Tangent zeros and asymptotes
$\tan x=0\text{ at }x=k\pi,\quad x=\frac{\pi}{2}+k\pi\text{ are vertical asymptotes}$
Sine and cosine range
$d-|a|\le y\le d+|a|$
Extreme values
$y_{\max}=d+|a|,\quad y_{\min}=d-|a|$
Sine and cosine amplitude
$A=|a|$
Sine and cosine period
$T=\frac{2\pi}{|b|}$
Tangent period
$T=\frac{\pi}{|b|}$
Midline
$y=d$
Transformed sine and cosine
$y=a\sin(b(x-c))+d,\quad y=a\cos(b(x-c))+d$
Transformed tangent
$y=a\tan(b(x-c))+d$
Shift from expanded inside
$bx+q=b\left(x+\frac{q}{b}\right)=b\left(x-\left(-\frac{q}{b}\right)\right)$
Tangent centre and asymptotes
$x_{\mathrm{centre}}=c+kT,\quad x_{\mathrm{asymptote}}=c+\left(k+\frac12\right)T$
Five-point x-values
$c,\quad c+\frac{T}{4},\quad c+\frac{T}{2},\quad c+\frac{3T}{4},\quad c+T$
Midline from extrema
$d=\frac{M+m}{2}$
Amplitude from extrema
$|a|=\frac{M-m}{2}$
Frequency coefficient from period
$|b|=\frac{2\pi}{T}$
Quarter-period table step
$\Delta x=\frac{T}{4}$
Sine and cosine models
$y=a\sin(b(x-c))+d,\quad y=a\cos(b(x-c))+d$
Tangent model
$y=a\tan(b(x-c))+d$
Amplitude and midline
$A=|a|,\quad y=d$
Phase shift from expanded form
$a\sin(bx+q)+d\text{ has shift }-\frac{q}{b}$
Topic 3.8Not started
Solving trigonometric equations
Trigonometric equations usually have repeated solutions. Students combine algebra, the unit circle, reference angles, identities and graphical methods to find every valid answer in a stated interval.
230 min estimated
Topic 3.8 progress0%
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Learning objectives
Recognize the difference between a trigonometric identity and an equation to be solved.
Solve basic sine, cosine and tangent equations exactly.
Use inverse trigonometric functions to obtain reference or principal angles.
Use the unit circle, quadrants and signs to find all solutions in an interval.
Work fluently with equations stated in degrees or radians.
Use periodicity to write general solutions and extend solutions across intervals.
Solve trigonometric equations requiring algebraic rearrangement or factorization.
Use Pythagorean and double-angle identities to reduce equations to solvable forms.
Find approximate solutions to an appropriate stated accuracy.
Solve equations graphically and verify solutions with a GDC.
Present complete IB-style solutions with interval, mode, exactness and domain checks.
Key formulas
Solution check
$\text{substitute each proposed }x\text{ into the original equation}$
Sine and cosine feasibility
$-1\le a\le1$
Tangent target range
$a\in\mathbb R$
Positive sine pair
$x=\alpha,\quad x=\pi-\alpha$
Positive cosine pair
$x=\alpha,\quad x=2\pi-\alpha$
Positive tangent pair
$x=\alpha,\quad x=\pi+\alpha$
Related angles in one turn
$\alpha,\quad\pi-\alpha,\quad\pi+\alpha,\quad2\pi-\alpha$
Common sine magnitudes
$0,\quad\frac12,\quad\frac{\sqrt2}{2},\quad\frac{\sqrt3}{2},\quad1$
Common tangent magnitudes
$0,\quad\frac{\sqrt3}{3},\quad1,\quad\sqrt3$
Sine second solution
$x_2=\pi-x_1\quad\text{for a positive sine value in }[0,2\pi)$
Cosine reflected solution
$x_2=2\pi-x_1\quad\text{when }x_1\in[0,\pi]$
Interval transformation
$L\le x<U\implies nL\le nx<nU\quad\text{for }n>0$
Full-turn equivalence
$360^\circ=2\pi\text{ radians}$
Degree periods
$T_{\sin}=T_{\cos}=360^\circ,\quad T_{\tan}=180^\circ$
Radian periods
$T_{\sin}=T_{\cos}=2\pi,\quad T_{\tan}=\pi$
Sine general solutions
$x=\alpha+2k\pi\quad\text{or}\quad x=\pi-\alpha+2k\pi$
Cosine general solutions
$x=\pm\alpha+2k\pi$
Tangent general solution
$x=\alpha+k\pi$
Zero-product rule
$AB=0\implies A=0\text{ or }B=0$
Pythagorean identity
$\sin^2x+\cos^2x=1$
Sine double angle
$\sin2x=2\sin x\cos x$
Cosine double angle
$\cos2x=2\cos^2x-1=1-2\sin^2x$
Intersection method
$f(x)=g(x)\iff f(x)-g(x)=0$
Quadrant-related angles
$\alpha,\quad\pi-\alpha,\quad\pi+\alpha,\quad2\pi-\alpha$
Double-angle identities
$\sin2x=2\sin x\cos x,\quad\cos2x=2\cos^2x-1=1-2\sin^2x$
Period conversion
$2\pi\text{ radians}=360^\circ,\quad\pi\text{ radians}=180^\circ$
Unit 4
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Statistics and Probability
Statistics and probability develop the ability to analyze data, quantify uncertainty and use probability distributions. Technology is important, but interpretation and model choice remain central.
12 topics
Unit 4 progress0%
Topics in this unit
Topic 4.1Not started
Data concepts, sampling and reliability
Learn how data type, sampling design, bias, reliability and validity determine whether a statistical conclusion deserves to be trusted.
150 min estimated
Topic 4.1 progress0%
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Learning objectives
Classify data as categorical or numerical and distinguish nominal, ordinal, discrete and continuous variables.
Distinguish a population, census, sample, sampling frame, parameter and statistic.
Compare primary and secondary data and evaluate the advantages and limitations of each source.
Design and carry out simple random, systematic and stratified samples.
Recognize when convenience or voluntary-response sampling is likely to be biased.
Calculate systematic sampling intervals and proportional stratified allocations.
Distinguish sampling variation from selection, non-response, response and measurement errors.
Evaluate reliability, validity, representativeness and the suitability of data collection methods.
Choose a defensible sampling method for a stated population and research question.
Critically evaluate statistical claims using source, method, sample and conclusion evidence.
Key formulas
Sample proportion
$\hat p=\frac{x}{n}$
Sampling fraction
$f=\frac{n}{N}$
Systematic sampling interval
$k=\frac{N}{n}$
Selected positions
$r,r+k,r+2k,\ldots,r+(n-1)k$
Proportional stratum allocation
$n_h=n\frac{N_h}{N}$
Sampling calculation checklist
$\text{define population}\to\text{choose frame}\to\text{select method}\to\text{evaluate limitations}$
Core sampling formulas
$\hat p=\frac{x}{n},\qquad k=\frac{N}{n},\qquad n_h=n\frac{N_h}{N}$
Systematic interval
$k=\frac{N}{n}$
Stratified allocation
$n_h=n\frac{N_h}{N}$
Topic 4.2Not started
Presentation of data
Learn to organize, construct, interpret and compare statistical displays, from frequency tables and stem-and-leaf diagrams to histograms, cumulative-frequency graphs and box plots.
165 min estimated
Topic 4.2 progress0%
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Learning objectives
Construct and interpret frequency, relative-frequency and cumulative-frequency tables.
Distinguish grouped from ungrouped data and identify class boundaries, widths and midpoints.
Choose appropriately between bar charts, histograms, stem-and-leaf diagrams, cumulative-frequency graphs and box plots.
Construct and interpret histograms, including unequal class widths and frequency density.
Construct and read cumulative-frequency graphs using upper class boundaries.
Estimate medians, quartiles, percentiles and interquartile ranges from cumulative information.
Construct and interpret box-and-whisker plots from a five-number summary.
Use stem-and-leaf diagrams to preserve and compare ordered raw data.
Compare distributions using center, spread, shape and unusual values.
Use a GDC or spreadsheet efficiently while retaining correct scales, labels and statistical interpretation.
Key formulas
Relative frequency
$\text{relative frequency}=\frac{f}{n}$
Percentage frequency
$\text{percentage frequency}=\frac{f}{n}\times100\%$
Cumulative frequency
$F_i=\sum\limits_{j=1}^{i}f_j$
Class width
$w=U-L$
Class midpoint
$m=\frac{L+U}{2}$
Frequency density
$\text{frequency density}=\frac{f}{w}$
Histogram area
$f=(\text{frequency density})(\text{class width})$
Frequency between two values
$f(a<x\le b)\approx F(b)-F(a)$
Quartile levels
$Q_1:\frac{N}{4},\qquad Q_2:\frac{N}{2},\qquad Q_3:\frac{3N}{4}$
Percentile level
$\text{CF level}=\frac{p}{100}N$
Interquartile range
$\operatorname{IQR}=Q_3-Q_1$
Range
$\text{range}=\max-\min$
Outlier fences
$Q_1-1.5\operatorname{IQR},\qquad Q_3+1.5\operatorname{IQR}$
Difference in medians
$\Delta\widetilde{x}=\widetilde{x}_A-\widetilde{x}_B$
Display workflow
$\text{classify data}\to\text{calculate}\to\text{construct}\to\text{interpret}$
Core presentation formulas
$\frac{f}{n},\qquad \frac{f}{w},\qquad \operatorname{IQR}=Q_3-Q_1$
Relative frequency
$\frac{f}{n}$
Frequency density
$\frac{f}{w}$
Quartile levels
$\frac{N}{4},\quad\frac{N}{2},\quad\frac{3N}{4}$
Topic 4.3Not started
Measures of center and spread
Learn to calculate, interpret and choose measures of typical value and variability for raw, frequency and grouped data, including the effects of outliers and linear transformations.
170 min estimated
Topic 4.3 progress0%
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Learning objectives
Calculate and interpret mean, median and mode for ungrouped data.
Calculate weighted means from frequency tables and estimate means from grouped data.
Determine quartiles, percentiles, range and interquartile range using an explicitly stated convention.
Calculate and interpret population and sample variance and standard deviation.
Estimate variance and standard deviation from grouped class midpoints.
Explain how skewness and outliers affect measures of center and spread.
Choose suitable paired summaries such as mean with standard deviation or median with IQR.
Compare distributions quantitatively using center, spread and contextual units.
Determine how adding or multiplying every data value changes center, variance and spread.
Use one-variable GDC statistics correctly and distinguish population from sample output.
Key formulas
Sample mean
$\bar{x}=\frac{1}{n}\sum\limits_{i=1}^{n}x_i$
Zero-sum deviations
$\sum\limits_{i=1}^{n}(x_i-\bar{x})=0$
Odd-n median position
$\frac{n+1}{2}$
Even-n median
$\widetilde{x}=\frac{x_{(\frac{n}{2})}+x_{(\frac{n}{2}+1)}}{2}$
Quartile cumulative levels
$\frac{N}{4},\qquad\frac{N}{2},\qquad\frac{3N}{4}$
Percentile cumulative level
$\frac{p}{100}N$
Range
$\text{range}=x_{\max}-x_{\min}$
Interquartile range
$\operatorname{IQR}=Q_3-Q_1$
Population variance
$\sigma^2=\frac{1}{N}\sum\limits_{i=1}^{N}(x_i-\mu)^2$
Population computational form
$\sigma^2=\frac{1}{N}\sum\limits_{i=1}^{N}x_i^2-\mu^2$
Sample variance
$s^2=\frac{1}{n-1}\sum\limits_{i=1}^{n}(x_i-\bar{x})^2$
Standard deviations
$\sigma=\sqrt{\sigma^2},\qquad s=\sqrt{s^2}$
Frequency mean
$\bar{x}=\frac{\sum\limits_{i=1}^{k}f_ix_i}{\sum\limits_{i=1}^{k}f_i}$
Grouped mean estimate
$\bar{x}\approx\frac{\sum\limits_{i=1}^{k}f_im_i}{\sum\limits_{i=1}^{k}f_i}$
Grouped population variance estimate
$\sigma^2\approx\frac{\sum\limits_{i=1}^{k}f_im_i^2}{\sum\limits_{i=1}^{k}f_i}-\bar{x}^{\,2}$
Transformed mean
$\bar{y}=a\bar{x}+b$
Transformed variance
$\operatorname{Var}(Y)=a^2\operatorname{Var}(X)$
Transformed standard deviation
$\sigma_Y=|a|\sigma_X$
Transformed IQR
$\operatorname{IQR}_Y=|a|\operatorname{IQR}_X$
Common potential-outlier fences
$Q_1-1.5\operatorname{IQR},\qquad Q_3+1.5\operatorname{IQR}$
Exam workflow
$\text{identify data role}\to\text{calculate}\to\text{interpret}\to\text{check units}$
Core center-and-spread formulas
$\bar{x}=\frac{1}{n}\sum\limits_{i=1}^{n}x_i,\qquad \operatorname{IQR}=Q_3-Q_1,\qquad \sigma=\sqrt{\frac{1}{N}\sum\limits_{i=1}^{N}(x_i-\mu)^2}$
Mean
$\bar{x}=\frac{1}{n}\sum\limits_{i=1}^{n}x_i$
IQR
$\operatorname{IQR}=Q_3-Q_1$
Topic 4.4Not started
Bivariate data and linear correlation
Bivariate analysis studies paired values for two variables. Scatter diagrams, Pearson’s correlation coefficient and regression describe linear association, while careful interpretation separates association from causation.
175 min estimated
Topic 4.4 progress0%
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Learning objectives
Recognize bivariate data as paired observations of two variables measured on the same individuals or objects.
Construct and interpret scatter diagrams using appropriate axes, units and scales.
Describe the direction, form and strength of an association and identify unusual observations.
Use technology to calculate Pearson’s product-moment correlation coefficient and interpret its sign and magnitude.
Explain why Pearson’s correlation coefficient measures only linear association.
Assess how outliers and influential points can change a correlation or regression model.
Distinguish correlation from causation and suggest plausible confounding variables or reverse causation.
Draw or interpret a line of best fit that passes through the mean point.
Use the regression line of y on x for interpolation and judge when extrapolation is unreliable.
Use a GDC efficiently and communicate statistical conclusions with variables, units and appropriate precision.
Key formulas
Pearson’s product-moment correlation coefficient
$r=\frac{\displaystyle\sum\limits_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\displaystyle\left[\sum\limits_{i=1}^{n}(x_i-\bar{x})^2\right]\left[\sum\limits_{i=1}^{n}(y_i-\bar{y})^2\right]}}$
Range of r
$-1\le r\le1$
Strength comparison
$|r_1|>|r_2|\quad\Longrightarrow\quad r_1\text{ represents the stronger linear association}$
Mean point
$(\bar{x},\bar{y})=\left(\frac{1}{n}\sum\limits_{i=1}^{n}x_i,\frac{1}{n}\sum\limits_{i=1}^{n}y_i\right)$
Gradient from two points on a line
$m=\frac{y_2-y_1}{x_2-x_1}$
Regression line of y on x
$\hat{y}=ax+b$
Regression coefficients
$a=r\frac{s_y}{s_x},\qquad b=\bar{y}-a\bar{x}$
Residual
$e=y-\hat{y}$
Correlation and regression toolkit
$-1\le r\le1,\qquad \hat{y}=ax+b,\qquad e=y-\hat{y}$
Pearson correlation coefficient
$r=\frac{\displaystyle\sum\limits_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\displaystyle\left[\sum\limits_{i=1}^{n}(x_i-\bar{x})^2\right]\left[\sum\limits_{i=1}^{n}(y_i-\bar{y})^2\right]}}$
Regression and residual
$\hat{y}=ax+b,\qquad e=y-\hat{y}$
Topic 4.5Not started
Probability concepts
Probability provides a mathematical language for uncertainty. Experiments, sample spaces, events, complements and relative frequency connect theoretical models with repeated real-world observations.
165 min estimated
Topic 4.5 progress0%
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Learning objectives
Distinguish experiments, trials and outcomes and construct complete sample spaces.
Represent sample spaces using lists, tables, grids or tree-style organization when appropriate.
Define events as subsets of a sample space and use basic set notation accurately.
Interpret probabilities on the scale from impossible to certain.
Calculate theoretical probabilities when outcomes are equally likely.
Use complementary events to simplify probability calculations.
Recognize mutually exclusive events and apply the appropriate basic addition rule.
Calculate and interpret experimental probability and relative frequency.
Explain long-run stabilization and use simulations to estimate probabilities.
Calculate expected numbers of occurrences and communicate conclusions with appropriate context and precision.
Key formulas
Counting outcomes in successive choices
$n(U)=n_1n_2\cdots n_k$
Event containment
$A\subseteq U$
Probability bounds
$0\le P(A)\le1$
Impossible and certain events
$P(\varnothing)=0,\qquad P(U)=1$
Equally likely outcomes
$P(A)=\frac{n(A)}{n(U)}$
Complement rule
$P(A')=1-P(A)$
Complement total
$P(A)+P(A')=1$
Mutually exclusive condition
$A\cap B=\varnothing\qquad\Longleftrightarrow\qquad P(A\cap B)=0$
Addition rule for mutually exclusive events
$P(A\cup B)=P(A)+P(B)$
Relative frequency
$\widehat{P}(A)=\frac{f_A}{N}$
Expected number of occurrences
$E(\text{number of occurrences of }A)=NP(A)$
Total probability for exhaustive outcomes
$\sum\limits_{i=1}^{k}P(A_i)=1$
Core rules
$P(A')=1-P(A),\qquad P(A\cup B)=P(A)+P(B)\ \text{if }A\cap B=\varnothing$
Foundational probability toolkit
$P(A)=\frac{n(A)}{n(U)},\qquad P(A')=1-P(A),\qquad \widehat{P}(A)=\frac{f_A}{N}$
Probability scale and equally likely outcomes
$0\le P(A)\le1,\qquad P(A)=\frac{n(A)}{n(U)}$
Complement, relative frequency and expectation
$P(A')=1-P(A),\qquad \widehat{P}(A)=\frac{f_A}{N},\qquad E(\text{count})=NP(A)$
Topic 4.6Not started
Probability diagrams and combined events
Probability diagrams organize overlapping and multi-stage events. Venn diagrams, outcome tables and trees make addition, multiplication, conditional probability, independence and replacement effects visible.
190 min estimated
Topic 4.6 progress0%
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Learning objectives
Represent combined events using Venn diagrams, tree diagrams, sample-space diagrams and tables of outcomes.
Translate verbal conditions into union, intersection and complement notation.
Use the general addition rule without double-counting an intersection.
Recognize mutually exclusive events and distinguish them from independent events.
Use tree diagrams to organize sequential probabilities and combine complete paths.
Apply the general multiplication rule using conditional probability.
Calculate and interpret conditional probabilities from data, diagrams or formulas.
Test whether two events are independent using product or conditional-probability conditions.
Update branch probabilities correctly in experiments with and without replacement.
Choose an efficient representation and solve multi-step combined-event problems.
Communicate IB-style solutions with defined events, exact working and contextual conclusions.
Key formulas
Two-set region counts
$n(A\text{ only})=n(A)-n(A\cap B),\qquad n((A\cup B)')=n(U)-n(A\cup B)$
De Morgan’s laws
$(A\cup B)'=A'\cap B',\qquad (A\cap B)'=A'\cup B'$
General addition rule
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$
Mutually exclusive condition
$A\cap B=\varnothing\qquad\Longleftrightarrow\qquad P(A\cap B)=0$
Disjoint addition
$P(A\cup B)=P(A)+P(B)$
Tree-diagram strategy
$P(E)=\sum\limits_{j=1}^{m}\prod_{k=1}^{s_j}p_{jk}$
Conditional probability
$P(A\mid B)=\frac{P(A\cap B)}{P(B)},\qquad P(B)>0$
Conditional probability from counts
$P(A\mid B)=\frac{n(A\cap B)}{n(B)}$
General multiplication rule
$P(A\cap B)=P(A)P(B\mid A)=P(B)P(A\mid B)$
Three-stage path
$P(A\cap B\cap C)=P(A)P(B\mid A)P(C\mid A\cap B)$
Independent-events product condition
$P(A\cap B)=P(A)P(B)$
Equivalent conditional condition
$P(A\mid B)=P(A),\qquad P(B)>0$
Combined-events toolkit
$P(A\cup B)=P(A)+P(B)-P(A\cap B),\qquad P(A\mid B)=\frac{P(A\cap B)}{P(B)}$
Addition and complement
$P(A\cup B)=P(A)+P(B)-P(A\cap B),\qquad P(A')=1-P(A)$
Conditional and multiplication rules
$P(A\mid B)=\frac{P(A\cap B)}{P(B)},\qquad P(A\cap B)=P(A)P(B\mid A)$
Independence test
$A\text{ and }B\text{ independent}\quad\Longleftrightarrow\quad P(A\cap B)=P(A)P(B)$
Topic 4.7Not started
Discrete random variables
Discrete random variables turn countable outcomes into numerical models. Learn to build and validate probability distributions, calculate expectation, variance and standard deviation, and interpret those results in IB-style contexts.
175 min estimated
Topic 4.7 progress0%
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Learning objectives
Define a random variable and distinguish a discrete random variable from a continuous one.
Translate outcomes from an experiment into values of a discrete random variable.
Construct, read and graph discrete probability distributions given by a table, formula or context.
Verify that a proposed probability distribution has valid probabilities that total 1.
Find unknown probabilities using the total-probability condition.
Calculate expected value from a discrete probability distribution.
Interpret expected value as a long-run mean rather than a guaranteed single outcome.
Use expected gain to assess whether a game is fair and to compare decisions.
Calculate and interpret variance and standard deviation for a discrete random variable.
Use a calculator or GDC efficiently while retaining exact structure and sufficient precision.
Present complete IB-style working with notation, units and contextual conclusions.
Key formulas
Valid probability values
$0\le P(X=x_i)\le1\quad\text{for }i=1,2,\ldots,n$
Total probability
$\sum\limits_{i=1}^{n}P(X=x_i)=1$
Normalization equation
$\sum\limits_{i=1}^{n}p_i=1$
Expected value
$E(X)=\mu=\sum\limits_{i=1}^{n}x_iP(X=x_i)$
Fair-game condition
$E(X)=0\quad\text{when }X\text{ is the player's net gain}$
Second moment
$E(X^2)=\sum\limits_{i=1}^{n}x_i^2P(X=x_i)$
Variance
$\operatorname{Var}(X)=\sum\limits_{i=1}^{n}(x_i-\mu)^2P(X=x_i)=E(X^2)-[E(X)]^2$
Standard deviation
$\sigma_X=\sqrt{\operatorname{Var}(X)}$
Expectation under a linear transformation
$E(aX+b)=aE(X)+b$
Variance under a linear transformation
$\operatorname{Var}(aX+b)=a^2\operatorname{Var}(X)$
Standard deviation under a linear transformation
$\sigma_{aX+b}=|a|\sigma_X$
Probability total
$\sum\limits_{i=1}^{n}P(X=x_i)=1$
Expected value
$E(X)=\sum\limits_{i=1}^{n}x_iP(X=x_i)$
Variance shortcut
$\operatorname{Var}(X)=E(X^2)-[E(X)]^2$
Fair-game condition
$E(X)=0\quad\text{for signed net gain }X$
Topic 4.8Not started
Binomial distribution
The binomial distribution models the number of successes in a fixed set of independent trials with two outcomes and constant success probability. Learn to validate the model, calculate exact and cumulative probabilities, and interpret its mean and spread.
180 min estimated
Topic 4.8 progress0%
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Learning objectives
Recognize the four conditions required for a binomial model and explain why each matters.
Define a binomial random variable as the number of successes in a fixed number of trials.
Interpret the parameters n, p and q and use the notation X distributed as B of n and p.
Identify the support of a binomial random variable as the integers from 0 to n.
Calculate an exact binomial probability using technology and understand its probability formula.
Translate verbal probability statements into exact, cumulative and interval notation.
Use complements efficiently for at-least and more-than probabilities.
Calculate and interpret the mean, variance and standard deviation of a binomial distribution.
Use GDC binomial PDF and CDF functions accurately, including boundary adjustments.
Apply binomial models to real-world contexts and identify situations where the assumptions fail.
Present complete IB-style solutions with model conditions, calculator input, accuracy and contextual conclusions.
Key formulas
Binomial notation
$X\sim B(n,p)$
Failure probability
$q=1-p$
Binomial support
$x\in\{0,1,2,\ldots,n\}$
Exact binomial probability
$P(X=x)=\binom{n}{x}p^x(1-p)^{n-x}$
Combination
$\binom{n}{x}=\frac{n!}{x!(n-x)!}$
GDC exact-probability input
$\operatorname{binompdf}(n,p,x)=P(X=x)$
GDC cumulative-probability input
$\operatorname{binomcdf}(n,p,x)=P(X\le x)$
At least r successes
$P(X\ge r)=1-P(X\le r-1)$
More than r successes
$P(X>r)=1-P(X\le r)$
Inclusive interval
$P(a\le X\le b)=P(X\le b)-P(X\le a-1)$
Binomial mean
$E(X)=\mu=np$
Binomial variance
$\operatorname{Var}(X)=\sigma^2=np(1-p)=npq$
Binomial standard deviation
$\sigma=\sqrt{np(1-p)}$
Binomial model
$X\sim B(n,p)$
Exact probability
$P(X=x)=\binom{n}{x}p^x(1-p)^{n-x}$
At least r
$P(X\ge r)=1-P(X\le r-1)$
Mean
$E(X)=np$
Variance
$\operatorname{Var}(X)=np(1-p)$
Standard deviation
$\sigma=\sqrt{np(1-p)}$
Topic 4.9Not started
Normal distribution
The normal distribution models continuous data with a symmetric bell-shaped curve. Students interpret its parameters, calculate areas as probabilities, and use normal and inverse-normal GDC functions.
210 min estimated
Topic 4.9 progress0%
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Learning objectives
Describe the defining properties of a normal distribution and interpret its mean and standard deviation.
Use normal-distribution notation correctly and connect areas under a density curve with probabilities.
Apply symmetry and the 68–95–99.7 empirical rule to estimate and check probabilities.
Use normal CDF technology to calculate lower-tail, upper-tail, and interval probabilities.
Use inverse-normal technology to find values associated with specified cumulative probabilities.
Find unknown boundary values from stated cumulative, upper-tail, and central probabilities.
Judge whether a normal model is reasonable and communicate IB solutions with suitable diagrams, notation, and rounding.
Key formulas
Total probability
$P(-\infty<X<\infty)=1$
Center of a normal distribution
$\text{mean}=\text{median}=\text{mode}=\mu$
Area on either side of the mean
$P(X<\mu)=P(X>\mu)=\frac{1}{2}$
Normal-distribution notation
$X\sim N(\mu,\sigma^2)$
Expected value
$E(X)=\mu$
Variance
$\operatorname{Var}(X)=\sigma^2$
Standard deviation
$\operatorname{SD}(X)=\sigma=\sqrt{\operatorname{Var}(X)}$
Within one standard deviation
$P(\mu-\sigma<X<\mu+\sigma)\approx0.68$
Within two standard deviations
$P(\mu-2\sigma<X<\mu+2\sigma)\approx0.95$
Within three standard deviations
$P(\mu-3\sigma<X<\mu+3\sigma)\approx0.997$
Complement rule
$P(X>a)=1-P(X\le a)$
Interval from cumulative probabilities
$P(a<X<b)=P(X<b)-P(X\le a)$
Probability at one point
$P(X=a)=0$
Mirror-tail symmetry
$P(X<\mu-d)=P(X>\mu+d)$
Each tail outside a central area
$P(X<L)=P(X>U)=\frac{1-c}{2}$
Interval probability on a GDC
$P(a<X<b)=\operatorname{normalcdf}(a,b,\mu,\sigma)$
Lower-tail probability
$P(X<b)=\operatorname{normalcdf}(-\infty,b,\mu,\sigma)$
Upper-tail probability
$P(X>a)=\operatorname{normalcdf}(a,\infty,\mu,\sigma)$
Quantile definition
$P(X\le x_q)=q$
Inverse-normal command
$x_q=\operatorname{invNorm}(q,\mu,\sigma)$
Convert an upper-tail area
$P(X>x)=p\quad\Longrightarrow\quad P(X\le x)=1-p$
Highest p cutoff
$P(X>k)=p\quad\Longrightarrow\quad k=\operatorname{invNorm}(1-p,\mu,\sigma)$
Central interval tail area
$\text{each tail}=\frac{1-c}{2}$
Central interval boundaries
$L=\operatorname{invNorm}\!\left(\frac{1-c}{2},\mu,\sigma\right),\quad U=\operatorname{invNorm}\!\left(\frac{1+c}{2},\mu,\sigma\right)$
Normal density
$f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$
Inflection points
$x=\mu-\sigma\quad\text{and}\quad x=\mu+\sigma$
Expected count
$\text{expected count}=nP(\text{event})$
Normal model
$X\sim N(\mu,\sigma^2)$
Total area
$P(-\infty<X<\infty)=1$
Symmetry
$P(X<\mu-d)=P(X>\mu+d)$
Normal CDF
$P(a<X<b)=\operatorname{normalcdf}(a,b,\mu,\sigma)$
Inverse normal
$P(X\le x_q)=q\quad\Longrightarrow\quad x_q=\operatorname{invNorm}(q,\mu,\sigma)$
Topic 4.10Not started
Regression line of x on y
Regression models a linear relationship for prediction. This topic focuses on choosing the correct response variable, finding the regression line of x on y with technology, and judging whether a prediction is reliable.
195 min estimated
Topic 4.10 progress0%
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Learning objectives
Explain what a regression line represents and distinguish association from an exact functional rule.
Choose between regression of y on x and regression of x on y from the variable being predicted.
Use a GDC to enter paired data and obtain the appropriate linear regression equation.
Interpret the slope and intercept of a regression equation with correct units and context.
Use regression equations for interpolation and identify the additional risk of extrapolation.
Explain how correlation, scatter, outliers, and the data range affect prediction reliability.
Avoid rearranging one regression equation as a substitute for the opposite regression direction.
Present current-syllabus IB solutions with suitable notation, technology evidence, rounding, and contextual conclusions.
Key formulas
Correlation range
$-1\le r\le1$
Regression of y on x
$\hat y=a+bx$
Vertical residual
$e_y=y-\hat y$
Least-squares objective for y on x
$\min\sum\limits_{i=1}^{n}(y_i-\hat y_i)^2$
Regression of x on y
$\hat x=c+dy$
Horizontal residual
$e_x=x-\hat x$
Least-squares objective for x on y
$\min\sum\limits_{i=1}^{n}(x_i-\hat x_i)^2$
Centroid property
$(\bar x,\bar y)\text{ lies on both regression lines}$
Slope of y on x
$b_{yx}=r\frac{s_y}{s_x}$
Slope of x on y
$b_{xy}=r\frac{s_x}{s_y}$
Product of regression slopes
$b_{yx}b_{xy}=r^2$
Slope units
$\text{slope units}=\frac{\text{response units}}{\text{explanatory units}}$
Rearrangement that is not a new regression
$\hat x=c+dy\quad\Longrightarrow\quad y=\frac{\hat x-c}{d}$
Exact-inverse special case
$|r|=1\quad\Longrightarrow\quad b_{yx}b_{xy}=1$
Predict y from x
$\hat y=a+bx$
Predict x from y
$\hat x=c+dy$
Residual directions
$e_y=y-\hat y,\qquad e_x=x-\hat x$
Centroid
$(\bar x,\bar y)\text{ lies on both lines}$
Slope check
$b_{yx}b_{xy}=r^2$
Topic 4.11Not started
Formal conditional probability and independence
Conditional probability restricts attention to outcomes consistent with known information. Independence means that learning one event occurred does not change the probability of the other.
180 min estimated
Topic 4.11 progress0%
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Learning objectives
Interpret conditional probability as a probability calculated within a restricted sample space.
Use the formal definition P(A given B) = P(A intersection B) divided by P(B), with the correct non-zero condition.
Rearrange conditional probability to calculate intersections through the multiplication rule.
Represent conditional and combined events with Venn diagrams, two-way tables, and tree diagrams.
Decide whether two events are independent using conditional probabilities or the product test.
Use complements when testing independence and explain why independence is not the same as mutual exclusivity.
Solve multi-step IB-style probability problems and communicate assumptions, working, and conclusions clearly.
Key formulas
Complement rule
$P(A')=1-P(A)$
General addition rule
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$
Conditional probability
$P(A\mid B)=\frac{P(A\cap B)}{P(B)},\quad P(B)>0$
Conditional probability from counts
$P(A\mid B)=\frac{n(A\cap B)}{n(B)}$
Multiplication rule
$P(A\cap B)=P(B)P(A\mid B)$
Equivalent order
$P(A\cap B)=P(A)P(B\mid A)$
Three-event chain rule
$P(A\cap B\cap C)=P(A)P(B\mid A)P(C\mid A\cap B)$
Complementary branches
$P(A\mid B)+P(A'\mid B)=1$
Conditional independence test
$P(A\mid B)=P(A)$
Reverse conditional test
$P(B\mid A)=P(B)$
Product test for independence
$P(A\cap B)=P(A)P(B)$
IB independence statement
$P(A\mid B)=P(A)=P(A\mid B')$
Complement product
$P(A\cap B')=P(A)P(B')$
Mutual exclusivity
$P(A\cap B)=0$
General multiplication rule
$P(A\cap B)=P(B)P(A\mid B)$
Independent product rule
$P(A\cap B)=P(A)P(B)$
Independent conditional rule
$P(A\mid B)=P(A)=P(A\mid B')$
Mutually exclusive rule
$P(A\cap B)=0$
Topic 4.12Not started
Standardization of normal variables
Standardization converts a raw normal value into a z-score measuring its signed distance from the mean in standard-deviation units. This common scale supports probability calculations and fair comparisons across different distributions.
180 min estimated
Topic 4.12 progress0%
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Learning objectives
Explain standardization as a change of scale that preserves relative position and cumulative probability.
Describe the standard normal distribution and interpret positive, zero, and negative z-scores.
Convert raw values to z-scores using the population mean and standard deviation.
Convert z-scores back to raw values and interpret the result with units and context.
Find left-tail, right-tail, and interval probabilities using standardized values.
Compare observations drawn from normal distributions with different means, standard deviations, or units.
Use normal CDF and inverse-normal GDC functions with correct parameters and bounds.
Solve IB-style problems involving percentiles, unknown cutoffs, means, or standard deviations.
Key formulas
Standard normal notation
$Z\sim N(0,1)$
Standard normal center
$E(Z)=0,\qquad \operatorname{SD}(Z)=1$
Standardization formula
$z=\frac{x-\mu}{\sigma},\qquad \sigma>0$
Random-variable transformation
$Z=\frac{X-\mu}{\sigma}$
Inverse standardization
$x=\mu+z\sigma$
Equal-distance symmetry
$P(Z<-a)=P(Z>a),\qquad a>0$
Standardized distribution
$X\sim N(\mu,\sigma^2)\quad\Longrightarrow\quad Z=\frac{X-\mu}{\sigma}\sim N(0,1)$
Preserved cumulative probability
$P(X\le x)=P\!\left(Z\le\frac{x-\mu}{\sigma}\right)$
Raw interval to standard interval
$P(a<X<b)=P\!\left(\frac{a-\mu}{\sigma}<Z<\frac{b-\mu}{\sigma}\right)$
Right-tail complement
$P(Z>z)=1-P(Z\le z)$
Separate standardizations
$z_A=\frac{x_A-\mu_A}{\sigma_A},\qquad z_B=\frac{x_B-\mu_B}{\sigma_B}$
Percentile conversion
$x_p=\mu+z_p\sigma$
Mean from a standardized cutoff
$\mu=x-z\sigma$
Standard deviation from a cutoff
$\sigma=\frac{x-\mu}{z},\qquad z\ne0$
Raw value to z-score
$z=\frac{x-\mu}{\sigma}$
z-score to raw value
$x=\mu+z\sigma$
Standard normal transformation
$X\sim N(\mu,\sigma^2)\Longrightarrow Z=\frac{X-\mu}{\sigma}\sim N(0,1)$
Matching cumulative areas
$P(X\le x)=P\!\left(Z\le\frac{x-\mu}{\sigma}\right)$
Raw interval conversion
$P(a<X<b)=P\!\left(\frac{a-\mu}{\sigma}<Z<\frac{b-\mu}{\sigma}\right)$
Unit 5
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Calculus
Calculus is a major emphasis of AA SL. The topic connects rates of change, graph behavior, optimization, motion and accumulation through differentiation and integration.
11 topics
Unit 5 progress0%
Topics in this unit
Topic 5.1Not started
Limits, gradients and rates of change
Calculus begins with what a function approaches and how quickly one quantity changes relative to another. Limits connect secant gradients, tangent gradients, difference quotients and derivatives.
240 min estimated
Topic 5.1 progress0%
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Learning objectives
Interpret finite limit notation and distinguish a limiting value from the value of the function at the point.
Evaluate simple limits by direct substitution or algebraic simplification and identify when a two-sided limit does not exist.
Estimate limits and gradients from graphs and numerical tables.
Calculate and interpret average rates of change as secant gradients.
Explain instantaneous rate of change as the gradient of a tangent.
Construct and simplify a difference quotient.
Use the definition of a derivative as a limit in simple first-principles problems.
Interpret derivatives with correct signs, units, technology evidence and real-world context.
Key formulas
Finite limit
$\lim_{x\to a}f(x)=L$
Continuity at a point
$\lim_{x\to a}f(x)=f(a)$
Direct-substitution rule for a continuous function
$\lim_{x\to a}f(x)=f(a)$
Difference-of-squares limit
$\lim_{x\to a}\frac{x^2-a^2}{x-a}=2a$
Condition for a two-sided limit
$\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=L\quad\Longrightarrow\quad\lim_{x\to a}f(x)=L$
Average rate of change
$\frac{f(b)-f(a)}{b-a},\qquad a\ne b$
Secant gradient using an increment
$\frac{f(x+h)-f(x)}{h},\qquad h\ne0$
Instantaneous rate at x
$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$
Difference quotient
$\frac{f(x+h)-f(x)}{h},\qquad h\ne0$
Derivative from first principles
$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$
Derivative at a specific point
$f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$
Equivalent derivative notation
$f'(x)=y'=\frac{dy}{dx}$
Tangent equation at x=a
$y-f(a)=f'(a)(x-a)$
Rate units
$\text{derivative units}=\frac{\text{output units}}{\text{input units}}$
Velocity from displacement
$v(t)=\frac{ds}{dt}$
Centered-difference estimate
$f'(a)\approx\frac{f(a+h)-f(a-h)}{2h}$
Average rate
$\frac{f(b)-f(a)}{b-a}$
Difference quotient
$\frac{f(x+h)-f(x)}{h}$
Derivative
$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$
Tangent line
$y-f(a)=f'(a)(x-a)$
Topic 5.2Not started
Increasing and decreasing functions
The sign of the first derivative reveals where a graph rises or falls. Sign diagrams connect derivative calculations to increasing intervals, decreasing intervals and the classification of stationary points.
220 min estimated
Topic 5.2 progress0%
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Learning objectives
Describe increasing and decreasing behavior from a graph, table or context.
Use $f'(x) > 0$ to identify increasing intervals and $f'(x) < 0$ to identify decreasing intervals.
Find critical values by solving $f'(x) = 0$ and checking points where the derivative is undefined.
Construct and interpret a first-derivative sign diagram.
Classify stationary points as local maxima, local minima or stationary points of inflection.
Interpret the graph of f′ to describe the behavior of f.
Use a GDC to locate stationary points and verify intervals of increase or decrease.
Communicate interval notation, endpoint restrictions and contextual interpretations accurately.
Key formulas
Increasing test
$f'(x)>0$
Decreasing test
$f'(x)<0$
Stationary-value condition
$f'(c)=0$
Critical-value search
$f'(x)=0\quad\text{or}\quad f'(x)\text{ is undefined}$
Local maximum sign change
$f':\ +\longrightarrow-$
Local minimum sign change
$f':\ -\longrightarrow+$
No extremum
$f':\ +\longrightarrow+\quad\text{or}\quad f':\ -\longrightarrow-$
Increasing
$f'(x)>0$
Decreasing
$f'(x)<0$
Stationary value
$f'(c)=0$
Local maximum
$f':\ +\longrightarrow-$
Local minimum
$f':\ -\longrightarrow+$
Topic 5.3Not started
Derivatives of polynomials
Polynomial differentiation turns a function into a gradient function. The power, constant, sum and difference rules make it possible to find tangent gradients, stationary points and contextual rates efficiently.
220 min estimated
Topic 5.3 progress0%
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Learning objectives
Explain differentiation as the process of finding an instantaneous rate-of-change or gradient function.
Use derivative notation accurately and distinguish a derivative function from its value at one input.
Apply the power rule to monomials with nonnegative integer powers.
Differentiate constants, constant multiples, sums and differences.
Differentiate and simplify complete polynomial functions term by term.
Find gradients and equations of tangents at specified points.
Find and classify stationary points of polynomial functions.
Interpret polynomial derivatives in graphical and real-world applications with correct units.
Key formulas
Derivative from first principles
$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$
Equivalent derivative notation
$f'(x)=y'=\frac{dy}{dx}$
Gradient at x=a
$m_{\text{tangent}}=f'(a)$
Power rule
$\frac{d}{dx}\left(x^n\right)=nx^{n-1}$
Power rule with a constant coefficient
$\frac{d}{dx}\left(ax^n\right)=anx^{n-1}$
Derivative of a constant
$\frac{d}{dx}(c)=0$
Constant multiple rule
$\frac{d}{dx}\left[af(x)\right]=af'(x)$
Derivative of a linear term
$\frac{d}{dx}(ax)=a$
Sum rule
$\frac{d}{dx}\left[f(x)+g(x)\right]=f'(x)+g'(x)$
Difference rule
$\frac{d}{dx}\left[f(x)-g(x)\right]=f'(x)-g'(x)$
General polynomial derivative
$\frac{d}{dx}\left(\sum\limits_{k=0}^{n}a_kx^k\right)=\sum\limits_{k=1}^{n}ka_kx^{k-1}$
Gradient at a point
$m=f'(a)$
Inputs with gradient m
$f'(x)=m$
Tangent equation
$y-f(a)=f'(a)(x-a)$
Normal gradient
$m_{\text{normal}}=-\frac{1}{f'(a)}$
Stationary condition
$f'(c)=0$
Stationary point coordinates
$\left(c,f(c)\right)$
Velocity from displacement
$v(t)=s'(t)$
Acceleration from displacement
$a(t)=v'(t)=s''(t)$
Power rule
$\frac{d}{dx}\left(ax^n\right)=anx^{n-1}$
Constant rule
$\frac{d}{dx}(c)=0$
Gradient at x=a
$m=f'(a)$
Topic 5.4Not started
Tangents and normals
The derivative gives the tangent gradient at a point on a curve. A normal passes through the same point perpendicular to the tangent, linking calculus, coordinate geometry and gradient conditions.
220 min estimated
Topic 5.4 progress0%
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Learning objectives
Interpret the derivative as the gradient of a tangent to a curve.
Find the point of contact and tangent gradient at a specified input.
Write tangent equations using point-slope form.
Explain a normal as the line perpendicular to the tangent at the contact point.
Use the negative-reciprocal relationship between finite nonzero tangent and normal gradients.
Find equations of normals, including horizontal and vertical special cases.
Find unknown contact points from tangent or normal gradient conditions.
Use algebra and a GDC to solve multi-step tangent and normal problems accurately.
Key formulas
Tangent gradient
$m_{\text{tangent}}=f'(a)$
Point-slope form
$y-y_1=m(x-x_1)$
Tangent at x=a
$y-f(a)=f'(a)(x-a)$
Perpendicular-gradient condition
$m_1m_2=-1$
Normal gradient
$m_{\text{normal}}=-\frac{1}{f'(a)}$
Normal at x=a
$y-f(a)=-\frac{1}{f'(a)}(x-a)$
Given tangent gradient
$f'(x)=m$
Given normal gradient
$f'(x)=-\frac{1}{n}$
Horizontal tangent
$f'(a)=0\quad\Longrightarrow\quad y=f(a)$
Corresponding vertical normal
$x=a$
Parallel-gradient condition
$m_1=m_2$
Tangent approximation
$f(x)\approx f(a)+f'(a)(x-a)$
Tangent equation
$y-f(a)=f'(a)(x-a)$
Normal equation
$y-f(a)=-\frac{1}{f'(a)}(x-a)$
Perpendicular gradients
$m_{\text{tangent}}m_{\text{normal}}=-1$
Topic 5.5Not started
Introduction to integration
Integration reverses differentiation, reconstructs functions from their rates of change and measures accumulated change. Students learn the power rule, the role of the constant of integration and how initial conditions select one antiderivative.
220 min estimated
Topic 5.5 progress0%
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Learning objectives
Explain integration as reverse differentiation and distinguish an antiderivative from a derivative.
Interpret the notation $\int f(x)\,dx$ , identifying the integrand, variable of integration and constant of integration.
Use the power rule to integrate powers and polynomial expressions, including constant multiples, sums and differences.
Explain why every indefinite integral requires $+C$ and describe the resulting family of antiderivatives.
Find a particular function from its derivative and a boundary or initial condition.
Check an antiderivative by differentiating the result.
Apply elementary integration to motion, marginal-rate and accumulation contexts.
Use a GDC appropriately to check graphs and evaluate simple definite integrals while showing required analytical working.
Key formulas
Antiderivative condition
$F'(x)=f(x)$
Reverse relationship
$\frac{d}{dx}\left(\int f(x)\,dx\right)=f(x)$
Indefinite integral notation
$\int f(x)\,dx=F(x)+C\quad\text{when }F'(x)=f(x)$
Definite integral notation
$\int_a^b f(x)\,dx$
Integration power rule
$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C,\qquad n\neq-1$
Constant multiple rule
$\int kf(x)\,dx=k\int f(x)\,dx$
Sum and difference rule
$\int\bigl(f(x)\pm g(x)\bigr)\,dx=\int f(x)\,dx\pm\int g(x)\,dx$
Integral of a constant
$\int k\,dx=kx+C$
Family of antiderivatives
$F(x)+C,\qquad C\in\mathbb{R}$
Polynomial rule
$\int\left(\sum\limits_{k=0}^{m}a_kx^k\right)dx=\sum\limits_{k=0}^{m}\frac{a_k}{k+1}x^{k+1}+C$
Condition from a point
$F(a)=b\quad\text{for a point }(a,b)$
Verification statement
$\frac{d}{dx}\bigl(F(x)+C\bigr)=f(x)$
Initial-value pattern
$y'=f(x),\ y(a)=b\quad\Longrightarrow\quad y=\int f(x)\,dx\text{ and then use }y(a)=b$
Fundamental Theorem of Calculus
$\int_a^b f(x)\,dx=\bigl[F(x)\bigr]_a^b=F(b)-F(a)$
Motion hierarchy
$a(t)=v'(t),\qquad v(t)=s'(t)$
Displacement from velocity
$s(t)=\int v(t)\,dt$
Antiderivative definition
$F'(x)=f(x)\quad\Longleftrightarrow\quad\int f(x)\,dx=F(x)+C$
Power rule
$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C,\quad n\neq-1$
Constant rule
$\int k\,dx=kx+C$
Linearity
$\int(af(x)+bg(x))\,dx=a\int f(x)\,dx+b\int g(x)\,dx$
Definite integral
$\int_a^b f(x)\,dx=F(b)-F(a)$
Topic 5.6Not started
Derivatives of elementary and composite functions
Extend differentiation from polynomials to power, exponential, logarithmic and trigonometric functions. Use the chain, product and quotient rules to differentiate composite and combined functions accurately.
260 min estimated
Topic 5.6 progress0%
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Learning objectives
Differentiate positive, negative and rational powers on intervals where the functions are defined.
Differentiate natural and general exponential functions.
Differentiate natural and general logarithmic functions while respecting domain restrictions.
Differentiate sine, cosine and tangent functions in radians.
Identify inner and outer functions in a composite expression.
Apply the chain rule to powers, exponentials, logarithms and trigonometric composites.
Use constant multiple, sum, product and quotient rules accurately.
Apply derivative rules to gradients, tangents, stationary points and contextual rates of change.
Use a GDC to evaluate and check derivatives without replacing required analytical working.
Key formulas
Power rule
$\frac{d}{dx}(x^n)=nx^{n-1}$
Constant multiple power
$\frac{d}{dx}(kx^n)=knx^{n-1}$
Natural exponential derivative
$\frac{d}{dx}(e^x)=e^x$
General exponential derivative
$\frac{d}{dx}(a^x)=a^x\ln a,\qquad a>0,\ a\neq1$
Natural logarithm derivative
$\frac{d}{dx}(\ln x)=\frac{1}{x},\qquad x>0$
General logarithm derivative
$\frac{d}{dx}(\log_a x)=\frac{1}{x\ln a},\qquad x>0$
Derivative of sine
$\frac{d}{dx}(\sin x)=\cos x$
Derivative of cosine
$\frac{d}{dx}(\cos x)=-\sin x$
Derivative of tangent
$\frac{d}{dx}(\tan x)=\sec^2x$
Composite notation
$y=f(g(x))=(f\circ g)(x)$
Chain rule
$\frac{d}{dx}\bigl(f(g(x))\bigr)=f'(g(x))g'(x)$
Leibniz chain notation
$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$
Composite power rule
$\frac{d}{dx}\bigl([g(x)]^n\bigr)=n[g(x)]^{n-1}g'(x)$
Exponential chain rule
$\frac{d}{dx}\bigl(e^{g(x)}\bigr)=e^{g(x)}g'(x)$
General exponential chain rule
$\frac{d}{dx}\bigl(a^{g(x)}\bigr)=a^{g(x)}\ln(a)\,g'(x)$
Logarithmic chain rule
$\frac{d}{dx}\bigl(\ln(g(x))\bigr)=\frac{g'(x)}{g(x)}$
Sine chain rule
$\frac{d}{dx}\bigl(\sin(g(x))\bigr)=\cos(g(x))g'(x)$
Cosine chain rule
$\frac{d}{dx}\bigl(\cos(g(x))\bigr)=-\sin(g(x))g'(x)$
Tangent chain rule
$\frac{d}{dx}\bigl(\tan(g(x))\bigr)=\sec^2(g(x))g'(x)$
Linearity of differentiation
$\frac{d}{dx}\bigl(af(x)+bg(x)\bigr)=af'(x)+bg'(x)$
Product rule
$\frac{d}{dx}(uv)=u'v+uv'$
Quotient rule
$\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{u'v-uv'}{v^2},\qquad v\neq0$
Tangent gradient
$m=f'(a)$
Stationary-point condition
$f'(x)=0$
Natural exponential
$\frac{d}{dx}(e^x)=e^x$
Natural logarithm
$\frac{d}{dx}(\ln x)=\frac{1}{x}$
Trigonometric derivatives
$(\sin x)'=\cos x,\quad(\cos x)'=-\sin x,\quad(\tan x)'=\sec^2x$
Chain rule
$\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$
Product rule
$(uv)'=u'v+uv'$
Quotient rule
$\left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^2}$
Topic 5.7Not started
Second derivative
The second derivative measures how the first derivative changes. Use it to interpret concavity, relate the graphs of a function and its derivatives, classify stationary points and model acceleration.
230 min estimated
Topic 5.7 progress0%
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Learning objectives
Explain the second derivative as the derivative of the first derivative.
Use $f''(x)$ , $y''$ , $\frac{d^2y}{dx^2}$ and related notation correctly.
Find second derivatives of polynomial, exponential, logarithmic, trigonometric and composite functions.
Interpret the sign and value of the second derivative as information about changing gradient and concavity.
Relate the behavior and graphs of $f$ , $f'$ and $f''$ .
Use the second-derivative test to classify stationary points when it is conclusive.
Explain why $f''(a)=0$ is inconclusive and test for a genuine change of concavity.
Interpret the second derivative as acceleration in motion contexts.
Use a GDC to evaluate and check second derivatives while showing required analytical working.
Key formulas
Second derivative definition
$f''(x)=\frac{d}{dx}\bigl(f'(x)\bigr)$
Equivalent notation
$f''(x)=y''=\frac{d^2y}{dx^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)$
Successive derivative process
$f(x)\xrightarrow{\frac{d}{dx}}f'(x)\xrightarrow{\frac{d}{dx}}f''(x)$
Exponential pattern
$\frac{d^2}{dx^2}\bigl(e^{kx}\bigr)=k^2e^{kx}$
Sine pattern
$\frac{d^2}{dx^2}\bigl(\sin(kx)\bigr)=-k^2\sin(kx)$
Cosine pattern
$\frac{d^2}{dx^2}\bigl(\cos(kx)\bigr)=-k^2\cos(kx)$
Concavity conditions
$f''(x)>0\Rightarrow\text{concave up},\qquad f''(x)<0\Rightarrow\text{concave down}$
Derivative hierarchy
$f\xrightarrow{\text{gradient}}f'\xrightarrow{\text{gradient}}f''$
Example derivatives
$f(x)=x^3-3x,\qquad f'(x)=3x^2-3,\qquad f''(x)=6x$
Second-derivative test
$f'(a)=0,\quad f''(a)>0\Rightarrow\text{local minimum},\quad f''(a)<0\Rightarrow\text{local maximum}$
Inflection requirement
$f''\text{ changes sign at }x=a$
Motion derivative hierarchy
$v(t)=s'(t),\qquad a(t)=v'(t)=s''(t)$
Second derivative
$f''(x)=\frac{d}{dx}\bigl(f'(x)\bigr)=\frac{d^2y}{dx^2}$
Concavity
$f''>0\Rightarrow\text{concave up},\qquad f''<0\Rightarrow\text{concave down}$
Stationary-point test
$f'(a)=0:\quad f''(a)>0\Rightarrow\min,\quad f''(a)<0\Rightarrow\max$
Motion hierarchy
$v=s',\qquad a=v'=s''$
Topic 5.8Not started
Local maxima, minima and points of inflection
Use first and second derivatives to locate and classify stationary points, analyze concavity, identify genuine points of inflection and solve optimization problems within their allowed domains.
260 min estimated
Topic 5.8 progress0%
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Learning objectives
Find stationary points by solving $f'(x)=0$ and calculate their coordinates.
Distinguish local extrema from global extrema and from stationary points that are not extrema.
Classify stationary points using the first-derivative sign test.
Classify stationary points using the second-derivative test when it is conclusive.
Use the sign of $f''$ to determine concavity.
Identify and justify stationary and non-stationary points of inflection.
Relate key features of the graphs of $f$ , $f'$ and $f''$ .
Check endpoints when finding absolute extrema on a closed interval.
Build and solve one-variable optimization models with appropriate constraints.
Use a GDC to locate and verify graph features while communicating analytical reasoning.
Key formulas
Stationary-point condition
$f'(a)=0$
Local minimum test
$f'(a)=0,\quad f''(a)>0$
Local maximum test
$f'(a)=0,\quad f''(a)<0$
Concave up
$f''(x)>0$
Concave down
$f''(x)<0$
Inflection inputs for the visual
$x=\pm\sqrt{\frac{2}{3}}$
Inflection coordinates
$\left(\pm\sqrt{\frac{2}{3}},-\frac{20}{9}\right)$
Stationary condition
$f'(a)=0$
First-derivative maximum pattern
$f':\quad +\;\to\;-$
First-derivative minimum pattern
$f':\quad -\;\to\;+$
Second-derivative minimum test
$f'(a)=0,\quad f''(a)>0$
Second-derivative maximum test
$f'(a)=0,\quad f''(a)<0$
Inflection requirement
$f''\text{ changes sign at }x=a$
Topic 5.9Not started
Kinematics
Model one-dimensional motion by connecting displacement, velocity and acceleration through differentiation and integration, interpreting signs and turning times, and distinguishing net displacement from total distance travelled.
280 min estimated
Topic 5.9 progress0%
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Learning objectives
Define displacement, velocity, acceleration and distance with correct signs and units.
Use $v(t)=s'(t)$ and $a(t)=v'(t)=s''(t)$ to differentiate motion functions.
Integrate acceleration and velocity and use initial conditions to determine constants.
Find when a particle is at rest and determine whether it changes direction.
Use velocity signs to identify positive and negative directions of motion.
Find and classify maximum and minimum displacement values.
Calculate displacement over an interval using positions or a definite integral of velocity.
Calculate total distance travelled by splitting at every change of direction or integrating $|v|$ .
Interpret displacement–time, velocity–time and acceleration–time graphs using gradients and signed areas.
Use a GDC to solve motion equations and evaluate derivatives or integrals while showing analytical reasoning.
Key formulas
Velocity from displacement
$v(t)=\frac{ds}{dt}=s'(t)$
Acceleration from velocity
$a(t)=\frac{dv}{dt}=v'(t)=s''(t)$
Average velocity
$\bar v=\frac{s(t_2)-s(t_1)}{t_2-t_1}$
Speed
$\text{speed}=|v(t)|$
Velocity change
$v(t_2)-v(t_1)=\int_{t_1}^{t_2}a(t)\,dt$
Displacement change
$s(t_2)-s(t_1)=\int_{t_1}^{t_2}v(t)\,dt$
Displacement maximum test
$v(r)=0,\quad a(r)<0$
Displacement minimum test
$v(r)=0,\quad a(r)>0$
Net displacement
$\Delta s=s(t_2)-s(t_1)$
Total distance
$D=\int_{t_1}^{t_2}|v(t)|\,dt$
Derivative chain
$s(t)\xrightarrow{\frac{d}{dt}}v(t)\xrightarrow{\frac{d}{dt}}a(t)$
Total distance on an interval
$D=\int_{t_1}^{t_2}|v(t)|\,dt$
Velocity
$v(t)=s'(t)=\frac{ds}{dt}$
Acceleration
$a(t)=v'(t)=s''(t)=\frac{d^2s}{dt^2}$
Topic 5.10Not started
Indefinite integrals and reverse chain rule
Build families of antiderivatives, apply core integration rules, recognize reverse-chain patterns, determine constants from conditions, and verify every result by differentiation.
300 min estimated
Topic 5.10 progress0%
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Learning objectives
Explain an indefinite integral as a family of antiderivatives.
Use the constant of integration correctly and determine it from an initial condition.
Apply constant, sum, difference and power rules for integration.
Recall standard antiderivatives for reciprocal, exponential and trigonometric functions.
Recognize an inner function and its derivative inside a composite integrand.
Apply the reverse chain rule to powers, logarithmic, exponential and trigonometric forms.
Adjust for a missing constant factor without changing the integrand.
Distinguish reverse-chain patterns from expressions that require another method.
Check an indefinite integral by differentiating the proposed antiderivative.
Communicate IB-style working with exact notation, restrictions and a final constant.
Key formulas
Indefinite integral definition
$\int f(x)\,dx=F(x)+C\quad\text{when }F'(x)=f(x)$
Power rule
$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C,\quad n\ne-1$
Constant multiple
$\int kf(x)\,dx=k\int f(x)\,dx$
Sum and difference
$\int(f(x)\pm g(x))\,dx=\int f(x)\,dx\pm\int g(x)\,dx$
Integral of a constant
$\int k\,dx=kx+C$
Reciprocal
$\int\frac{1}{x}\,dx=\ln|x|+C$
Natural exponential
$\int e^x\,dx=e^x+C$
Cosine
$\int\cos x\,dx=\sin x+C$
Sine
$\int\sin x\,dx=-\cos x+C$
Secant squared
$\int\sec^2x\,dx=\tan x+C$
General reverse-chain form
$\int g'(x)f(g(x))\,dx=F(g(x))+C\quad\text{where }F'(u)=f(u)$
Composite power
$\int g'(x)[g(x)]^n\,dx=\frac{[g(x)]^{n+1}}{n+1}+C,\quad n\ne-1$
Linear-composite power
$\int(ax+b)^n\,dx=\frac{(ax+b)^{n+1}}{a(n+1)}+C,\quad a\ne0,\ n\ne-1$
Logarithmic reverse chain
$\int\frac{g'(x)}{g(x)}\,dx=\ln|g(x)|+C$
Natural-exponential reverse chain
$\int g'(x)e^{g(x)}\,dx=e^{g(x)}+C$
Linear exponential
$\int e^{ax+b}\,dx=\frac{1}{a}e^{ax+b}+C,\quad a\ne0$
Cosine reverse chain
$\int g'(x)\cos(g(x))\,dx=\sin(g(x))+C$
Sine reverse chain
$\int g'(x)\sin(g(x))\,dx=-\cos(g(x))+C$
Secant-squared reverse chain
$\int g'(x)\sec^2(g(x))\,dx=\tan(g(x))+C$
Verification principle
$\frac{d}{dx}\left(\int f(x)\,dx\right)=f(x)$
Composite power
$\int g'(x)[g(x)]^n\,dx=\frac{[g(x)]^{n+1}}{n+1}+C$
Logarithmic pattern
$\int\frac{g'(x)}{g(x)}\,dx=\ln|g(x)|+C$
Exponential pattern
$\int g'(x)e^{g(x)}\,dx=e^{g(x)}+C$
Cosine pattern
$\int g'(x)\cos(g(x))\,dx=\sin(g(x))+C$
Sine pattern
$\int g'(x)\sin(g(x))\,dx=-\cos(g(x))+C$
Topic 5.11Not started
Definite integrals and areas between curves
Evaluate definite integrals as signed accumulation, distinguish signed integral from geometric area, locate intersections, and split regions so every required area is counted positively.
300 min estimated
Topic 5.11 progress0%
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Learning objectives
Explain a definite integral as signed accumulation over an interval.
Identify and interpret lower and upper limits of integration.
Evaluate definite integrals using an antiderivative and the fundamental theorem of calculus.
Distinguish signed area from geometric area.
Find areas between a curve and the x-axis, including regions below the axis.
Find total area by splitting at every x-intercept.
Solve for curve intersections that define integration limits.
Find area between two curves using upper function minus lower function.
Split an area calculation whenever the sign or upper curve changes.
Use a GDC to find intersections and evaluate definite integrals while communicating IB-style reasoning.
Key formulas
Signed-area interpretation
$\int_a^b f(x)\,dx=\text{area above the }x\text{-axis}-\text{area below the }x\text{-axis}$
Fundamental theorem of calculus
$\int_a^b f(x)\,dx=[F(x)]_a^b=F(b)-F(a)\quad\text{where }F'(x)=f(x)$
Reverse limits
$\int_b^a f(x)\,dx=-\int_a^b f(x)\,dx$
Equal limits
$\int_a^a f(x)\,dx=0$
Split an interval
$\int_a^b f(x)\,dx=\int_a^c f(x)\,dx+\int_c^b f(x)\,dx$
Area above the x-axis
$A=\int_a^b f(x)\,dx\quad\text{when }f(x)\ge0$
Area below the x-axis
$A=-\int_a^b f(x)\,dx=\int_a^b[-f(x)]\,dx\quad\text{when }f(x)\le0$
Total area
$A_{\mathrm{total}}=\int_a^b|f(x)|\,dx$
Intersection condition
$f(x)=g(x)$
Area between two curves
$A=\int_a^b[\text{upper}(x)-\text{lower}(x)]\,dx$
Fundamental theorem
$\int_a^b f(x)\,dx=F(b)-F(a)$
Split interval
$\int_a^b f(x)\,dx=\int_a^c f(x)\,dx+\int_c^b f(x)\,dx$
Area above axis
$A=\int_a^b f(x)\,dx\quad\text{if }f(x)\ge0$
Area below axis
$A=-\int_a^b f(x)\,dx\quad\text{if }f(x)\le0$
Area between curves
$A=\int_a^b[\text{upper}(x)-\text{lower}(x)]\,dx$

Mathematics: Analysis and Approaches

Course Overview

Continue where you left off

Course units

Number and Algebra

Topics in this unit

Scientific notation and operations with numbers

Learning objectives

Key formulas

Arithmetic sequences and series

Learning objectives

Key formulas

Geometric sequences and series

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Key formulas

Financial applications

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Key formulas

Integer exponents and introduction to logarithms

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Key formulas

Simple deductive proof

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Key formulas

Rational exponents and logarithm laws

Learning objectives

Key formulas

Infinite geometric series

Learning objectives

Key formulas

Binomial theorem

Learning objectives

Key formulas

Functions

Topics in this unit

Equations of Straight Lines

Learning objectives

Key formulas

The Concept of a Function

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Key formulas

Graphs of Functions

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Key formulas

Key Features of Graphs

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Key formulas

Composite and Inverse Functions

Learning objectives

Key formulas

Quadratic Functions

Learning objectives

Key formulas

Quadratic Equations and Inequalities

Learning objectives

Key formulas

Reciprocal and Rational Functions

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Key formulas

Exponential and logarithmic functions

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Key formulas

Solving equations graphically and analytically

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Key formulas

Transformations of graphs

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Key formulas

Geometry and Trigonometry

Topics in this unit

Three-dimensional geometry

Learning objectives

Key formulas

Trigonometry in triangles

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Key formulas

Applications of trigonometry

Learning objectives

Key formulas

Radian measure, arcs and sectors