Discovering Fixed Points and Cycles from a Composition Graph
MEMBERLogin required
Worked Examples
Learn TMUA problem-solving techniques through carefully selected worked examples organised by Concept Bridge topic.
Availability follows each Topic, Worked Example, and Practice access level. Full solutions require membership unless marked public.
Reading and Iterating Functions from Graphs
1 Worked Examples
MEMBERLogin requiredView Topic
Geometric Series and Scaling
1 Worked Examples
MEMBERLogin requiredView Topic
Recovering a Missing Geometric Block
MEMBERLogin required
Choosing a Useful Form of the Expansion
1 Worked Examples
MEMBERLogin requiredView Topic
Rewriting Before Expanding
MEMBERLogin required
Following Function Relationships
Follow the information contained in an indirect function relationship without unnecessarily finding the whole function.
1 Worked Examples
PUBLICView Topic
Beyond the Discriminant
1 Worked Examples
PUBLICView Topic
How Turning Points Control the Number of Solutions
MEMBERLogin required
Modulus and Symmetry in Graphs
1 Worked Examples
PUBLICView Topic
When Does the Reflected Graph Exist?
MEMBERLogin required
Optimising Geometric Distances
1 Worked Examples
MEMBERLogin requiredView Topic
Reducing a Circle-to-Curve Distance
MEMBERLogin required
Relating Coefficients and Term Positions
1 Worked Examples
MEMBERLogin requiredView Topic
Using Term Position and Coefficient Ratios
MEMBERLogin required
Manipulating Sums and Indices
1 Worked Examples
MEMBERLogin requiredView Topic
Cancellation After Rewriting the Summand
MEMBERLogin required
Divisibility in Binomial Expansions
1 Worked Examples
MEMBERLogin requiredView Topic
Counting Coefficients Divisible by 72
PREMIUMLogin required
Building New Graphs from a Known Graph
1 Worked Examples
MEMBERLogin requiredView Topic
Building a Graph Through Two Successive Operations
MEMBERLogin required
Optimising with a Substitution
1 Worked Examples
MEMBERLogin requiredView Topic
Finding the Range After a Substitution
MEMBERLogin required
Functional Equations
Use relationships between function values to create new equations and extract the information you need.
1 Worked Examples
MEMBERLogin requiredView Topic
Choosing Useful Substitutions in a Functional Equation
PREMIUMLogin required
Finding a Required Term or Coefficient
1 Worked Examples
MEMBERLogin requiredView Topic
Finding a Coefficient Without Full Expansion
MEMBERLogin required
Using Substitution to Find Coefficient Sums
1 Worked Examples
MEMBERLogin requiredView Topic
Finding an Even-Power Coefficient Sum Without Expanding
MEMBERLogin required
Index Shifts and Subsequence Structure
1 Worked Examples
MEMBERLogin requiredView Topic
Matching Terms at Different Positions
MEMBERLogin required
Composing Graph Transformations
1 Worked Examples
PUBLICView Topic
Composing Transformations of an Already Transformed Function
MEMBERLogin required
Partial Sums and Blocks
1 Worked Examples
MEMBERLogin requiredView Topic
Comparing Equal-Length Blocks
MEMBERLogin required
Optimising with Restricted Inputs
1 Worked Examples
PUBLICView Topic
Combining a Hidden Restriction with Stationary Points
MEMBERLogin required
Building an Expression to Optimise
1 Worked Examples
MEMBERLogin requiredView Topic
Building an Area Expression from Symmetry
MEMBERLogin required
Beyond Polynomial Functions
1 Worked Examples
MEMBERLogin requiredView Topic
Finding the Critical Level of a Modulus Graph
MEMBERLogin required
Following Repeated Relationships
Follow a relationship through several steps to reveal cycles, fixed behaviour and repeating structure.
1 Worked Examples
MEMBERLogin requiredView Topic
Closing a Cycle of Inputs and Function Values
MEMBERLogin required
Working Backwards Through a Function
Decide when information about an output can be traced backwards to determine the input.
1 Worked Examples
MEMBERLogin requiredView Topic
Recovering an Input from Indirect Information
MEMBERLogin required
When a Function Loses Information
Recognise when several inputs can produce the same output and determine what information remains when a relationship is reversed.
1 Worked Examples
MEMBERLogin requiredView Topic
When Equal Outputs Do Not Give Equal Inputs
MEMBERLogin required
Recurrence Relations
Use relationships between successive terms to uncover patterns, cycles and structures without calculating every term.
1 Worked Examples
MEMBERLogin requiredView Topic
Using Consecutive Terms to Determine the Rule
MEMBERLogin required
From Critical Points to Edge Cases
1 Worked Examples
MEMBERLogin requiredView Topic
Finding Boundaries Before Solving
MEMBERLogin required
Functions Defined by Rules
Recognise the different ways unfamiliar functions and mathematical rules can be defined in TMUA problems.
3 Worked Examples
MEMBERLogin requiredView Topic
Reading a Defined Output as an Input Interval
MEMBERLogin required
Understanding Indexed Functions
MEMBERLogin required
Recognising Structure in a Defined Rule
PREMIUMLogin required
Counting Solutions in Repeating Patterns
Use periodicity and interval restrictions to count repeated solutions efficiently.
1 Worked Examples
MEMBERLogin requiredView Topic
Using One Period to Count Solutions
MEMBERLogin required
When the Domain Changes the Answer
Understand how intervals and domain restrictions can change the number of valid solutions.
1 Worked Examples
PREMIUMLogin requiredView Topic
Counting Solutions on a Restricted Domain
PREMIUMLogin required
Counting Solutions of Composite Functions
1 Worked Examples
MEMBERLogin requiredView Topic
Counting Solutions Using Two Functions
MEMBERLogin required