The modules, one by one
Each area below lists the concepts named in the course document, what the TASC exam asks of them, and the mistake that most often costs marks.
- Function notation, domains and ranges
- Polynomial factorisation, binomial expansion and graph features
- Exponential and logarithmic functions and modelling
- Transformations, composites, inverses and piecewise functions
- Radians, the unit circle and exact values
- Trigonometric identities and symmetry
- Circular graphs, transformations and equations
- Periodic modelling and interpretation
- Limits, first principles and differentiability
- Product, quotient and chain rules
- Tangents, normals and derivative graphs
- Stationary points, optimisation and motion
- Antiderivatives and boundary conditions
- Definite integrals and the fundamental theorem
- Areas under and between curves
- Recovering functions and modelling displacement
- Discrete random variables, expectation and variance
- Binomial distributions and modelling assumptions
- Normal probabilities, quantiles and inverse parameters
- Sample proportions and confidence intervals
Area 1 of 20
Function notation, domains and ranges
A function includes its domain. Find every algebraic and contextual input restriction, then determine outputs actually attained, checking vertices, endpoints and excluded values.
What the course document lists under this area · 6 points
- A function is a rule with an input set
- Finding a maximal real domain
- Range is an output set, not another domain
- Reading domain and range from a graph
- Domains in practical models
- Communicating sets and checking conclusions
What the exam asks
External Criterion 4: functions, algebra and graphs, assessed in both calculator-free Section A and calculator-permitted Section B. Apply the specific knowledge in this note with working and justified conclusions; internal communication, organisation and tool-use requirements remain separate.
Where marks go missing
Treating f(a+1) as f(a)+1.
Area 2 of 20
Polynomial factorisation, binomial expansion and graph features
Use factors to reveal zeros and multiplicities, expanded form to reveal coefficients, and derivative signs when stationary-point classification is required. Check reconstructed expressions against every condition.
What the course document lists under this area · 6 points
- Factors connect algebra to intercepts
- Factor and remainder reasoning
- Multiplicity and sign charts
- Using the binomial theorem efficiently
- Building a graph from algebra
- Parameters and constructing polynomial models
What the exam asks
External Criterion 4: functions, algebra and graphs, assessed in both calculator-free Section A and calculator-permitted Section B. Apply the specific knowledge in this note with working and justified conclusions; internal communication, organisation and tool-use requirements remain separate.
Where marks go missing
Reading x+a as a root at a rather than −a.
Area 3 of 20
Exponential and logarithmic functions and modelling
Exponentials describe multiplicative change; logarithms recover exponents. Preserve domain restrictions, isolate the exponential component, and distinguish limiting levels from attained outputs.
What the course document lists under this area · 6 points
- Exponential change and the role of the base
- Logarithms reverse exponential rules
- Solving equations with logarithms
- Graphs, asymptotes and transformations
- Estimating an exponential model from data
- Decay toward a baseline and threshold times
What the exam asks
External Criterion 4: functions, algebra and graphs, assessed in both calculator-free Section A and calculator-permitted Section B. Apply the specific knowledge in this note with working and justified conclusions; internal communication, organisation and tool-use requirements remain separate.
Where marks go missing
Using logarithm laws on a sum.
Area 4 of 20
Transformations, composites, inverses and piecewise functions
Track coordinates through transformations, check both stages of a composite, and exchange domain and range when inverting. Preserve excluded inputs and distinguish matching heights from matching gradients.
What the course document lists under this area · 6 points
- Tracking points through transformations
- Composing functions and checking existence
- Finding an inverse with a restricted domain
- Reciprocal functions and inherited holes
- Piecewise definitions and boundary values
- Combining domains, inverses and transformed rules
What the exam asks
External Criterion 4: functions, algebra and graphs, assessed in both calculator-free Section A and calculator-permitted Section B. Apply the specific knowledge in this note with working and justified conclusions; internal communication, organisation and tool-use requirements remain separate.
Where marks go missing
Treating a horizontal dilation as multiplication of x-coordinates by the internal coefficient.
Area 5 of 20
Radians, the unit circle and exact values
Use radians consistently, derive exact magnitudes from special triangles, and use the unit circle for signs and periodicity. Quadrant information resolves square-root choices in identity calculations.
What the course document lists under this area · 6 points
- Radians measure angle through arc length
- The unit circle defines sine and cosine
- Deriving the standard exact values
- Reference angles and coterminal rotations
- Using the Pythagorean identity with quadrant data
- Exact evaluation, notation and meaningful checks
What the exam asks
External Criterion 5: circular functions, assessed in both calculator-free Section A and calculator-permitted Section B. Apply the specific knowledge in this note with working and justified conclusions; internal communication, organisation and tool-use requirements remain separate.
Where marks go missing
Substituting degrees into an arc-length formula requiring radians.
Area 6 of 20
Trigonometric identities and symmetry
Derive signs from unit-circle symmetry, use identities within their domains, and preserve excluded and zero cases when simplifying or solving.
What the course document lists under this area · 6 points
- An identity holds across its domain
- Reflection and rotation identities
- Quarter-turn relationships
- Simplifying expressions without losing restrictions
- Using identities to solve equations
- Checking identities and detecting false rules
What the exam asks
External Criterion 5: circular functions, assessed in both calculator-free Section A and calculator-permitted Section B. Apply the specific knowledge in this note with working and justified conclusions; internal communication, organisation and tool-use requirements remain separate.
Where marks go missing
Treating an equation as true for every angle.
Area 7 of 20
Circular graphs, transformations and equations
Use amplitude, period and phase to sketch circular functions, transform the angle domain before solving, and justify that every branch and allowed endpoint has been considered.
What the course document lists under this area · 6 points
- Reading the parameters of sine and cosine
- Tangent graphs and asymptotes
- Solving basic circular equations on an interval
- Transforming the domain before solving
- Equations with both sine and cosine
- Checking graphs and numerical roots
What the exam asks
External Criterion 5: circular functions, assessed in both calculator-free Section A and calculator-permitted Section B. Apply the specific knowledge in this note with working and justified conclusions; internal communication, organisation and tool-use requirements remain separate.
Where marks go missing
Calling a negative multiplier a negative amplitude.
Area 8 of 20
Periodic modelling and interpretation
Build amplitude and midline from extrema, determine the true cycle interval, anchor the phase to an event, and turn threshold roots into contextual time intervals.
What the course document lists under this area · 6 points
- Choosing a sinusoidal model
- Determining phase from an event
- Recovering a period from incomplete cycle information
- Solving a threshold and finding a duration
- Comparing changed periodic conditions
- Interpreting a model within its limits
What the exam asks
External Criterion 5: circular functions, assessed in both calculator-free Section A and calculator-permitted Section B. Apply the specific knowledge in this note with working and justified conclusions; internal communication, organisation and tool-use requirements remain separate.
Where marks go missing
Using the maximum as the amplitude.
Area 9 of 20
Limits, first principles and differentiability
Derive rates by simplifying a nonzero-step difference quotient before taking its limit. Check both sides at joins and distinguish function values, slopes and stationary-point classifications.
What the course document lists under this area · 6 points
- Average and instantaneous rates
- What a limit does and does not say
- Deriving the quadratic derivative from first principles
- Cubic first principles and interpreting the result
- Continuity is necessary but not sufficient
- Derivative graphs and local behaviour
What the exam asks
External Criterion 6: differential calculus, assessed in both calculator-free Section A and calculator-permitted Section B. Apply the specific knowledge in this note with working and justified conclusions; internal communication, organisation and tool-use requirements remain separate.
Where marks go missing
Using output divided by input as the instantaneous rate.
Area 10 of 20
Product, quotient and chain rules
Identify the outer operation, apply its rule, and differentiate each nested piece with its own chain factors. Preserve domains and check representative slopes before interpreting tangents or stationary points.
What the course document lists under this area · 6 points
- Elementary derivatives and linear combinations
- The chain rule follows nested dependence
- Products require two contributions
- Quotients and the order of subtraction
- Combining rules in a structured calculation
- Verification and using a derivative result
What the exam asks
External Criterion 6: differential calculus, assessed in both calculator-free Section A and calculator-permitted Section B. Apply the specific knowledge in this note with working and justified conclusions; internal communication, organisation and tool-use requirements remain separate.
Where marks go missing
Multiplying factor derivatives instead of using the product rule.
Area 11 of 20
Tangents, normals and derivative graphs
Use the original function for contact points and its derivative for directions. Handle vertical normals explicitly, solve for unknown contact inputs, and distinguish derivative slopes from original heights.
What the course document lists under this area · 6 points
- A tangent needs a point and a gradient
- Normals and exceptional gradients
- Finding where a tangent has a specified direction
- Tangents through an external point
- Constructing a derivative graph from known slopes
- Using a tangent as a local approximation
What the exam asks
External Criterion 6: differential calculus, assessed in both calculator-free Section A and calculator-permitted Section B. Apply the specific knowledge in this note with working and justified conclusions; internal communication, organisation and tool-use requirements remain separate.
Where marks go missing
Using f′(a) as the contact height.
Area 12 of 20
Stationary points, optimisation and motion
Classify stationary candidates with local signs, compare all feasible candidates for global extrema, and interpret motion through velocity signs rather than position alone.
What the course document lists under this area · 6 points
- Stationary candidates and sign changes
- Local extrema versus interval extrema
- Building an optimisation function from constraints
- Optimising a volume with a meaningful domain
- Position, velocity and acceleration
- Distance, displacement and a complete motion account
What the exam asks
External Criterion 6: differential calculus, assessed in both calculator-free Section A and calculator-permitted Section B. Apply the specific knowledge in this note with working and justified conclusions; internal communication, organisation and tool-use requirements remain separate.
Where marks go missing
Classifying every derivative zero as a maximum or minimum.
Area 13 of 20
Antiderivatives and boundary conditions
Integrate to a family, apply every supplied condition to the correct function, and differentiate the completed answer as a check. Preserve domain restrictions and distinguish position from change in position.
What the course document lists under this area · 6 points
- Antidifferentiation recovers a family of functions
- The power rule and its exception
- Linear inner functions and reverse chain factors
- Determining a constant from a point
- Recovering unknown parameters with multiple conditions
- Reconstructing position from velocity
What the exam asks
External Criterion 7: integral calculus, assessed in both calculator-free Section A and calculator-permitted Section B. Apply the specific knowledge in this note with working and justified conclusions; internal communication, organisation and tool-use requirements remain separate.
Where marks go missing
Omitting the constant from an indefinite integral.
Area 14 of 20
Definite integrals and the fundamental theorem
Evaluate signed accumulation through antiderivative endpoint differences, preserve orientation, and use additivity or supplied derivative relationships before attempting unnecessary algebra.
What the course document lists under this area · 6 points
- Accumulation and signed contributions
- Evaluating with an antiderivative
- Additivity, reversed limits and constant factors
- Accumulation functions and changing upper limits
- Solving for an unknown boundary or parameter
- Using a supplied derivative relationship
What the exam asks
External Criterion 7: integral calculus, assessed in both calculator-free Section A and calculator-permitted Section B. Apply the specific knowledge in this note with working and justified conclusions; internal communication, organisation and tool-use requirements remain separate.
Where marks go missing
Calling every definite integral a positive area.
Area 15 of 20
Areas under and between curves
Locate boundaries and crossings first, integrate non-negative vertical separations piece by piece, and check both the geometry and the units before reporting total area.
What the course document lists under this area · 6 points
- Geometric area and signed integral
- Area below one positive curve
- Area between two curves uses their vertical difference
- Splitting where the upper curve changes
- Areas with a parameter
- Area models, units and final verification
What the exam asks
External Criterion 7: integral calculus, assessed in both calculator-free Section A and calculator-permitted Section B. Apply the specific knowledge in this note with working and justified conclusions; internal communication, organisation and tool-use requirements remain separate.
Where marks go missing
Taking the absolute value of a cancelled whole integral as total area.
Area 16 of 20
Recovering functions and modelling displacement
Recover changes through definite integrals, attach initial levels for positions or amounts, and split signed motion or piecewise rates where interpretation requires it.
What the course document lists under this area · 6 points
- A rate determines change before it determines level
- Recovering a curve from tangent information
- Displacement from a velocity function
- Reading displacement from a velocity graph
- Piecewise rates and continuity of accumulated quantity
- Average rate and checking a reconstructed model
What the exam asks
External Criterion 7: integral calculus, assessed in both calculator-free Section A and calculator-permitted Section B. Apply the specific knowledge in this note with working and justified conclusions; internal communication, organisation and tool-use requirements remain separate.
Where marks go missing
Confusing a change with a final level.
Area 17 of 20
Discrete random variables, expectation and variance
Build a valid probability table, calculate weighted moments accurately, and distinguish long-run averages from individual outcomes and event probabilities.
What the course document lists under this area · 6 points
- A random variable assigns numbers to outcomes
- Completing and using a probability table
- Expected value is a weighted long-run mean
- Variance measures spread around the mean
- Comparing and transforming scores
- Interpreting distributions and avoiding overclaims
What the exam asks
External Criterion 8: probability and statistics, assessed in both calculator-free Section A and calculator-permitted Section B. Apply the specific knowledge in this note with working and justified conclusions; internal communication, organisation and tool-use requirements remain separate.
Where marks go missing
Assigning equal probabilities just because values are distinct.
Area 18 of 20
Binomial distributions and modelling assumptions
Check the trial assumptions, define the count event precisely, and use complements and integer boundary checks to make binomial calculations complete and interpretable.
What the course document lists under this area · 6 points
- Recognising a binomial experiment
- Exact counts and the binomial formula
- Cumulative events and complements
- Mean, variance and the shape of a count
- Solving for trial numbers or probabilities
- Checking independence and interpreting expected counts
What the exam asks
External Criterion 8: probability and statistics, assessed in both calculator-free Section A and calculator-permitted Section B. Apply the specific knowledge in this note with working and justified conclusions; internal communication, organisation and tool-use requirements remain separate.
Where marks go missing
Using a binomial model without fixed trials or constant independent probabilities.
Area 19 of 20
Normal probabilities, quantiles and inverse parameters
Use density areas for continuous probabilities, standardise with the correct spread, and translate tail statements carefully before cumulative or inverse-normal calculations.
What the course document lists under this area · 6 points
- A continuous normal model
- Standardising and interpreting z-scores
- Calculating interval and tail probabilities
- Quantiles and inverse normal calculations
- Finding an unknown mean or standard deviation
- Continuous density and model checks
What the exam asks
External Criterion 8: probability and statistics, assessed in both calculator-free Section A and calculator-permitted Section B. Apply the specific knowledge in this note with working and justified conclusions; internal communication, organisation and tool-use requirements remain separate.
Where marks go missing
Entering variance where technology asks for standard deviation.
Area 20 of 20
Sample proportions and confidence intervals
Distinguish the unknown population parameter from the sample estimate, use the appropriate sampling spread, and interpret confidence as procedural coverage while checking sample quality and approximation conditions.
What the course document lists under this area · 6 points
- Population parameters and sample statistics
- The sampling distribution of a proportion
- Probabilities for sample proportions
- Constructing a confidence interval
- Interpreting confidence correctly
- Confidence, precision and sample size
What the exam asks
External Criterion 8: probability and statistics, assessed in both calculator-free Section A and calculator-permitted Section B. Apply the specific knowledge in this note with working and justified conclusions; internal communication, organisation and tool-use requirements remain separate.
Where marks go missing
Treating the sample proportion as the known population proportion.