Interdisciplinary Foundations

Determine whether a relation defines a function under specified domains and codomains.

Determine the domain and range of a function from its formula or graph.

Evaluate and graph piecewise-defined and absolute-value functions.

Form sums, differences, products, and quotients of functions and determine their domains.

Compose two functions and determine the domain of the composition.

Classify a function as injective, surjective, bijective, or none of these.

Find an inverse function after making any necessary domain restriction.

Apply translations, reflections, and scalings to construct a transformed function graph.

Determine parameter conditions for a quadratic equation to have specified types of roots.

Represent complex numbers on the Argand plane and calculate their sums, products, quotients, and moduli.

Solve quadratic equations with complex roots.

Solve polynomial and rational inequalities using a sign chart and represent the solution on a number line.

Determine terms and finite sums of arithmetic sequences from supplied conditions.

Determine terms and finite or convergent infinite sums of geometric sequences.

Translate finite sums into sigma notation and apply standard summation rules.

Construct and use a geometric-sequence model for compound growth or decay.

Convert angles between degrees and radians and determine coterminal and reference angles.

Compute exact trigonometric values using the unit circle, quadrant signs, and symmetry.

Sketch one period of the sine, cosine, and tangent functions and identify each function’s domain, range, and period.

Graph transformations of the form a·sin(bx+c)+d and a·cos(bx+c)+d.

Simplify trigonometric expressions using reciprocal, quotient, and Pythagorean identities.

Apply compound-angle identities to calculate exact values and simplify expressions.

Apply double-angle and half-angle identities to solve trigonometric problems.

Solve trigonometric equations on a specified interval and express their general solutions.

Calculate the slope and inclination of a line from geometric or coordinate information.

Construct an equation of a line from specified points, slopes, intercepts, or angle conditions.

Determine whether two lines are parallel or perpendicular and calculate their point of intersection.

Calculate the angle between two lines from their slopes.

Calculate the perpendicular distance from a point to a line.

Solve locus problems whose conditions produce a line or circle.

Calculate distances and internal section points in three-dimensional coordinates.

Select an efficient line equation form to solve a multi-constraint coordinate problem.

Derive the standard equation of a parabola from its focus-and-directrix definition.

Determine the vertex, focus, axis, directrix, and latus rectum of a parabola from its equation.

Derive a standard equation of an ellipse from its focal definition.

Determine the centre, vertices, foci, axes, and eccentricity of an ellipse from its equation.

Derive a standard equation of a hyperbola from its focal definition.

Determine the centre, vertices, foci, asymptotes, and eccentricity of a hyperbola from its equation.

Classify a standard-form equation as a circle, parabola, ellipse, or hyperbola and extract its defining features.

Construct the equation of a conic from a given set of geometric conditions.

Estimate a function’s limit at a point from a numerical table or graph.

Determine whether a two-sided limit exists by comparing the corresponding one-sided limits.

Evaluate limits by direct substitution and algebraic limit laws.

Evaluate indeterminate rational limits by factoring and cancellation.

Evaluate limits containing radicals by rationalisation.

Evaluate trigonometric limits using the standard limit of sin(x)/x as x approaches zero.

Evaluate polynomial and rational limits as the variable grows without bound.

Diagnose limit failure caused by a jump, unbounded behaviour, or oscillation.

Estimate and interpret a derivative from a graph or table of values.

Compute the derivative of a simple polynomial from first principles.

Differentiate polynomial combinations using the power, constant-multiple, and sum rules.

Differentiate products and quotients of functions.

Differentiate composite functions using the chain rule.

Differentiate expressions involving sine and cosine.

Construct the equation of a tangent line using a derivative.

Solve a basic single-variable optimisation problem using critical points.

Identify sampling, nonresponse, and measurement bias in a described data collection process.

Select an appropriate graph for a variable and detect misleading choices of scale or presentation.

Choose between the mean and median for a dataset based on its distribution and outliers.

Calculate and interpret the range, interquartile range, variance, and standard deviation of a dataset.

Compare two distributions using their centre, spread, skewness, and outliers.

Calculate and interpret standardised scores to compare observations from different distributions.

Evaluate whether a verbal claim is supported by a supplied set of descriptive statistics.

Design a simple data collection plan that aligns a population, sample, variable, and research question.

Represent a random experiment with a sample space and events.

Calculate equally likely probabilities using permutations and combinations.

Apply complements, unions, intersections, and the addition rule to event probabilities.

Calculate and interpret a conditional probability.

Test whether two events are independent using probability data.

Update a probability using Bayes’ rule in a two-stage scenario.

Construct a probability distribution for a finite discrete random variable.

Calculate the expected value and variance of a discrete random variable and use them to compare choices.

Construct and interpret a scatterplot for two quantitative variables.

Calculate and interpret Pearson’s correlation coefficient for a small dataset.

Assess how an outlier or linear transformation affects a correlation coefficient.

Distinguish evidence of association from evidence of causation in an observational claim.

Fit a simple least-squares regression line from summary statistics.

Interpret a regression slope, intercept, and residual in the context of a problem.

Calculate simple and weighted price index numbers from economic data.

Evaluate how the base year and weighting scheme affect an index-number comparison.

Distinguish a population parameter from a sample statistic and identify sources of sampling variability.

Calculate probabilities using a binomial probability model.

Calculate standard normal probabilities and quantiles using z-scores.

Describe the sampling distribution of a sample mean under repeated random sampling.

Calculate and interpret the standard error of a sample mean.

Construct a confidence interval for a population mean from sample summary statistics.

Interpret a confidence level without treating it as the probability that a fixed parameter lies in one computed interval.

Perform and interpret a one-sample hypothesis test for a population mean in a large-sample setting.