Grade 12

Use first- and second-derivative sign analysis to find critical numbers, determine intervals of increase and decrease, classify local extrema, and identify concavity and points of inflection. Use derivative information to write tangent and normal line equations and sketch standard polynomial, rational, exponential, logarithmic, and trigonometric curves using intercepts, domain restrictions, end behavior, and asymptotes where appropriate. Model and solve single-variable optimization and related-rates problems by applying derivative tests, endpoint analysis, implicit differentiation, constraints, feasible domains, units, and contextual interpretation.

Systematically count finite outcomes using organized lists, tables, and tree diagrams, applying the fundamental counting principle, addition and multiplication principles, permutations with or without repetition, and combinations while distinguishing ordered arrangements from unordered selections. Apply the binomial theorem for nonnegative integer powers, expanding $(a+b)^n$, identifying terms and coefficients, and interpreting binomial coefficients as combination counts. Calculate probabilities in equally likely finite sample spaces using counting techniques, conditional probabilities, and complementary events, and determine whether events are independent or mutually exclusive using multiplication and conditional-probability rules.

Students determine derivatives from first principles using the difference quotient and apply linearity, power, trigonometric, exponential, and logarithmic differentiation rules to familiar functions, including appropriate domains and radians for trigonometric functions. They differentiate products, quotients, and composite functions using the product, quotient, and chain rules, including standard algebraic, trigonometric, exponential, and logarithmic combinations. They compute second, third, and higher-order derivatives when patterns are clear, interpreting the second derivative through concavity and acceleration while excluding multivariable, inverse, hyperbolic, variable-base, and advanced generalized differentiation.

Apply exponent and logarithm laws to simplify, expand, and condense expressions, including zero, negative, and rational exponents where defined, and convert accurately between exponential and logarithmic forms while respecting domain restrictions. Graph and interpret transformed exponential and logarithmic functions, identifying growth or decay, asymptotes, domain, range, intercepts, transformations, and parameters in models such as $y=ab^x$, $y=Ae^{kx}$, and $y=a+b\log_c(x-h)$. Model exponential growth and decay and solve exponential equations using common bases or logarithms, as well as logarithmic equations, verifying solutions and rejecting those that violate positivity or introduce extraneous values.

Apply compound-interest models with varied compounding periods, convert among periodic, nominal, and effective annual rates, and determine present and future values for lump sums and regular payment streams. Calculate ordinary and due annuities, model regular deposits, determine fixed loan and mortgage payments, and create amortization schedules showing interest, principal reduction, remaining balances, total repayment, and total interest. Compare investment and borrowing options and evaluate long-term financial plans by aligning rates, cash-flow timing, fees, inflation, taxes, and time horizons using present value, future value, effective rates, and total costs.

Determine the real domain and range of functions, zeros and $x$- and $y$-intercepts, intervals of positivity and negativity, vertical and horizontal asymptotes, and end behavior for common polynomial, rational, exponential, and logarithmic functions. Interpret graphs, tables, formulas, and first-derivative information to identify intervals of increase and decrease, local and absolute extrema, intervals of concavity, and points of inflection over the stated domain. Sketch coherent function graphs by integrating these characteristics, including symmetry, discontinuities, periodicity where applicable, and qualitative behavior while distinguishing exact features from approximations.

Students describe, represent, and graph reflections across the coordinate axes, vertical and horizontal stretches or compressions, and translations using $g(x)=a f(b(x-h))+k$, interpreting changes to key points, domain, range, and other features across equations, graphs, tables, and verbal descriptions. They form, evaluate, and interpret composite functions from equations, tables, graphs, and transformation rules, recognizing that order matters and determining composite domains by enforcing restrictions on both successive evaluations. They determine and verify inverse functions for one-to-one real-valued functions by interchanging inputs and outputs, applying necessary domain restrictions, using composition to check results, and relating inverse graphs through reflection across $y=x$ rather than reciprocal operations.

Represent lines in two- and three-dimensional coordinate geometry using vector, parametric, and symmetric equations, and represent planes in scalar form $ax+by+cz=d$ using normal vectors obtained from coefficients or cross products. Determine intersections by solving linear systems, including the point of intersection or parallel/coincident cases for two-dimensional lines, and intersecting, parallel, coincident, or skew relationships among three-dimensional lines and planes, with plane intersections expressed parametrically when appropriate. Calculate point-to-point, point-to-line, point-to-plane, and parallel-line or parallel-plane distances as minimum Euclidean lengths, identifying zero distance for intersecting objects and common perpendiculars for parallel or skew objects.

Construct and interpret matrices from data, identify their dimensions, rows, columns, and elements, test matrix equality and multiplication compatibility, and perform addition, subtraction, scalar multiplication, and transposition using their properties. Multiply matrices and vectors, determine product dimensions, apply multiplication properties and the identity matrix, and use matrices to model, combine, and interpret transformations and systems of linear equations. Calculate 2-by-2 and 3-by-3 determinants using minors and cofactors, determine invertibility, find and verify inverses using the adjugate, and solve matrix equations and linear systems with inverse matrices.

Students factor higher-degree real-coefficient polynomials using common factors, grouping, identities, substitution, the Rational Root Theorem, the Factor Theorem, and polynomial division, and solve equations for real and nonreal complex roots, including repeated roots and multiplicities. They perform long division and synthetic division by linear divisors, using placeholders for missing powers, and apply the Remainder Theorem to evaluate polynomials and connect remainders, factors, and zeros. They determine rational-function restrictions, holes, vertical, horizontal, and slant asymptotes, solve rational equations and inequalities through domain and sign analysis, and sketch polynomial functions using zeros, multiplicities, degree, leading coefficient, intercepts, and end behavior.

Define discrete random variables and construct or verify valid probability distributions by listing possible values and assigning nonnegative probabilities that sum to 1. Calculate and interpret expected value, variance, and standard deviation for finite discrete distributions, including their meanings, units, and implications for typical outcomes and variability. Model repeated independent trials with binomial distributions, calculate exact and range probabilities, and use distributions and summary measures to compare risks, costs, payoffs, and fairness.

Calculate and interpret average rates of change as secant slopes, and estimate instantaneous rates of change numerically and graphically as tangent slopes with appropriate signs and units. Evaluate finite and infinite one-sided and two-sided limits from graphs, tables, and algebraic expressions using substitution, factoring, canceling, or rationalizing, distinguishing limiting behavior from the value at the point. Identify removable, jump, and infinite discontinuities and relate the limit of difference quotients to instantaneous rate of change and the derivative when it exists, without formal $\varepsilon$-$\delta$ proofs or advanced generalizations.

Use normal distributions to calculate and interpret probabilities, percentiles, and z-scores, compare relative positions across distributions, and recognize that probabilities represent areas under a continuous curve. Construct and interpret single-predictor linear regression models using slope, intercept, residuals, and $R^2$, and interpret correlation coefficients in context while distinguishing association from causation. Evaluate sampling methods, sources of bias, confidence intervals, margins of error, significance, variability, and statistical claims, distinguishing statistical significance from practical importance and recognizing the limits of generalization.

Use the unit circle to determine exact trigonometric values at quadrantal angles and angles with reference angles $\pi/6$, $\pi/4$, and $\pi/3$, apply reciprocal, quotient, Pythagorean, and compound-angle identities, and verify identities while respecting signs, domains, and undefined values. Transform and graph sine, cosine, and tangent functions, interpreting amplitude, period, phase shift, vertical displacement, midline, asymptotes, key features, and standard real-valued models of periodic phenomena. Solve trigonometric equations over specified degree or radian intervals using algebraic manipulation, identities, factoring, substitution, periodicity, quadrant relationships, inverse-trigonometric values, and verification of all solutions.

Students represent vectors geometrically and in Cartesian components in two dimensions, and in three dimensions where appropriate, determining their magnitude, direction, and unit vector and multiplying them by real scalars. They add and subtract vectors, resolve them into perpendicular components, and use the dot product to determine alignment, perpendicularity, and the angle between nonzero vectors. They apply these skills to two-dimensional force, displacement, velocity, and acceleration problems, interpreting resultant vectors and recognizing equilibrium when the net force is zero.