Mathematical Foundations II

The course covers quadratics, including solving quadratic equations using factoring, completing the square, the quadratic formula, analyzing the discriminant, applying relationships between roots and coefficients, graphing and transforming quadratic functions, determining vertices, axes of symmetry, roots, domains, and ranges, solving nonlinear systems, finding intersections between lines and parabolas, and modeling vertical motion.

Higher-degree polynomial topics include analyzing polynomial structure, factoring higher-order expressions, solving cubic and higher-degree polynomial equations, performing synthetic and long division, applying the factor, remainder, and rational roots theorems, determining root multiplicities, and graphing polynomial functions using their roots, degree, leading coefficient, and end behavior.

The study of number systems includes distinguishing rational and irrational numbers, working with imaginary and complex numbers, performing arithmetic with complex numbers, solving quadratic equations with complex roots, using the cyclic properties of the imaginary unit, and representing complex numbers geometrically in the complex plane using magnitude and argument.

In functions, students work with function arithmetic and composition, analyze local extrema, identify one-to-one and invertible functions, calculate inverses, interpret inverse graphs, study periodic and even or odd functions, investigate unbounded behavior and asymptotes, and transform graphs using translations, stretches, compressions, and reflections.

The course covers exponentials and logarithms, along with radical and rational functions. Students convert between exponential and logarithmic forms, apply logarithm laws, solve exponential and logarithmic equations, analyze transformed exponential and logarithmic graphs, simplify rational expressions, solve rational equations, identify rational-function roots and asymptotes, simplify radicals, solve radical equations, and analyze transformed square root and cube root functions.

Work with sequences includes defining recursive and explicit sequences, working with arithmetic and geometric sequences, using sigma notation, and analyzing finite sums. In geometry, students apply coordinate methods, distance and midpoint formulas, geometric transformations, congruence, similarity, and three-dimensional measurements involving polyhedra, cylinders, spheres, and other solids.

Trigonometry topics include solving right triangles using trigonometric ratios, applying the law of sines and law of cosines, working with radian measure, using the unit circle to extend trigonometry beyond acute angles, evaluating special trigonometric ratios, graphing all six trigonometric functions, analyzing their properties, and transforming trigonometric graphs. Students also study vectors, including vector addition, scalar multiplication, magnitudes, unit vectors, linear combinations, and two-dimensional Cartesian representations.

The course introduces the concepts that underpin calculus through limits and continuity. Students estimate limits from graphs, work with one-sided limits and limits at infinity, apply algebraic limit laws, evaluate limits involving different function families, and determine continuity at points and over intervals.

In introductory calculus, students connect average rates of change to instantaneous rates of change, define derivatives, apply differentiation rules to algebraic, exponential, logarithmic, and trigonometric functions, find tangent and normal lines, calculate higher-order derivatives, differentiate composite functions using the chain rule, and find elementary antiderivatives.

Statistics, probability, and combinatorics are also covered. Students calculate measures of centrality and spread, analyze covariance and z-scores, interpret scatter plots, trend lines, correlation, regression, and residuals, and develop probability reasoning using sample spaces, events, complements, unions, intersections, counting principles, permutations, and combinations.

Quadratics

• Identifying quadratic equations and writing them in standard form.
• Solving perfect square quadratic equations.
• Applying the zero product rule.
• Solving quadratic equations using differences of squares and factoring.
• Completing the square in quadratic expressions and equations.
• Solving quadratic equations by completing the square.
• Applying the quadratic formula.
• Using the discriminant to determine the number and type of solutions.
• Determining unknown parameters in quadratic equations.
• Equating coefficients in polynomial identities.
• Applying Vieta's formulas to relationships between roots and coefficients.
• Graphing and transforming quadratic functions.
• Determining vertices, axes of symmetry, roots, and intercepts of parabolas.
• Using the average of the roots to find an axis of symmetry.
• Writing quadratic functions in vertex form.
• Determining domains and ranges of quadratic functions.
• Solving systems involving quadratic equations and functions.
• Finding intersections of lines and parabolas.
• Determining inverses of quadratic functions on suitable domains.
• Modeling downward vertical motion using quadratic functions.

Polynomials

• Applying closure properties and degree rules for polynomials.
• Finding least common multiples of monomials and polynomials.
• Determining roots of polynomial equations.
• Solving higher-order polynomial equations using greatest common factors and grouping.
• Factoring polynomials using differences of squares, sums and differences of cubes, and biquadratic forms.
• Using synthetic division to divide polynomials.
• Dividing polynomials using long division.
• Manipulating rational expressions arising from polynomial division.
• Applying the factor theorem to determine factors and unknown coefficients.
• Applying the remainder theorem.
• Using the rational roots theorem to identify possible rational roots.
• Factoring cubic and quartic polynomials using known factors or roots.
• Determining multiplicities of polynomial roots.
• Graphing elementary cubic functions and their transformations.
• Analyzing polynomial end behavior.
• Graphing cubic and higher-degree polynomials using roots, multiplicities, and leading coefficients.

Number Systems

• Classifying rational, irrational, and real numbers.
• Reasoning about sums and products of rational and irrational numbers.
• Evaluating square roots of negative numbers.
• Solving quadratic equations with purely imaginary solutions.
• Identifying the real and imaginary parts of complex numbers.
• Adding, subtracting, and multiplying complex numbers.
• Solving quadratic equations with complex roots.
• Evaluating powers of the imaginary unit using its cyclic property.
• Representing complex numbers in the complex plane.
• Calculating the magnitude of a complex number.
• Determining the argument of a complex number.

Functions

• Adding, subtracting, multiplying, and dividing functions.
• Evaluating and constructing composite functions.
• Solving equations involving function composition.
• Identifying local extrema and distinguishing local from global extrema.
• Determining whether a function is one-to-one.
• Evaluating inverse functions using diagrams, tables, graphs, and definitions.
• Calculating inverses of functions.
• Determining domains and ranges of inverse functions.
• Identifying intervals on which functions are invertible.
• Interpreting and graphing inverse functions.
• Plotting relationships in which x is expressed as a function of y.
• Analyzing periodic functions.
• Classifying even and odd functions.
• Identifying unbounded behavior and vertical asymptotes.
• Transforming functions using horizontal and vertical translations.
• Stretching, compressing, and reflecting functions.
• Combining multiple graph transformations.
• Constructing transformed function expressions.
• Determining points, extrema, domains, and ranges on transformed curves.

Exponentials & Logarithms

• Converting between exponential and logarithmic forms.
• Evaluating logarithms using their definitions and exponent rules.
• Evaluating common and natural logarithms.
• Simplifying logarithmic expressions.
• Applying the product, quotient, and power rules for logarithms.
• Combining logarithm laws to rewrite expressions.
• Using the change-of-base formula.
• Solving exponential equations using logarithms.
• Solving equations involving the natural exponential function.
• Solving exponential equations with different bases.
• Solving exponential equations using factoring and the zero-product property.
• Solving logarithmic equations using algebraic methods and logarithm laws.
• Graphing and transforming exponential growth and decay functions.
• Determining properties of transformed exponential functions.
• Finding inverses of exponential and logarithmic functions.
• Graphing and transforming logarithmic functions.
• Determining domains, ranges, roots, and asymptotes of transformed logarithmic functions.

Radical & Rational Functions

• Simplifying rational expressions using polynomial factorization.
• Adding and subtracting rational expressions.
• Multiplying and dividing rational expressions.
• Expressing rational functions as sums of partial fractions.
• Solving rational equations with multiple terms.
• Graphing reciprocal functions.
• Transforming reciprocal functions using translations, stretches, and reflections.
• Determining domains and ranges of transformed reciprocal functions.
• Finding inverses of reciprocal functions.
• Finding intersections of lines and reciprocal functions.
• Determining roots of rational functions.
• Identifying vertical and horizontal asymptotes of rational functions.
• Rationalizing algebraic denominators containing one or two terms.
• Solving radical equations.
• Finding inverses of radical functions.
• Graphing square root and cube root functions.
• Transforming radical functions.
• Determining domains, ranges, and roots of transformed radical functions.

Sequences

• Identifying and evaluating terms of sequences.
• Defining and evaluating recursive sequences.
• Analyzing Fibonacci sequences.
• Using sigma notation to express finite sums.
• Applying properties of finite series.
• Identifying arithmetic sequences and calculating common differences.
• Writing recursive and explicit formulas for arithmetic sequences.
• Translating between recursive and explicit formulas.
• Determining terms and indexes in arithmetic sequences.
• Identifying geometric sequences and calculating common ratios.
• Writing recursive and explicit formulas for geometric sequences.
• Determining terms and indexes in geometric sequences.

Geometry

• Calculating equations of parallel and perpendicular lines.
• Finding midpoints and distances in the coordinate plane.
• Calculating the shortest distance between a point and a line.
• Applying the distance formula in three dimensions.
• Translating, rotating, reflecting, dilating, and stretching geometric figures.
• Representing transformations using coordinate rules.
• Combining geometric transformations.
• Connecting rigid motions to congruence.
• Analyzing similar polygons and similarity transformations.
• Identifying reflective and rotational symmetry.
• Identifying three-dimensional shapes and their properties.
• Counting faces, vertices, and edges of polyhedra.
• Using nets to identify polyhedra and calculate surface areas.
• Calculating volumes and surface areas of cubes, rectangular solids, cylinders, and spheres.
• Applying Euler's formula for polyhedra.
• Identifying the five Platonic solids.

Trigonometry

• Identifying opposite, adjacent, and hypotenuse sides in right triangles.
• Evaluating sine, cosine, tangent, and reciprocal trigonometric ratios.
• Calculating side lengths and angle measures using right-triangle trigonometry.
• Applying trigonometric relationships involving complementary angles.
• Evaluating special trigonometric ratios.
• Using trigonometry to calculate areas of right triangles.
• Converting between degrees and radians.
• Applying the law of sines and law of cosines.
• Calculating the area of a general triangle.
• Representing positive and negative angles in the coordinate plane.
• Identifying coterminal and reference angles.
• Using the unit circle to extend trigonometric ratios to all quadrants.
• Evaluating trigonometric ratios of quadrantal and special angles.
• Applying the Pythagorean identity.
• Graphing sine, cosine, tangent, cotangent, secant, and cosecant functions.
• Determining domains, ranges, periods, extrema, and asymptotes of trigonometric functions.
• Transforming trigonometric functions using translations, stretches, compressions, and reflections.
• Combining transformations of trigonometric functions.
• Determining zeros, extrema, and other properties of transformed trigonometric functions.

Vectors

• Distinguishing vector quantities from scalar quantities.
• Representing vectors geometrically.
• Adding vectors using the triangle law.
• Calculating vector magnitudes.
• Solving problems using vector diagrams.
• Identifying and working with parallel vectors.
• Determining unit vectors.
• Forming linear combinations of vectors.
• Expressing position vectors using known vectors.
• Representing two-dimensional Cartesian vectors in component form.
• Adding and scaling Cartesian vectors.
• Calculating magnitudes of two-dimensional Cartesian vectors.

Limits & Continuity

• Estimating finite limits from graphs.
• Evaluating left-hand and right-hand limits.
• Determining whether a limit exists using one-sided limits.
• Estimating limits at infinity and infinite limits from graphs.
• Applying the constant, sum, product, quotient, power, and root rules for limits.
• Evaluating limits at infinity of polynomial functions.
• Evaluating limits of reciprocal, exponential, logarithmic, radical, and trigonometric functions.
• Evaluating limits of piecewise functions.
• Calculating limits of rational functions using factoring.
• Identifying continuity and discontinuity from graphs.
• Determining continuity at a point.
• Analyzing left and right continuity.
• Determining continuity over intervals.

Introduction to Calculus

• Calculating average rates of change of functions.
• Interpreting average rates of change over varying intervals.
• Understanding instantaneous rates of change.
• Defining and interpreting the derivative.
• Applying the power rule for differentiation.
• Applying the sum and constant multiple rules.
• Calculating slopes of tangent lines using derivatives.
• Determining equations of tangent and normal lines.
• Interpreting derivatives in real-world contexts.
• Differentiating exponential, logarithmic, and trigonometric functions.
• Calculating second and higher-order derivatives.
• Applying the product rule and quotient rule for differentiation.
• Estimating derivatives from tables and graphs.
• Differentiating reciprocal trigonometric functions.
• Applying the chain rule to composite functions.
• Differentiating composite exponential, logarithmic, and trigonometric functions.
• Selecting appropriate differentiation procedures.
• Understanding antiderivatives and indefinite integrals.
• Applying the sum and constant multiple rules for indefinite integrals.
• Integrating reciprocal, exponential, and trigonometric functions.

Statistics

• Calculating the mean of a data set.
• Calculating variance and standard deviation.
• Estimating means and variances for grouped data.
• Calculating and interpreting covariance.
• Using sums of squares in statistical calculations.
• Calculating and interpreting z-scores.
• Reading and interpreting scatter plots.
• Identifying and estimating trend lines.
• Using trend lines to make predictions.
• Determining and interpreting linear correlation.
• Calculating and interpreting correlation coefficients.
• Constructing and interpreting linear regression models.
• Analyzing residuals and residual plots.
• Selecting appropriate regression models.

Probability & Combinatorics

• Identifying sets and representing their elements.
• Calculating experimental probabilities.
• Using experimental probability to make predictions.
• Identifying probability experiments, sample spaces, and events.
• Describing and comparing likelihoods.
• Calculating probabilities of single events.
• Using probability to make predictions.
• Calculating probabilities of complementary events.
• Representing probability relationships using Venn diagrams.
• Calculating unions and intersections of sets.
• Calculating probabilities of compound events.
• Using Venn diagrams to calculate probabilities involving complements.
• Applying the rules of sum and product in counting problems.
• Evaluating factorials.
• Calculating arrangements and permutations.
• Calculating combinations.
• Solving probability problems using combinatorics.