Online Algebra 2 Course for Grades 9 and 10

Linear inequalities and absolute value, functions and lines, piecewise functions and transformations, systems of equations and inequalities with optimization, then matrices, determinants, Cramer's Rule and inverses.

• Linear Equations, Inequalities and Absolute ValueAge problems and linear models, then inequalities: graphing them, interval notation, compound AND/OR inequalities and absolute-value inequalities, including the no-solution and all-real-numbers cases.
• Relations and FunctionsRelations and functions: mapping diagrams, the vertical line test, domain and range, and evaluating at any expression. Then the three forms of a line, average rate of change, slope, and equations of parallel and perpendicular lines.
• Special Functions, Transformations, Inequalities and SystemsPiecewise, greatest-integer and absolute-value functions, then shifts, reflections, stretches and compressions of any graph. Linear and absolute-value inequalities are graphed, and systems are solved by graphing, substitution and elimination.
• Systems of Inequalities and OptimizationSystems of inequalities solved by graphing, the vertices of the solution region, and linear programming: maximizing or minimizing over a feasible region in real problems. Ends with systems of three equations in three variables.
• Matrices I - Operations and DeterminantsReading and building matrices; adding, subtracting, scaling and multiplying them, and which properties hold. Also reflection and rotation matrices, 2×2 and 3×3 determinants, and the area of a triangle from its vertices.
• Matrices II - Inverse and Cramers RuleCramer's Rule for two and three variables, the identity matrix and inverse matrices (2×2 formula; 3×3 by cofactors and adjugate), and systems written as A·X = B and solved with X = A⁻¹B. Includes real-data problems and secret codes.

Four ways to solve quadratics, complex numbers, quadratic functions and their graphs, then polynomial division, the Factor and Rational Zero theorems, and radical equations.

• Quadratic EquationsFour ways to solve a quadratic: the square-root method, completing the square, the quadratic formula (derived in class) and factoring. The discriminant predicts the number of roots, and an equation can be built from its roots.
• Complex NumbersThe imaginary unit i, the 4-step cycle of its powers, and arithmetic with complex numbers, including division by the conjugate. Then the complex plane and |z|, quadratics with a negative discriminant, and the square root of a complex number.
• Quadratic Functions and TransformationsReading a parabola: vertex, axis, zeros, range, maximum or minimum. Vertex form by completing the square, graphing with shifts, reflections and stretches, quadratic inequalities, and systems of a parabola and a line.
• Polynomials and Radical EquationsPolynomial operations, long and synthetic division, and reading end behavior and zeros from a graph. The Factor and Rational Zero theorems factor cubics. Then nth roots, radical equations with extraneous roots, and radical inequalities.

Exponent and radical rules, operations on functions and inverses, exponential and logarithmic functions and equations, then rational expressions, rational equations and the graphs of rational functions.

• Exponents and Rational ExponentsThe seven properties of exponents, simplifying nth roots (and when absolute value is required), and converting between radicals and fractional exponents. Radical expressions are simplified and denominators rationalized in mixed problems.
• Operations with Functions and InversesAdding, subtracting, multiplying and dividing functions, then composition, with pricing problems (markups and coupons) where order matters. Then inverse functions: when one exists, the 5-step method, and the mirror-image graph across y = x.
• Exponential Functions and Intro to LogarithmsThe basic function families, then graphing exponential growth and decay with transformations, modeling with a·bˣ and compound interest, and solving exponential equations by matching bases. Ends by introducing logarithms.
• Logarithms Properties Graphs and EquationsThe product, quotient and power properties of logarithms (with a proof of the product property), condensing and expanding, change of base, logarithmic graphs, and logarithmic equations solved with a domain check. Uses the Richter scale.
• Rational Expressions and EquationsExcluded values, simplifying by factoring and canceling, and the four operations on rational expressions. Mixed expressions, complex fractions and conjugate rationalizing, then rational equations (with extraneous solutions) and work-rate problems.
• Rational Functions and GraphsGraphing y = 1/x and hyperbolas a/(x − h) + k with their asymptotes; the three degree cases for horizontal asymptotes; holes; Boyle's Law. The second half is mixed review of exponent rules, inverses, logarithms and complex fractions.

One week each on parabolas, circles, ellipses and hyperbolas: definitions, equations, graphs, completing the square and real-world applications.

• Conic Sections I ParabolasThe four conic sections, and how to classify one from its general equation. Then the parabola from its focus and directrix: deriving its equation, vertical and horizontal forms, writing equations from information or a graph, and the satellite dish.
• Conic Sections II CirclesThe circle: its definition and equation; equations from a graph, a point, a diameter or tangent lines; centers in a given quadrant; and center and radius by completing the square. Parabolas and circles are mixed together in review.
• Conic Sections III EllipsesThe ellipse as the points with a constant sum of distances to two foci: center, vertices, co-vertices and foci from an equation, c² = a² − b², horizontal and vertical forms, writing equations from data, completing the square, and eccentricity.
• Conic Sections IV HyperbolasThe hyperbola as the points with a constant difference of distances to two foci, with c² = a² + b². Graphing with the box and asymptotes, writing equations, tangent circles, eccentricity, LORAN navigation, and systems of conic inequalities.

Arithmetic sequences and series, then geometric sequences, finite and infinite geometric series, and sigma notation.

• Arithmetic Sequences and SeriesSequences and series, and arithmetic sequences as linear functions: the nth term formula aₙ = a₁ + (n − 1)d and the sum formula Sₙ = n(a₁ + aₙ)/2, with the story of young Gauss. Applications include seating, savings and distance.
• Geometric Sequences and SeriesGeometric sequences as exponential functions, aₙ = a₁·rⁿ⁻¹, the finite sum formula and the geometric mean. Then growth and decay applications (halving, depreciation), infinite series when |r| < 1, and sigma notation with its properties.

Trigonometric ratios, radians and the unit circle, graphs of trig functions, angle formulas and identities, trig equations, and the Laws of Sines and Cosines.

• Trigonometry I Ratios and Angle MeasureThe six trigonometric ratios from a right triangle (SOH-CAH-TOA), exact values for 30°, 45° and 60°, and the basic identities. Then coterminal angles, radians, and ratios of any angle using a point (x, y, r) and reference angles.
• Trigonometry II Graphs and Angle FormulasGraphs of sine, cosine, tangent and the reciprocal functions, with amplitude, period, phase shift and vertical shift (a Ferris wheel model). Then derived sum/difference, double-angle and half-angle formulas, and reduction formulas.
• Trigonometric Identities Equations and Law of SinesStrategies for verifying identities (convert to sine and cosine, work the harder side, multiply by the conjugate), then trig equations that are linear or quadratic in one function or need an identity. Ends with the Law of Sines.
• More Trig Equations Reductions and Law of CosinesMore trigonometric equations: multiple angles, equations mixing two functions, and Pythagorean substitutions. Reduction formulas are practiced on new angles, then the Law of Cosines is derived.

Probability rules, permutations, combinations and the binomial distribution, then descriptive statistics, the normal distribution, study design and margin of error.

• Probability FundamentalsSample spaces, the counting principle, theoretical and geometric probability, and the complement rule. Then two-way tables, conditional probability, independent and dependent events, sampling without replacement, and disjoint and overlapping events.
• Permutations Combinations and Binomial DistributionPermutations (including repeated items) and combinations, and deciding which one a problem needs, used inside probability problems such as poker hands. Then the Binomial Theorem, Pascal's Triangle and the binomial distribution.
• Data Analysis and StatisticsMean, median, mode, range and standard deviation; the normal distribution, Empirical Rule, z-scores and skewed shapes. Then sampling methods, bias, experiments versus observational studies, margin of error and judging a claim from a sample.

• SAT Practice I Heart of AlgebraSAT-style practice in linear equations (including clearing fractions), word problems, systems, absolute-value equations, inequalities and their graphs, and mixture problems, with the strategies of backsolving and testing numbers.
• SAT Practice II Passport to Advanced MathSAT-style practice in polynomial operations, exponent rules, dividing by a binomial, special products and factoring, radicals and rational exponents, "disguised" quadratics, quadratic graphs, and radical and exponential equations.
• SAT Practice III Problem Solving and Data AnalysisSAT-style practice in ratio and percent-mixture problems, successive percent change, averages and weighted averages, reading charts and data tables, and scatterplots with lines of best fit.
• SAT Practice IV Additional TopicsSAT-style practice in complex numbers (powers of i, arithmetic, division, modulus) and in geometry: volume and surface area, circle-in-square diagrams, arc length and sector area, and special right triangles.
• SAT Practice V Full Mixed Practice and Course Wrap-UpA mixed SAT-style set across every category in the course, a backsolving bonus, a course wrap-up with a skill map by week, and a final quiz with one question from every unit.