Algebra II
Algebra II extends algebraic reasoning to a full catalog of function families — polynomial, rational, exponential, logarithmic, radical, and piecewise — plus complex numbers, matrices, conic sections, sequences and series, and combinatorics with probability.
Who Should Take This
Designed for students who have completed Algebra I and Geometry and are heading toward Pre-Calculus or college-level math. Also valuable for adults returning to STEM fields who need to rebuild their algebraic toolkit before taking Calculus or Statistics.
What's Covered
1Functions and Function Operations
2Polynomial Functions
3Rational Functions
4Exponential and Logarithmic Functions
5Complex Numbers
6Systems of Equations and Inequalities
7Matrices
8Sequences and Series
9Conic Sections
10Probability and Combinatorics
What's Included in AccelaStudy® AI
Adaptive Knowledge Graph
Practice Questions
Lesson Modules
Console Simulator Labs
Exam Tips & Strategy
13 Activity Formats
Course Outline
1Functions and Function Operations 9 topics
Describe a function's domain and range using interval notation, set-builder notation, and inequality notation, and identify domain restrictions from denominators equal to zero and expressions under even radicals
Apply function composition f∘g by substituting g(x) into f(x), evaluate compositions for given values, and determine the domain of the composed function based on both component functions' restrictions
Apply inverse function methods by swapping x and y, solving for the new y, verifying using composition that f(f⁻¹(x)) = x, and graphing f and f⁻¹ as reflections across y = x
Describe piecewise functions by identifying each sub-function's rule and its restricted domain interval, evaluate piecewise functions at given inputs, and graph them accurately including open and closed endpoints
Apply transformations of parent functions (vertical/horizontal shifts, stretches/compressions, reflections) to graph transformed functions and write equations from transformed graphs
Analyze absolute value functions by rewriting them as piecewise functions, determining vertices, graphing V-shapes, and solving absolute value equations and inequalities using case analysis
Apply even and odd function identification by checking whether f(-x) = f(x) (even, symmetric about y-axis) or f(-x) = -f(x) (odd, symmetric about origin) and using this to predict graph symmetry
Analyze restricted-domain functions and greatest-integer (floor) function behavior, sketch step functions, and evaluate the floor function ⌊x⌋ for positive, negative, and non-integer inputs
Apply radical function graphing by identifying the domain restriction x ≥ k for √(x-k), sketching the graph as a half-parabola on its side, and solving radical equations by isolating the radical and squaring both sides checking for extraneous solutions
2Polynomial Functions 6 topics
Describe polynomial degree, leading coefficient, and end behavior using the rules that even-degree polynomials have both ends pointing the same direction and odd-degree polynomials have ends pointing opposite directions
Apply advanced factoring techniques including factoring by grouping, sum and difference of cubes (a³ ± b³), completing the square to convert to vertex form, and the quadratic formula with discriminant analysis
Apply the Factor Theorem and Remainder Theorem to perform synthetic division, test potential rational roots using the Rational Root Theorem, and fully factor higher-degree polynomials
Apply multiplicity rules to predict whether a graph crosses or touches the x-axis at each zero and sketch polynomial graphs using intercepts, end behavior, and turning points
Analyze polynomial function models for real-world phenomena including projectile motion, volume optimization, and profit functions, solving for critical values and interpreting in context
Apply partial fraction decomposition to rational expressions with linear non-repeated denominators by writing the fraction as a sum of simpler fractions and solving the resulting system for the unknown numerator coefficients
3Rational Functions 5 topics
Describe vertical asymptotes as values where the denominator equals zero and the numerator does not, horizontal asymptotes from the ratio of leading coefficients, and holes from common factors in numerator and denominator
Apply simplification of rational expressions by factoring numerator and denominator, canceling common factors, and noting domain restrictions at all original zeros of the denominator
Apply multiplication, division, addition, and subtraction of rational expressions by finding the least common denominator for addition/subtraction and simplifying fully, checking that results maintain original domain restrictions
Analyze rational equations by multiplying through by the LCD to clear fractions, solving the resulting polynomial equation, and checking for extraneous solutions introduced by the clearing step
Apply graphing of rational functions by plotting vertical asymptotes, horizontal or oblique asymptotes, holes, x-intercepts, y-intercepts, and test points in each region created by the vertical asymptotes to determine sign
4Exponential and Logarithmic Functions 6 topics
Describe exponential functions f(x) = abˣ by identifying the initial value a, base b, growth vs. decay behavior (b > 1 vs. 0 < b < 1), and the horizontal asymptote at y = 0
Apply exponential growth and decay models including compound interest A = P(1 + r/n)^(nt), continuous growth A = Pe^(rt), and half-life formulas to solve for unknown time, rate, or amount
Describe logarithms as the inverse of exponentials, convert between logarithmic and exponential form logb(x) = y ↔ bʸ = x, evaluate common and natural logarithms, and state properties (product, quotient, power rules)
Apply logarithm properties to expand and condense logarithmic expressions, use the change-of-base formula to evaluate logarithms with non-standard bases, and solve exponential equations by taking logarithms of both sides
Analyze logarithmic and exponential models for pH, earthquake magnitude (Richter scale), sound intensity (decibels), and population growth, solving for unknowns and interpreting the logarithmic scale's meaning
Describe the natural number e ≈ 2.71828 as the base of natural exponential and logarithmic functions, compute limits (1 + 1/n)ⁿ as n → ∞ conceptually, and apply ln and eˣ in continuous growth and decay models
5Complex Numbers 5 topics
Describe the imaginary unit i = √(-1), simplify powers of i using the cyclic pattern (i, -1, -i, 1), and write complex numbers in standard form a + bi identifying real and imaginary parts
Apply arithmetic operations to complex numbers including addition, subtraction, multiplication using FOIL and i² = -1, and division by multiplying by the complex conjugate to obtain a real denominator
Apply the quadratic formula to equations with negative discriminants, express solutions as complex conjugate pairs a ± bi, and verify solutions by substitution into the original equation
Analyze the complex number plane (Argand diagram), compute the modulus |a + bi| = √(a² + b²), and interpret conjugate pairs as reflections across the real axis in the context of polynomial roots
Apply DeMoivre's Theorem at an introductory level to compute integer powers of complex numbers written in polar form r(cosθ + i sinθ), explaining how the modulus is raised to the power and the argument is multiplied
6Systems of Equations and Inequalities 5 topics
Apply substitution and elimination methods to solve systems of two linear equations, classify systems as consistent-independent, consistent-dependent, or inconsistent, and interpret solutions graphically
Apply Gaussian elimination to solve 3×3 systems of linear equations by writing the augmented matrix, performing row operations, and back-substituting from row echelon form
Apply graphing and substitution to solve systems involving one linear and one quadratic equation, identifying zero, one, or two intersection points and interpreting results in context
Analyze linear programming by graphing a system of linear inequalities, identifying the feasible region, evaluating an objective function at corner points, and determining maximum or minimum values
Apply Cramer's Rule to solve 2×2 and 3×3 systems of equations using ratios of determinants and identify when Cramer's Rule fails because the coefficient matrix determinant is zero
7Matrices 5 topics
Describe matrix dimensions, elements, and notation, perform matrix addition, subtraction, and scalar multiplication, and explain when matrix addition is defined by matching dimensions
Apply matrix multiplication by computing each entry as a dot product of a row and column, verify that matrix multiplication is not commutative in general, and determine when multiplication is defined by inner dimensions
Apply determinant formulas for 2×2 matrices (ad - bc) and 3×3 matrices by cofactor expansion, and use the inverse matrix formula or row reduction to find A⁻¹ and verify that AA⁻¹ = I
Analyze systems of linear equations represented as AX = B by computing X = A⁻¹B when the determinant is nonzero, and explain why det(A) = 0 signals no unique solution exists
Apply matrix transformations to represent geometric transformations in the plane including rotation, reflection, scaling, and translation (using augmented coordinates), connecting linear algebra to coordinate geometry
8Sequences and Series 6 topics
Describe arithmetic sequences by their first term and common difference d, write explicit formulas aₙ = a₁ + (n-1)d and recursive formulas aₙ = aₙ₋₁ + d, and compute any specified term
Apply the arithmetic series sum formula Sₙ = n/2(a₁ + aₙ) to compute finite sums and solve problems involving total salary over n years, total distance, and stepped accumulation patterns
Describe geometric sequences by their first term and common ratio r, write explicit formulas aₙ = a₁rⁿ⁻¹, and apply the geometric series sum formula Sₙ = a₁(1 - rⁿ)/(1 - r) for finite sums
Apply the infinite geometric series sum formula S = a₁/(1 - r) for |r| < 1, determine whether a geometric series converges or diverges, and model repeating decimals as convergent geometric series
Apply sigma notation to write and evaluate finite sums, expand summation expressions into individual terms, and rewrite series in sigma notation identifying the index variable, bounds, and general term
Apply recursive sequence analysis to Fibonacci-like sequences by computing terms from a given recurrence relation, recognizing that the ratio of consecutive terms of the Fibonacci sequence converges to the golden ratio φ ≈ 1.618
9Conic Sections 6 topics
Describe conic sections (circle, ellipse, parabola, hyperbola) as cross-sections of a double cone and identify each from its standard equation by the signs and equality of squared terms
Apply completing the square to convert general form Ax² + Bxy + Cy² + Dx + Ey + F = 0 to standard form for circles and ellipses, identifying center and radii
Apply parabola equations in vertex form and standard form, identify the vertex, focus, directrix, and axis of symmetry, and graph both upward/downward and left/right opening parabolas
Apply hyperbola equations to identify transverse and conjugate axes, vertices, foci, and asymptote equations y = ±(b/a)x or y = ±(a/b)x, and graph both horizontal and vertical opening hyperbolas
Analyze conic section applications including satellite dish parabolic reflectors, planetary elliptical orbits, hyperbolic navigation, and cooling tower hyperbolic shapes, connecting geometric properties to physical behavior
Apply the eccentricity definition to classify conic sections — e = 0 for circle, 0 < e < 1 for ellipse, e = 1 for parabola, e > 1 for hyperbola — and compute eccentricity from the given equation's parameters
10Probability and Combinatorics 7 topics
Apply the Fundamental Counting Principle and factorial notation to count ordered arrangements and explain why n! counts the number of ways to arrange n distinct objects
Apply permutation formula P(n,r) = n!/(n-r)! for ordered selections and combination formula C(n,r) = n!/[r!(n-r)!] for unordered selections, distinguishing which applies based on whether order matters
Apply the Binomial Theorem (a + b)ⁿ = ΣC(n,k)aⁿ⁻ᵏbᵏ to expand binomial expressions, identify specific terms using the general term formula, and connect binomial coefficients to Pascal's triangle
Apply probability rules for mutually exclusive events (P(A∪B) = P(A)+P(B)) and independent events (P(A∩B) = P(A)·P(B)), and compute conditional probability P(A|B) = P(A∩B)/P(B)
Analyze binomial probability distributions by computing P(X = k) = C(n,k)pᵏ(1-p)ⁿ⁻ᵏ for exactly k successes in n independent Bernoulli trials and interpreting expected value np as the long-run average
Apply Bayes' theorem in discrete contexts using total probability to compute the denominator, solve classic problems including false positive medical testing and email spam filtering, and interpret the often counterintuitive results
Analyze the difference between permutations with repetition (nʳ), permutations without repetition P(n,r), combinations C(n,r), and combinations with repetition C(n+r-1,r), selecting the correct counting formula based on whether order matters and whether repetition is allowed
Scope
Included Topics
- Functions as mappings (domain, range, function notation, composition, inverses), polynomial functions (degree, end behavior, zeros, factoring, polynomial division), rational functions (domain restrictions, asymptotes, simplifying, operations), exponential functions (growth and decay, graphs, solving exponential equations), logarithmic functions (definition, properties, change of base, solving log equations), radical functions (domain, graphs, simplifying), piecewise and absolute value functions, complex numbers (imaginary unit, arithmetic, conjugates, modulus), systems of linear equations/inequalities in two and three variables, matrices (operations, determinants, inverse matrices, solving systems with matrices), sequences and series (arithmetic, geometric, sigma notation, finite sums), conic sections (circles, ellipses, hyperbolas, parabolas), probability and combinatorics (permutations, combinations, Binomial theorem), introduction to right triangle trigonometry, advanced factoring techniques (grouping, sum/difference of cubes, completing the square, quadratic formula review)
Not Covered
- Calculus limits and derivatives (covered in Calculus I)
- Polar coordinates and parametric equations (typically Pre-Calculus or above)
- Trigonometric identities and Law of Sines/Cosines beyond intro right-triangle trig (covered in Trigonometry)
- Multivariable functions and 3D graphing beyond introduction
- Statistical inference (covered in Statistics)
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