Algebra 2 Curriculum
Below are skills needed, with links to resources to help with that skill. We also encourage plenty of exercises and book work. Curriculum Home
Important: this is a guide only.
Check with your local education authority to find out their requirements.
Algebra 2 | Numbers
Perform arithmetic operations (addition, subtraction, multiplication, division) with expressions containing irrational numbers in radical form
Rationalize a denominator containing a radical expression
Understand the meaning of algebraic numbers and transcendental numbers.
Investigate advanced concepts of prime numbers and factors, including: Coprimes, Mersenne primes, Perfect numbers, Abundant numbers, Deficient numbers, Amicable numbers, Euclid's proof that the set of prime numbers is endless, and Goldbach's conjecture.
Investigate numbers that are Pythagorean triples.
Be familiar with well-known trancendental numbers, such as e, pi and the Liouville Constant.
Know that Euler's Totient Function (The Phi function) is the number of integers from 1 to n that don't share any prime factors with n. Know how to calculate it in special cases and the in the general case.
Use a variety of simple codes as an introduction to RSA.
Use RSA Cryptography to encode and decode messages.
Algebra 2 | Complex Numbers
Write square roots of negative numbers in terms of i, and solve simple equations whose solutions are powers of i
Simplify powers of i
Determine the conjugate of a complex number
Perform arithmetic operations on complex numbers and write the answer in the form "a+bi"
Note: This includes simplifying expressions with complex denominators.
Represent a complex number on the complex plane (Argand diagram).
Know how to calculate the magnitude and angle of a complex number, and express a complex number in polar form
Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane;
Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.
Factor polynomial expressions as the product of complex factors.
For example x^2 + y^2 = (x + yi)(x - yi)
Be familiar with Euler's Formula for Complex Numbers and convert complex numbers between the forms
a + bi and re^(ix)
Algebra 2 | Measurement
Be familiar with the metric (SI) units used in Mathematics and Physics.
Algebra 2 | Algebra
Solve absolute value equations and inequalities involving linear expressions in one variable
Simplify radical expressions
Perform addition, subtraction, multiplication, and division of radical expressions
Rationalize denominators involving algebraic radical expressions
Perform arithmetic operations on rational expressions and rename to lowest terms
Solve radical equations
Solve rational equations and inequalities
Use direct and inverse variation to solve for unknown values
Understand what is meant by the terms and the degree of a polynomial and the degree of a rational expression.
Understand how mathematical modelling can be used to "model", or represent, how the real world works; but taking into account any possible constraints.
Know how to decompose a rational expression into partial fractions.
Determine whether a given value is a solution to a given radical equation in one variable.
Algebra 2 | Exponents
Analyze and solve verbal problems that involve exponential growth and decay
Rewrite algebraic expressions with fractional exponents as radical expressions
Rewrite algebraic expressions in radical form as expressions with fractional exponents
Evaluate exponential expressions, including those with base e
Solve exponential equations with or without common bases
Graph exponential functions of the form y = a^x or y = -a^x for positive values of a, including a = e
Solve an application which results in an exponential function
Apply the rules of exponents to simplify expressions involving negative and/or fractional exponents
Rewrite algebraic expressions that contain negative exponents using only positive exponents
Evaluate numerical expressions with negative and/or fractional exponents, without the aid of a calculator (when the answers are rational numbers)
Algebra 2 | Inequalities
Solve quadratic inequalities in one and two variables, algebraically and graphically (includes higher degree - graphically only).
Know open and closed interval notation and how they relate to points on the number line and the solution of inequalities.
Know the properties of inequalities, including the Transitive Property, the Reversal Property, and the Law of Trichotomy.
Algebra 2 | Linear Equations
Solve systems of three linear equations in three variables algebraically, using the substitution method or the elimination method.
Find the general solution of a Linear Diophantine Equation
Algebra 2 | Quadratic Equations
Use the discriminant to determine the nature of the roots of a quadratic equation
Determine the sum and product of the roots of a quadratic equation by examining its coefficients.
Determine the quadratic equation, given the sum and product of its roots
Know and apply the technique of completing the square
Solve quadratic equations, using the quadratic formula
Solve systems of equations involving one linear equation and one quadratic equation algebraically
Note: This includes rational equations that result in linear equations with extraneous roots.
Solve systems of equations involving one linear equation and one quadratic equation graphically
Solve quadratic equations by factoring
Apply quadratic equations to examples from the real world
Algebra 2 | Logarithms
Evaluate logarithmic expressions in any base
Apply the properties of logarithms to rewrite logarithmic expressions in equivalent forms
Solve a logarithmic equation by rewriting as an exponential equation
Graph logarithmic functions, using the inverse of the related exponential function
Understand that Euler's number, e, is the base of the Natural Logarithms and the Natural Exponential Function.
Write a logarithmic expression in exponential form and vice versa
Algebra 2 | Polynomials
Find the solutions to polynomial equations of higher degree that can be solved using factoring and/or the quadratic formula
Approximate the solutions to polynomial equations of higher degree by inspecting the graph
Factor polynomial expressions completely, using any combination of the following techniques: common factor extraction, difference of two perfect squares, quadratic trinomials
Perform arithmetic operations with polynomial expressions containing rational coefficients
Identify and factor the difference of two cubes or the sum of two cubes.
Know and understand the Fundamental Theorem of Algebra.
Divide a polynomial by a monomial or binomial, where the quotient has a remainder. Use Polynomial long division.
Investigate ways to search for all real roots (zeros) of a polynomial expression.
Know the rule of signs for polynomials.
Understand and apply The Remainder Theorem and The Factor Theorem.
Determine the sum and product of the roots of a cubic and higher polynomials by examining its coefficients.
Algebra 2 | Sets
Introduction to groups.
Understand what is meant by a Power Set of a given set, and that the power set for a set with n members has 2^n members.
Algebra 2 | Logic
Determine the negation of a statement and establish its truth value
Know and apply the conditions under which a compound statement (conjunction, disjunction, conditional, biconditional) is true
Identify and write the inverse, converse, and contrapositive of a given conditional statement and note the logical equivalences
Write a proof arguing from a given hypothesis to a given conclusion
Understand the principle of Mathematical Induction as a method of proof.
Understand what is meant by each of the terms: Theorems, Corollaries and Lemmas.
Algebra 2 | Functions
Determine the domain and range of a function from its equation
Write functions in functional notation
Use functional notation to evaluate functions for given values of the domain
Find the composition of functions
Define the inverse of a function
Determine the inverse of a function and use composition to justify the result
Perform transformations with functions and relations: f(x+a), f(x)+a, f(-x), -f(x), af(x), f(ax)
Determine the domain and range of a function from its graph
Identify relations and functions, using graphs
Introduction to functions
Types of function
Understand the meaning of an asymptote and distinguish between the three types - horizontal asymptote, vertical asymptote and oblique asymptote.
Find the equations of the horizontal, vertical and oblique asymptotes for a rational expression.
Give the correct domain for the composition of two functions.
Recognize the properties, shape and symmetry of the graph of a cubic function.
Understand the difference between Range and Codomain.
Understand that a function can be even, odd or neither even nor odd, and know how to determine whether a given function is even, odd or neither even nor odd.
Define and understand the 'floor', 'ceiling', 'integer' and 'fractional part' functions, and investigate their graphs.
Add, subtract, multiply and divide functions; and find the Domain of the sum, difference, product or quotient respectively.
Understand what is meant by a 'Piecewise' function, how to define the various pieces, and how to determine the domain and range for such a function.
Write a domain or range of a function using Set Builder notation.
Compare properties of two or more functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
Algebra 2 | Sequences and Sums
Identify an arithmetic or geometric sequence and find the formula for its nth term
Determine the common difference in an arithmetic sequence
Determine the common ratio in a geometric sequence
Determine a specified term of an arithmetic or geometric sequence
Specify terms of a sequence, given its recursive definition
Represent the sum of a series, using sigma notation
Determine the sum of the first n terms of an arithmetic or geometric series
Apply the binomial theorem to expand a binomial and determine a specific term of a binomial expansion
Know and apply sigma notation
Define the Fibonacci sequence and the Golden ratio and investigate the relationship between them.
Know the names of special sequences such as Triangular Numbers, Square Numbers, Cube Numbers, Tetrahedral Numbers and Fibonacci numbers; and how they are generated.
Know the formulae for:
1. The sum of the first n natural numbers.
2. The sum of the squares of the first n natural numbers.
3. The sum of the cubes of the first n natural numbers.
Investigate Pascal's Triangle and its properties; including its relationship to sets of numbers (such as triangular numbers and Fibonacci numbers), and the Binomial coefficients.
Use differences to find the rule for a sequence
Express an arithmetic sequence or a geometric sequence as a function:
either 1. Recursively.
or 2. As an explicit linear function (arithmetic sequence) or an explicit exponential function ( geometric sequence).
Algebra 2 | Vectors
Understand what is meant by a vector
Know how to add and subtract vectors, and how to break a vector into two pieces
Understand what is meant by the magnitude of a vector and how to multiply a vector by a scalar
Calculate the magnitude and direction of a vector from its x and y lengths, or vice versa
Understand unit vectors
Know the two ways to find the dot product of two vectors (in 2 or 3 dimensions)
Know the two ways to find the cross product of two vectors (in 2 or 3 dimensions)
Solve problems involving velocity, force and other quantities that can be represented by vectors.
Algebra 2 | Matrices
Know how to add and subtract matrices, how to find the negative of a matrix, how to multiply a matrix by a constant, and how to find the transpose of a matrix.
Know the conditions under which two matrices can be multiplied, and how to perform the multiplication.
Understand that multiplication of matrices is not commutative.
Know what is meant by different types of matrix: square, identity, diagonal, scalar, triangular, zero, symmetric and Hermitian matrices.
Evaluate the determinant of a 2 by 2 matrix or a 3 by 3 matrix.
Know the conditions under which a matrix has a multiplicative inverse and what is meant by a singular matrix.
Find the inverse of a matrix (if it exists) by swapping around the elements and multiplying by the reciprocal of the determinant.
Find the inverse of a matrix (if it exists) using elementary row operations.
Find the inverse of a matrix (if it exists) using Minors, Cofactors and Adjugate.
Solve a system of linear equations using matrices.
Represent and manipulate data using matrices, e.g. the sales of different types of pie by a shop on different days of the week.
Multiply a matrix by a column vector to produce another vector - a matrix equation.
Represent:
1. Transformations (reflections, rotations and dilations)
2. Systems of linear equations
as the product of a square matrix with a column vector.
Know how to find the eigenvalues and eigenvectors of 2 X 2 and simple 3 X 3 matrices.
Know how to find the rank of a matrix; understand linear dependence, linear independence and basis vectors.
Algebra 2 | Graphs
Given the equation of a circle in Standard Form, or its center and radius, write its equation in General Form.
Write the equation of a circle, given its center and a point on the circle, or given the endpoints of a diameter
Write the equation of a circle from its graph.
Note: The center is an ordered pair of integers and the radius is an integer.
Graph and solve compound loci in the coordinate plane
Find the center and/or radius of a circle given its equation in Standard Form
Convert the equation of a circle in General Form to Standard Form
Find the center and/or radius of a circle given its equation in General Form
Graph circles of the form (x - h)^2 + (y - k)^2 = r^2
Understand Conic Sections (circle, ellipse, parabola, hyperbola)
Find the x and y intercepts for a graph given its equation.
Investigate various approximate formulae for finding the perimeter of an ellipse, and compare them.
Determine the equation of a curve given some points on the curve.
Derive the equation of a parabola given a focus or directrix.
Derive the equations of ellipses and hyperbolas given the foci.