# 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

☐ Surds

☐ Perform arithmetic operations on irrational expressions

☐ Surds

☐ Rationalize a denominator containing a radical expression

☐ Understand the meaning of algebraic numbers and transcendental numbers.

☐ Pi

☐ 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.

☐ Pi

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).

☐ Vectors

☐ 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

☐ Simplify complex fractional expressions

☐ 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.

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

☐ Triplets

☐ 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

☐ Vectors

☐ Know how to add and subtract vectors, and how to break a vector into two pieces

☐ Vectors

☐ Understand what is meant by the magnitude of a vector and how to multiply a vector by a scalar

☐ Vectors

☐ Calculate the magnitude and direction of a vector from its x and y lengths, or vice versa

☐ Vectors

☐ Understand unit vectors

☐ Vectors

☐ Know the two ways to find the dot product of two vectors (in 2 or 3 dimensions)

☐ Vectors

☐ Know the two ways to find the cross product of two vectors (in 2 or 3 dimensions)

☐ Vectors

☐ Solve problems involving velocity, force and other quantities that can be represented by vectors.

☐ 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.

☐ Matrices

☐ 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.

☐ 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

☐ Ellipse

☐ Circle

☐ 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)

☐ Ellipse

☐ Parabola

☐ Circle

☐ 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.

☐ Ellipse

☐ Determine the equation of a curve given some points on the curve.

☐ Derive the equation of a parabola given a focus or directrix.

☐ Parabola

☐ Derive the equations of ellipses and hyperbolas given the foci.

☐ Ellipse