Algebra II covers linear and quadratic functions, polynomial and rational functions, exponential and logarithmic functions, analytic geometry, graphing techniques, using matrices for solving systems of equations and inequalities, conic sections, and complex numbers, using selected units of the Sullivan Precalculus text. Students will work closely with the expressions that define functions and continue to improve their ability to model situations and solve equations. Students will build on their foundation of Algebra and Geometry in this class to be able to use their new skills for new problem-solving situations, exhibit increased confidence in their ability to solve mathematical problems, and be prepared for Precalculus and beyond.
Unit 1: Graphs
Students review characteristics of linear equations, relationships, rates, and graphs to generalize these aspects of all types of equations. Equations and graphs of circles are presented, and students apply generalizations about graphing in this context.
Unit 2: Functions and Their Graphs
Students are introduced to the definition of a function, function notation, and methods for determining whether a relation is a function. Students study increasing, decreasing, and constant rate intervals, intercepts, and domain and range. The Difference Quotient is introduced as an extension of rate, and also for algebra skills practice; an in-depth discussion is saved for future courses. Students continue to generalize characteristics of equations and graphs through graphs of composite functions.
Unit 3: Linear and Quadratic Functions
Even though students have had experience with these topics, some new topics are introduced, including: rate as a linear function test and line of best fit from data. Students use quadratic graphs as the context for examining shift, compression, stretch, and reflection transformations. In this unit, quadratic equations are used to show characteristics and processes that are general to all polynomials, such as determining equations from minimal given information, and analyzing meaning from x or y coordinates of the vertex.
Unit 4: Polynomial and Rational Functions
Students extend their knowledge of function behavior to include end behavior and whether a graph crosses or touches the x-axis at each intercept, and use all known characteristics to sketch graphs of polynomial functions. Students determine asymptotes of rational functions and use them to graph. New factoring tools are introduced, such as the Intermediate Value Theorem, the Rational Zeros theorem, and synthetic division.
Unit 5: Introduction to Exponential and Logarithmic Functions
This unit includes a brief and separate section on complex numbers, so students get exposure to the concept, format, conjugates, and operations with complex numbers, and students are expected to perform algebraic proofs of some properties. The rest of the unit focuses on composite and inverse functions, followed by the introduction of exponential and logarithmic functions. Some basic properties allow students to solve some equations, but the more sophisticated techniques will be mastered in the following unit.
Unit 6: Further Work with Exponential and Logarithmic Functions
Once students have mastered basic manipulations with exponential and logarithmic functions, the focus shifts in this unit to learning properties that allow equations to be rearranged into fewer terms for solving. Students also learn methods for evaluating logs whose base is neither 10 nor e. Application problems are introduced, including models built from given data for exponential growth and compound interest.
Unit 7: Systems of Equations and Inequalities
In this unit, students are introduced to matrices as a new method for solving systems of equations. Before the study of augmented matrices, students first learn operations with matrices, determinants, and inverses of matrices. There is also a brief review of substitution and elimination method but the unit focuses mainly on augmented matrices and Cramer’s rule.
Unit 8: Conics
In this unit, students take their knowledge about equations of circles and parabolas and apply the general principles to ellipses and hyperbolas, including the focus and directrix, which are added to their previous understanding of parabolas. In addition to algebraic manipulation for graphing, real-world application problems are included for each conic section.