3. Linear Equations and Functions in Two Variables
A useful application of Algebra is to explore the Rectangular (Cartesian) System including how to graph line, functions and inverse functions. This topic explores the Rectangular (Cartesian) Coordinate System and graphing linear equations in two variables. We continue by exploring slopes of lines (including parallel and perpendicular lines), the distance and midpoint formulas; given information, writing equations in the Point-Slope, Slope-Intercept and the General Forms; exploring relations, functions, function motation, and the algebra of functions as well as inverse functions.
4. Systems of Linear Equations and Inequalities
After exploring one linear equation in two variables, we explore the aspects of more linear equations in two variables called a system of equations. We explore how to solve a system of equations using the Graphing, Substitution, and Elimination Methods. Extending our concepts from Elementary Algebra, we explore how to solve systems of linear equations in three or more variables; solving a system of equation using determinants and Cramer's Rule; and solving systems of linear equations using matrices. We explore graphing linear inequalities in two variables; linear programmings and finish with apply these skills to real world situations.
5. Laws of Exponents, Polynomials and Polynomial Functions
Since Algebra is the generalization of Arithmetic, we explore general operations with algebraic expressions. In this topic, we explore exponents and Scientific Notation; how to add subtract, multiply and divide polynomials. We finish this topic with explore a different way to perform long division of polynomials, synthetic division.
6. Factoring Polynomials; Factoring by Grouping
This topic is completely devoted to the reverse operation from multiplying, factoring. In this topic, we explore many more advanced problems than Elementary Algebra in factoring polynomials. Sections covered within this topic include factoring monomial factors and factoring by grouping; factoring trinomials with leading coefficient 1 or not 1; factoring special types of polynomials such as the difference of two squares, perfect square trinomials and the sum and difference of two cubes as well as more factoring by grouping; general methods of factoring; solving nonlinear equations by factoring and solving applied problems by factoring.
7. Rational Expressions, Equations, and Functions
Combining the skills learned from operations with fractions and with factoring, we learn how to reduce, multiply, divide, add and subtract rational expressions. Further we learn how to simplify complex rational expressions; solving rational equations; and applying these skills to real world applications.
8. Radicals and Rational Exponents
In this topic, we turn our attention to writing radicals and roots to rational exponents. Then, using the laws of exponents, we extend our knowledge of simplifying radicals. We further investigae arithmetic operations involving radicals, division of radicals including rationalize the denominator; solving equations involving radicals; explore in more detail complex numbers and where to apply these skills to real world scenarios.
9. Quadratic Equations, Functions, and Inequalities
Extending our knowledge of solving linear equations, we now focus on solving quadratic equations. In this topic, we explore solving polynomial equations by factoring and the square root property; solving quadratic equations by completing the square and by using the quadratic formula; solving equations that are quadratic in form; solving inequalities involving quadratic forms; exploring variation and applications to the real world.
10. Conic Sections
At this point, we now investigate other forms of equations to graph. In this topic, we explore equations, now to graph and information about circles; parabolas; ellipses; and hyperbolas. We further explore nonlinear systems of equations and inequalities.
11. Exponential and Logarithmic Functions
Turning to expressions with variables in the powers, we explore exponential functions and their inverse functions, logarithmic functions. In this topic, we investigate exponential functions; logarithmic functions; properties of logarithms; solving exponential and logarithmic equations; solving compound interest problems and applications to real world scenarios.
12. Sequences, Series, and the Binomial Theorem
We conclude this subject with the notion of natural patterns. To this end, we investigate sequences, series, and summation notation; arithmetic and geometric sequences and series; and the Binomial Theorem.
