ALGEBRA II

Students will:

•Solve equations and inequalities involving absolute value

•Solve systems of linear equations and inequalities (in two or three variables) by substitution, with graphs, or with matrices

•Be adept at operations on polynomials, including long division

•Factor polynomials representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes

•Demonstrate knowledge of how real and complex numbers are related both arithmetically and graphically. In particular, they can plot complex numbers as points in the plane

•Add, subtract, multiply, and divide complex numbers

•Add, subtract, multiply, divide, reduce, and evaluate rational expressions with monomial and polynomial denominators and simplify complicated rational expressions, including those with negative exponents in the denominator

•Solve and graph quadratic equations by factoring, completing the square, or using the quadratic formula. Apply these techniques in solving word problems. They also solve quadratic equations in the complex number system

•Demonstrate and explain the effect that changing a coefficient has on the graph of quadratic functions; that is, students can determine how the graph of a parabola changes as a, b, and c vary in the equation y = a(x-b)2+c

•Graph quadratic functions and determine the maxima, minima, and zeros of the function

•Prove simple laws of logarithms.

a) Understand the inverse relationship between exponents and logarithms and use this relation ship to solve problems involving logarithms and exponents

b) Judge the validity of an argument according to whether the properties of real numbers, exponents, and logarithms have been applied correctly at each step

•Know the laws of fractional exponents, understand exponential functions, and use these functions in problems involving exponential growth and decay

•Use the definition of logarithms to translate between logarithms in any base

•Understand and use the properties of logarithms to simplify logarithmic numeric expressions and to identify their approximate values

•Determine whether a specific algebraic statement involving rational expressions, radical expressions, or logarithmic or exponential functions is sometimes true, always true, or never true

•Demonstrate and explain how the geometry of the graph of a conic section (e.g., asymptotes, foci, eccentricity) depends on the coefficients of the quadratic equation representing it

•Given a quadratic equation of the form ax2 + by2 + cx + dy + e = 0, students can use the method for completing the square to put the equation into standard form and can recognize whether the graph of the equation is a circle, ellipse, parabola, or hyperbola. Students can then graph the equation

•Use fundamental counting principles to compute combinations and permutations

•Use combinations and permutations to compute probabilities

•Know the binomial theorem and use it to expand binomial expressions that are raised to positive integer powers

•Apply the method of mathematical induction to prove general statements about the positive integers

•Find the general term and the sums of arithmetic series and both finite and infinite geometric series

•Derive the summation formulas for arithmetic series and for both finite and infinite geometric series

•Solve problems involving functional concepts, such as composition, defining the inverse function and performing arithmetic operations on functions

•Use properties from number systems to justify steps in combining and simplifying functions