Algebra II Topics for 1st Semester Exam
1st 9-weeks:
Variables and expressions (base, power, exponent, evaluating for given values of the variable) Order of Operations (PEMDAS) Categories of Numbers (Real, Rational, Irrational, Integers, Whole, Natural)
Properties: Additive Identity ( a+0 = a ) Multiplicative Identity ( a�1 a ) Multiplicative Property of Zero ( a �0 0 ) Additive Inverse Prop. ( a+( - a ) = 0 ) a b Multiplicative Inverse Prop. ( � 1) b a Reflexive ( a= a ) Symmetric ( if a = b, then b = a) Transitive (if a = b, and b = c, then a = c) Substitution Addition/Subtraction Properties of Equality Multiplication/Division Properties of Equality Trichotomy Prop. (exactly one of these is true: either a< b or a> b a= b BIG 3: Distributive a( b+ c ) = ab + ac Commutative a+ b = b + a or a� b b a (Order) Associative a+( b + c ) = ( a + b ) + c or (a鬃 b ) c= a 鬃 ( b c ) (Grouping)
Solving Linear Equations - using addition and subtraction - using multiplication and division Questions to ask yourself: 1. What is being done to the variable? 2. How can I “un-do” that? 3. Perform operation. 4. Check solution. Solving Multi-step equations (work backward through the Order of Operations) Solving equations with variable on both sides (get all variables on one side, all constants on the other)
Literal Equations (solving for one variable in terms of another) Absolute Value Equations Recall that absolute value is always positive! Set up 2 cases: if the number inside the abs.value symbol is positive OR if it’s negative You must ALWAYS CHECK your solution (your correct solving MAY produce an invalid solution).
Solving Inequalities Remember that when you multiply or divide by a negative number, you must flip the sign! Show solutions in set notation. Open circles vs. closed circles- bold arrows, when?
Solving Absolute Value Inequalities Compound inequalities AND ( , < ) vs. OR ( , > ) (graph goes in 2 directions) Set up 2 cases: if the number inside the abs.value symbol is positive or if it’s negative
Working Backwards from Graph to state Absolute Value Inequality
Coordinate Geometry: Cartesian coordinate plane, terms and definitions, labeling axes, origin, quadrants, abscissa, ordinate, domain, range, mapping
Relation vs. Function- (vertical line test, discrete vs. continuous function)
Linear Equations: (graph is a line) (degree of polynomial must be 1) Both graphing and writing equations of lines Standard Form (Ax + By = C) Slope-intercept form ( y= mx + b )
Point-slope form y- y1 = m( x - x 1 ) Graphing: using slope and y-intercept Using x-intercept and y-intercept Using a T-table Review Best practices for graphing HOY VUX Perpendicular lines: slopes are negative reciprocals Parallel lines: have equal slopes Solving Systems (show solutions as ordered pairs) Graphing Substitution Elimination Cramer’s Rule (set up coefficient matrix and substitute your constants in place of the variable you’re solving for in numerator) Terminology: Independent, Dependent, Inconsistent Recall “special cases”: (CD = same line, infinite # solutions; Incon.= no solution = parallel lines)
Graphing Systems of Linear Inequalities Graph line (solid or dotted?), shade on the appropriate side by picking a test point (the origin works really well); the solution to the system is where the graphs overlap
Linear Programming, including word problems Graph system, shading only the overlapping region (region of feasibility) Potential max & min at the vertices of the region Define function first based on variables determined by what you’re asked to find Evaluate function to find maximum or minimum
Matrices: Dimensions: row x column Scalar multiplication Adding/subtracting matrices Determinants of 2x2 matrix Evaluating 3x3 determinants by diagonals AND by expansion by minors (don’t forget alternating signs) Matrix multiplication Finding the area of a triangle using matrices Inverse matrices Solving systems by use of matrices
Honors: step functions (greatest integer function f( x ) = [ x]
Piecewise functions and their graphs
------2nd Nine Weeks Monomials and Polynomials
Monomials: Multiplying (add powers)
Dividing (subtract powers)
Power to a Power (multiply exponents) Zero Exponent (anything to the zero power is 1)
1 Negative Exponents a-n = an
Scientific Notation (be able to go from standard notation to scientific and back; Find products and quotients of number expressed in scientific notation)
Polynomials: type (monomial, binomial, trinomial) Degree (the degree of each monomial term is the sum of its exponents- The degree of the polynomial is the biggest degree of all of the monomial pieces). Descending powers of a variable
Adding and subtracting polynomials (combine like-terms; add/subtract the coefficients) Subtract by adding its opposite. Multiplying a Polynomial by a Monomial (distribute) Multiplying a Polynomial by a Polynomial (distribute as many times as necessary) To multiply two binomials together, FOIL (distribute twice) “Special Products” (square of a sum, square of a difference, difference of squares) – still, FOIL!
Factoring
Prime, composite numbers Prime factorization Greatest common factor (GCF) Factoring = “un-Distributing” 7 Rules for Factoring: 1) GCF 2) Difference of Two Squares 3) Trinomial with Leading Coeff. = 1 4) Sum/Difference of Two Cubes 5) Perfect Square Trinomials 6) Factoring by Grouping (4 terms) 7) Leading Coefficient > 1 (Guess and check) Factoring “Flow Chart” Solving Equations by Factoring (using zero product property)
Radicals Radical expressions Rational Exponents Solving Radical Equations and Inequalities
Complex Numbers a+ bi Cycle of powers of i Quadratics:
Solving Quadratic Equations by Factoring Solving Quadratic Equations by Completing the Square Quadratic Formula and the Discriminant
Additional Honors Topics:
More extensive work with rational exponents Deriving Quadratic Formula Simplifying radical expressions involving the use of absolute value symbols