ALGEBRA II: CONCEPTS AND APPLICATIONS
ABOUT THIS BOOK CONVENTIONS USED IN THIS BOOK FOOLISH ASSUMPTIONS HOW THIS BOOK IS ORGANIZED PART I: HOMING IN ON BASIC SOLUTIONS PART II: FACING OFF WITH FUNCTIONS PART III: CONQUERING CONICS AND SYSTEMS OF EQUATIONS PART IV: SHIFTING INTO HIGH GEAR WITH ADVANCED CONCEPTS PART V: THE PART OF TENS ICONS USED IN THIS BOOK WHERE TO GO FROM HERE OUTLINING ALGEBRA PROPERTIES KEEPING ORDER WITH THE COMMUTATIVE PROPERTY MAINTAINING GROUP HARMONY WITH THE ASSOCIATIVE PROPERTY DISTRIBUTING A WEALTH OF VALUES CHECKING OUT AN ALGEBRAIC ID SINGING ALONG IN-VERSES ORDERING YOUR OPERATIONS EQUIPPING YOURSELF WITH THE MULTIPLICATION PROPERTY OF ZERO EXPOUNDING ON EXPONENTIAL RULES MULTIPLYING AND DIVIDING EXPONENTS GETTING TO THE ROOTS OF EXPONENTS RAISING OR LOWERING THE ROOF WITH EXPONENTS MAKING NICE WITH NEGATIVE EXPONENTS IMPLEMENTING FACTORING TECHNIQUES FACTORING TWO TERMS TAKING ON THREE TERMS FACTORING FOUR OR MORE TERMS BY GROUPING LINEAR EQUATIONS: HANDLING THE FIRST DEGREE TACKLING BASIC LINEAR EQUATIONS CLEARING OUT FRACTIONS ISOLATING DIFFERENT UNKNOWNS LINEAR INEQUALITIES: ALGEBRAIC RELATIONSHIP THERAPY SOLVING BASIC INEQUALITIES INTRODUCING INTERVAL NOTATION COMPOUNDING INEQUALITY ISSUES ABSOLUTE VALUE: KEEPING EVERYTHING IN LINE SOLVING ABSOLUTE-VALUE EQUATIONS SEEING THROUGH ABSOLUTE-VALUE INEQUALITY SOLVING SIMPLE QUADRATICS WITH THE SQUARE ROOT RULE FINDING SIMPLE SQUARE-ROOT SOLUTIONS DEALING WITH RADICAL SQUARE-ROOT SOLUTIONS DISMANTLING QUADRATIC EQUATIONS INTO FACTORS FACTORING BINOMIALS FACTORING TRINOMIALS FACTORING BY GROUPING RESORTING TO THE QUADRATIC FORMULA FINDING RATIONAL SOLUTIONS STRAIGHTENING OUT IRRATIONAL SOLUTIONS FORMULATING HUGE QUADRATIC RESULTS COMPLETING THE SQUARE: WARMING UP FOR CONICS SQUARING UP TO SOLVE A QUADRATIC EQUATION COMPLETING THE SQUARE TWICE OVER GETTING PROMOTED TO HIGH-POWERED QUADRATICS (WITHOUT THE RAISE) HANDLING THE SUM OR DIFFERENCE OF CUBES TACKLING QUADRATIC-LIKE TRINOMIALS SOLVING QUADRATIC INEQUALITIES KEEPING IT STRICTLY QUADRATIC SIGNING UP FOR FRACTIONS INCREASING THE NUMBER OF FACTORS ACTING RATIONALLY WITH FRACTION-FILLED EQUATIONS SOLVING RATIONAL EQUATIONS BY TUNING IN YOUR LCD SOLVING RATIONAL EQUATIONS WITH PROPORTIONS RIDDING YOURSELF OF A RADICAL SQUARING BOTH SIDES OF A RADICAL EQUATION CALMING TWO RADICALS CHANGING NEGATIVE ATTITUDES ABOUT EXPONENTS FLIPPING NEGATIVE EXPONENTS OUT OF THE PICTURE FACTORING OUT NEGATIVES TO SOLVE EQUATIONS FOOLING AROUND WITH FRACTIONAL EXPONENTS COMBINING TERMS WITH FRACTIONAL EXPONENTS FACTORING FRACTIONAL EXPONENTS SOLVING EQUATIONS BY WORKING WITH FRACTIONAL EXPONENTS COORDINATING YOUR GRAPHING EFFORTS IDENTIFYING THE PARTS OF THE COORDINATE PLANE PLOTTING FROM DOT TO DOT STREAMLINING THE GRAPHING PROCESS WITH INTERCEPTS AND SYMMETRY FINDING X- AND Y-INTERCEPTS REFLECTING ON A GRAPH’S SYMMETRY GRAPHING LINES FINDING THE SLOPE OF A LINE FACING TWO TYPES OF EQUATIONS FOR LINES IDENTIFYING PARALLEL AND PERPENDICULAR LINES LOOKING AT 10 BASIC FORMS LINES AND QUADRATICS CUBICS AND QUARTICS RADICALS AND RATIONALS EXPONENTIAL AND LOGARITHMIC CURVES ABSOLUTE VALUES AND CIRCLES SOLVING PROBLEMS WITH A GRAPHING CALCULATOR ENTERING EQUATIONS INTO GRAPHING CALCULATORS CORRECTLY LOOKING THROUGH THE GRAPHING WINDOW DEFINING FUNCTIONS INTRODUCING FUNCTION NOTATION EVALUATING FUNCTIONS HOMING IN ON DOMAIN AND RANGE DETERMINING A FUNCTION’S DOMAIN DESCRIBING A FUNCTION’S RANGE BETTING ON EVEN OR ODD FUNCTIONS RECOGNIZING EVEN AND ODD FUNCTIONS APPLYING EVEN AND ODD FUNCTIONS TO GRAPHS FACING ONE-TO-ONE CONFRONTATIONS DEFINING ONE-TO-ONE FUNCTIONS ELIMINATING ONE-TO-ONE VIOLATORS GOING TO PIECES WITH PIECEWISE FUNCTIONS DOING PIECEWORK APPLYING PIECEWISE FUNCTIONS COMPOSING YOURSELF AND FUNCTIONS PERFORMING COMPOSITIONS SIMPLIFYING THE DIFFERENCE QUOTIENT SINGING ALONG WITH INVERSE FUNCTIONS DETERMINING IF FUNCTIONS ARE INVERSES SOLVING FOR THE INVERSE OF A FUNCTION INTERPRETING THE STANDARD FORM OF QUADRATICS STARTING WITH “A” IN THE STANDARD FORM FOLLOWING UP WITH “B” AND “C” INVESTIGATING INTERCEPTS IN QUADRATICS FINDING THE ONE AND ONLY Y-INTERCEPT FINDING THE X-INTERCEPTS GOING TO THE EXTREME: FINDING THE VERTEX LINING UP ALONG THE AXIS OF SYMMETRY SKETCHING A GRAPH FROM THE AVAILABLE INFORMATION APPLYING QUADRATICS TO THE REAL WORLD SELLING CANDLES SHOOTING BASKETBALLS LAUNCHING A WATER BALLOON TAKING A LOOK AT THE STANDARD POLYNOMIAL FORM EXPLORING POLYNOMIAL INTERCEPTS AND TURNING POINTS INTERPRETING RELATIVE VALUE AND ABSOLUTE VALUE COUNTING INTERCEPTS AND TURNING POINTS SOLVING FOR POLYNOMIAL INTERCEPTS DETERMINING POSITIVE AND NEGATIVE INTERVALS USING A SIGN-LINE INTERPRETING THE RULE FINDING THE ROOTS OF A POLYNOMIAL FACTORING FOR POLYNOMIAL ROOTS SAVING YOUR SANITY: THE RATIONAL ROOT THEOREM LETTING DESCARTES MAKE A RULING ON SIGNS SYNTHESIZING ROOT FINDINGS USING SYNTHETIC DIVISION TO TEST FOR ROOTS SYNTHETICALLY DIVIDING BY A BINOMIAL WRINGING OUT THE REMAINDER (THEOREM) EXPLORING RATIONAL FUNCTIONS SIZING UP DOMAIN INTRODUCING INTERCEPTS ADDING ASYMPTOTES TO THE RATIONAL POT DETERMINING THE EQUATIONS OF VERTICAL ASYMPTOTES DETERMINING THE EQUATIONS OF HORIZONTAL ASYMPTOTES GRAPHING VERTICAL AND HORIZONTAL ASYMPTOTES CRUNCHING THE NUMBERS AND GRAPHING OBLIQUE ASYMPTOTES ACCOUNTING FOR REMOVABLE DISCONTINUITIES REMOVAL BY FACTORING EVALUATING THE REMOVAL RESTRICTIONS SHOWING REMOVABLE DISCONTINUITIES ON A GRAPH PUSHING THE LIMITS OF RATIONAL FUNCTIONS EVALUATING LIMITS AT DISCONTINUITIES GOING TO INFINITY CATCHING RATIONAL LIMITS AT INFINITY PUTTING IT ALL TOGETHER: SKETCHING RATIONAL GRAPHS FROM CLUES EVALUATING EXPONENTIAL EXPRESSIONS EXPONENTIAL FUNCTIONS: IT’S ALL ABOUT THE BASE, BABY OBSERVING THE TRENDS IN BASES MEETING THE MOST FREQUENTLY USED BASES: 10 AND E SOLVING EXPONENTIAL EQUATIONS MAKING BASES MATCH RECOGNIZING AND USING QUADRATIC PATTERNS SHOWING AN “INTEREST” IN EXPONENTIAL FUNCTIONS APPLYING THE COMPOUND INTEREST FORMULA LOOKING AT CONTINUOUS COMPOUNDING LOGGING ON TO LOGARITHMIC FUNCTIONS MEETING THE PROPERTIES OF LOGARITHMS PUTTING YOUR LOGS TO WORK SOLVING LOGARITHMIC EQUATIONS SETTING LOG EQUAL TO LOG REWRITING LOG EQUATIONS AS EXPONENTIALS GRAPHING EXPONENTIAL AND LOGARITHMIC FUNCTIONS EXPOUNDING ON THE EXPONENTIAL NOT SEEING THE LOGS FOR THE TREES CUTTING UP A CONE OPENING EVERY WHICH WAY WITH PARABOLAS LOOKING AT PARABOLAS WITH VERTICES AT THE ORIGIN OBSERVING THE GENERAL FORM OF PARABOLA EQUATIONS SKETCHING THE GRAPHS OF PARABOLAS CONVERTING PARABOLIC EQUATIONS TO THE STANDARD FORM GOING ROUND AND ROUND IN CONIC CIRCLES STANDARDIZING THE CIRCLE SPECIALIZING IN CIRCLES PREPARING YOUR EYES FOR SOLAR ELLIPSES RAISING THE STANDARDS OF AN ELLIPSE SKETCHING AN ELLIPTICAL PATH FEELING HYPER ABOUT HYPERBOLAS INCLUDING THE ASYMPTOTES GRAPHING HYPERBOLAS IDENTIFYING CONICS FROM THEIR EQUATIONS, STANDARD OR NOT LOOKING AT THE STANDARD LINEAR-SYSTEMS FORM AND ITS POSSIBLE SOLUTIONS GRAPHING SOLUTIONS OF LINEAR SYSTEMS PINPOINTING THE INTERSECTION TOEING THE SAME LINE TWICE DEALING WITH PARALLEL LINES ELIMINATING SYSTEMS OF TWO LINEAR EQUATIONS WITH ADDITION GETTING TO AN ELIMINATION POINT RECOGNIZING SOLUTIONS FOR PARALLEL AND COEXISTING LINES SOLVING SYSTEMS OF TWO LINEAR EQUATIONS WITH SUBSTITUTION VARIABLE SUBSTITUTING MADE EASY IDENTIFYING PARALLEL AND COEXISTING LINES USING CRAMER’S RULE TO DEFEAT UNWIELDY FRACTIONS SETTING UP THE LINEAR SYSTEM FOR CRAMER APPLYING CRAMER’S RULE TO A LINEAR SYSTEM RAISING LINEAR SYSTEMS TO THREE LINEAR EQUATIONS SOLVING THREE-EQUATION SYSTEMS WITH ALGEBRA SETTLING FOR A GENERALIZED SOLUTION FOR LINEAR COMBINATIONS UPPING THE ANTE WITH INCREASED EQUATIONS APPLYING LINEAR SYSTEMS TO OUR 3-D WORLD USING SYSTEMS TO DECOMPOSE FRACTIONS CROSSING PARABOLAS WITH LINES DETERMINING THE POINT(S) WHERE A LINE AND PARABOLA CROSS PATHS DEALING WITH A SOLUTION THAT’S NO SOLUTION INTERTWINING PARABOLAS AND CIRCLES MANAGING MULTIPLE INTERSECTIONS SORTING OUT THE SOLUTIONS PLANNING YOUR ATTACK ON OTHER SYSTEMS OF EQUATIONS MIXING POLYNOMIALS AND LINES CROSSING POLYNOMIALS NAVIGATING EXPONENTIAL INTERSECTIONS ROUNDING UP RATIONAL FUNCTIONS PLAYING FAIR WITH INEQUALITIES DRAWING AND QUARTERING INEQUALITIES GRAPHING AREAS WITH CURVES AND LINES USING YOUR IMAGINATION TO SIMPLIFY POWERS OF I UNDERSTANDING THE COMPLEXITY OF COMPLEX NUMBERS OPERATING ON COMPLEX NUMBERS MULTIPLYING BY THE CONJUGATE TO PERFORM DIVISION SIMPLIFYING RADICALS SOLVING QUADRATIC EQUATIONS WITH COMPLEX SOLUTIONS WORKING POLYNOMIALS WITH COMPLEX SOLUTIONS IDENTIFYING CONJUGATE PAIRS INTERPRETING COMPLEX ZEROS DESCRIBING THE DIFFERENT TYPES OF MATRICES ROW AND COLUMN MATRICES SQUARE MATRICES ZERO MATRICES IDENTITY MATRICES PERFORMING OPERATIONS ON MATRICES ADDING AND SUBTRACTING MATRICES MULTIPLYING MATRICES BY SCALARS MULTIPLYING TWO MATRICES APPLYING MATRICES AND OPERATIONS DEFINING ROW OPERATIONS FINDING INVERSE MATRICES DETERMINING ADDITIVE INVERSES DETERMINING MULTIPLICATIVE INVERSES DIVIDING MATRICES BY USING INVERSES USING MATRICES TO FIND SOLUTIONS FOR SYSTEMS OF EQUATIONS UNDERSTANDING SEQUENCE TERMINOLOGY USING SEQUENCE NOTATION NO-FEAR FACTORIALS IN SEQUENCES ALTERNATING SEQUENTIAL PATTERNS LOOKING FOR SEQUENTIAL PATTERNS TAKING NOTE OF ARITHMETIC AND GEOMETRIC SEQUENCES FINDING COMMON GROUND: ARITHMETIC SEQUENCES TAKING THE MULTIPLICATIVE APPROACH: GEOMETRIC SEQUENCES RECURSIVELY DEFINING FUNCTIONS MAKING A SERIES OF MOVES INTRODUCING SUMMATION NOTATION SUMMING ARITHMETICALLY SUMMING GEOMETRICALLY APPLYING SUMS OF SEQUENCES TO THE REAL WORLD CLEANING UP AN AMPHITHEATER NEGOTIATING YOUR ALLOWANCE BOUNCING A BALL HIGHLIGHTING SPECIAL FORMULAS REVEALING SET NOTATION LISTING ELEMENTS WITH A ROSTER BUILDING SETS FROM SCRATCH GOING FOR ALL (UNIVERSAL SET) OR NOTHING (EMPTY SET) SUBBING IN WITH SUBSETS OPERATING ON SETS CELEBRATING THE UNION OF TWO SETS LOOKING BOTH WAYS FOR SET INTERSECTIONS FEELING COMPLEMENTARY ABOUT SETS COUNTING THE ELEMENTS IN SETS DRAWING VENN YOU FEEL LIKE IT APPLYING THE VENN DIAGRAM USING VENN DIAGRAMS WITH SET OPERATIONS ADDING A SET TO A VENN DIAGRAM FOCUSING ON FACTORIALS MAKING FACTORIAL MANAGEABLE SIMPLIFYING FACTORIALS HOW DO I LOVE THEE? LET ME COUNT UP THE WAYS APPLYING THE MULTIPLICATION PRINCIPLE TO SETS ARRANGING PERMUTATIONS OF SETS MIXING UP SETS WITH COMBINATIONS BRANCHING OUT WITH TREE DIAGRAMS PICTURING A TREE DIAGRAM FOR A PERMUTATION DRAWING A TREE DIAGRAM FOR A COMBINATION