Math Handbook Of Formulas, Processes And Tricks
Math Handbookof Formulas, Processes and Tricks(www.mathguy.us)Algebra and PreCalculusPrepared by: Earl L. Whitney, FSA, MAAAVersion 3.2July 10, 2019Copyright 2008‐19, Earl Whitney, Reno NV. All Rights Reserved
Algebra HandbookTable of ContentsPageDescription91011121314Chapter 1: BasicsOrder of Operations (PEMDAS, Parenthetical Device)Graphing with Coordinates (Coordinates, Plotting Points)Linear Patterns (Recognition, Converting to an Equation)Identifying Number PatternsCompleting Number PatternsReal Number Sets (Sets of Numbers, Real Number Set Tree)1516Chapter 2: OperationsOperating with Real Numbers (Absolute Value, Add, Subtract, Multiply, Divide)Properties of Algebra (Addition & Multiplication, Zero, Equality)1819Chapter 3: Solving EquationsSolving Multi‐Step EquationsTips and Tricks in Solving Multi‐Step Equations20212223Chapter 4: Probability & StatisticsProbability and OddsProbability with DiceCombinationsStatistical Measures2425262728293031323334Chapter 5: FunctionsIntroduction to Functions (Definitions, Line Tests)Special Integer FunctionsOperations with FunctionsComposition of FunctionsInverses of FunctionsTransformation – TranslationTransformation – Vertical Stretch and CompressionTransformation – Horizontal Stretch and CompressionTransformation – ReflectionTransformation – SummaryBuilding a Graph with TransformationsCover art by Rebecca Williams,Twitter handle: @jolteonkittyVersion 3.2Page 2 of 187July 10, 2019
Algebra HandbookTable of ContentsPageDescription35363738394041Chapter 6: Linear FunctionsSlope of a Line (Mathematical Definition)Slope of a Line (Rise over Run)Slopes of Various Lines (8 Variations)Various Forms of a Line (Standard, Slope‐Intercept, Point‐Slope)Slopes of Parallel and Perpendicular LinesParallel, Perpendicular or NeitherParallel, Coincident or Intersecting42434445464748Chapter 7: InequalitiesProperties of InequalitiesGraphs of Inequalities in One DimensionCompound Inequalities in One DimensionInequalities in Two DimensionsGraphs of Inequalities in Two DimensionsAbsolute Value Functions (Equations)Absolute Value Functions (Inequalities)49505152535455Chapter 8: Systems of EquationsGraphing a SolutionSubstitution MethodElimination MethodClassification of Systems of EquationsLinear DependenceSystems of Inequalities in Two DimensionsParametric Equations56575859Chapter 9: Exponents (Basic) and Scientific NotationExponent FormulasScientific Notation (Format, Conversion)Adding and Subtracting with Scientific NotationMultiplying and Dividing with Scientific NotationVersion 3.2Page 3 of 187July 10, 2019
Algebra HandbookTable of hapter 10: Polynomials – BasicIntroduction to PolynomialsAdding and Subtracting PolynomialsMultiplying Binomials (FOIL, Box, Numerical Methods)Multiplying PolynomialsDividing PolynomialsFactoring PolynomialsSpecial Forms of Quadratic Functions (Perfect Squares)Special Forms of Quadratic Functions (Differences of Squares)Factoring Trinomials – Simple Case MethodFactoring Trinomials – AC MethodFactoring Trinomials – Brute Force MethodFactoring Trinomials – Quadratic Formula MethodSolving Equations by Factoring737475767779Chapter 11: Quadratic FunctionsIntroduction to Quadratic FunctionsCompleting the SquareTable of Powers and RootsThe Quadratic FormulaQuadratic Inequalities in One VariableFitting a Quadratic through Three Points808182838485Chapter 12: Complex NumbersComplex Numbers ‐ IntroductionOperations with Complex NumbersThe Square Root of iComplex Numbers – Graphical RepresentationComplex Number Operations in Polar CoordinatesComplex Solutions to Quadratic Equations86878889Chapter 13: RadicalsRadical RulesSimplifying Square Roots (Extracting Squares, Extracting Primes)Solving Radical EquationsSolving Radical Equations (Positive Roots, The Missing Step)Version 3.2Page 4 of 187July 10, 2019
Algebra HandbookTable of 103104105106107Chapter 14: MatricesMatrix Addition and Scalar MultiplicationMatrix MultiplicationIdentity Matrices and Inverse MatricesInverse of a 2x2 MatrixMatrix Division and InversesDeterminants – The General CaseCalculating Inverses – The General Case (Gauss‐Jordan Elimination)Calculating Inverses Using Adjoint MatricesCramer’s Rule – 2 EquationsCramer’s Rule – 3 EquationsAugmented Matrices2x2 Augmented Matrix Examples3x3 Augmented Matrix ExampleCharacteristic Equation and EigenvaluesEigenvectors2x2 Eigenvalues and Eigenvectors – General CaseCalculating Inverses Using Characteristic 4125Chapter 15: Exponents and LogarithmsExponent FormulasLogarithm FormulaseTable of Exponents and LogsConverting Between Exponential and Logarithmic FormsExpanding Logarithmic ExpressionsCondensing Logarithmic ExpressionsCondensing Logarithmic Expressions – More ExamplesGraphing an Exponential FunctionFour Exponential Function GraphsGraphing a Logarithmic FunctionFour Logarithmic Function GraphsGraphs of Various FunctionsApplications of Exponential Functions (Growth, Decay, Interest)Solving Exponential and Logarithmic EquationsVersion 3.2Page 5 of 187July 10, 2019
Algebra HandbookTable of 135136Chapter 16: Polynomials – IntermediatePolynomial Function GraphsFinding Extrema with DerivativesFactoring Higher Degree Polynomials – Sum and Difference of CubesFactoring Higher Degree Polynomials – Variable SubstitutionFactoring Higher Degree Polynomials – Synthetic DivisionComparing Synthetic Division and Long DivisionZeros of Polynomials – Developing Possible RootsZeros of Polynomials – Testing Possible RootsIntersections of Curves (General Case, Two Lines)Intersections of Curves (a Line and a Parabola)Intersections of Curves (a Circle and an Ellipse)137138139139140141143145146147Chapter 17: Rational FunctionsDomains of Rational FunctionsHoles and AsymptotesGraphing Rational FunctionsSimple Rational FunctionsSimple Rational Functions ‐ ExampleGeneral Rational FunctionsGeneral Rational Functions ‐ ExampleOperating with Rational ExpressionsSolving Rational EquationsSolving Rational 0161162Chapter 18: Conic SectionsIntroduction to Conic SectionsParabola with Vertex at the Origin (Standard Position)Parabola with Vertex at Point (h, k)Parabola in Polar FormCirclesEllipse Centered on the Origin (Standard Position)Ellipse Centered at Point (h, k)Ellipse in Polar FormHyperbola Centered on the Origin (Standard Position)Hyperbola Centered at Point (h, k)Hyperbola in Polar FormHyperbola Construction Over the Domain: 0 to 2πGeneral Conic Equation ‐ ClassificationGeneral Conic Formula – Manipulation (Steps, Examples)Parametric Equations of Conic SectionsVersion 3.2Page 6 of 187July 10, 2019
Algebra HandbookTable of 172173174175176177Chapter 19: Sequences and SeriesIntroduction to Sequences and SeriesFibonacci SequenceSummation Notation and PropertiesSome Interesting Summation FormulasArithmetic SequencesArithmetic SeriesPythagorean Means (Arithmetic, Geometric)Pythagorean Means (Harmonic)Geometric SequencesGeometric SeriesA Few Special Series (π, e, cubes)Pascal’s TriangleBinomial ExpansionGamma Function and n !Graphing the Gamma Function178IndexUseful WebsitesMathguy.us – Developed specifically for math students from Middle School to College, based on theauthor's extensive experience in professional mathematics in a business setting and in mathtutoring. Contains free downloadable handbooks, PC Apps, sample tests, and more.http://www.mathguy.us/Wolfram Math World – Perhaps the premier site for mathematics on the Web. This site containsdefinitions, explanations and examples for elementary and advanced math topics.http://mathworld.wolfram.com/Purple Math – A great site for the Algebra student, it contains lessons, reviews and homeworkguidelines. The site also has an analysis of your study habits. Take the Math Study Skills Self‐Evaluation to see where you need to improve.http://www.purplemath.com/Math.com – Has a lot of information about Algebra, including a good search htmlVersion 3.2Page 7 of 187July 10, 2019
Algebra HandbookTable of ContentsSchaum’s OutlinesAn important student resource for any high school math student is a Schaum’s Outline. Each bookin this series provides explanations of the various topics in the course and a substantial number ofproblems for the student to try. Many of the problems are worked out in the book, so the studentcan see examples of how they should be solved.Schaum’s Outlines are available at Amazon.com, Barnes & Noble, Borders and other booksellers.Note: This study guide was prepared to be a companion to most books on the subject of HighSchool Algebra. In particular, I used the following texts to determine which subjects to includein this guide. Algebra 1 , by James Schultz, Paul Kennedy, Wade Ellis Jr, and Kathleen Hollowelly. Algebra 2 , by James Schultz, Wade Ellis Jr, Kathleen Hollowelly, and Paul Kennedy.Although a significant effort was made to make the material in this study guide original, somematerial from these texts was used in the preparation of the study guide.Version 3.2Page 8 of 187July 10, 2019
Chapter 1BasicsAlgebraOrder of OperationsTo the non‐mathematician, there may appear to be multiple ways to evaluate an algebraicexpression. For example, how would one evaluate the following?3 4 76 5You could work from left to right, or you could work from right to left, or you could do anynumber of other things to evaluate this expression. As you might expect, mathematicians donot like this ambiguity, so they developed a set of rules to make sure that any two peopleevaluating an expression would get the same answer.PEMDASIn order to evaluate expressions like the one above, mathematicians have defined an order ofoperations that must be followed to get the correct value for the expression. The acronym thatcan be used to remember this order is PEMDAS. Alternatively, you could use the mnemonicphrase “Please Excuse My Dear Aunt Sally” or make up your own way to memorize the order ofoperations. The components of PEMDAS are: PAnything in Parentheses is evaluated first. E Items with Exponents are evaluated next. M Multiplication and D Division are performed next. A Addition and S Subtraction are performed last.Note: When there are multipleoperations in the same category,for example, a division and twomultiplications, the operationsare performed from left to right.Parenthetical Device. A useful device is to use apply parentheses to help you rememberthe order of operations when you evaluate an expression. Parentheses are placed around theitems highest in the order of operations; then solving the problem becomes more natural.Using PEMDAS and this parenthetical device, we solve the expression above as follows:Initial Expression:3 4 7Add parentheses/brackets:3 4 7Solve using PEMDAS:846 2584150Final AnswerVersion 3.26 56 5234Page 9 of 187Note: Any expression which isambiguous, like the one above, ispoorly written. Students should striveto ensure that any expressions theywrite are easily understood by othersand by themselves. Use of parenthesesand brackets is a good way to makeyour work more understandable.July 10, 2019
Chapter 1BasicsAlgebraGraphing with CoordinatesGraphs in two dimensions are very common in algebra and are one of the most commonalgebra applications in real life.yCoordinatesThe plane of points that can be graphed in 2 dimensions iscalled the Rectangular Coordinate Plane or the CartesianCoordinate Plane (named after the French mathematicianand philosopher René Descartes).Quadrant 2Quadrant 1xQuadrant 3Quadrant 4 Two axes are defined (usually called the x‐ and y‐axes). Each point on the plane has an x value and a y value, written as: (x‐value, y‐value) The point (0, 0) is called the origin, and is usually denoted with the letter “O”. The axes break the plane into 4 quadrants, as shown above. They begin with Quadrant 1where x and y are both positive and increase numerically in a counter‐clockwise fashion.Plotting Points on the PlaneWhen plotting points, the x‐value determines how far right (positive) or left (negative) of the origin the point isplotted. The y‐value determines how far up (positive) or down (negative) from the origin the point isplotted.Examples:The following points are plotted in the figure tothe right:A (2, 3)B (‐3, 2)C (‐2, ‐2)D (4, ‐1)O (0, 0)Version 3.2in Quadrant 1in Quadrant 2in Quadrant 3in Quadrant 4is not in any quadrantPage 10 of 187July 10, 2019
Chapter 1BasicsAlgebraLinear PatternsRecognizing Linear PatternsThe first step to recognizing a pattern is to arrange a set of numbers in a table. The table canbe either horizontal or vertical. Here, we consider the pattern in a horizontal format. Moreadvanced analysis generally uses the vertical format.Consider this pattern:x‐valuey‐value0619212315418521To analyze the pattern, we calculate differences of successive values in the table. These arecalled first differences. If the first differences are constant, we can proceed to converting thepattern into an equation. If not, we do not have a linear pattern. In this case, we may chooseto continue by calculating differences of the first differences, which are called seconddifferences, and so on until we get a pattern we can work with.In the example above, we get a constant set of first differences, which tells us that the patternis indeed linear.x‐valuey‐valueFirst Differences061932123315341835213Converting a Linear Pattern to an EquationCreating an equation from the pattern is easy if you haveconstant differences and a y‐value for x 0. In this case, The equation takes the form 𝒚 𝒎𝒙 𝒃, where “m” is the constant difference from the table, and “b” is the y‐value when x 0.In the example above, this gives us the equation: 𝒚𝟑𝒙𝟔.Note: If the table does not have avalue for x 0, you can still obtainthe value of “b”. Simply extend thetable left or right until you have anx‐value of 0; then use the firstdifferences to calculate what thecorresponding y‐value would be.This becomes your value of “b”.Finally, it is a good idea to test your equation. For example, if 𝑥 4, the above equation gives𝑦3 46 18, which is the value in the table. So we can be pretty sure our equation iscorrect.Version 3.2Page 11 of 187July 10, 2019
BasicsADVANCEDChapter 1AlgebraIdentifying Number PatternsWhen looking at patterns in numbers, is is often useful to take differences of the numbers youare provided. If the first differences are not constant, take differences again.n‐3‐11357 n2510172637 2357911222222222When first differences are constant, the pattern represents alinear equation. In this case, the equation is: y 2x ‐ 5 . Theconstant difference is the coefficient of x in the equation.When second differences are constant, the pattern represents aquadratic equation. In this case, the equation is: y x 2 1 . Theconstant difference, divided by 2, gives the coefficient of x2 in theequation.When taking successive differences yields patterns that do not seem to level out, the patternmay be either exponential or recursive.n5711193567n23581321Version 3.2 2248163224816 2123581123In the pattern to the left, notice that the first and seconddifferences are the same. You might also notice that thesedifferences are successive powers of 2. This is typical for anexponential pattern. In this case, the equation is: y 2 x 3 .In the pattern to the left, notice that the first and seconddifferences appear to be repeating the original sequence. Whenthis happens, the sequence may be recursive. This means thateach new term is based on the terms before it. In this case, theequation is: y n y n‐1 y n‐2 , meaning that to get each new term,you add the two terms before it.Page 12 of 187July 10, 2019
BasicsADVANCEDChapter 1AlgebraCompleting Number PatternsThe first step in completing a number pattern is to identify it. Then, work from the right to the left, filling inthe highest order differences first and working backwards (left) to complete the table. Below are twoexamples.Example 1Example 2n‐162562123214n‐162562123214n‐162562123214 271937619112182430666 2 71937619112182430n ‐176192537626112391214127341169510217727 33666666 2 312182430364248666666Consider in the examples the sequences of sixnumbers which are provided to the student. You areasked to find the ninth term of each sequence.n23581321Step 1: Create a table of differences. Take successivedifferences until you get a column of constantdifferences (Example 1) or a column that appears torepeat a previous column of differences (Example 2).n23581321Step 2: In the last column of differences you created,continue the constant differences (Example 1) or therepeated differences (Example 2) down the table.Create as many entries as you will need to solve theproblem. For example, if you are given 6 terms andasked to find the 9th term, you will need 3 ( 9 ‐ 6)additional entries in the last column.n23581321Step 3: Work backwards (from right to left), filling ineach column by adding the differences in the columnto the right.n23581321345589In the example to the left, the calculations areperformed in the following order:2Column : 30 6 36; 36 6 42; 42 6 48Column : 91 36 127; 127 42 169; 169 48 217Column n: 214 127 341; 341 169 510; 510 217 727 2 3123581123011 2 123581123 2 312358132134112358130112353011235The final answers to the examples are the ninth items in each sequence, the items in bold red.Version 3.2Page 13 of 187July 10, 2019
Chapter 1BasicsAlgebraReal Number SetsNumber SetSymbolNatural (or,Counting) NumbersN orZ DefinitionExamplesNumbers that you wouldnormally count with.1, 2, 3, 4, 5, 6, Whole NumbersWAdd the number zero to theset of Natural Numbers0, 1, 2, 3, 4, 5, 6, IntegersZWhole numbers plus the setof negative Natural Numbers ‐3, ‐2, ‐1, 0, 1, 2, 3, Any number that can beRational NumbersQexpressed in the formAll integers, plus fractions andmixed numbers, such as:,2 174,, 3365where a and b are integersand 𝑏 0.Real NumbersRAll rational numbers plus rootsand some others, such as:Any number that can bewritten in decimal form,even if that form is infinite. 2 , 12 , π, eReal Number Set TreeReal ersion 3.2IrrationalFractions andMixed NumbersNegativeIntegersZeroPage 14 of 187July 10, 2019
Chapter 2OperationsAlgebraOperating with Real NumbersAbsolute ValueThe absolute value of something is the distance it is from zero. The easiest way to get theabsolute value of a number is to eliminate its sign. Absolute values are always positive or 0. 5 5 3 3 0 1.5 01.5Adding and Subtracting Real NumbersAdding Numbers with the Same Sign: Adding Numbers with Different Signs: Add the numbers without regardto sign.Give the answer the same sign asthe original numbers.Examples:63912 6 18 Ignore the signs and subtract thesmaller number from the larger one.Give the answer the sign of the numberwith the greater absolute value.Examples:633711 4Subtracting Numbers: Change the sign of the number or numbers being subtracted.Add the resulting numbers.Examples:6363313 4 1349Multiplying and Dividing Real NumbersNumbers with the Same Sign: Version 3.2Numbers with Different Signs: Multiply or divide the numberswithout regard to sign.Give the answer a “ ” sign.Examples:6 318 1812 34 4 Page 15 of 187Multiply or divide the numbers withoutregard to sign.Give the answer a “-” sign.Examples:6 3181234July 10, 2019
Chapter 2OperationsAlgebraProperties of AlgebraProperties of Addition and Multiplication. For any real numbers a, b, and c:PropertyDefinition for Addition𝑎Closure Property𝑏 is a real number𝑎Identity Property0𝑎Inverse Property𝑎0𝑎𝑎Commutative PropertyAssociative PropertyDefinition for Multiplication𝑏𝑎𝑎𝑎𝑏𝑏𝑐𝑎𝑎 1
Chapter 14: Matrices 90 Matrix Addition and Scalar Multiplication 91 Matrix Multiplication 92 Identity Matrices and Inverse Matrices 93 Inverse of a 2x2 Matrix 94 Matrix Division and Inverses 95 Determinants – The General Case 96 Calculating Inverses – The General Case (Gauss‐Jordan Elimination) .
MATH FORMULAS The most powerful EMPOWER tactic for the Quantitative section of the GMAT 1. Including Geometry, more than half the questions in the Quant section involve a specific math formula (or more than 1), so knowing the formulas is a must to solving the problem. 2. Math formulas can
SHOW FORMULAS If you ever want to see formulas in a spreadsheet’s cells, rather than the calculated answer to the formula, click the Show Formulas button in the Formula Auditing group of the Formulas tab. Formulas will appear in the spreadsheet
With the formulas! Unlike the SAT, the ACT does not give you a list of formulas at the beginning of the math test. That means that you absolutely must memorize some formulas. This PDF contains a list of math formulas the the ACT commonly tests. Print this out and carry it with you . That way, you can study a bit here and there and -
see what the data, formulas, and functions that are/were entered. The tutorial will work with student grades to provide a context for the calculations. Formulas (and cell references): Formulas allow us to do math with the values in our spreadsheet. In addition, we can use formulas combined
Math 5/4, Math 6/5, Math 7/6, Math 8/7, and Algebra 1/2 Math 5/4, Math 6/5, Math 7/6, Math 8/7, and Algebra ½ form a series of courses to move students from primary grades to algebra. Each course contains a series of daily lessons covering all areas of general math. Each lesson
[-] GED Math o [-] GED Test Overview [-] GED Math Test Overview Why Math? Math formulas for geometry [-] GED Math Test Overview Managing your time on the GED Math Test Introduction to GED Math Test [-] GED Test Math Overview GED Test Answer Grid o [-] Number Operations and Number Sense [-] Recognizing Numbe
The ACT Math tests students on high school math that should be learned between grades 9-11. Unlike the SAT, the ACT Math section does not provide a list of formulas that can be used on the test. Understanding and memorizing the majority of these formulas we've provided in this PDF are vital for success. Print this out for easy studying! ACT .
Naming Compounds / Writing Formulas / Mole Math / % Composition / Empirical & Molecular Formulas . Steps for Writing Binary Formulas Step1: Find the charges of the element’s ions. Step 2: Swap and drop the ion charges. Step 3: Don’t write ones(as subscripts) or charges(as
