Reference Journal

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Table of ContentsChapter 1 – Review of Algebra…page 5

Chapter 2 – Word Problems… page

Chapter 3 – Factoring …………page

Chapter 4- Algebraic Fractions… page

Chapter 5- Graphing Equations…page

Chapter 6- Systems of Equations... Page

Chapter 7- Radicals ……………….Page

Chapter 8- Quadratic Equations…..Page

Table of Contents

Chapter 9 – Proofs

Chaoter 10- Transformations

Chapter 1- Algebra Review

Evaluating ExpressionsRemember to put fractions in ( ) in calc

Remember to put any negative numbers in ( ) in the calc… if there is an exponent that goes after the close )

Chapter 1 Algebra Review

Order of Operations PEMDAS(x +y)2 … in order to solve this re-write twice and foilIf there is a negative in front of ( ) make sure to multiply all terms by a negative

–ex. 40 –(3x+2) = 40 -3x -2

When multiplying polynomials with exponents…add the exponentsWhen dividing polynomials with exponents… subtract the exponents

Chapter 1 Algebra Review

Solving Equations

Multiply on either side of the equation to get rid of the ( )… DO NOT write ( ) again!

Combine like terms on each side of the equation separately

Begin to move terms from one side of the equation to the other using inverse operations

Chapter 2- Word Problems

Number Word problems5 less than twice a number… 2x-5

Product = *, Quotient = /,

Sum = +, Difference= -

Chapter 2 - Word ProblemsConsecutive Integer Problems

Motion Problems

Coin Problems

Percent Problems

Investment

Age

Chapter 3 Factoring

LOOK FOR GCF!!!!!!!!!!

Look for a Difference of 2 Perfect Squares (a + b) (a –b)

Look for a trinomial and do factoring by grouping!!!!

Factoring By Grouping

Multiply first number (not variable) by the last number

Look for factors of this new number that add to the center term

Re-write trinomial splitting the center term into these two factors with the variable!!!

Draw (), factor out GCF from each group

If two binomials match, smile and write that binomial once and the other numbers and variables as the other binomial

71.(3y-2)(y-2) 87. (r-5s)(r+2s)73.(2x+3)(x-1)89.(m+n)(5m-2n)75.(3x+5)(x-1)77.(5x-8)(x+1) 79.(2x-5)(2x-1)81.(3x+4)(2x-1) 83.(x+5)(10x-1)85.(x+2y)(x+y)

6. s(t-1)(t+1) 24. 4(5x+3y)(5x-3y)8. 2(x-4)(x+4) 26. 3(x+1)(x+1)10.2(3m-2)(3m+2) 28. x(x+5)(x+2)12. 7(3c-1)(3c+1) 30. 2a(x-3)(x+2)14.y(y-5)(y+5) 32. (z3-z)(z3+z) 16.a(2a+b)(2a-b) 34. (x2+2)(x+1)(x-1)18.d(3b+1)(3b-1)36. (y+3)(y-3)(y-2)(y+2)20 (x2+1)(x-1)(x+1)22. 5(r+R)(r-R)

Chapter 4- Algebraic Fractions

Multiplication and Division of EXPRESSIONS-

Factor all numerators and denominators

For division: flip the second fraction

Cancel any like terms if one is in the top and one is in the bottom

Multiply across (top times top and bottom times bottom) and simplify

Adding and Subtracting EXPRESSIONS

Find the lowest common denominator…sometimes you have to factor the bottoms to do this

Multiply each fraction by the missing terms of the denominator

Add the tops together…make sure to put in () and distribute if there is subtraction

*DO NOT CANCEL BOTTOMS

Algebraic EquationsFollow the steps for adding and subtracting… except after you create like denominators you can cancel them away

Finish solving the equation with only the tops

Inequality EquationsFollow steps for regular equations (with or without fractions)

If you are multiplying or dividing by a negative number, you MUST flip the inequality sign

Solving Equations with many Variables

Always get all the terms that include an (x) on one side of the equation and all the other terms on the opposite side

Usually you factor afterwards

Chapter 5- Graphing Equations Y = mx + b … this is the form that all

Linear equations should be in if you are going to graph them

M = slope = (y2-y1)/(x2-x1)= numerator is the amount you rise, the denominator is the amount you run (left)

B= y-intercept- point at which the line crosses the y-axis

To graph: 1) plot b 2) then use slope for rise and run

Graphing in Calc:Press Y=Make sure the Y= menu is clear Type in the given equation in y=mx +b formPress graph to see the graph, if you don’t see the graph, press zoom and then hit # 6 zoom standard, then hit graph again…now u should see the graphPress 2nd, then hit graph, this shows the table for your given equation

Types of Slopes Positive –rises from left to right (m is pos)

Negative- falls from left to right (m is neg)

Zero slope – horizontal line… y=?

Undefined slope or No slope – vertical line… x = ?

Parallel lines have the same slope

Given two Points: Write the equation of a line

Plug the two points into the slope formula

Plug the slope into y = mx +b, for m

Plug in one of the two given points into the equation with the slope, solve for b

Then plug the slope and y-intercept into y = mx+ b

Absolute Value Graphs

Y = |x|… this graph is a v that has its center point on the origin

Y = |x| + a … this moves the v up or down on the y-axis

Y = |x + a|… this moves the v left or right… it moves in the opposite direction of a

Y = a|x|…this makes the v wider or thinner, if a is a fraction =wider, if a is a number greater than 1 = thinner

Absolute Value Graphs ContinuedY = -|x|…v flips upside down

For the Calculator:

Press y=, press MATH, press right arrow to NUM, press #1 abs,

put the part of the equation that is inside the bars inside (), put all other parts outside the ()… Y = 2|x-3|+4… the x-3 goes inside the ()

x = |y|… sideways in the 1st and 4th quadrant

Graphing InequalitiesPut the equation in y = mx + b form, remember if you are dividing or multiplying by a negative you must flip the inequality sign

< or > then the line is dashed

≥ or ≤ then the line is solid

To see shading on the calc:

Press y=, move cursor to the far left so that the line next to Y1 is blinking, then press ENT either 3 or 4 times, 3 times for greater than, 4 times for less than

Chapter 6- Systems of EquationsConsistent System-

Inconsistent System-

Dependent System-

The 3 methods of solving a systemGraphically

Addition Method

Substitution Method

Word Problems

Chapter 7 – Radicals

How do I add or subtract radicals?

How do I multiply radicals?Monomial*monomial

Binomial *binomial

How do I divide with radicals?

How do I solve Radical Equations?

How do I add/subtract with Radicals?Simplify all radicals

List the factor sets for the number under the radical sign and break the number into the factor set that has the largest perfect square

Then take the square root of the perfect square and write it on the outside of the radical sign

Variables- even exponents divide by 2 and write the variable with that exponent on the outside of the radical sign

Odd exponents- subtract one from the exponent and leave this on the inside of the radical, then divide the even exponent by 2 and write on the outside of the radical sign

ONLY ADD/SUB if numbers/variables under the radical are exactly the same!!!

How do I multiply/divide radicals?

Multiply/divide the numbers/variables outside the radical sign by other numbers/variables outside the radical sign and only multiply/divide interior numbers/variables by interior numbers/variables

Ex. (3√5)(4 √6) = 12 √30

Ex. 2 * √5= 2 √5

FOIL FOR BINOMIAL*BINOMIAL

Ex. (3√5 + 2)(4 √6 + 4)= 12 √30+12 √5+8√6+8

Rationalizing a FractionNEVER leave a radical in the denominator of a fraction

Multiply the numerator and denominator of the fraction by the radical number… this will make the radical go away in the denominator because you are squaring it

Ex. (3√5)/(4 √6)…multiply by√6/√6…answer is √30/8

Perfect Squares to 15 and 20

4,9,16,25,36,49,64,81,100,121,144,

169,196,225…400

Solving Radical Equations

Isolate the Radical portion of the equation

Square both sides of the equationIf a binomial is being squared, make sure to write twice and FOIL

Now the radical should have disappeared… solve like a regular equation…you should always check your answers…you can get an erroneous answer

Chapter 8- Quadratic EquationsHow do I solve for the Roots of an equation? (find the zeros of the equation)

Using the calculator:

Using Factoring: Put equation in standard form so that it is equal to zero

When taking the square root of both sides of an equation you get a positive and negative answer

See chapter 3 for factoring steps

Using the Quadratic Formula: X= -b +/- √b2 – 4ac

2a

Using the Calculator:TO find the Roots: Type equation into Y=Press zoom, press zoom standardPress 2nd, trace, #2 for ZeroUse cursor to go to the left side of the first root, press ENTUse cursor to go to the right side of the first root, press ENTPress ENT again…answer appearsRepeat for second root on rightAlso you can use these steps to find a minimum or maximum

Using the Calculator:TO Graph a parabola with the vertex:

To find the vertex without the calc…….x= -b/2a

Enter the equation in Y=, press graph

Press 2nd, Graph for Table

Look for symmetry in the y chart

The number in the center of the symmetry is your vertex

Plot those 7 points on your graph!

Word Problems

Consecutive Integer, Even and OddX, x+1, x+2Even/odd- x, x+2, x+4

Rectangle Problems: Perimeter = 2L +2w Area= L*W

Remember to always put a binomial variable in () in an equationEx. The sum of the squares of two consecutive integers… X2 + (x+1)2

ProofsAddition and Subtraction Proofs

Partition… a part + a part = whole

Substitution… the two pieces you are trying to prove congruent

Use addition if you have 4 small pieces or 3 small pieces (then you need reflexive)

Use subtraction if you have 2 big pieces and 2 small pieces or 2 big pieces and 1 small piece (then you need reflexive)

List of TheoremsDefinition of Midpoint: a line has only one midpt which cuts it in half

Definition of a Angle Bisector: it is a line that cuts an ANGLE in half

Definition of a Line Bisector: it is a line that goes to the MIDPT

Definition of a Median: a line drawn from a vertex of a triangle to the midpt of a side

Definition of Altitude: a line drawn from a vertex that is perpendicular to the side (makes right angles)

List of TheoremsPerpendicular lines form right angles

All right angles are congruent

Vertical angles are congruent

Definition of Supplementary Angles

Supplements of congruent angles are congruent

If two angles are congruent and supplementary then they are right angles

Definition of Complementary angles

Complements of congruent angles are congruent

List of Theorems Isos. Triangle Theorem: if 2 base angles are congruent, then 2 sides are congruent

In an isos. Triangle, if the line drawn from the vertex angle is an altitude then it is also the median and angle bisector

Definition of an Equilateral Triangle: all sides and angles are congruent

Perpendicular Bisector Theorem: if given a picture that looks like a kite with the 2 top sides congruent and 2 bottom sides congruent, then there is a perp. Bis.

Proving TrianglesSAS, ASA, SSSCPCTC… use this after you prove 2 triangles congruent to prove other sides or angles congruentIf you are given 2 sets of lines or angles that are cut in half and you want to prove that half of one line is congruent to half of the other line then ….Use Division, or halves of equal quantities are equal or multiplicationReflexive: a side or angle is congruent to itselfAll right angles are 90All straight lines are 180 degrees

Chapter 10- TransformationsLine Reflection… r name of the line you reflect over

Point Reflection… Ro for origin or another point (x,y)

Rotation… Rdegrees

Dilation… Dnumber that you multiply the preimage by

Translation…T(x,y) that you add to the preimage

Glide Reflection… composition of translation and reflectionLine Symmetry Point SymmetryOrientation, Direct Isometry and Opposite Isometry

Line and Point Reflections

r x-axis…(x,y) …(x, -y)

r y-axis…(x,y)…(-x,y)

r y=x …(x,y) …(y,x)

r y=-x…(x,y)…(-y,-x)

Ro…reflect over origin (x,y)…(-x,-y)

Count the boxes to the origin and then go that distance from the origin in the other direction

RotationsR90=R-270…(x,y)…(-y,x)

R180=R-180…(x,y)…(-x,-y)

R270=R-90…(x,y)…(y,-x)

Count the distance that the preimage is from the x-axis and make the image this distance from the y-axis in the correct quadrant. Count the distance the preimage is from the y-axis and make the image this distance from the x-axis.

Dilation and TranslationDilation – Da...(x,y)…(ax,ay)

Translation-T(a,b)…(x,y)…(x+a,y+b)

Glide Reflection… either T(x,y) ◦ r y=x

Or ry=x ◦ T(x,y)

◦….then… but you work from RIGHT to LEFT

Defintions:Orientation- the letters going around a figure from right to left or left to rightDirect Isometry- is a transformation that preserves distance and orientation…translation, rotation, point reflectionOpposite Isometry- preserves distance but not orientation…line reflectionDilation – DOES NOT PRESERVE DISTANCE

Definitions Line Symmetry- you can cut it in half in a direction or you can fold the figure on top of itself perfectly in some direction

Point Symmetry- turn your piece of paper 180 degrees and the picture should look the same!!!