Reference Journal Your Name
Jan 01, 2016
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
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
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