Algebra

=5/4/11= Maggie Sheehan =  Add/Subtract   = =  ·  Only like terms    = =  ·  Exponents don’t change    = =  Multiply   = =  ·  Multiply __anything__    = =  ·  Exponents and same variables are added    = =  Multiplying – 3 methods   = =  1) Distributive (best for monomial x polynomial)    = =     = =   Ex. 3x^2 (4x^2-2x+1)   = =   12x^4-6x^3+3x^2   = =     = =   Ex. (5x^2-4) (3x^2-5x+10)   = =   15x^4-25x+50x^2-12x^2+20x-40   = =   15x^4-25x^3+38x^2+20x-40   = =     = =     = =   2) FOIL (only used bino x bino)    = =  Stands for First Outer Inner Last   = =    = =   Ex. (3x-7) (4x+11)   = =  12x^2+33x-28x-77   = =  12x^2+5x-77   = =    = =   3) Tables (best for big)   = =     = =   Ex. (x^2-2x+1) (4x^2-8x+6)   = =     = =||     =     = =   4x^4-16x^3+26x^2-20x+6   = = = =5/1/11= Maria Ezzell __Polynomial__ -math expression containing numbers, variables, and/or exponents in 1 or more terms.
 * 4x^2  ||   -8x   ||   6   || ||   x^2   ||   4x^4   ||   -8x^3   ||   6x^2   || ||   -2x   ||   -8x^3   ||   16x^2   ||   -12x   || ||   1   ||   4x^2   ||   -8x   ||   6   ||=
 * Can't have negative exponents and variables.
 * Topically no.

__Monomial__ - Has one term. Examples: -7, 2xy, 14abcde, 8x^4y^7
 * Degree of a monomial= sumof its exponents. Examples: 3x^2y= Degree 3, 4x^4y^8= Degree 12, 17= Degree 0
 * No variable= Degree 0.

__Binomial__ - Has two terms. Examples: 5x-1, 3xy+1/2, 4x^5-3x

__Trinamial__ - Has three terms. Examples: 4x^2-11x+5, 3xy-6x^2+7z

__Degree of a polynomial__ - Degree of it's highest term. Examples: 3x^2y+11xy-6xy^4 (2+1) (1+1) (1+4) = Degree 5 polynomial

4+11x^2y-6x^3y^4 =Degree 7

4x^3-8x^5+x-11x^2-5 (in descending order would be)= -8x^5+4x^3-11x^2+x-5
 * write in descending order.

Adding and Subtracting

 * combine like terms (same variable and same expoient)
 * Add:The subtract sign gets distributed Example:(4x^2-2x+11)+(5x-6+2x^3) changed to: 2x^3+4x^2+3x+5
 * Subtract- change to a postive than the rest following to there oppisite. Example: (3x^4x+1)-(x^3-4x+8) changed to: (3x^4x+1)+(x^3+4x-8)

4/4/11 Skyler Isch =*Compound Probability is the chance of 2 or more events happenening= =3/14/11= Will Schweitzer
 * __Compound Probability__**
 * __--->Independent Probability__**
 * The result of the 1st event will not affect the 2nd event
 * Ex. flipping a coin and then rolling a dice, spinning a spinner and flipping a coin, picking a marble out of a bag and then putting the marble back in the bag the second time you pick a marble
 * __--->Dependent Probability__**
 * =The result of the 1st event will have an effect the 2nd event=
 * =Ex. picking a marble out of a bag and not replacing it afterwards=
 * =Ex. picking a marble out of a bag and not replacing it afterwards=



=3/4/11= Sarah Draper Scientific Notation Word Problems Hint: a(b)^x (a times b to the power of x) b= 1+% for increase =1-% for decay

Examples: A 1938 comic book was sold in 2005. In 1980 it sold for $55. If value increases at a rate of 2.8% per year what did it cost in 2005? Step 1) Find the equation: y=55(1.028)^x Step 2) Plug/Solve: 55(1.028)^25 x=25 because the number of years from 1980 to 2005 is 25. $109.70

You put $125 in a savings account with 2% interest yearly. In 5 years what do you have? 1) y=125(1.02)^x 2) y=125(1.02)^5 $138.01

A farmer bought a tractor in 1999 for $30,000. The value decreases 18% per year. What is the value in 2005? 1) y=30,000(.82)^x **Remember decay: 1-%!! 2)y=30,000(.82)^6 $9,120.20

=2/28/11= Ann Wolf Scientific Notation

-Only one digit to the left of the decimal point-

__Scientific Notation to Standard form__ + move decimal point to the right - move to the left

__Standard form to Scientific Notation__ +move decimal point to the left -move to the right

__How to use SN on a calculator__ Use EE button

__Practice Problems__ 1) 3.6 x 10^4 =36000 2) 1.57 x 10^-5 =0.0000157 3) 47100000 = 4.71 x 10^ 7 4) 0.0000187 =1.87 x 10^-5

=2/16/11= Megan Lobring =**Exponents**=

**Divide**- example 1.) __a^m__ = a^mn __x^5__ = x^2 you get the x^2 because you subtract the exponents 5-3 and you get 2
a^n = x^3 =


 * example 2.) __12x^9__ = 4x^8**
 * 3x =**

( b ) = b^m
 * Power of Quotients-** example 1.) ( __a )__ in parenthesis to the m power = __a^m__.

example 2.) (__x__) in parenthesis to the 7 power = __x^7__ (y) = y^7

example 3.) __(4x^3)__ in parenthesis to the 3 power = __64x^9__ then simplify = __8x^9__ (2y^7) = 8y^21 = y^21 **2/1/11** Maddy Carroll = SUBSTITUTION =


 * involves 2 substitution to find (x,y)
 * used most by when a variable is already by itself
 * if in solving, the x's disappear, and you get a true statement (4=4) then there is an infinate solution....
 * if you get a false statement, such as (3=9), there is no solution.

EX:

y=2x-3 x+3y=5

those are the given equations.

x+3(2x-3)=5 x+6x-9=5 7x-9=5

7x=14

x=2

plug the x value into any of the given equations to find the value of y.

y=2(2)-3

y=4-3

y=1

(2,1)

EX:

3x+y=-7 ---> equals: y=3x-7 -2x+4y=0

-2x+4(3x-7)=0 -2x+-12x-28=0

-14x-28=0

-14x=28

x=-2

y=-3(-2)-7 y=6-7

y=-1

(-2,-1)

=1/31/2011 = =Elizabeth Dorsey = =__ELIMINATION __= ex.
 * =add or subtract to eliminate a variable =
 * add if coefficents are opp.
 * subtract if coefficents are the same
 * sometimes we need to multiply before we add or subtract

3x+4y=8 -3x+5y=10

9y=18 y=2

3x+8=8 3x=0 x=0

(0,2)

ex. 5x+6y=4 7x+6y=8

-2x=-4 x=2

10+6y=4 6y=-6 y=-1

(2,-1)

ex. 9x-3y=18 3y=-7x+30

9x-3y=18 7x+3y=30

16x=48 x=3

27-3y=18 -3y=-9 y=3

(3,3)

ex. 4x-3y=8 5x-2y=-11

5(4x-3y=8) -4(5x-2y=-11)

20x-15y=40 -20x=8y=44

-7y=84 y=-12

4x-3(-12)=8 4x+36=8 4x=-28 x=-7

(-7,-12)

ex. 2x+y=-9 4x+11y=9

-2(2x+y=-9) -4x+-2y=18

-4x+-2y=18 4x+11y=9

9y=27 y=3

2x+3=-9 2x=-12 x=-6

(-6,3)

=Sammie Miller 1/19/11 = == =Graphing Absolute Value= __Sam Shockley__ All make v shape, every vertex is on x-axis, vetex is at (#,0) EX: y=lx-7l--> vertex=(7,0) slope=1 All make v shape, vetex (0,0), negative # is same graph as positive #, # is slope,y=l2xl is the same as y=l-2xl All make v shape, vertex (0,0), # is slope, if # is neagative it makes an upside down v. All make v shape, vertex is (0,#) EX: y=lxl+8-> vertex=(0,8) slope=1 y=l2xl-3->vertex(0,8) slope=2 y=-2/3lx-4l+1--->vertex(4,1) slope=-2/3 y=-l2xl+4->vertex(0,4) slope=-2 //__Graphing Inequalities__// Carrie Gleason //1/3/11// =**__//Absolute Value Inequalities://__**=
 * 1/5/11**
 * Section 1**- y=lx+#l
 * Section 2**- y=l#xl
 * Section 3**-y=#lxl
 * Section 4**- y=lxl+#
 * Section 5 Examples**
 * //__ADAM TOERNER __//**

- Must have absolute value alone before changing to negative.

- When looking at a negative case, switch inequality sign. EXAMPLE: Postive Case: 3x + 5 < 4 ---> x <3 Negative Case: 3x + 5 > -14 --> x > 6 1/3

- < and =< always mean "and" 3 < x < 5 - > and => always mean "or" 6 > x OR x > 2 =Compound inequalities lily ganote= = = =**Solving Inequalities** //Zachary Ball//= (<, >, <=, =>)

Solve just like an equation.

With 2 exceptions.

1) The variable and Inequality sign must remain in the answer

2) If multiplying or dividing by a negative, flip the inequality sign. (coefficient of X is negative) =Parallel and perpendicular lines=||||||||~  || parallel lines= same slope
 * || [|zacharybernard]

perpendicular lines=opposite recipicles for the slope

write and equation for a line going through (-3,3) and parallel to y=-2x+5

(-3,3) m=-2 y=mx+b

3=-2(-3)+b 3=6+b -3=6

equation would be- y=-2x+3

write an equation for a line going through the points (-1,7) and perpendicular to y=-1/3x+2

(-1,7) m=3 its perpendictular to -1/3

7=3(-1)+b 7=-3+b 10=b

final equation - y=3x+10

equations y=-3x+1 and y=1/3x+1/3 are perpendictular to eachother.

equations y=1/3x+1/3 and 2x+ -6y+4 are parallel to eachother ||

11/11/10 Jasmine Jay



11/2/10 ross osborne

Standard form of a graphing equation ax+by=c (not slope intercept form y=mx+b) 1.a and b are all integers (=, -, and whole #) 2.a must always be positive

Equation for how to find the x and y intercepts: sample: 2x+5y=10 y=2 because when x is zero, 2 is the # that fits to make the equation correct. That is the same for x is 5, y=0.
 * x || y ||  ||
 * 0 || 2 || y intercept ||
 * 5 || 0 || x intercept ||

Here is how you switch to standard form from slope intercept form: sample: 1/2x+1=y -1/2x -1/2x -2 [1=((-1/2x)+y)] -2=x+(-2y)

10/28/10 ashleigh sherman **1.** A general equation in slope intercept formis mx+b=y **2.** This tells us the slope and the y intercept. **3**. You can find this information in the coeffecient and the constant. **4.** These key pieces of information can help us graph the line because you can start at the y intercept than do rise over run to find the next point. **8.** The steps for finding an equation of a line in slope intercept form: 1. Find the slope y-y x-x 2. y=mx+b, plug in for m,x and y. 3. Write an equation by filling in m and b.

10/27/10
Scott Boggess Slope = rise/run

Slope of a Horizontal line
rise/run = #/0 = 0 (y = #)

Slope of a vertical Line
rise/run = #/0 = UNDEFINED x = #

Slope Intercept Form
y=mx+b m = slope b= y-intercept (x=0 this is the # on the graph) ex. y=2/3x-1 Slope=2/3 y-intercept= -1 ex. y=3x+2 Slope= 3/1 y-intercept= 2

10/27/10
Scott Boggess Slope = rise/run

Slope of a Horizontal line
rise/run = #/0 = 0 (y = #)

Slope of a vertical Line
rise/run = #/0 = UNDEFINED x = #

Slope Intercept Form
y=mx+b m = slope b= y-intercept (x=0 this is the # on the graph) ex. y=2/3x-1 Slope=2/3 y-intercept= -1 ex. y=3x+2 Slope= 3/1 y-intercept= 2

LINEAR EQUATIONS- SLOPE AND Y-INTERCEPT 10/26/10 ABBY WINTERNITZ LINEAR EQATUONS- · Contains 2 variables (x,y) · Infinite number of solutions (why it’s a line on a graph)

SLOPE- · Tells us the steepness of a line · Ratio of rise to the run · Rise/run = y1-y2/x1-x2 =__^__y/__^__x (  Example:  (-3, 7) (5, -1)  7 - -1/-3 - 5 = 8/-8= -1   · Positive slope- line will go uphill    · Negative slope- line will go downhill    · The bigger the absolute value of the Slope, the steeper the line will be

FINDING SLOPE- Given 1 line- · Choose 2 points · Count rise and run as you move from left point to right point Given 2 lines- · Subtract your y’s over x’s in the same order.

Y-INTERCEPT- · Tells us where the line crosses the y- axis. · //__ Sometimes __// //the// starting point.

10/16/10Solving Literal Equations

Example
 * Using equation solving steps
 * Get the requested variable by itself
 * The answer will look funny


 * 1) ax + by = c Get y by itself by first subtracting ax
 * 2) by = c - ax Then divide by b
 * 3) y = c - ax / b

Solving Percent Problems Using Proportions:
 * 1) 3x / 4 = y / z Multiply both sides by 4
 * 2) 3x = 4y / z Divide both sides by 3
 * 3) x = 4y / 3z

Solve then x/78 = 80/100 and x = 62.4 and finally add 62.4 + 78 to get 140.4
====  Writing and solving ratios, proportions, and percents. A ratio uses division to compare 2 quantities. You can write ratios in 3 ways: x to y, x:y, x/y **A ratio should be written in simplest form** ** A volley ball team plays 14 home matches and 10 away matches. **** a. Find the ratio of home games to away games:14/10=7/5 **** b. Find the ratio of away games to home games:10/14=5/7 **** Proportions: **** A proportion is a equation that states 2 ratios are equivalent.(remember cross, multiply, divide) **** Practice **** 11/6=x/30 z/54=5/9 6r/10=36/15 **** 6x=330 9z=270 ({10x36}/15)/6=r **** X=55 z=30 r=4 **** 5m/6=10/12 -49/7=a+7/6 11/w=33/w=24 18/13+d=6/d-13 **** ([6x10]/12)/5=m (-49x6)/7=-42 33w=11w+4 6d+78=18d+-234  ****  M=1 -42-7=-49 a=-49 22w=264 12d+78=-234  **** 12d=312 d=26 **** Percents ****  Key terms: is part of whole  **** What percent of 25 is 17 What number is 18% of 150 What number is 15% of 88 **** 17/25=x/100 x/150=18%/100% x/88=15%/100% **** 17x100/25=x x=68% 150x18/100=x x=27 88x15/150=x x=13.2 **** 71.5 is 52% of what number 81is 54% of what number What percent of 225 is 300 **** 71.5/x=52%/100% 81/x=54%/100% 300/225=x/100% **** 71.5x100/52=x x=137.5 81x100/54=x x=150 300x100/225=x x=133.33% ** ====

10/11/10 Notes: Solving equations checklist: 1. Switch subtraction. (from minus a positive to plus a negative) 2. Use dist. prop. to remove parentheses. (In 3(X+2) You would distribute the 3 to the X and the 2 and get rid of parentheses) 3. Combine like terms (In 3+2 you would combine the 3 and the 2 and write that down instead) 4. Move ALL variables to one side (left) by adding the opp. (In 3X+2=2X+4 you would subtract the 2X from both variables) 5. Move ALL numbers to other side (right) by adding the opp. (In 3X+2=2X+4 you would subtract the 2 from both numbers) 6. Divide by coefficient of X EXAMPLE: 3(X+4)=2(X-8) You would then change the X-8 to X+-8 3(X+4)=2(X+-8) You would then distribute the 3 to the X and the 4 and the 2 to the X and the -8 3X+12=2X+-16 You would then combine the like terms (there aren't any) 3X+12=2X+-16 You would then subtract 2X from both variables X+12=-16 You would then subtract 12 from both sides X=-28 And there is your answer!! :)

10/4/10

Solving multi step equations.

8x+(-3x)+(-10)= 20 3/2(3x+5)=-24

5x +(-10)= 20 9/2x+15/2=-48/2

+10 +10 -15/2 -15/2

5x=30 9/2x=(-63/2)

<span style="display: block; margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">X=6 X=-7

<span style="display: block; margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">7x+2(x+6)=39 9x+(-5)=1/4(16x+60)

<span style="display: block; margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">7x+2x+12=39 9x+(-5)=4x+15

<span style="display: block; margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">9x+12=39 -4x -4x

<span style="display: block; margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">-12 -12 5x+(-5)=15

<span style="display: block; margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">9x=27 +5 +5

<span style="display: block; margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">x=3 5x=20

<span style="display: block; margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">X=4

<span style="display: block; margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">Equations do not always have one solution. An equation that is true for all values of the variable is an Identity.

<span style="display: block; margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">The solution for an identity is all real numbers. Some equations have lR.

<span style="display: block; margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">__WHEN SOLVING EQUATIONS SHOW ALL WORK!!!__

<span style="display: block; margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">3x-17=2(x+-5)+x+(-7) 5(x+1)=2x+11+3x <span style="display: block; margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">2x+-10+x+-7 5x+5=5x+11 <span style="display: block; margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">3x+-17=3x+-17 -5x -5x <span style="display: block; margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">-3x -3x 5=11 <span style="display: block; margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">-17=-17 false <span style="display: block; margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">true no solution <span style="display: block; margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">all real numbers 0+/

10/3/10 Chapter 3 Notes: Solving One and Two Step Equations:

The inverse of addition is subtraction. The inverse of multiplication is division. Inverse operations are opposite.

Addition Property of Equality- if a+b=c+d then a+b+x=c+d+x: x=5 x+2=5+2 Subtraction Property of Equality- if a+b=c+d the a+b-x=c+d-x: x=5 x-2= 5-2 Multiplication Property of Equality- if a+b=c+d then x(a+b)=x(c+d): x=5 2x=2x5 Division Property of Equality- if a+b=c+d then a+b/x=c+d/x: x=5 x/2=5/2

Chapter 2 Notes:

9/16/10

Prerequisite Knowledge

Integers
 * Adding: If signs are alike you add; if signs are different you subtract
 * Subtracting: Switch subtraction sign to addition and next number to its opposite then follow adding rules
 * Multiplying and Dividing: multiply and divide as usual and then if signs are alike you get a positive answer and if signs are different you get a negative answer

Decimals
 * Line up decimal points to add and subtract
 * Multiply as usual and then count the number of numbers to the right of the decimal in the problem and move decimal same number of spots in from the right
 * Divide by moving decimal out of number outside of divide sign and then move decimal same number of spots inside. Finally, bring decimal straight up into the answer.

Fractions
 * Get a common denominator (change numerator also). Add numerators and put it over the common denominator Reduce if possible
 * To multiply just multiply the tops and then the bottoms. Reduce if possible.
 * To divide flip the second fraction and then follow rules for multiplying. Reduce if possible.
 * It is easiest to do all operations if you switch from a mixed number to an improper (top heavy) fraction

9/17/10

Classifying Real Numbers:
 * Natural Numbers - counting numbers
 * Whole Numbers - all natural numbers plus zero
 * Integers - all natural numbers, all whole numbers, plus negative counting numbers
 * Rational Numbers - all natural numbers, all whole numbers, all integers, any fraction, and any decimal that ends or repeats
 * Irrational Numbers - any decimal that does not end or repeat

9/20/10

Properties of Real Numbers
 * Commutative Property (add and multiply) - you can switch the order
 * Associative Property (add and multiply) - you can switch the way they are grouped
 * Additive Identity - you can add 0 and not change the identity of a number
 * Multiplicative Identity - you can multiply by 1 and not change the identity of a number
 * Additive Inverse - you add the opposite of a number to get an answer of 0 (the identity)
 * Multiplicative Inverse - you multiply the reciprocal of a number to get an answer of 1 (the identity)
 * Property of Zero - multiplying by 0 will give an answer of 0
 * Property of -1 - Multiplying by negative one will make a number its opposite

Sometimes - give an example of true and one of false Always - no examples needed Never - give an example of false

9/22/10

Distributive Property a(b + c) = ab + ac or a(b - c) = ab - ac

Uses of distributive property:
 * 1) Simplify expressions 3(x + 5) = 3x + 15
 * 2) Combine like terms 4x + 11x = (4 + 11)x = 15x
 * 3) Mental math 6(53) = 6(50) + 6(3) = 300 + 18 = 318

9/23/10

To compare and order real numbers always convert them to decimals (top of fraction divided by bottom of fraction) Remember the farther right on a number line the bigger it is.