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1. Definition
2. Example
3. Levels of usage
A. Review Algebra I (Chapter 1-2)
reflexive propert- Everything is equal to itself
symmetric property- you can change the order of the problem as long as the signs are the same a-b+c=d
transitive property- If a=b and b=c then a must equal c
closure property- two real numbers equal a real number 8+7=15
communitive property of multiplication- order doesnt matter ex. axb=bxa
coefficient- a number in front of a variable that it multiplies by. ex: 3x 3 is the cofficient
absolute value- the absolute value of lxl,is its distance from zero on a number line addition property of equality for all numbers a,b,and c, if a=c then a+c=b+c addition property of inequality for all number a,b,c if a
Natural Numbers
How we count (no decimals/fractions)
{1,2,3,4,5… ect.}
Whole Numbers
Natural Numbers- contain a 0 (no decimals/fractions)
{0,1,2,3,4… ect.}
Integers
Whole Numbers- contain negatives (no decimals/fractions)
{…-3,-1,-1,0,1,2,3…}
Rational Numbers
"Predictable" numbers that can be written as a fraction
(this includes non-terminating, repeating decimals) (5 as a fraction is 5/1)
Irrational Numbers
A decimal that does not end and is not predictable (aka non-terminating, non-repeating)
Any non Perfect Square root or pi
Real Numbers
Any number you can think of that's not in the imaginary category
Real Imaginary
Naturals Complex numbers i=√-1
Integers
Rational
Irrational
Operations
Add, subtract, multiply, divide
Variable
Any letter in a problem (a, b, x)
Coefficient
A number that multiplies the variable (10x+25 coefficient of x is 10)
Simplify
The act of combining like terms or reducing to lowest terms
This is no equal sign (4/2 -> 2)
Solve
The act of getter whatever variable you are looking for alone
All other numbers and variables will be on the other side of an equation
There is an equal sign (2x=b divide by two on each side: x=b/2)
absolute value-the distance a number is from zero l2l= 2 l-2l=2
additive inverse - the additive inverse is the oppisite of a number or what needs to be added to a number to make it zero 2+-2=0
theorem- a statement that can be proved is a theorem
multiplication inverse- what you multiply a number by to get one. Is also known as reciprical
4x1/4=1
evaluation- to get the value of the expression is to evaluate
communitive property of addition- a+b=b+a
communitive property of multiplication- axb=bxa
Reflexive Property-Everything is equal to itself EX: ab=ab
Symmetric Property- you can switch sides only if the signs stay the same EX: a-b+c=d
Transitive Property: If a=b and b=c then a and c are also equal
Closure Property: The sum of two real numbers equals a real number EX: 9+9=18
Associative Property of Addition: The grouping of the addens doesn't change the sum EX: 3(5+6)= 14 or 6(5+3)=14
Associative Property of Multiplication: The grouping of the factors doesn't change the product
EX: 2x(4x6)= 48 or 4x(6x2)=48
Distributive Property: The number that is on the outside of the parthenses gets distributed to every number on the inside EX: 4(2+7)=8+28= 36
Identity Property of Addition: A number added to 0 is that number EX: 10+0=10
Identity Property of Multiplication: A number multiplied by 1 is that number EX: 6x1=6
Inverse Property: When you multiply by the respical you get 1 EX: 1/4 x 4/1= 4/4= 1
Inequalities- Any mathematical sentence containing <, >, ≤, ≥
Ex. x+3> 6
Coordinate- On a number line, the number that corresponds to a point
Distance- the space between two points on a number line having coordinates a and b
Ex. |a-b| or |b-a|
For example the distance from 2 to -3 is |-3-2| = |-5| = 5
B. Graphing (Chapter 3)
How to Find the Slope:
Formula when having an equation with 2 points
y2-y1
______
x2-x1
EXAMPLE: (4,6) (10,5)
5-6 -1
___ = ____ -1/6 is the slope
10-4 6
Ordered Pair- A pair of numbers in a particular order; the coordinates of a point in a plane
Ex. (6,7) (1,2)
Solution- A replacement for a variable that makes an equation or inequality true
Ex. Y=3x-1
Solution = (-1, -4)
Y-coordinate: Y-coordinate is the second number of an ordered pair
Ex. (3,1)
X-coordinate- The first number of an ordered pair
Ex. (9,4)
Slope- A number that tells how steeply the line slants, the ratio of rise to run. We use “m” to designate slope.
Ex. 4
---
2
From the example, you take go down to up. You move your point over 2 and up 4.
Point-slope equation- To find the equation of a line, when you have the slope and the coordinates of a point
Formula: (y-y1)= m(x- x1)
Ex. (2,1) m=5
(y-1)=5(x-2)
y-1=5x-10
+1 +1
----------------
Y=5x-9
C. Systems of Equations (Chapter 4)
*Find out what one variable equals, then back substitute to figure out the other variables*
*Use combination to get your equations in triangular form*
Ax+By+Cz=D
Ey+Fz=G
Hz=I
Equation 1: x+y+z=4
Equation 2: x-2y-z=1
Equation 3: 2x-y-z=-1
In this equation...multiply equation 2 by -1, then add Equation 1 and Equation 2 so the "x"s cancel out.
x+y+z=4
+-x+2y+z=-1
-----------------
3y+2z=3 ---------> replace row 2
Equation 1:x+y+z=4
Equation 2:3y+2z=3
Equation 3:2x-y-z=-1
Multiply equation 1 by -2, then add equation 1 and equation 3 so the "x"s cancel out.
-2x-2y-2z=8
+2x-y-2z=-1
------------------
-3y-4z=-9 ---------->replace with row 3
Equation 1: x+y+z=4
Equation 2: 3y+2z=3
Equation 3: -3y-4z=-9
Add equation 2 and equation 3 to cancel out the "y"s.
3y+2z=3
+-3y-4z=-9
-------------------
-2z=-6 --------->replace with row 3
Equation 1: x+y+z=4
Equation 2: 3y+2z=3
Equation 3: -2z=-6
Solve for "z" in equation 3
-2z=-6
____ ___
-2 -2
z=3
Solve for Equation 2
3y+2(3)=3
3y+6=3
3y=-3
y=-1
Solve for Equation 1
x+(-1)+(3)=4
x+2=4
x=2
(2,-1,3) is your answer!
Substitution method- useful technique for solving systems in which a variable has a coefficient of 1
2x+y=6
3x+4y=4
Find a variable with a coefficient of 1, in this case, you can choose “y” in equation 1
Isolate the variable, and solve for y
2x+y=6
-2x -2x
Y= 6-2x
You then substitute this for “y” in the second equation
3x+4(6-2x)=4
3x+24-8x=4
-5x+24=4
-24 -24
-5x=-20
You then solve for “x”
-5x=-20
---- -----
-5 -5
X= 4
Now substitute 4 for “x” in either equation to solve for “y”
2x+y=6
2(4)+y=6
8+y=6
-8 -8
Y=-2
Your answer ends up becoming: (4,-2)
Chapter 7
Square root- a number that when multiplied by itself equals a given number
EX. C2=A
22=2 x 2 = 4
Cube root- A number that when multipled 3 times by itself equals a given number
EX. Find the cube root of 27
C3=A
33=3x3x3=27
IMAGINARY AND COMPLEX NUMBERS
Imaginary numbers- were invented so that negative numbers would have square roots and certain equations would have solutions. They consist of all numbers bi, where b Is a real number and iis the imaginary unit, with the property that i2= -1
i = √-1
i 1= i
i 2= -1
i 3= -i
i 4= 1
Examples:
√-5= √-1*5
= √-1√5
= i√5, or √5i
-√-7= -√-1*7= -√-1√7= -i√7
√-99= √-1*9*11= i√9√11= 3i√11
To multiply imaginary number or an imaginary number by a real number, it is important first to express the imaginary numbers in terms of i.
Examples
47i*2 = 94i
√-5*2i = i√5*2i = 2i2√5= -2√5
-√-3*√-7= -i√3 * i√7 = -i2√21 = -(-1)√21= √21
Complex numbers
Complex numbers consist of all sums a + bi, where a and b are real numbers and I is the imaginary unit. The real part is a , and the imaginary part is bi.
Adding and subtracting
7i + 9i = 16i
(-5 + 6i) + (2 - 11i) = -5 + 2 + 6i - 11i = -3 -5i
(2 + 3i) - (4 + 2i) = 2 + 3i - 4 - 2i = -2 + i
TRIG
Right Triangle- a triangle with a 90 degree (right) angle
3 Sides:
a. leg
b. leg
c. hypotenuse- opposite of the right angle
Opposite- across from
Adjacent- next to
Sine- The ratio of the opposite side of Theta to the hypotenuse
Sine= Oppsite over hypotenuse
Cosine- The ratio of the adjacent side of Theta to the hypotenuse
Cos- Adjacent over hypotenuse
Tangent- The ratio of the oppsite side to the adjacent side
Tan- opposite over adjacent
Cosecent- respical of the Sine
Hypotenuse over opposite
Secant- respical of Cosine
SECO- Hypotenuse over adjacent
Contangent- respical of tangent
COT- adjacent over opposite
co-terminal angles- two angles with the same initial and terminal sides
AKA two angles that start and stop in the same place
Reference angles- a positive angle made from the terminal side of an angle to the X-axis
Positive angles- Angles that move in a counter clockwise rotation on a graph
Negative angles- Angles that move in a clockwise rotation on a graph
Quadrantal angles- An angle whose terminal side ends directly on either the X or Y axis
Terminal Side- the rotating ray in an angle
Initial side- The positive half of the x-axis
Theta- To define a trigonometric function of an angle, you assign “theta” to one of the angles in a triangle to show its angular variable
Pythagorean theorem : the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse ex. a2+b2=c2 Hypotenuse- the largest side in a right triangle and is always opposite the right angle Opposite- across from Adjacent- next to SINE- The fraction of the opposite side over the hypotenuse COS- The fraction of the adjacent side over the hypotenuse TAN-The fraction of the oppsite side over adjacent side CSC- Flipped sine hypotenuse over opposite SECO- Flipped cosine hypotenuse over adjacent COTO- Flipped tangent adjacent over opposite
- Square root- a number a is a number c such that c=a
25 has a square root of 5 because 5=25
- Principal square root- a nonnegative number is it's nonnegative square root. The symbol √a represents the principal square root of a. √25=5
- Multiplying binomials
Multiply the firsts, then the outside terms, then the inside terms, then the last terms. We abbreviate this as FOIL
(A+B)(C+D) = AC + AD + BC + BD
- Squaring Binomials
(A+B)2 = A2 + 2AB + B2 (A-B)2 = A2 - 2AB + B2
The square of a binomial is the square of the first expression, plus or minus twice the product of the expressions, plus the square of the second expression
- Multiplying sums and differences
(A+B)(A-B) = A2 - B2
The product of the sums and difference of two expressions is the square of the first expression minus the square of the second.
Factoring- The process of finding multiples of a product
Process of Factoring
1. GCF
2. Diff. of Squares
3. Sum and Difference of cubes
4. Perfect square trinomials
5. Factor by grouping
6. Standard Factoring
1. x+6x
2. x-4 (x+2)(x-2)
3. x+8y (x+2y)(x-2xy+4y)
4. X-8 perfect cube + perfect cube
(x-2)(x+2x+4)
a3-b ---> (A-b)(a2+ab+b2)
a3+b ---> (3a+b)(a2-ab+b2)
5. Factor by grouping
ac+ad+bc+bd
a(c+d)+b(c+d)
(a+b)(c+d)
6. x+8x+15
Find 2 binomials when you multiply together give you the given trinomial
( )( )=x+8x+15
(x+5)(x+3)
Axis of symmetry
-vertical like cutting a parabola in 2
x=-b/2a
The opposite of b over 2a
Imaginary Numbers
Invented so that negative numbers would have square roots and certain equations would have answers
i=imaginary numbers
i=√-1
√-36
/ \
√-1 x √36
/ \
i 6
6i = imaginary number
Complex number
Combination of an imaginary number and a real number
4+6i
4(6i)=24i 4i(6i)=24i i=√-1 i=-1 i=1