Download LINES OF BEST FIT and LINEAR REGRESSION and

ALGEBRA I FORMULAS AND FACTS FOR EOC
Commutative Property:
Order of Operations:
Percent of Change:
a+b=b+a
Associative Property: (a + b) + c = a + (b + c)
PEMDAS
new  original
change
 100  % of change or
 100  % of change
original
original
Distribute (multiply) a term to each individual term inside the
Distributive Property: parentheses. Be careful of negative and positive values.
-3y (5 + x - 2y) = (-3y)*5 + (-3y)*x + (-3y)*(-2y) = -15y-3yx + 36y2
To combine like terms, the variables and exponents must be same.
Combining Like Terms: 3x2 + 4x – 6 + 6x + 14 – 5x2 = (3 - 5)x2 + (4 + 6)x +( –6 + 14)
= -2x2 + 10x + 8
CALCULATOR: Left Side = Y1. Right Side = Y2. [GRAPH].
Find Intersection: [2nd], [Trace], [5:Intersect], [ENTER], [ENTER],
Solving equations:
[ENTER]
By HAND: Use SADMEP – cancel operations
Independent Variable:
Domain:
Range:
Greatest Common
Factor (GCF)
x-values
Dependent Variable:
y-values
All x-values used an equation, function, or graph (left and right)
All y-values used an equation, function, or graph (up and down)
Largest Integer into all numbers and smallest exponent of variable
Exp: 10x2yz4 and 15x5y3; GCF = 5x2y
Least Common Multiple Smallest integer multiply to equal, largest exponent each variable
Exp: 10x2yz4 and 15x5y3; LCM = 30x5y3z4
(LCM)
y  y1
rise
Slope:
m 2
m
or
between (x1, y1) and (x2, y2)
x2  x1
run
Parallel Lines:
Perpendicular Lines:
Slope – Intercept form:
Same slope
Opposite (Negative) reciprocal slope.
y = mx + b
m = slope and b = y-intercept
Horizontal Line:
Vertical Line:
y – y1 = m (x – x1)
point: (x1, y1) and m = slope
Ax + By = C
1) GCF of A, B, and C = 1
2) NO FRACTIONS
3) A is positive
Equation: y = #; Zero (No) Slope
Equation: x = #; Undefined Slope
Direct Variation:
y = kx ; y varies directly with x; multiply from X to Y or vice versa
Point-Slope Form:
Standard Form:
Midpoint Formula:
between (x1, y1) and (x2, y2)
 x1  x2 y1  y2 
,


2 
 2
Matrices:
rows go across,
columns go down.
CALCULATOR: Create - [2nd], [Matrix], EDIT, Select a Matrix to use ([A],
[B], etc), Input Rows and Columns, Input Elements
Use Operations – [2nd], [Matrix], NAMES, Select a Matrix ([A], [B],…)
GRAPHING: Solve for slope intercept form
 Plug into y = and find the intersection of the lines
MATRIX: Write both equations in STANDARD FORM of lines
Solving a system of
equations:
ax  by  c
a
b
c
MATRIX = 

dx  ey  f
d e f 
Plug coefficients and constants matrix and perform RREF operation.


If bottom row is 0 1 #, the last column is your answer.
If bottom row is 0 0 1, there is no solution.
 If bottom row is 0 0 0, there are infinitely many solutions.
Graph Inequalities:
Distance Formula:
between (x1, y1) and (x2, y2)
Quadratic Formula:
ax2 + bx + c = 0
Step #1: Solve for slope intercept form. If you multiply or divide by
a negative, then flip direction of inequality.
Step #2: Solid Line when ≥ or ≤ AND Dotted Line when > or <
Step #3: Shade Up (right) when ≥ or > AND Shade Down (left)
when ≤ or <
2
2
DISTANCE = ( x 2  x1 )  ( y2  y1 ) ; Draw a right triangle
 b  b 2  4ac
x
2a
Pythagorean Theorem:
a2 + b2 = c2
Exponential Functions:
y = abx
b = base or pattern of multiplication, a = initial value
Exponential Growth:
y = a(1 + r)x
(Increases/ Appreciates)
a = initial value, r = rate of percent increase (4.5%; r = 0.045)
Exponential Decay:
y = a(1 - r)x
(Decrease/ Depreciates)
a = initial value, r = rate of percent decrease (5.7%’ r = 0.057)
Common Shapes’ Area and/or Perimeter Formulas
Perimeter of a Figure: add up all sides
Area of a Rectangle: A = l*w
Area of a Circle: A = r2
r
Perimeter of a rectangle:
Circumference of Circle:
P = 2l + 2w
C = 2r
Area of Triangle:
A = ½ bh
Perimeter of a triangle:
P = s1 + s2 + s3
h
b
Area of a Square:
A = s2
Perimeter of a square:
P = 4s
b1
Area of a Trapezoid:
A = ½ (b1 + b2)h
h
b2
Volume of a Cylinder:
V = r2h
w
l
s
s
CALCULATOR COMMANDS
Graphing: [Y=] enter in the equation, [ZOOM], [6: ZStandard]
If the graph is not shown, then the change the window by: Make sure xmin < xmax and ymin < ymax
[WINDOW] and adjust…
YMAX (see farther up)
XMIN (see more left)
XMAX (see more right)
YMIN
(see farther down)
To find the MAXIMUM VALUE: [2nd], [TRACE], [4] (maximum)
Left Bound: move the cursor to the left of the maximum (top of hill) ENTER
Right Bound: move the cursor to the right of the maximum (top of hill) ENTER
Guess: move the cursor to the maximum (top of the hill) ENTER
To find the MINIMUM VALUE: [2nd], [TRACE], [3] (minimum)
Left Bound: move the cursor to the left of the minimum (bottom of valley) ENTER
Right Bound: move the cursor to the right of the minimum (bottom of valley) ENTER
Guess: move the cursor to the minimum (bottom of valley)ENTER
To find the ROOTS/ ZEROS/ X-INTERCEPTS:
[Y =] make Y1 = Equation and Y2 = 0 [GRAPH], [2nd], [TRACE], [5] (intersect)
Move cursor to intersection [ENTER], [ENTER], [ENTER]
To find X when Y = #:
[Y =] make Y1 = Equation and y2 = #, [GRAPH], [2nd], [TRACE], [5] (intersect)
Move cursor to intersection ENTER, ENTER, ENTER
Make sure that your window shows the intersection
To find Y when X = #.
Option #1: [2nd], [TRACE], [1] (value), X= #, [ENTER]
Option #2: [2nd], [WINDOW] let TblStart = #, [2nd], [Graph]
To find INITIAL VALUE or Y-INTERCEPT, look when x = 0.
LINES OF BEST FIT and LINEAR REGRESSION and PREDICTION EQUATIONS
INPUTTING DATA:
1) STAT -> EDIT
2) L1 –X VALUES, L2 - Y VALUES
Make sure that rows represent points/ ordered pairs
MAKE A SCATTER PLOT:
1) 2ND, STAT PLOT [Y=], ENTER
2) TURN STAT PLOT ON (ENTER)
3) TYPE: HIGHLIGHT FIRST GRAPH
4) X-LIST: L1
5) Y-LIST: L2
FINDING THE LINE OF FIT:
1) [STAT] -> Scroll to CALC
2) 4 [LINREG(ax + b)]
 a = slope (m) and b = y-intercept
3) VARS, Y-VARS, FUNCTION, Y1,
[ENTER]
4) GO TO [Y =];
You can now see the equation of the line
5) GRAPH;
You can see the stat plot and the line together
SEE THE ENTIRE SCATTER PLOT:
1) WINDOW
2) XMIN: # < smallest number in L1
3) XMAX: # > biggest number in L1
4) YMIN: # < smallest number in L2
5) YMAX: # > biggest number in L2
Now have an equation to help predict values by
1) Searching the TABLE:
2ND, WINDOW, TblStart = value you want
2ND, GRAPH [TABLE]
2) Finding a specific VALUE:
2ND – TRACE [CALC] – 1 - ENTER
TOUGH VERBAL TRANSLATIONS
1) 4 less a number: n – 4
2) 4 less than a number: 4 – n
3) 4 is less than a number: 4 < n
4) a number is at least 10: n ≥10
5) 4 less ½ of the difference of a number and 5:
4 – ½ (n – 5)
6) 5 more a number: 5 + n
7) 5 is more than a number: 5 > n
8) 5 more than a number: n + 5
9) a number is at most 10: n ≤ 10
10) 2 more than ½ the sum of a number and 12:
½ (n + 12) + 2
Chapter 9 Factoring Review
Section 9.1: Is a number prime or composite?
Find the Prime Factorization: All Prime numbers that multiply together to equal a number. Draw your tree.
Example: 180 = 2*2*3*3*5

Find the Greatest Common Factor (GCF)
between numbers: pair up all common prime
factors of the numbers and multiply together
Example: GCF = 2*3 or 6
72 = 2 * 2 *2 * 3* 3
36 = 2 * 2 * 3* 3
42 = 2 * 3 * 7

Find the Greatest Common Factor (GCF)
between monomials: find the GCF of the
coefficients and then for any common variable
pick the smallest exponent for each variable
Example: GCF = 2*x2*y3 = 2x2y3
6x2y6 = 2*3*x2*y6
32x3y4 = 2*2*2*2*x3*y4
10x5y3 = 2*5*x5*y3
FACTORING TECHNIQUE
2 or
more
terms
2
terms
GREATEST COMMON FACTOR (GCF):
(1) Find the GCF of all terms of the polynomial,
(2) Divide the polynomial by the GCF, and
(3) Write the factors as the GCF times polynomial from
step 2.
Special Case: DIFFERENCE OF SQUARES:
a2 – b2 = (a + b) (a – b)
ax2 + bx + c
Step #1:Greatest Common Factor of a, b, and c
3
terms
Step #2: Factor-Sum Tree to find a pair M and N
 Multiply to equal = product of a and c
 Add to equal = b
Step #3: Split the middle term, bx = Mx + Nx
Step #4: Factor by Grouping (See below)
Factor by Grouping:
Factor the 1 2 terms and the last 2 terms separately by
their respective GCFs.
ax + bx + ay + by
st
4
terms
GCF of ax + bx is x
GCF of ay + by is y
x (a + b) + y(a + b)
(x + y)(a + b)
EXAMPLES
3x3
6x2
+
+ 15x
(1) GCF = 3x
(2) (3x3 + 6x2 + 15x)/(3x) = x2 + 2x + 5
(3) = 3x(x2 + 2x + 5)
4x2 – 25
(2x)2 – (5)2
(2x – 5) (2x + 5)
x2 + 11x + 24
3*8 = 24 and 3 + 8 = 11
x2 + 3x + 8x + 24
= (x + 3)(x + 8)
6x2 – x – 2
3*-4 = -12 and 3 + - 4 = -1
= 6x2 + 3x – 4x – 2
= 3x(2x + 1) – 2(2x + 1)
= (3x – 2) (2x + 1)
3x2– 6x + 5x – 10
= (3x2 – 6x) + (5x – 10)
GCF = 3x
GCF = 5
= 3x(x – 2) + 5(x – 2)
= (3x + 5) (x – 2)
Hint: If x (a - b) + y(-a + b), then make a
subtraction for y x (a - b) – y(a - b)
CHAPTER 8 MONOMIALS and POLYNOMIALS REVIEW
MULTIPLYING MONOMIALS
DIVIDING MONOMIALS
When multiplying monomials of the same
base, we ADD the exponents and MULTIPLY
the coefficients.
 See the base more than once (Left to
Right) and combine the powers to make
one base.
Example: (4m3n4)(5m5n3) = (4*5) m3+5 n4+3
Solution: 20m8n7
When dividing monomials of the same base, we
SUBTRACT the exponents and SIMPLIFY the
coefficients.
 Subtract the bigger MINUS smaller power
and place the new base in the location of
the bigger.
Example:
7x5 y2
1x 2
;
Solution:

21 x 3 y 8
3 y6
7 1
 ; x: 5 – 3 = 2 in top; y: 8 – 2 = 6 in bottom
21 3
POWERS OF MONOMIALS
When we raise a monomial to an exponent, we MULTIPLY the exponents in the monomial and
COEFFICIENTS gains the exponent.
Example 1: (-2m2n5)3 = (-2)3 m2*3 n5*3
SOLUITION: -8 m6 n15
SPECIAL MONOMIAL CASES:
3
3
 3 x   31 x 1 
33 x 3
27 x 3
EXAMPLE 2:  2    1 2   3 2*3 
4 y
64 y 6
 4y   4 y 
POLYNOMIAL INTRODUCTION:
A ZERO EXPONENT: CANCELS THE
BASE
POLYNOMIAL: A monomial or the sum
(addition or subtraction) of monomials.
Algebraically: If a = base, then a0 = 1
Degree of a monomial: the sum of the
exponents of all its variables.
 Degree of 5mn2 is 1 + 2 = 3
 3x
2
 4y
Example: ( 2 x )
0

  ( 2 x ) * 1  2 x

NEGATIVE EXPONENTS: Change the
exponent to a positive and switch it’s location
(OR bottom to top)
1
am
1
an
(Bottom to Top): If a = base,  n 
a
1
m

(Top to Bottom): If a = base, a
Examples:
3s 2
3

#1:
4r
4rs 2
3 x 2 3 y 3

#3:
4 y 3 4 x 2
5s
5sr 5

#2:
7
7 r 5
Degree of a polynomial: the greatest degree of
any term in the polynomial.
 Degree of -4x2y2 + 3x2 + 5y is 4.
ADDING AND SUBTRACTING
POLYNOMIALS:
 Combine Like Terms between polynomials.
 Do not change the exponents of your variable
only the coefficients from your addition or
subtraction.
Examples:
1. (3x2 – 4x + 8)+(2x – 7x2 – 5) = - 4x2 - 2x + 3
3x2 + -7x2 + -4x + 2x + 8 + -5
2. (3n2 + 13n3 + 5n)–(7n + 4n3) = 9n3 + 3n2 –2n
3n2 + 13n3 – 4n3 + 5n – 7n
DISTRIBUTIVE PROPERTY:
 Circle the ENTIRE term in front of
parentheses
 Draw Arrows to each term in polynomial
include plus or minus sign
 Multiply to get new terms
 Add all terms
FOIL
(Arrow or Box Method)
 Circle the EACH term in FIRST
Binomial include plus or minus sign
 Draw Arrows from each circled term to
each term in 2nd Binomial
 Multiply to get 4 new terms
 Combine any Like Terms
-3x2 (6xy2 – 3x + 5x2y + 2y - 8)
(5x – 3) (2x + 7)
(-3x2)(6xy2) = -18x3y2
(-3x2)(– 3x) = 9x3
(-3x2)( 5x2y) = -15x4y
(-3x2)( 2y) = -6x2y
(-3x2)(- 8) = 24x2
F: (5x)(2x) = 10x2
O: (5x)(7) = 35x Like Terms
I: (-3)(2x) = -6x
L: (-3)(7) = -21
Solution:
3 2
Solution: 10x2 + 29x - 21
3
-18x y + 9x - 15x4y - 6x2y + 24x2
MULTIPLYING POLYNOMIALS
SPECIAL PRODUCTS
POLYNOMIAL times
POLYNOMIAL
Special products are shortcuts for FOIL
 SQUARE OF A SUM: The square of a + b
is the square of a plus twice the product of a
and b plus the square of b.
(a  b)2  (a  b)(a  b)  a 2  2ab  b2
 Circle the EACH term in FIRST
polynomial include plus or minus sign
 Draw Arrows from each circled term to
each term in polynomial
 Multiply to get new terms
 Combine Like Terms
 SQUARE OF A DIFFERENCE: The
square of a + b is the square of a minus twice
the product of a and b plus the square of b.
(3x – 2) (6x2 + 7x - 8)
(a  b)2  (a  b)(a  b)  a 2  2ab  b2

PRODUCT OF SUM AND
DIFFERENCE: The product of a + b and a
– b is the square of a minus the square of b.
(3x)(6x2) = 18x3
(3x)(7x) = 21x2
(3x)(-8) = -24x
(a  b)( a  b)  a 2  b2
REMINDER for BOX METHOD: If the
arrows don’t work for you, use the BOX METHOD
to organize your multiplications. After filling in
your boxes, you will add together all the terms by
combining like terms.
(-2)(6x2) = -12x2
(-2)(7x) = -14x
(-2)(-8) = 16
Solution: 18x3+ 9x2 - 38x +16
2x
x
x(2x) = 2x2
-7 -7(2x) = -14x
+3
x(3x) = +3x
-7(3) = -21
(x – 7) (2x + 3) = 2x2 – 11x – 21