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Math A - Regents Review |
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Math A - Regents Review |
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1 Scientific Notation
2 Properties
i Commutative Property �order in which the 2 #�s are (+) or (x) does not matter.
ii Associative Property � order in which the 3 #�s are grouped when (+) or (x) does not matter.
iii Identity Property � for (+) it is 0 and for (x) it is 1.
iv Inverse Property � for (+) it is the negative of the # and for (x) it is the reciprocal.
v Distributive Property � x (y + z) = xy + yz
3 Logic
Logically Equivalent Statements � when 2 sentences have the same truth-value (conditional and its contra positive.)
i Open Sentence � cannot be judged on true and false
ii Statement � a sentence that can be true or false
A True Statement (root or solution)
B Compound Statement � 2 regular statements
(a) Conjunction (&) true only when both are true
(b) Disjunction (or) true when one or both are true
(c) Conditional (if, then) True except when 1 is true and 1 is false
� Converse � switch 1st and 2nd statements
� Inverse � negate (not) both sentences
� Contra positive � negate (not) and switch the 1st and 2nd statements
(d) Bi conditional (if, only if) True when both is true or both is false
+ + = + +
- - = - +
+ - = + -
- + = --
4 Factoring
i Type 1 � find the GCF and take it out and leave the rest �> ab + ac = a (b+c)
ii Type 2 � find the difference of 2 perfect squares �> 36x2 � 49 = (6x-7) (6x+7) *use FOIL if you have patience to check it*
iii Type 3 � find factors of last that equal middle �> x2 +9x +20 = (x+4) (x+5)
5 Inequalities (<, >, <, >)
i You solve the equation like always but flip the sign around when you divide the inequality by a negative number. (If it was < flip it to >)
ii When you need to show your inequality on a number line you darken the circle when it is included in the interval (> or <) and when its not shaded its not included. (< or >)
6 Quadratic Equations
i Remember that you always wind up with 2 answers and you usually reject the negative one when working with geometry.
7 Angles (Triangles can also be classified by their angles)
i Acute = less then 90o
ii Right Angle = 90o
iii Obtuse Angle = 90o to 180o
iv Supplementary Angle = When 2 angles measure up to 180 in degrees.
v Complementary = When 2 angles measure up to 90 in degrees.
vi Vertical Angles = the opposite angles formed when 2 lines intersect are equal.
vii Alternate Interior Angles are = (remember the Z)
viii Corresponding Angles are = (remember the F)
8 Sides
i Scalene Triangle = all sides have different measures
ii Isosceles Triangle = 2 equal sides
iii Equilateral Triangle = all sides are equal
9 Rules of Triangles
i If a triangle has 2 equal sides (angles), the angles (sides) opposite them are also equal.
ii If a triangle has 3 equal sides then each angle is 60o
iii Each side of the triangle has to be shorter then the sum of the to other sides and longer then their difference.
iv A triangle is congruent to another triangle when either SSS=SSS, SAS=SAS, ASA=ASA, AAS=AAS, HL=HL are true. When two triangles are = you can prove their parts are congruent by saying corresponding parts of equal triangles are = (CPCTC)
v To find the length of a side of a right triangle you can use the Pythagorean theorem (A2+B2=C2)
vi Sin A = Opposite leg Cosine A = Adjacent leg Tangent A = Opposite leg
Hypotenuse Hypotenuse Adjacent leg
*SOH, CAH, TOA
*When you are missing the side of a triangle you use the regular button but when you are missing the angle of a triangle you need to push 2nd button (yellow 1) then the regular button.
10 Rules of Quadrilaterals
i Parallelogram
A Opposite sides have same lengths
B Opposite angles have equal measures
C Diagonals bisect each other
ii Rectangle
A All rules of Parallelogram
B Has 4 right angles
C Diagonals have same length
iii Rhombus
A All rules of Parallelogram
B All sides have same length
C Diagonals make right angles
D Diagonals bisect the angles
E Diagonals are perpendicular
iv Square
A All rules of Parallelogram
B Has 4 right angles
C All sides have same length
D Diagonals bisect the angles
E Diagonals have the same length
F Diagonals are perpendicular
v Trapezoid
A One pair of sides are parallel (bases)
B One pair of sides aren�t parallel (legs)
vi Isosceles Trapezoid
A Legs have same length
B The base angles are equal and the upper angles are also equal
C Diagonals are =
11 Perimeter Formulas (distance around object)\
i Square � Add up all sides
ii Rectangle � Add up all sides
12 Area Formulas (the space inside the object)
i Square � (S)(S)
ii Rectangle � (Length)(Width)
iii Triangle � � (Base)(Height)
iv Trapezoid - � h (b1+b2)
v Circle � pie r2
13 Perimeter and Volume
i To find the perimeter you need to add up all the sides (like a fence or gate)
ii Circumference of a circle � 2pie r
iii Volume � (L)(W)(H) for mostly all shapes except Cylinder that is area of the base multiplied by the height.
14 Locus (Loci) � the set of points that satisfy a given condition.
i The locus of points at a fixed distance, d, from point P is a circle with the given point P as its center and d as its radius.
A Think of as a dog attached to a leash on pole (P) and the leash is 2 feet long (D) the if he tries to move the biggest shape he can make is a circle making that the locus of points.
ii The locus of points at a fixed distance, d, from a line, l, is a pair of parallel lines d distance from l and on either side of l.
A Think of a teacher putting tape on the floor (L) and she asks you to line up sour sticks (D) 3 feet away from the tape on both sides of the tape. You wind up making 2 parallel lines and the lines are the locus of points.
iii The locus of points equidistant from 2 fixed points is the perpendicular bisector of the line segment whose endpoints are the fixed points.
A Think of a fire drill your teachers stand by the 2 sides of the exit and they ask you to line up single file in between them heading down the stairs. Eventually everyone becomes the perpendicular bisector, forming the locus of points.
iv The locus of points equidistant from 2 parallel lines is the parallel midway between them.
A Think of someone running away from the cops on a bike they somehow wind up driving between 2 buildings that are parallel to each other. You need to drive the bike in the middle of the 2 buildings because you are scared the cops are going to jump out of a building to grab you. The path you take is called the locus of points because it forms a straight line in the middle of the 2 buildings.
v The locus of points equidistant from 2 intersecting lines is the bisector of each pair of vertical angles.
A Think of a race between 3 people but since the roads are narrow the judges asked that you run in 3 separate directions if you all running at the same pace the middle person will be the locus of points that is equidistant from the 2 other teammates.
vi Compound Locus
A It�s the same as regular locus but they have more then one condition. The points where the 2 loci meet are called compound locus.
15 Symmetry
i Line Symmetry � if you can fold the picture in � and the 2 sides are the same it is symmetrical (B, D, E, A, M, W, Y, �)
*There can be more then one line of symmetry to a picture
ii Point Symmetry � if you can turn the figure upside down and it looks exactly the same it has point symmetry. (X, S)
16 Transformations � moves the figure according to the rule
i Reflection � flips the figure over (mirror image) Example: (2,4)
A Y- axis - negate the 1st # = (-2, 4)
B X �axis - negate the 2nd # = (2, -4)
C Y = X � flip the #�s around = (4,2)
D Origin � negate both numbers (-2, -4)
ii Translation � slides the figure up, down, sideways, or both
iii Rotation � turns or rotates the figure
iv Dilation � changes the size but not shape of the figure.
Quadrants
II | I
--------|----------
III | IV
17 Formulas
i Midpoint Formula � (x,y) = X1 + X2 , y1 + y2
2 2
ii
Distance Formula �
iii Slope of a line (the steepness of a line) � m = y2-y1
x2-x1
A / = Positive slope
B \ = Negative slope
C | = No slope or undefined
D � = Zero slope
iv Equation of a line � y = mx+b
A M = Slope of the line
B B = the y-intercept or the point at which the line crosses the y-axis
� Parallel lines have the same slope (y = 5x+2 and y = 5x-3)
-b/2a
Perpendicular lines have slopes that are negative reciprocals of each other (y = 5x+2 and y = -1/5x-3)
v Measure of interior angles � (n-2)180�n = # of sides
vi Measure of exterior angles � 360/n
vii Equation of a circle
A When the origin is (0,0) = x2 + y2 = r2
B When the origin is different (a, b) = (x-a)2 + (y-b)2 = r2
18 Parabola � the �U� shaped thing on the graph � (y = ax2 + bx + c)
i Axis of Symmetry � is the line that cuts the parabola into 2 equal halves. =
ii If the �a� is positive the parabola is like a U (smiling) and it is a minimum point
iii If the �a� is negative the parabola is line an n (frowning) and it is a maximum point
iv The turning point or vertex is the point where the parabola �turns direction�
v The roots of a equation are the points where the parabola crosses the x-axis
*Don�t forget to make the chart with x then equation in middle and y on other side.
19 Linear Equations
i Its basically making that chart that you do with parabolas so you always need to solve for y.
ii Once you solve for Y you need to plot the points on graph
iii Draw a solid line if the inequality is < or > and a dotted line if the inequality is a < or >.
iv Pick a point on the graph that is not on the line and see if it is true or false
A True � shade the part that has the point you picked in it
B False � shade the part that doesn�t have the point you picked in it
20 Statistics
i Mean � the average of data (add all numbers and divide but how many numbers you have)
ii Median - is the middle position of a list of data that is arranged in size order (put numbers in size order pick out middle number, if there�s 2 middles fine the middle of the 2 numbers or just leave the answer as the 2 numbers)
iii Mode � the number in a list of data that occurs most often (the number that appears the most)
21 Methods of Organizing Data
i Whisker Box Plot
A Make the following box:
| ---------|---------|
|----------- | | |--------------|
| ---------|---------|
(a) First Line � the lowest number
(b) Second Line � the median of the lower numbers
(c) Third Line � the median of all numbers
(d) Fourth Line � median of higher numbers
(e) Fifth Line � the largest number
ii Frequency Tables / Histograms
A It�s a bar graph that has all the pars connected. You make it like you always make a regular bar graph but u leave a small space before you start to show that it does not start at �0�
22 Probability
i ! = how many diff times the number can be written
A Example: 5! = 120 = 5 x 4 x 3 x 2 x 1
ii Probability of an event = number of times it can occur
Total possible # of outcomes
i Permutation(nPr) � when the order matters
A N = how many numbers you have
B R = how many numbers you need
ii Combination (nCr) � order of numbers don�t matter
23 Exponents
i Multiplying � add exponents
ii Divide � subtract exponents
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