Download Math A Review Sheet - Xaverian Math Department

1. MATHEMATICAL REASONING
Conditional:
p q
Inverse:
~p  ~q
Converse:
q p
Contrapositive: ~q  ~p
the conditional and contrapositive have the same truth value
the inverse and the converse have the same truth value
Venn Diagram:
2. NUMBERS AND NUMBERATION
Rational Numbers: numbers that can be expressed as the quotient/ratio of two integers OR terminating/repeating decimals
counting numbers (1,2,3,….)
whole numbers (0,1,2,3,…)
integers (…-3,-2,-1,0,1,2,3…)
Irrational Numbers: numbers that neither repeat nor terminate
Closure: A set of numbers is said to be closed under an operation if that operation always results in exactly one number that
is a member of the set
Commutative Property: the order in which the operations are performed does not affect the result [addition/multiplication]
Associative Property: the way in which the numbers are grouped does not affect the outcome [(addition)/(multiplication)]
Distributive Property: a(b + c) = ab + ac
Additive Inverses: two numbers that add together to equal zero
Identity element for addition is zero
Multiplicative Inverses: two numbers that multiply to equal one
Identity element for multiplication is one
3. OPERATIONS
Isometry: original figure is congruent to image
Transformations: movement of a point or figure in the plane according to a rule
ry  axis (-x,y)
rorigin (-x,-y)
Reflection rx axis (x,-y)
Translation: add or subtract to x and y
Glide Reflection
Rotation: counterclockwise turn R90 (-y,x)
ry  x
R180 (-x,-y)
(y,x)
R270 (y,-x)
Dilation: multiply x and y by the constant *** not isometric
Absolute value: distance of a number from zero
Order of Operations: Parenthesis Exponents Multiplication Division Addition Subtraction
Monomial (term): a number, a variable or the product of a number and one or more variables
Like terms: contain the same variable(s) to the identical power
5 Rules of Exponents:
r m r n  r m n
rm
 r mn
to divide like bases, subtract the exponents
rn
to multiply like bases, add the exponents
to take a power of a power, multiply the exponents
r 
m n
to multiply different bases, distribute the exponents rs 
m
 r mn
 rmsm
m
rm
r
to divide different bases, distribute the exponents    m
s
s
0
Zero Exponent: any variable to the zero power is one a  1
1
a
Negative Exponent: x  a
x
Factoring Methods:
GCF:
Difference of Two Perfect Squares:
a 2  b 2  (a  b)(a  b)
Quadratic Trinomial: ax  bx  c
(When a = 1): 1. look for factors of the third term (c)
2. choose the factors that will combine to equal the second term (b)
Factoring Completely: most often using the GCF method to factor, then factoring the parenthesis
2
Radicals:
only add/subtract like radicals
ab  a b
a

b
a
b
Equations: To get the variable alone on one side of the equals sign. Use ‘opposite’ operations. Whatever is done to one
side, is done to the other.
1. clear parenthesis 2. combine like terms 3. addition/subtraction 4. multiplication/division
Inequalities: Follow the same rules as equations, EXCEPT: multiply/divide by a negative number you must flip the sign
< less than
> greater than
these are open circles on the number line
these are closed circles on the number line
 less than or equal to  greater than or equal to
Rational Fractions (factor if possible!!):
Addition/Subtraction: must have the same denominator
Multiplication: cancel top to bottom/crosswise, then multiply the numerators and multiply the denominators
Divide: multiply by the reciprocal
Equations: multiply each side of the equals sign by the LCD to ‘get rid’ of the denominators
Zero Product Property: If ab= 0, then either a = 0 or b = 0
Solving Quadratic Equations: when the equation is set to zero, factor and use the zero product property to solve
Roots: solutions of a quadratic equation, ‘x’ intercepts of a parabola (graph of a quadratic equations)
4. MODELLING/MULTIPLE REPRESENTATIONS
Complementary angles: two angles whose sum is 90°
Supplementary angles: two angles whose sum is 180°
Vertical angles: two pairs of angles formed by two intersecting lines
Perpendicular lines: two lines that intersect to form 4 right angles
Perpendicular bisector of a line: line that is perpendicular to the segment at its midpoint
Parallel lines: lines in the same plane that do not intersect
If two parallel lines are cut by a third line (a transversal), then the following pairs of congruent angels are formed:
Alternate interior angles (congruent pairs) 3,6 and 4,5
Corresponding angles (congruent pairs)
1,5 3,7 2,6 4,8
Alternate exterior angles (congruent pairs) 1,8 and 2,7
Interior angles on the same side of the transversal are supplementary 3,5 and 4,6
***vertical angles 1,4 2,3 5,8 6,7
***supplementary angles
Polygon: simple closed figure consisting only of line segments. The point where line segments meet is called a vertex.
Name
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
n-gon
Number of Sides
3
4
5
6
7
8
9
10
n
Equilateral: all sides congruent
Equiangular: all angles congruent
Interior angles of a polygon with n sides is 180(n-2)°
Exterior angles of a polygon with n sides is 360°
Triangles:
Scalene: no equal sides
Isosceles: two equal sides
Equilateral: three equal sides
Acute: three acute angles
Equiangular: three equal angles
Right: one right angle
Obtuse: one obtuse angle
Quadrilaterals:
Parallelogram: opposite sides are parallel and congruent
a diagonal creates two congruent triangles
opposite angles are congruent
consecutive angles are supplementary
diagonals bisect each other
Rectangle:
Rhombus:
all angles are right angles
diagonals are congruent
Square: has all the properties of both a
rectangle and a rhombus
all sides are congruent
diagonals are perpendicular to each other
diagonals bisect angles of the rhombus
5. MEASUREMENT
Slope-intercept form of a line: y = mx + b where m is the slope and b is the ‘y’-intercept
Slope: a ratio that compares the change in the vertical distance along a line to the change in the horizontal distance
y 2  y1
x2  x1
when moving left to right:
horizontal line has a slope of zero
vertical line has an undefined slope
lines that rise have a positive slope
lines that fall have a negative slope
Midpoint Formula:
 x 2  x1 y 2  y1 
,


2 
 2
x2  x1 2   y 2  y1 2
Distance Formula:
Perimeter: sum of the lengths of the sides of a polygon
Area:
Rectangle: A = lw
2
Square: A = s
Parallelogram: A = bh
Trapezoid: A =
Triangle: A =
1
(b1 + b2)
2
1
bh
2
Volume:
3
Cube: e
Rectangular Solid: lwh
Cylinder:
r 2 h
1 2
r h
3
2
Circle: C = 2 r and A = r
( x  h) 2  ( y  k ) 2  r 2 where (h,k) are the coordinates of the center
Cone:
Loci:
1. Equidistant from 2 points
2.
Equidistant from 2 lines
3.
Point and a
4.
Line and a distance
5.
Equidistant from intersecting lines
distance
Ratio: comparison of two numbers using
following ways:
x to y or
x:y or
division. A ratio is usually expressed in one of the
x
y
Proportion: an equation showing that two ratios are equivalent.
extreme
mean

mean
extreme
Similar Polygons: have congruent angles and their corresponding sides are in proportion
Trigonometry: the relationship between two sides and an angle of a right triangle
Soh Cah Toa: mnemonic device used to help remember the trig ratios
 : the symbol for angle
adj
hyp
2
2
2
Pythagorean Theorem: a  b  c
Sin 
opp
hyp
Cos 
Tan 
opp
adj
Isometry: original figure is congruent to image
Transformations: movement of a point or figure in the plane according to a rule
Reflection
Dilation
Translation
Glide Reflection
Rotation
Locus (plural loci): is the set of all points that satisfy a given condition or a set of conditions
Mean: the average when the total of the scores is divided by the number of scores
Median: the middle score when the scores are arranged numerically from least to greatest
Mode: the score or scores that occur the greatest number of times
6. UNCERTAINTY
Counting Principle: If an event can happen in ‘m’ ways, followed by an event that can happen in ‘n’ ways, the total number
of ways the two events can happen is ‘mn’
P(A and B) is found by finding the probability of all outcomes that are in both event A and event B
P(A or B) is found by finding the probability of all outcomes that are in either event A or event B or in both events
P(A or B) = P(A) + P(B) – P(A and B)
Permutation: an arrangement of a set of objects in a particular order nPr
Combination: an arrangement of a set of objects in which their order is not important nCr
7. PATTERNS/FUNCTIONS
System of Equations: two or more equations
Three methods to solve: graphing, substitution, addition