College Mathematics CLEP
vertices/vertex
"corners"/angles of a closed shape
absolute value rules
(presuming that B does Not = 0) |A ÷ B| = |A| ÷ |B| |A × B| = |A| × |B| |A|² = A² -*when there is a variable inside absolute value brackets and/or if you are solving for an equation when one side has absolute value brackets the value of what is inside the absolute value brackets may be positive or negative so you have to solve for both*, ex. |5 - 3x| = 7 [or -7] -3x = 2 OR -3x = -12 x = -2/3 OR x = 4 (note this equation would have No solution: |5x + 4| = -3 [*an absolute value can't be negative*])
median
(after the numbers are put in order) the middle number (when there are two middle numbers it is their average)
mean
(average) add the numbers together then divide by how many numbers there were
area
(note: height = altitude) -triangles: 1/2 × base [to which the altitude is drawn] × height -parallelograms: base × height [drawn perpendicular to the base] -rectangles [& squares]: length × width -square (alternative): 1/2 × (length of diagonal)² -rhombus: 1/2 × (diagonal a × diagonal b) -trapezoid: 1/2 × height × (base a + base b) -circle: π × radius²
combination
(without repetition) *n*/[total number of objects] objects taken *r*/[size of group] at a time (the order doesn't matter, ex. groups) formula: n! ÷ r! × (n-r)! (note: after cancelling out almost everything you are left with: the first r factors of n! ÷ r!, ex. if n = 8 and r = 3 then it would become (8 × 7 × 6) ÷ (3 × 2 × 1) which can further cancel out to 8 × 7 = 56 [the denominator is always able to cancel out, ex. (10 × 9 × 8) ÷ (3 × 2 × 1) = 10 × 3 × 4])
Cartesian product
(written: M × W) the set of all ordered pairs of elements (where the first component/x coordinate is a member of M and the second component/y coordinate is a member of W) ex. M × W = {1, 3, 5} × {2, 8} = {(1, 2), (1, 8), (3,2), (3, 8), (5, 2), (5, 8)} note: making a table/graph with the first set as the rows and the second set as the columns may help -to find out how the # of members/elements the set will have when you multiply the sets together you multiply the # of elements in set 1 with the # of elements in set 2
permutation
-(without repetition) *n*/[total number of objects] objects taken *r*[size of group] at a time (the order does matter/make a difference, ex. lists) formula: n! ÷ (n-r)! (note: ÷ (n-r)! cancels out part of n!, ex. 6!/4! = 6 × 5 = 30 & 12!/9! = 12 × 11 × 10 = 1320) -(with repetition) n^r
measurements/formatting numbers
-*unit conversion*: 1 ft. = 12 in. 1 yd. = 3 ft. 1 mi. = 1,760 yd. 1 in. = 2.54 cm. 1 ft. = 30.48 cm. 1 meter/m. = 3.2808 ft. 1 m. = 39.37 in. 1 km = 1000 m. 1 km = 0.621 mi. 1 qt. = 2 pints 1 peck = 8 qt. 1 bushel = 4 pecks 1 pint = 2 cup 1 qt. = 2 pints 1 gl. = 4 qt. 1 gl. = 8 pints 1 lb. = 16 oz. 1 ton = 2000 lb. 1 oz. = 28.35 gr. 1 kg = 1000 gm 1 lb. = 0.4536 kg. (9/5 x °C) + 32 = 1° F -*metric prefixes*: y, z, a, f, p, nano (n) = 0.000000001 micro (µ) = 0.000001 milli (m) = 0.001 centi (c) = 0.01 deci (d) = 0.1 deca (da) = 10 hecto (h) = 100 kilo (k) = 1,000 mega (M) = 1,000,000 giga (G) = 1,000,000,000 tera (T) = 1,000,000,000,000 P, E, Z, Y -*scientific notation*: a way to abbreviate very large or small numbers which have many digits, ex. 5.6 x 10^-9 = 0.0000000056 (note: you move the decimal point so that the # times 10 is less than 10, the exponent indicates how many places the decimal point was moved, and a negative exponent means the # is a small # and a positive exponent means the # is big) -*significant figures*: all non-zero digits, zeros between non-zero digits, trailing zeros/those to the right of the last non-zero digit in a number with a decimal point, and sometimes trailing zeros in a number without a decimal point
properties of real numbers
-commutative property: (applies to addition and multiplication) the numbers can be rearranged in the equation without changing the answer -associative property: (applies to addition and multiplication) the numbers can be grouped/associated in any order, ex. 2 + (3 + 4) = (2 +3) + 4 -*distributive property: the first number gets distributed to the ones in parentheses, ex. 2 × (3 + 4) = (2 × 3) + (2 × 4)* -identity property: adding 0 or multiplying by 1 does Not change the original value -inverse property: the inverse of + is - (ex. 3 + (-3) = 0) & the inverse of × is ÷ (ex. 3 × 1/3 = 1) & note: the multiplicative inverse does Not work for 0 because *a number ÷ 0 = not defined* -order property: Only 1 of the following is true x > y, x = y, or x < y -*to find the lowest common multiple/LCM of two numbers first find their prime factorizations, then find any factors which they have in common and eliminate all except for exactly one set of them (ex. the prime factorizations of 27 and 90 are 3 × 3 × 3 and 3 × 3 × 2 × 5 respectively so they have exactaly two 3s in common so you eliminate all the duplicates so you only have one set of the factors which they have in common, thus you are left with 3, 2, and 5) then multiply all of the factors which you haven't eliminated together to equal the LCM [note: for 3 or more #s multiply each prime factor the maximum number of times it appeared in any number, ex. 100, 81, 36 = (2, 2, 5, 5), (3, 3, 3, 3), (3, 3, 2, 2) = multiply 5, 5, 2, 2, 3, 3, 3, 3]
equations
-equivalent equations: equations with the same solution -an equation without a solution is said to have a solution set that is the empty/null set -you can combine like terms, ex. 3x + y + x + 2y = 4x + 3y & 2y × x × y × y × 4y = 2y × x × 4y × y² -you need to perform the same operations to *both sides* of the equation
sets
-intensional definition = using a rule or semantic/words description (sometimes abbreviated, ex. A = {k | 0 < k < 10, k a whole number} = {1, 2, 3, ... 8, 9}) & extensional definition = denoted by enclosing the list of members in {} -sets which differ only in that one has duplicate members are in fact exactly identical (ex. the set {11, 6, 6} is exactly identical to the set {11, 6} & {6, 11} = {11, 6} = {11, 6, 6, 11}) -the order of the elements in a set does Not matter -a finite set has a countable number of members (including ∅) & an infinite set has an infinite amount of members -∈ = belongs to/is a member/element of (ex. x ∈ B) -∅ = empty/null set (∅ ⊆ every set & every set is a subset of itself) -the difference of two sets: A - B = all elements in A but not in B
logarithms
-logarithmic function (which is the inverse of an exponential function): f^-1(x) = (log⌄b) × x [f^-1( ) indicates that it is the inverse Not that it is a negative exponent] because a logarithm is the inverse of an exponent (b^x = y) = ([log⌄b] × y = x) [note: argument = whatever is inside the logarithm (ex. [log⌄b] × (argument) = x)] -if there isn't a base to a logarithm then it is implied that the base is 10 -natural logarithm: a logarithm with the base of e (there is a button for e on a calculator but e ≈ 2.72), sometimes abbreviated with a fancy lowercase LN with the argument after it -when the argument is negative or 0 then the logarithm is undefined (because it doesn't work) -(if N, M, p, and b are positive and b = 1): log⌄b (1) = 0 log⌄b (b) = 1 log⌄b (bⁿ) = n (note: these rules are just a different way of saying the same rules that apply to exponents, you can turn the logarithm into the exponent form to verify this) -product rule: (they must have the same base) log⌄a (3) + log⌄a (5) = log⌄a (3 × 5) -quotient rule: (they must have the same base) log⌄a (3) - log⌄a (5) = log⌄a (3 ÷ 5) -power rule: log⌄b (M^p) = p × log⌄b (M) = p × b^answer to logarithm = M
graphed functions
-note: the x is also called the abscissa and the y is also called the ordinate -linear: diagonally going up (bottom left to top right), they are first degree functions (they don't have any powers of x) -constant: horizontal line -quadratic: [f(x) = x²] a "U" shape anchored at 0 on the x-axis -cubic: [f(x) = x³] a curved somewhat vertical line which goes through quadrant III (-,-) and quadrant I (+,+) -square root: [f(x) = √x] a ray which starts at 0 on the x-axis and goes through quadrant I (+.+) -absolute value: [f(x) = |x|] a "V" shape anchored at 0 on the x-axis -increasing/positive: if the function goes up from left to right (lower x value,lower y value & higher x value, higher y value) -decreasing/negative: if the function goes down from left to right (lower x value,higher y value & higher x value, lower y value) -0 slope = a horizontal line & no slope = a vertical line -slope (sometimes abbreviated as m) = rise ÷ run = y from 2nd point - y from 1st point ÷ x from 2nd point - x from 1st point -note that in a function in the form y = ax + b the a = the slope & b = the y intercept -the perpendicular slope/the negative reciprocal of the slope is a fraction with one over the slope (don't forget to change the positive/negative sign that was originally there) -even: if replacing x with -x does Not change the answer (on a graph it is mirrored/symmetric around the y-axis) -odd: if replacing x with -x results in changing every sign of every term of the original function (symmetric around the origin/(0,0) point) -greatest integer function: the greatest integer (...-4, -3, -2, -1, 0, 1, 2, 3, 4...) that is less than or equal to x (basically if it is already an integer you don't change it but if it is a decimal you round it Down, ex. 3 = 3; 5.7 = 5; -3.4 = -3, then you plot it on a graph with (x,x), ex. (7,7) or (-5,-5), etc.) -to find a function's x intercept/where it crosses the x-axis set y = 0 then solve; to find the y intercept set the x = 0 then solve -to find the domain (all possible x values) from a graph look for the highest and lowest points on the x-axis which the function touches (note that if a function continues indefinitely on that axis in a certain direction the domain will be all numbers above/below the part which does Not continue indefinitely [depending on which direction the function goes from the finite side]); to find the range (all possible y coordinates which the function may be on) on a graph is similar to finding the domain (except you look on the y-axis, thus you look for the highest and the lowest points) -to find a functional value (the y value of a coordinate) with a graph look at the x value they want it to correspond with and find where the function crosses that x value on the graph to find your corresponding y value
fractions
-proper fractions: between -1 and +1 (the numerator/top is smaller than the denominator/bottom) -improper fractions: the numerator is bigger than the denominator -mixed numbers: when (after you simplify an improper fraction) you have a whole number and a fraction left
volume & surface area
-rectangular solid: length × width × height -sphere: (4/3) × π × radius³ -circular cylinder: π × radius² × height -circular cone: (1/3) × π × radius² × height & -rectangular solid: 2(L × W) + 2(H × W) + 2(H × L) -sphere: 4 × π × radius² -circular cylinder: 2 × π × radius × height -circular cone: π × radius × [r + √(radius² + height²)]
sequential sentences/"calculus of sentences"
-sentence: an expression which can be labeled true or false -sentences can be connected using AND, OR, NOT, and IF-THEN -consistent: if it is possibly true -logically true: if it is impossible for it to be false/inconsistent -inconsistent/logically false: if it is impossible for it to be true -logically indeterminate/contingent: if it is neither logically true nor false -logically equivalent (denoted by ↔): if it is impossible for one of the sentences to be true and the other to be false at the same time (they have to both be have the same truth value [both true or both false]), thus "if P then Q"'s equivalent would be "not P or Q" -statement: it is either true or false not both -conjunction: "a and b" (denoted: a ^ b) -disjunction: "a or b" (denoted: a ⌄ b) -negation: "not a" (denoted: ~ a) -implication: 1. conditional statement: "if a, then b" (denoted: a → b, note: if a is true/T but b is false/F then a → b is false/F), 2. hypothesis/premise of the implication/antecedent: "if a", 3. conclusion of the implication/consequent: "then b", [a = a sufficient condition & b = a necessary condition & in a biconditional statement both a and b are sufficient and necessary conditions] -converse: the converse of a → b is b → a -contrapositive: the contrapositive of a → b is ~b → ~a -contraposition: a law that says that a conditional statement is logically equivalent to its contrapositive -inverse: the inverse of a → b is ~a → ~b -biconditional: "a if and only if b" (denoted: a ↔ b) -counterexample: an example that disproves a universal/"for all" statement -valid: an argument is valid if the truth of the premises means that the conclusions must also be true -intuition: making generalizations based on insight -syllogism: three statements of which you can use the major premise (the first statement, which is a general statement about a whole group) and the minor premise (the second statement, which is a specific statement which indicates that a certain individual/thing is a member of the group that the major premise refers to) to deduce the third statement (called the deduction, which is a statement that basically says/implies that the major premise also applies to the individual/thing which is a member of the group), however even if the major and the minor premises are true the deduction may be false if the minor premise is inappropriate (ex. all who vote are at least 18, Jane is at least 18, Jane votes [although the premises are true, Jane could choose to not vote if she felt like it was a waste of time]) -a truth table: an exhaustive list of the possible values (T/true or F/false) of a sentence (IFF = "if and only if") -(theorem 1) double negation = identity: ~~X [not-not] ↔ X -(theorems 2 & 3) properties of conjunction (^) or disjunction (⌄) [note all the signs in a given equation must be either conjunction ^ or disjunction ⌄ for these rules to apply]: 1. commutativity: X ^/⌄ Y ↔ Y ^/⌄ X 2. associativity: X ^/⌄ (Y ^/⌄ Z) ↔ (X ^/⌄ Y) ^/⌄ Z -(theorem 4) distributive laws: 1. X ⌄ (Y ^ Z) ↔ (X ⌄ Y) ^ (X ⌄ Z) 2. X ^ (Y ⌄ Z) ↔ (X ^ Y) ⌄ (X ^ Z) -(theorem 5) De Morgan's laws for sentences: 1. ~(X ^ Y) ↔ (~X) ⌄ (~Y) 2. ~(X ⌄ Y) ↔ (~X) ^ (~Y) -(theorem 6) two logical identities: 1. X ↔ (X ^ Y) ⌄ (X ^ ~Y) 2. X ↔ X ⌄ (Y ^ ~Y) 3. (X → Y) ↔ (~X ⌄ Y) -(theorem 7) proof by contradiction: X → Y ↔ ~X → ~Y
Interest
-simple: only based on the original amount of money invested/borrowed formula: principal [aka the amount of money invested/borrowed] × interest rate × number of years = interest -compound: you must calculate the interest each year and add it to the balance before you can calculate the next year's interest payment (because the interest from previous years is used to calculate the new interest, however the interest rate does Not change) long formula: 1. principal × interest rate = interest for 1st year, then 2. (principal + interest earned) × interest rate = interest for 2nd year, then 3. you do that as many years as necessary (note: sometimes the interest is added more or less often) short compound interest formula: future value = current value × {[1 + (i ÷ c)]^[c × # of years]} c = # of times that interest is compounded annually i = annual interest rate as a decimal -continuous interest formula: principle × e^(interest rate × # of years) note: there is a button on the calculator for e but e ≈ 2.7183 -effective annual interest rate: an investment's annual rate of interest which takes into account the effects of compounding, formula: (1 + i ÷ n)ⁿ - 1 (terms: i = annual interest rate; n = number of compounding periods) -effective annual yield/annual percentage rate (APR): the cost per year of borrowing money including any additional fees involved (ex. banks and credit cards are required to provide this information)
triangles
-the interior angles measure a total of 180° -scalene: has No equal (length) sides -isosceles: has at least two equal sides (the unequal side is called the base & the angles next to the base are equal) -equilateral: all three sides are equal (this type is also equiangular = each angle equals 60°) -acute: has three acute (< 90°) angles -obtuse: has one obtuse (> 90°) angle -right: has one right (90°) angle -median: a line segment connecting a vertex/"corner" and the midpoint of the opposite side -perpendicular bisector [of the side which it is perpendicular]: a line that bisects (divides in two) and is perpendicular (makes a plus sign [as opposed to parallel]) to one side of the triangle -angle bisector: a line that bisects an angle and extends to the opposite side of the triangle -midline: a line segment that goes from the midpoint of one side to the midpoint of another side of the triangle -exterior angle: an angle formed outside a triangle by one side and an extension of an adjacent (having a common vertex) side
factoring
-to break a number up into numbers that can be multiplied together to get the original number (ex. 45 = 5 × 8; prime factorization: ex. 45 = 5 × 2 × 2 × 2), you can also use exponents to write your answer so it isn't as long and repetitive -a *prime number (ex. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97...)* is a positive whole number that only has two factors (itself and one), the opposite is composite numbers (which also must be positive), 0 and 1 are neither prime nor composite -*divisibility tricks: if the number's digits added together add up to a number that's divisible by 3 then the number itself is divisible by 3; if the number is divisible twice by 2 then it's divisible by 4; if it's divisible by 2 and by 3 then it's divisible by 6; and if it's divisible twice by 3 (or if the sum of the digits is divisible by 9) then it's divisible by 9*
percents
-to find a percent: x% of y = x in decimal form × y thus if x% × y = a and you are only given y and a you can do a ÷ y = x (which can then be turned into a %) -note: a percent can easily be turned into a decimal (by moving the decimal point of the % two places to the right [which makes it smaller], because a % is in the hundredth place) or a fraction (by putting the number over 100)
moving functions
-translation of a function: the same function shape which [each point] is moved a specified amount up/down (by adding how far it moves up or using a negative to move it down to the function) and/or left/right (by changing every x in the function to (x + # of how far moved), positive moves it left [which is towards the negative half of the graph] and negative moves it right) y = f(x) + C [C > 0 moves it up; C < 0 moves it down] y = f(x + C) [*C > 0 moves it left*; C < 0 moves it right] -you can stretch/make the function have a wider mouth or you can compress/make the function skinnier (taller/y direction) by multiplying the whole function by a constant y = Cf(x) [stretch/wider = # > 1; compress/skinnier = 0 < # < 1] -you can stretch or compress a function in the x-direction by multiplying x (Not the whole function) by a constant y = f(Cx) [compress = # > 1; stretch = 0 < # < 1] -rotating 90°: clockwise = each point (x, y) is changed to (y, -x) counterclockwise = each point (x, y) is changed to (-y, x) -reflection (the mirror image of the function) about/around/switching sides of the: 1. x-axis (flip upside down) = each point (x, y) is changed to (x, -y); multiply the whole function by −1; y = −f(x) 2. y-axis (flip left/right) = each point (x, y) is changed to (-x, y); multiply the x-value by −1; y = f(−x) 3. the (line y = x) = point (x, y) is changed to (y, x)
subsets
-⊆ = what is on the closed side is a subset of/contained in what is on the open side (ex. A ⊆ B & {1, 3} ⊆ {1, 2, 3, 4}) -⊂ = if what is on the closed side is a *proper subset of/contained in what is on the open side (ex. A ⊂ B = B contains at least one element/member not contained in A) -improper subsets: a subset that contains all of the elements in the set, this is calculated by: 2^[number of elements in the set] (a proper subset excludes at least one element of the superset, this is calculated in the same way except that you subtract 1) -two sets are equivalent if they have the same Number of elements -subsets are a superset of the set they are a subset to (ex. A subset of B = B superset of A) -the power set of a set is all of that set's subsets -a universal set/U is a set from which other sets draw their members/elements & the compliment of a set (ex. A') is all of the elements of the universal set which are Not in the set that it is the compliment of
reciprocal
[of a number] is 1 ÷ that number (as a fraction: 1/that number) [0 is an exception to this]
union/∪
a combination of the sets (excluding duplicates), ex. {3,7,8} ∪ {12,3,8,16} = {3,7,8,12,16}
factor & multiple
a counting number that divides into another number (which the first number is a factor of) with no remainder & a number that can be divided by another number with no remainder (ex. multiples of 20 are 20, 40, 60, 80, etc.)
standard deviation (σ)
a measure of how spread out numbers are and how close individual data points are to the mean/average of the sample, (0 = all numbers are the same), in a normal distribution of data about 68% of the numbers will be within +/- one standard deviation from the mean (95.5% of the data is within 2σ and over 99% is within 3σ) formula: 1. find the average/mean, 2. separately subtract each # in set from the mean (# - mean), 3. square each #, 4. add the squares, 5. divide by the # of #s in the group (before step 4) minus 1 (this new # is the variance), 6. the square root (of the variance) is the σ -formula: variance = σ² = Σ ([each #] - mean)² ÷ (# of #s in original group) - 1
rational & irrational numbers
a number that can be written as a ratio/fraction (includes any whole number, ex. 8 = 8/1) [note: when written as a decimal they may have either terminating or nonterminating digits but if they are nonterminating then they have to have a repeating block] & a number that can be written as a decimal but Not as a fraction (because it has endless/nonterminating non-repeating digits after the decimal point)
z score
a score's relationship to the mean in a group of scores (0 means the score is the same as the mean) formula: z = [raw score] - μ/mean ÷ σ
inequalities
a statement in which two numbers/equations are compared to each other using the following symbols: > (greater than), < (less than), ≥ (greater than or equal to), ≤ (less than or equal to), ≠ (not equal to) -transitive property: if a < b and b < c, then a < c & if a > b and b > c, then a > c -addition property: if a > b, then a + c > b + c and a - c > b - c -*graphing an inequality on the number line: represented by a ray or a line segment [or uncommonly a line] (< or > than a certain/specified number is indicated with an open/outline of a circle on that number & ≤ or ≥ than a certain/specified number is indicated with a filled in circle/dot on that number* & if they only specify one of the boundaries for what number the expression is then there is a line going off in the direction that the number is) -**when you divide or multiply by a negative number you mess up/reverse/flip the inequality sign** -*conditional inequality*: when certain values of the variables make that inequality true and other values do not make it true -*absolute inequality*: any real value for the variable will work (ex. x + 5 > x + 2) -*inconsistent inequality*: if the inequality is always false/not true when its variables are allowable values (ex. x + 10 < x + 5 & 5y < 2y + y [this doesn't work if y > 0]) -*two inequalities have the same sense if their signs of inequality point in the same direction* -if a > b then aⁿ > bⁿ and a^-n < b^-n -if x > y and q > p then x + q > y + p -if x > y > 0 and q > p > 0 then x × q > y × p
events/E
a subset (containing at least one outcome) of a sample space/S (which is the set of all possible outcomes of the experiment) -independent events: if the outcome of one event does not affect the outcome of the other event, P(E and F) = P(E)∩P(F) -mutually exclusive events: if events have no elements in common/no overlap (ex. when rolling a die the event of an even number and the event of an odd number are mutually exclusive), thus: probability(E or/∪ F) = P(E) + P(F), [however when E and F are Not mutually exclusive: P(E or/∪ F) = P(E) + P(F) - P(E and/∩ F)] -complementary events: when the event doesn't happen/the other outcomes, P(not E) = 1 − P(E)
using negative numbers
addition: - + - = - + + - = + or - [whichever value is larger] subtraction: - - + = + + + [it is the same as adding a +] = + or - [depending] multiplication & division: - × + = - - × - = +
quadrilaterals
any polygon (a closed, plane figure with at least three straight sides and angles) with four sides -parallelograms: a quadrilateral where each set of its opposite sides are parallel -consecutive angles: the two angles that are at the ends of the same side of a parallelogram -diagonal [of a polygon]: a line segment connecting any two nonconsecutive (not next to each other) vertices, (if the diagonals of a parallelogram are equal it is a rectangle) -rectangle: a parallelogram with right angles -rhombus/rhombi: a parallelogram that has equal sides, (the diagonals of a rhombus are perpendicular bisectors of each other, thus they create right angles where they meet) -square: a rhombus (equal sides) with right angles, (the measure of either diagonal can be calculated by: length of a side × √2) -trapezoid: a quadrilateral with exactly two parallel sides (which are called the bases), (the median [which is parallel to the bases and is half of their sum] is the line connecting the midpoints of the nonparallel sides), there are two pairs of base angles (the angles on either end of the same one of the parallel lines), (the altitude is a line that connects the two bases) -isosceles trapezoid (which has equal nonparallel sides): its base angles are equal, its diagonals are equal, and the opposite angles are supplementary (add up to 180°)
scatterplot patterns
bivariate data (data that has two variables): the explanatory variable (on the x-axis) and the response variable (on the y-axis), the explanatory variable explains or predicts the response variable and the response variable measures the outcomes that have been observed -the shape: either linear (straight) or nonlinear (curved) -the direction: this describes what happens to the y variable as the x variable increases [typically from left to right] (showing the overall slope of the data), this is either positive (if y increases) or negative (if y decreases) -the strength: describes how tight or spread out the points on the scatter plot are -when analyzing the data look for outliers (stray data points isolated from the majority), clusters, or gaps
circles
circles are often named by their center point & π = pi ≈ 3.14 -radius: a line segment from the center to the outside of a circle -circumference: the length of the outside of a circle = π × diameter = 2 × π × radius -secant: a line that intersects a circle at two points -chord: a line segment that connects two points of a circle -diameter: a chord that passes through the center of the circle -line of centers: a line that passes through the centers of two or more circles -inscribed angle: an angle whose vertex/"corner"/angle is on the outside edge of the circle and whose sides are chords of the angle -central angle: an angle whose vertex is at the center of the circle and whose sides are radii -an arc of the circle: the portion of a circle cut off by a central angle, (the measure of a minor arc is the same as the measure of the angle) -the measure of an arc intercepted by a central angle: v/360 × (2 × π × radius) [v = the measure of the center angle] -sector: the "slice of pie/the circle" between two radii -area of a sector: v/360 × (π × radius²) [v = the measure of the center angle] -tangent: a line that only intersects at one point/once with a circle (and the spot where they intersect is called the point of tangency) -congruent circles: circles which have congruent/same size radii -concentric circles: circles which have the same center but different size radii (they look like circles inside of each other) -circumscribed circle: a circle which passes through all the vertices of a polygon, (and the polygon inside the circle is said to be inscribed in the circle)
working with odd and even numbers
e = even number & o = odd number e + e = e o + o = e e + o = o e × e = e e × o = e o × o = o
factorial function (!)
ex. 5! = 5 × 4 × 3 × 2 × 1; 12! = 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 8! ÷ 5! = 8 × 7 × 6
reverse FOIL
ex. x² + 4x -12 = (x - 2)(x + 6) note: the two factors whose product equals the constant (-12, and in this case one is positive and one is negative in order to get a negative product) when added together they equal the coefficient of the second to last term/x (4) -this can be used to solve for x: ex. 4x³ + 12x² - 72x first factor out like terms so you can multiply the whole equation by what they have in common: 4x(x² + 3x - 18) [note: this is the same as dividing the whole expression by what is outside the parenthesis] then it can be reverse foiled: 4x(x + 6)(x - 3) now to solve for x you set each of the terms equal to zero: 4x = 0 and x + 6 = 0 and x - 3 = 0 then solve to find values for x: x = 0 and x = -6 and x = 3 [note: if you go back to the original equation and plug these numbers in for x the equation will equal 0]
linear equations
format: a × x + b = 0 (a and b aren't variables and a ≠ 0) to solve this means to change it into the form: x = -b ÷ a (then to simplify if possible) -other formats: ax + by = c (where neither a or b = 0), or mx + b = y (m = slope & b = y intercept), {note: the form y - y1 = m(x - x1) may also be a helpful form to have the linear equation in} -linear equations can Not have exponents on the variables/must be a 1st degree polynomial -when it is in the form of a fraction you must cross multiply it before solving it, ex. (3x + 4) ÷ 3 = (7x + 2) ÷ 5 after cross multiplying this turns into 3 × (7x + 2) = 5 × (3x + 4) -note: ex. (3/5)y = 10 multiply both sides by the reciprocal (which is 5/3) to cancel out the coefficient of the y thus getting x = 50/3 -simultaneous linear equations: a linear equation with two unknown variables, in order to solve these you must have the same number of equations as there are unknown variables -methods for solving systems of simultaneous linear equations (multiple equations solved at once): 1. substitution: find the value of one of the variables in terms of the other (by isolating one of the variables) then plug that equation in everywhere that variable is (thus leaving you with only one variable), then solve and plug in the answer to find the other variable 2. addition or subtraction: multiply each equation (make sure to multiply the whole equation) by numbers (# can be different for each equation) that will make the equations have the same coefficient attached to the same variable (Not necessarily like signs), then if the signs (+/-) of the equal coefficients are the same subtract one equation from the other but if they are different then add the equations together (thus leaving you with only one variable), solve this equation then plug the answer into the other equation as the variable which was not eliminated and solve for the variable which was eliminated 3. graph: graph both equations, the point where they intersect is a simultaneous solution for the equations (its coordinates correspond to the answer that would be found by the other methods of solving) -there are consistent (if there is only one solution) and inconsistent (if it doesn't have any solutions, ex. parallel lines) systems of linear equations -dependent equations: equations which represent the same line (thus all of the points on the lines are the same/are on the other line too, and any point which satisfies one equation works for the other too), to determine if two equations are dependent you put the linear equations in slope intercept form (y = mx + b) then compare them -parallel lines: if their slopes are equal (put into the form y = mx + b to compare [however, if the y intercept/b is the same they are dependent equations]) they are parallel, you can also try to solve simultaneously but they will have no simultaneous solution (place where they cross/intersect) -perpendicular lines: if the slopes of the lines are negative reciprocals (negative reciprocals ex. 2 and -1/2; -5/6 and 6/5) of each other then the lines are perpendicular
quadratic equations
format: ax² + bx + c = 0, [a ≠ 0 & x = the only variable] -factoring to solve: (if a/the coefficient of x² = 1) 1. find two numbers which satisfy the following: m × w = c & m + w = b [notes: if c is positive then m and w will have the same sign as b; if c is negative m and w are going to have opposite signs with the larger number having the same sign as b], 2. then insert the numbers into this: (x + m)(x + w) = 0, to finish 3. solve both (x + m) = 0 & (x + w) = 0 to find both possible solutions -difference of two squares: a² - b² = (a + b)(a - b) [typically a is the variable x, note: a is the √ of a²] -sum of two squares: a² + b² = (a + bi)(a - bi) (note: if there is a plus in between two perfect squares then you can Not find the difference/factor it normally) this works because i² = -1 thus ex. x² + 25 = (x + 5i)(x - 5i) = [after FOIL] x² -5xi +5xi -25i² = [the -5xi and +5xi cancel out and i² = -1] x² + 25 -quadratic formula: x = (-b ± √[b² - 4ac]) ÷ 2a [note: the (b² - 4ac) part under the square root sign is called the discriminant of the quadratic equation & if the discriminant is negative the square root is an imaginary number and becomes a complex number after the numerator of the formula is simplified & if the discriminant = 0 the roots are real and equal & if the discriminant is > 0 and a and b were rational numbers then the roots are rational and it is a perfect square (otherwise the roots are irrational)]
factor theorem
if f(x) is a polynomial and f(#) = 0 then x - # is a factor of f(x) [remember factor means you multiply it by another number/expression to get the number it's a factor of] -to find a factor of a given polynomial you must find a number (usually a factor of the last/constant part of the polynomial) to make the variable which makes the polynomial equal 0 (if you know all but one factor than you can insert the number you are testing to check it but don't forget to include the factor you are testing into the multiplication too) then put that number into the form (x - #) [don't forget to add the negative/minus sign, thus if the # is negative it will become positive in the factor] and that is a factor Or -to find the factors of an equation you do reverse foil Or -to check the factors you can divide the polynomial by the factor and if it is indeed a factor of the polynomial you will end up with a remainder of 0 -also note that if f(a/b) = 0 then b × x - a is a factor of f(x) [a and b are constants]
remainder theorem
if p(x) (a polynomial with the variable of x) is divided by (x - a) [a is any constant] then the remainder is p(a) [which is plugging a into the polynomial for x] thus we can calculate the remainder without doing the division ex. p(x) = x3 - 7x - 6, and divided by x - 4 (so a = 4), then the answer would be x2 + 4x + 9 with a remainder of 30 note: remember that (just as it is when dividing real numbers without variables) the remainder is smaller that the number which is being divided by, and just as with regular numbers this shows this relationship: 13 ÷ 5 = 2 R 3 is the same as 13 = 5 × 2 + 3 (thus both p(x) ÷ (x - a) = answer with a remainder, and p(x) = [(x - a) × answer] + remainder are equivalent) {links to explanations: 1.https://www.khanacademy.org/math/algebra2/arithmetic-with-polynomials/long-division-of-polynomials/v/polynomial-division ([based on approx. 6 min. into video] *ask yourself this × what = what is under the divided by bracket?* then follow the instructions given normally [don't forget to multiply all of the terms which you are dividing by and subtracting the product(s) from what is under the division bracket]) 2.https://www.khanacademy.org/math/algebra2/arithmetic-with-polynomials/polynomial-remainder-theorem/v/polynomial-remainder-theorem}
fundamental counting principle/counting rule
if there are m ways one experiment can be performed and w ways that a second experiment can be performed then there are m × w (ex. 6 ways to roll a die × 2 ways to flip a coin) different ways both experiments can be performed in this order/m then w
intersection/joint probability/∩
if two events (E and F) occur on a single performance of an experiment (ex. when rolling a die: odd # and #5), an intersection of two sets includes only the parts that are in both of the sets -disjoint sets: ex. A ∩ B = ∅
a Venn diagram
is a visual way to show the relationships among sets that share something in common
histograms & stem-and-leaf plot
looks similar to a bar graph however the bars are Not necessarily the same thickness (ex. if the x-axis denotes years then one bar [continuous block which goes to the same height] could span three years and a different block could span five years) & has a vertical line with a single column of numbers to the left and corresponding rows of numbers on the right, the column of numbers on the left typically represent the tens place and the numbers on the right typically represent the ones place
real numbers & complex numbers
numbers that can be found on the number line including rational and irrational numbers -integers: ... -4, -3, -2, -1, 0, 1, 2, 3, 4, ... -whole numbers: 0, 1, 2, 3, 4, ... -natural numbers: 1, 2, 3, 4, ... -the square of a real number can Not be negative (0² = 0) & however complex numbers [ex. a + b × i] include i [√-1 & i² = -1] which is an imaginary unit thus complex numbers are Not real numbers -raising imaginary numbers to exponents: (remembers any number with the exponent of 0 = 1) i^1 = i i² = -1 i^3 = -i i^4 = 1 and the pattern repeats... thus iⁿ = i^(4 × y) + z [this equals the original exponent/n] and z will indicate which of the first four in the pattern iⁿ equals -add/subtract the real parts and add/subtract the imaginary parts: (x + yi) + (z + wi) = (x + z) + i(y + w) (x + yi) - (z + wi) = (x - z) + i(y - w) (x + yi) × (z + wi) = (xz + wy) + i(xw + yz) [FOIL =(multiply) First #s, Outer #s, Inner #s, Last #s & you could also use the distributive property] -dividing complex numbers: 1. find the conjugate of the complex number (conjugate of [also indicated with a line above the complex number] of a + bi = a - bi [you just change the sign of the imaginary part]) 2. multiply both complex numbers by the conjugate of the denominator the same way you would multiply complex numbers (when you multiply the conjugate and the original complex number you get a real number & you can simplify multiplying the complex number and its conjugate by doing this if you want: a² + b²) [note: every real number is also a complex number because they can be expressed as: a + 0i & (if b = a positive real number) √-b = i√b]
PEMDAS
order of operations: parentheses, exponents/logarithms, multiply/divide, add/subtract
laws of set operations
presuming that U is the universal set and A, B, C are any subsets of U -identity laws: A ∪ ∅ = A & A ∩ ∅ = ∅ & A ∪ U = U & A ∩ U = A -indempotent laws: A ∪ A = A & A ∩ A = A -compliment laws: A ∪ A' = U & A ∩ A' = ∅ & ∅' = U & U' = ∅ -commutative laws: A ∪ B = B ∪ A & A ∩ B = B ∩ A -associative laws: (A ∪ B) ∪ C = A ∪ (B ∪ C) & (A ∩ B) ∩ C = A ∩ (B ∩ C) -*distributive laws: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) & A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)* -*De Morgan's laws: (A ∪ B)' = A' ∩ B' & (A ∩ B)' = A' ∪ B'*
intervals
represent sets of points (on a number line) that satisfy the conditions of an inequality (note: < and > [and their representations on the number line] do Not count as end points but ≤ and ≥ do count as [definite] end points) -open: does Not include any end points, symbol: (...) -closed: includes two end points, symbol: [...] -half-open: includes one end point, symbol: [...) or (...]
probability
represented by a number from 0 to 1 (thus it can easily be represented by a percentage, fraction, or decimal), the sum of the probabilities of all the possible outcomes of an experiment equals 1 (because it is certain that one of those outcomes is going to happen, ex. probability of E + probability of E' [the compliment of E/E not happening] = 1) -the probability of A and/or B (if the probability of one event does Not effect the probability of the other event): P(A) + P(B) - P(A ∩ B) [the times A and B overlap/both occur] (ex. getting a spade or an ace from a deck of cards: 13/52 + 4/52 - 1/52 [which is the ace of spades]) -probability of A or B (Not both): P(A) + P(B) - 2 × P(A ∩/and B) -the odds of an event happening is a ratio of the # of favorable outcomes : the # of unfavorable outcomes/the favorable event's complement, (thus the odds of an event happening versus not happening can be added together to determine the total # of outcomes) -finding the probability of an empirical event: # of times E/specific outcome occurs ÷ total # of observed situations in which E could occur -theoretical probability: finding the probability of events that come from an equiprobable space/a sample space with equally likely outcomes, # of outcomes in E ÷ # of number of outcomes in S, (aka # of favorable outcomes ÷ total # of number of outcomes) -conditional probabilities: the probability of one event/A, given the occurrence of some other event/B, P(A|B) = P(A ∩ B) ÷ P(B), thus P(A ∩ B) = P(A|B) × P(B) -expected value: all outcomes multiplied by their probability of occurring then added together, ex. -1(1) + 0(1/2) + 1(1/8) + 2(1/4) + 10(1/8) = -1 + 0 + 1/8 + 2/4 + 10/8 = 7/8 or 0.875 -multiplication rule: the probability that events A and B both occur = the probability that A occurs times the probability that B occurs (considering that that A has already occurred)
fundamental theorem of algebra
says that a polynomial (an expression of more than two algebraic terms, especially containing different powers of the same variable) to the n[>0]th degree (x^n + bx^(n-1) + ax^(n-2)...-y, ex. 7x³ - 2x² + 4x - 8 is a polynomial to the 3rd degree) is going to have exactly n roots [a root is a value of x which will make the expression equal to zero, on a graph this is where the function crosses the x-axis] which may be real or complex (note that complex roots always come in pairs [the complex root and its conjugate] thus only polynomials to even degrees can have all of their roots as complex roots, however they can also have a combination of complex and real roots but the # of complex roots will always be even) pg.76(E&F) {link to explanation: https://www.khanacademy.org/math/algebra2/polynomial-functions/fundamental-theorem-of-algebra/v/fundamental-theorem-of-algebra-intro}
fundamental theorem of arithmetic/unique [prime] factorization theorem
says that every integer greater than 1 is either a prime number or is the product of prime numbers
range
the difference between the lowest and highest numbers (high - low)
mode
the number that there is the most of (note that on a graph: if the it is distributed normally/evenly/bell-shaped the mean, median, and mode are all approximately the same, if it is skewed to the left/the mode [which is represented by the highest point] is on the right/high end then mean < median < mode, and vise versa)
present value & future value
the value of your money now compared to its value at a future time (ex. if you invest your money it will be worth more in the future or inflation may effect the buying power of your money, etc.) formula: cash flow in the future ÷ (1 + the discount rate)^number of years between now and the future time & the value of an asset at a specific date in the future formulas: 1) for simple interest: current value × (1 + [interest rate × # of years]); 2) for compound interest: current value × ([1 + interest rate]^# of years) note: the interest rate is typically a %
functions
they have/are a certain rule of what to do with the domain (the input/what you start with, independent variables, x) in order to get the range (the output/what you end up with, dependent variables, y, functional value) -a relation: a relationship between sets of information (all functions are relations but not all relations are functions, for a relation to be a function there must be *only one y/answer/range that corresponds to each x* [however there can be more than one x that leads to the same y]), when a function is "a well-behaved relation" we know how to find the range when we are given the domain -a one-to-one function: when each y value is paired to exactly one x value, to test this you can do the horizontal line test (if a horizontal line only passed through the graphed function once then it is a one-to-one function), if a function is Not a one-to-one function then its inverse will Not be a function -the vertical line test: if any two points on the function have the same x value, aka if you can draw a vertical line on the graph which goes through/intersects the "function" twice, it is Not a function -you can also check if it is a function by solving for y (it must have a unique/exactly one answer for y in order to be a function, thus if the answer could be +/- then it isn't), first put it in the form y = ax + b (note a = the slope & b = the y intercept) then solve to see if it has exactly one value for y -exponential functions: [b > 0 and Not the # 1] y = f(x) = b^x -combining functions: (f +/-/×/÷ g)(x) = f(x) +/-/×/÷ g(x) -composite function: (f ° g)(x) = f(g(x)) [meaning you calculate g(x) then you calculate f(x) while treating g(x) as the x in f(x), thus it matters which order you do it in] -inverse function: a function that "reverses" another function, if f is a function mapping x/the variable/starting point to y/the answer then the inverse function of f maps y back to x, the inverse of a function is denoted as f^-1, ex. f(x) = 3x + 2 to find the inverse set it up as y = 3x + 2 then isolate the x, leaving you with (y - 2) ÷ 3 = x, which is the same as y/3 - 2/3 = x, then you switch the x's and the y's and you are left with the inverse of f(x) [f^-1(x) = (x - 2) ÷ 3 = y thus you could alternatively just reverse/undo (with the proper operations in Reverse PEMDAS order) everything which is done to x] note: if f and g are inverses and (f ° g) = x and (g ° f) = x then to find g when given f you change y = f(x) into y = g(x) and solve -to quickly determine if the inverse of a function will be a function you just have to do the horizontal line test (like the vertical line test) because the inverse is "mirrored" so that those points will not pass the vertical line test -to find the domain: to do so remember that the denominator of a function can Not = 0 and there can Not be a negative under a radical (thus any values for x which would break these two conditions can Not be a possible value for x); to look for values of x which would make the denominator = 0 you take the denominator and set it to 0 then solve for x; to get rid of a negative underneath a radical you take the expression out and set it as an inequality which is greater than or equal to 0 then you solve the inequality (in order to find all values of x which would work) -piecewise defined/hybrid functions: a function which is defined by multiple sub-functions, each sub-function applying to a certain interval of the main function's domain (a sub-domain), thus once you know your what domain to plug in you check to see which formula applies to it and solve to find the corresponding range -polynomial functions: functions involving only non-negative integer powers of x (ex. a quadratic function, a cubic function, etc.) -numerical function: a function whose domain and range are subsets of R/the set of real numbers -direct variation: if the dependent/y and independent/x variables are directly proportional -linear growth: it grows by the same amount in each time step/at a constant rate (ex. a line always going up 3 units for every 2 units it goes over) -exponential growth: the increase in a quantity according to a certain law (kind of like compound interest), function ex. y = 3^x
similar polygons
they must have congruent/the same measurements for the corresponding angles and their corresponding sides must have ratios that are equivalent (the numbers you get when you cross multiplying the ratios should be equal) thus they are the same shape but may be different sizes [note: their areas have the same ratio as the measures of a pair of corresponding sides²]
SohCahToa
to find the sine, cosine, or tangent of an angle on a right triangle: sine = opposite ÷ hypotenuse cosine = adjacent ÷ hypotenuse tangent = opposite ÷ adjacent (terms: hypotenuse = the long side of a right triangle which is opposite of the right angle, opposite = side of triangle opposite of the angle in question, adjacent = the side of the triangle which is next to the angle in question [which is not the hypotenuse or the opposite])
plane & solid geometry
two dimensional shapes & three dimensional objects
Pythagorean Theorem
used on a *Right triangle: a² + b² = c² (c = the hypotenuse/the long side/the side opposite the right angle a & b = the legs/arms/the other two sides) note: this can also be used to find the length of a diagonal line on a graph (you simply measure two imaginary legs and solve for the line they want the measurement of [which is the hypotenuse])
normal distribution
when information forms the shape of a bell curve (mostly in the middle and diminishing on the left and right), when this is the case the mean, median, and mode are all the same and in the center of the information
exponents/powers
where the base (the number that the exponent is on) is multiplied by itself the number of times indicated by the exponent (ex. 7³ = 7 × 7 × 7) -a negative exponent implies a fraction (ex. 7^-3 = 1/7³) -exponents on negative numbers: (-2)^4 = 16, however -2^4 = -16 [the negative is added after calculating 2^4] -any number with the exponent of 0 = 1 [unless the base is 0] -fractional exponents: [if m and w are both positive integers] a^(m/w) = w√a^m [note: w indicates what √ to take], ex. 3^(4/2) = ²√3^4 = √81 = 9 -negative fractional exponents: take the reciprocal of the base then do the same as you would for a fractional exponent, ex. 27^(-2/3) = 1/27^(2/3) = 1/³√27² = 1/³√729 = 1/9 -exponents on fractions: you raise the numerator to the exponent and the denominator to the exponent (ex. (2/3)² = 4/9) -general laws (note that all of the bases are the same): a^m × a^w & a^(2 × 3) = a^(m + w) (a^m)^w = a^(m × w) [a ≠ 0:] a^m ÷ a^w = a^(m - w) (a × b)^m = a^m × b^m [a ≠ 0:] (a/b)^m = a^m ÷ b^m -note: exponents can undue radicals (if you need the square root of something and you raise it to the power of 2 you eliminate the radical, and like wise getting rid of a cubed root radical by raising what is under the radical to the power of 3, however you have to remember to do the same thing to both sides of an equation so you would square one side of and equation [the whole side] then you would square the other side too) -a square root is the same as what is under the radical raised to the 1/2 power -to simplify a radical you find factors which are a perfect square and you put the square root [likewise you would do cubed roots if it was a cubed root] of them multiplied by what is left in the radical, ex. to simplify √72 you realize that 36 × 2 = 72 and because 6 is the square root of 36 you can simplify it to 6 × √2
