SAT Math Facts & Formulas

Transcription

1 Numbers, Sequences, Factors SAT Mat Facts & Formuas Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reas: integers pus fractions, decimas, and irrationas ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses / Exponents / Mutipy / Divide / Add / Subtract) eac term is equa to te previous term pus d Sequence: t 1, t 1 + d, t 1 + 2d,... Te n t term is t n = t 1 + (n 1)d Number of integers from i n to i m = i m i n + 1 Sum of n terms S n = (n/2) (t 1 + t n ) (optiona) Geometric Sequences: eac term is equa to te previous term times r Sequence: t 1, t 1 r, t 1 r 2,... Te n t term is t n = t 1 r n 1 Sum of n terms S n = t 1 (r n 1)/(r 1) (optiona) Prime Factorization: break up a number into prime factors (2, 3, 5, 7, 11,...) 200 = 4 50 = = 2 26 = Greatest Common Factor: mutipy common prime factors 200 = = GCF(200, 60) = = 20 Least Common Mutipe: ceck mutipes of te argest number LCM(200, 60): 200 (no), 400 (no), 600 (yes!) Percentages: use te foowing formua to find part, woe, or percent part = percent 100 woe pg. 1

2 Averages, Counting, Statistics, Probabiity average = sum of terms number of terms average speed = tota distance tota time Fundamenta Counting Principe: sum = average (number of terms) mode = vaue in te ist tat appears most often median = midde vaue in te ist (wic must be sorted) Exampe: median of {3, 10, 9, 27, 50} = 10 Exampe: median of {3, 9, 10, 27} = (9 + 10)/2 = 9.5 If an event can appen in N ways, and anoter, independent event can appen in M ways, ten bot events togeter can appen in N M ways. (Extend tis for tree or more: N 1 N 2 N 3...) Permutations and Combinations (Optiona): Te number of permutations of n tings taken r at a time is n P r = n!/(n r)! Te number of combinations of n tings taken r at a time is n C r = n!/ ( (n r)! r! ) Probabiity: probabiity = number of desired outcomes number of tota outcomes Te probabiity of two different events A and B bot appening is P(A and B) = P(A) P(B), as ong as te events are independent (not mutuay excusive). Powers, Exponents, Roots x a x b = x a+b (x a ) b = x a b x 0 = 1 x a /x b = x a b (xy) a = x a y a xy = x y 1/x b = x b { ( 1) n +1, if n is even; = 1, if n is odd. If 0 < x < 1, ten 0 < x 3 < x 2 < x < x < 3 x < 1. pg. 2

3 Factoring, Soving (x + a)(x + b) = x 2 + (b + a)x + ab FOIL a 2 b 2 = (a + b)(a b) Difference Of Squares a 2 + 2ab + b 2 = (a + b)(a + b) a 2 2ab + b 2 = (a b)(a b) x 2 + (b + a)x + ab = (x + a)(x + b) Reverse FOIL You can use Reverse FOIL to factor a poynomia by tinking about two numbers a and b wic add to te number in front of te x, and wic mutipy to give te constant. For exampe, to factor x 2 + 5x + 6, te numbers add to 5 and mutipy to 6, i.e., a = 2 and b = 3, so tat x 2 + 5x + 6 = (x + 2)(x + 3). To sove a quadratic suc as x 2 +bx+c = 0, first factor te eft side to get (x+a)(x+b) = 0, ten set eac part in parenteses equa to zero. For exampe, x 2 +4x+3 = (x+3)(x+1) = 0 so tat x = 3 or x = 1. To sove two inear equations in x and y: use te first equation to substitute for a variabe in te second. E.g., suppose x + y = 3 and 4x y = 2. Te first equation gives y = 3 x, so te second equation becomes 4x (3 x) = 2 5x 3 = 2 x = 1, y = 2. Functions A function is a rue to go from one number (x) to anoter number (y), usuay written y = f(x). For any given vaue of x, tere can ony be one corresponding vaue y. If y = kx for some number k (exampe: f(x) = 0.5 x), ten y is said to be directy proportiona to x. If y = k/x (exampe: f(x) = 5/x), ten y is said to be inversey proportiona to x. Te grap of y = f(x ) + k is te transation of te grap of y = f(x) by (, k) units in te pane. For exampe, y = f(x + 3) sifts te grap of f(x) by 3 units to te eft. Absoute vaue: x = { +x, if x 0; x, if x < 0. x < n n < x < n x > n x < n or x > n pg. 3

4 Paraboas: A paraboa parae to te y-axis is given by y = ax 2 + bx + c. If a > 0, te paraboa opens up. If a < 0, te paraboa opens down. Te y-intercept is c, and te x-coordinate of te vertex is x = b/2a. Lines (Linear Functions) Consider te ine tat goes troug points A(x 1, y 1 ) and B(x 2, y 2 ). Distance from A to B: Mid-point of te segment AB: Sope of te ine: (x2 x 1 ) 2 + (y 2 y 1 ) 2 ( x1 + x 2 2, y ) 1 + y 2 2 y 2 y 1 = rise x 2 x 1 run Point-sope form: given te sope m and a point (x 1, y 1 ) on te ine, te equation of te ine is (y y 1 ) = m(x x 1 ). Sope-intercept form: given te sope m and te y-intercept b, ten te equation of te ine is y = mx + b. Parae ines ave equa sopes. Perpendicuar ines (i.e., tose tat make a 90 ange were tey intersect) ave negative reciproca sopes: m 1 m 2 = 1. a a b b a b b a a b m b a Intersecting Lines Parae Lines ( m) Intersecting ines: opposite anges are equa. Aso, eac pair of anges aong te same ine add to 180. In te figure above, a + b = 180. Parae ines: eigt anges are formed wen a ine crosses two parae ines. Te four big anges (a) are equa, and te four sma anges (b) are equa. pg. 4

5 Trianges Rigt trianges: c a b 30 2x x 3 60 x x 2 45 x 45 x a 2 + b 2 = c 2 Specia Rigt Trianges A good exampe of a rigt triange is one wit a = 3, b = 4, and c = 5, aso caed a rigt triange. Note tat mutipes of tese numbers are aso rigt trianges. For exampe, if you mutipy tese numbers by 2, you get a = 6, b = 8, and c = 10 (6 8 10), wic is aso a rigt triange. A trianges: b Area = 1 2 b Anges on te inside of any triange add up to 180. Te engt of one side of any triange is aways ess tan te sum and more tan te difference of te engts of te oter two sides. An exterior ange of any triange is equa to te sum of te two remote interior anges. Oter important trianges: Equiatera: Tese trianges ave tree equa sides, and a tree anges are 60. Isoscees: An isoscees triange as two equa sides. Te base anges (te ones opposite te two sides) are equa (see te 45 triange above). Simiar: Two or more trianges are simiar if tey ave te same sape. Te corresponding anges are equa, and te corresponding sides are in proportion. For exampe, te triange and te triange from before are simiar since teir sides are in a ratio of 2 to 1. pg. 5

6 Circes (, k) r r n Arc Sector Area = πr 2 Circumference = 2πr Fu circe = 360 Lengt Of Arc = (n /360 ) 2πr Area Of Sector = (n /360 ) πr 2 Equation of te circe (above eft figure): (x ) 2 + (y k) 2 = r 2. Rectanges And Friends w Rectange Paraeogram (Square if = w) (Rombus if = w) Area = w Area = Reguar poygons are n-sided figures wit a sides equa and a anges equa. Te sum of te inside anges of an n-sided reguar poygon is (n 2) 180. Soids w r w Rectanguar Soid Voume = w Area = 2(w + w + ) Rigt Cyinder Voume = πr 2 Area = 2πr(r + ) pg. 6