Properties and Formulas
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1 Popeties nd Fomuls Cpte 1 Ode of Opetions 1. Pefom ny opetion(s) inside gouping symols. 2. Simplify powes. 3. Multiply nd divide in ode fom left to igt. 4. Add nd sutt in ode fom left to igt. Identity Popety of Addition Fo evey el nume n, n + 0 = n. Invese Popety of Addition Fo evey el nume n, tee is n dditive invese -n su tt n + (-n) = 0. Identity Popety of Multiplition Fo evey el nume n,1? n = n. Multiplition Popety of Zeo Fo evey el nume n, n? 0 = 0. Multiplition Popety of 1 Fo evey el nume n, -1? n =-n. Invese Popety of Multiplition Fo evey nonzeo el nume, tee is 1 multiplitive invese su tt Q 1 R = 1. Distiutive Popety Fo evey el nume,, nd : ( + ) = + ( + ) = + ( - ) = - ( - ) = - Commuttive Popety of Addition Fo evey el nume nd, + = +. Commuttive Popety of Multiplition Fo evey el nume nd,? =?. Assoitive Popety of Addition ( + ) + = + ( + ). Assoitive Popety of Multiplition (? )? =? (? ). Cpte 2 Addition Popety of Equlity if =, ten + = +. Suttion Popety of Equlity if =, ten - = -. Multiplition Popety of Equlity if =, ten? =?. Division Popety of Equlity wit 0, if =, ten =. Cpte 3 Te following popeties of inequlity e lso tue fo $ nd #. Addition Popety of Inequlity if., ten +. + ; if,, ten +, +. Suttion Popety of Inequlity if., ten -. - ; if,, ten -, -. Multiplition Popety of Inequlity Fo evey el nume nd, nd fo. 0, if., ten. ; if,, ten,. Fo evey el nume nd, nd fo, 0, if., ten, ; if,, ten.. Division Popety of Inequlity Fo evey el nume nd, nd fo. 0, if., ten. ; if,, ten,. Fo evey el nume nd, nd fo, 0, if., ten, ; if,, ten Popeties nd Fomuls
2 Reflexive Popety of Equlity Fo evey el nume, =. Symmeti Popety of Equlity Fo evey el nume nd, if =, ten =. Tnsitive Popety of Equlity if = nd =, ten =. Tnsitive Popety of Inequlity if, nd,, ten,. Slope-Inteept Fom of Line Eqution Te slope-inteept fom of line eqution is y = mx +, wee m is te slope nd is te y-inteept. Stndd Fom of Line Eqution Te stndd fom of line eqution is Ax + By = C, wee A, B, nd C e el numes nd A nd B e not ot zeo. Point-Slope Fom of Line Eqution Te point-slope fom of te eqution of nonvetil line tt psses toug te point (x 1, y 1 ) wit slope m is y - y 1 = m (x - x 1 ). Cpte 4 Coss Poduts of Popotion If = d, ten d =. Peent Eo Fomul getest possile eo peent eo = mesuement Poility Fomul P(event) = nume of fvole outomes nume of possile outomes Poility of Complement Fomul P(event) + P(not event) = 1; P(not event) = 1 - P(event) Poility of Two Independent Events If A nd B e independent events, P(A nd B) = P(A)? P(B). Poility of Two Dependent Events If A nd B e dependent events, P(A ten B) = P(A)? P(B fte A). Slopes of Pllel Lines Nonvetil lines e pllel if tey ve te sme slope nd diffeent y-inteepts. Any two vetil lines e pllel. Slopes of Pependiul Lines Two lines e pependiul if te podut of tei slopes is -1. A vetil nd oizontl line e pependiul. Cpte 7 Solutions of Systems of Line Equtions A system of line equtions n ve one solution, no solution, o infinitely mny solutions: If te lines ve diffeent slopes, te lines inteset, so tee is one solution. If te lines ve te sme slopes nd diffeent y-inteepts, te lines e pllel, so tee e no solutions. If te lines ve te sme slopes nd te sme y-inteepts, te lines e te sme, so tee e infinitely mny solutions. Popeties nd Fomuls Cpte 5 Aitmeti Sequene Te fom fo te ule of n itmeti sequene is A(n) = + (n - 1)d, wee A(n) is te nt tem, is te fist tem, n - 1 is te tem nume, nd d is te ommon diffeene. Cpte 6 Slope slope = vetil nge oizontl nge = ise un Cpte 8 Zeo s n Exponent Fo evey nonzeo nume, 0 = 1. Negtive Exponent Fo evey nonzeo nume nd intege n, -n = 1. n Sientifi Nottion A nume in sientifi nottion is witten s te podut of two ftos in te fom 3 10 n, wee n is n intege nd 1 #, 10. Popeties nd Fomuls 747
3 Multiplying Powes wit te Sme Bse Fo evey nonzeo nume nd integes m nd n, m? n = m + n. Dividing Powes wit te Sme Bse Fo evey nonzeo nume nd integes m nd n, m = m - n. n Rising Powe to Powe Fo evey nonzeo nume nd integes m nd n, A mn B = mn. Rising Podut to Powe Fo evey nonzeo nume nd nd intege n, () n = n n. Rising Quotient to Powe Fo evey nonzeo nume nd nd intege n, Q = n Rn. n Geometi Sequene Te fom fo te ule of geometi sequene is A(n) =? n - 1, wee A(n) is te nt tem, is te fist tem, n - 1 is te tem nume, nd is te ommon tio. Exponentil Gowt nd Dey An exponentil funtion s te fom y =? x, wee is nonzeo onstnt, is gete tn 0 nd not equl to 1, nd x is el nume. Te funtion y =? x models exponentil gowt fo. 0 nd. 1. is te gowt fto. Te funtion y =? x models exponentil dey fo. 0 nd 0,, 1. is te dey fto. Cpte 9 Ftoing Speil Cses Fo evey nonzeo nume nd : 2-2 = ( + )( -) = ( + )( + ) = ( + ) = ( - )( - ) = ( - ) 2 Cpte 10 Gp of Qudti Funtion Te gp of y = x 2 + x +, wee 2 0, s te 2 line x = 2 s its xis of symmety. Te x-oodinte 2 of te vetex is. 2 Zeo-Podut Popety Fo evey el nume nd, if = 0, ten = 0 o = 0. Qudti Fomul If x 2 + x + = 0 nd 2 0, 2 4 " ten x = Popety of te Disiminnt Fo te qudti eqution x 2 + x + = 0, wee 0, te vlue of te disiminnt 2-4 tells you te nume of solutions. If , tee e two el solutions. If 2-4 = 0, tee is one el solution. If 2-4, 0, tee e no el solutions. Cpte 11 Multiplition Popety of Sque Roots Fo evey nume $ 0 nd $ 0,! =!?!. Division Popety of Sque Roots Fo evey nume $ 0 nd. 0,! =. Å! Te Pytgoen Teoem In igt tingle, te sum of te sques of te lengts of te legs is equl to te sque of te lengt of te ypotenuse = 2 Te Convese of te Pytgoen Teoem If tingle s sides of lengts,, nd, nd = 2, ten te tingle is igt tingle wit ypotenuse of lengt. Te Distne Fomul Te distne d etween ny two points (x 1, y 1 ) nd (x 2, y 2 ) is d = Î Ax 2 2 x 1 B2 1 Ay 2 2 y 1 B 2. Te Midpoint Fomul Te midpoint M of line segment wit endpoints A(x 1, y 1 ) nd B(x 2, y 2 ) is Q x 1 1 x 2 2, y 1 1 y 2 2 R. Tigonometi Rtios lengt of leg opposite /A sine of &A = lengt of ypotenuse osine of &A = tngent of &A = lengt of leg djent to /A lengt of ypotenuse lengt of leg opposite /A lengt of leg djent to /A 748 Popeties nd Fomuls
4 Cpte 12 Multiplition Counting Piniple If tee e m wys to mke fist seletion nd n wys to mke seond seletion, tee e m 3 n wys to mke te two seletions. Pemuttion Nottion Te expession n P stnds fo te nume of pemuttions of n ojets osen t time. = n(n - 1)(n - 2) } ftos n P Comintion Nottion Te expession n C stnds fo te nume of omintions of n ojets osen t time. n C n n(n 2 1)(n 2 2) = P = P ( 2 1)( 2 2) Popeties nd Fomuls Popeties nd Fomuls 749
5 Fomuls of Geomety You will use nume of geometi fomuls s you wok toug you lgeook. Hee e some peimete, e, nd volume fomuls. P 2 2w A w Retngle w P 4s A s 2 Sque s d C 2p o C pd A p 2 Cile A 1 2 Tingle A Pllelogm 1 2 A 1 2 ( 1 2 ) Tpezoid w V B V w Retngul Pism V 1 3 B Pymid se se se V B V p 2 Cylinde V 1 3 B V 1 3 p2 Cone V 4 3 p3 Spee 750 Popeties nd Fomuls
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