Mth 65 Prelulus Review Properties of Inequlities 1. > nd > >. > + > +. > nd > 0 > 4. > nd < 0 < Asolute Vlue, if 0, if < 0 Properties of Asolute Vlue > 0 1. < < <. > > or <. or Intervl Nottion, x: < x<, x: x, x: x<, x: < x (, ) { x : x> } [, ) { x : x } (, ) { x : x< } (, ] { x : x }, R ( ) { } [ ] { } [ ) { } ( ] { } ( ) Distne Formul etween P1( x1, y1) nd P( x, y ) d( P, P ) ( x x ) + ( y y ) 1 1 1 Midpoint Formul + + M x x, y y 1 1 Symmetries of Grphs 1. y xis: Sustitution of x for x leds to the sme eqution.. x xis: Sustitution of y for y leds to the sme eqution.. origin: Sustitution of oth x nd y for x nd y leds to the sme eqution. Center-rdius form of the eqution of irle enter C( h, k) nd rdius r ( ) ( ) x h + y k r If r 1 then the irle is lled unit irle. 1
Lines nd Slopes y y1 slope m ; x x 1 m > 0 line rises from left to right m < 0 line flls from left to right m 0 line is horizontl m undefined line is vertil m m lines re prllel 1 m m 1 lines re perpendiulr 1 Equtions of Lines point-slope form y y m( x x ) 1 1 slope-interept form y mx + stndrd or generl form x + y or x + y + d 0 horizontl line y k, konstnt vertil line x k, k onstnt Prols y ( x h) + k vertex ( hk, ); opens up if >0, down if <0 x y k + h vertex ( hk, ; opens right if >0, left if <0 ( ) ) y x + x + vertex ( hk, ); opens up if >0, down if <0 where h nd 4 k (or simply sustitute the vlue of h) 4 x y + y + vertex ( hk, ; opens right if >0, left if <0 Funtions where ) k nd 4 h 4 (or simply sustitute the vlue of k) A funtion f from set D to set E, f : D E, is orrespondene tht ssigns to eh element x of D extly one element y of E. The set D is the domin of the funtion. The rnge of f is the suset of E onsisting of ll possile funtion vlues f( x) for x in D. A funtion is usully defined y giving the formul or rule for finding f( x). The domin is then the set of ll vlues of x for whih f( x ) is rel. f is sid to e defined t x if f( x ) is rel or f( x) exists, i.e., x is in the domin of f; f is sid to e undefined t x if x is not in the domin of f. In the nottion vrile. y f( x), x is lled the independent vrile nd y is lled the dependent Grph of funtion is the grph of the eqution y f( x ) for x in the domin of f. Vertil Line Test: Every vertil line intersets the grph of funtion in t most one point. zeros of funtion: solutions of the eqution f( x ) 0; the x-interepts of the grph even funtion: if f( x) f( x) x Df ; grph is symmetri wrt y-xis
odd funtion: if f( x) f( x) x Df Bsi Grphs 1. f( x) x liner funtion (identity funtion) ; grph is symmetri wrt origin. f( x) x qudrti funtion. f( x) x ui funtion 4. f( x) 1 x reiprol funtion
5. f( x) x solute vlue funtion 6. f( x) x squre root funtion 7. f( x) x 1 8. f( x) x 4
Pieewise-defined funtions - funtions tht re defined y more thn one expression exmple: x if x< 0 f( x) 1 if x 0 gretest integer funtion ( ) f x x is the gretest integer n suh tht n x; on the oordinte line, n is the first integer to the left of (or equl to) x. Trnsformtions of Grphs 1. vertil shifts - result when positive onstnt is dded to or sutrted from f( x) ex. y x + y x 4. horizontl shifts ex. ( x ) ( x 5 y y + si prol shifted units down si prol shifted 4 units down si prol shifted units to the right ) si prol shifted 5 units to the left. refletions ex. y x refletion of the si prol long the x-xis 4. ompressions/expnsions 1 ex. y x expnded si prol y x ompressed si prol 5. omintion of the ove ex. ( 1) y x + + 5 Opertions on Funtions f + g f + g ( x) f( x) + g( x) 1. sum : ( ). differene : ( ) f g f g ( x) f( x) g( x). produt fg : ( fg )( x) f ( x) g( x) 4. quotient / : f g f g ( x) f( x) g( x) 5
The domin of f + g, f g, fg,nd f g is the intersetion of the domins of f nd g; for f g, the domin is the set of ll x in the intersetion for whih Composition of Funtions ( f g) ( x) f ( g( x )) gx ( ) 0. The domin of f g is the set of ll x in the domin of g suh tht g(x) is in the domin of f. Further Clssifition of Funtions polynomil f( x) x x x n n 1 n + n 1 + + 1 + 0 where 0, 1,, n re rel nd the exponents re nonnegtive integers If 0, then f hs degree n. n degree 0: f( x) onstnt funtion degree 1: f( x) x+ liner funtion degree : f ( x) x + x + qudrti funtion p rtionl ( ) ( x f x ), where p nd g re polynomil funtions gx ( ) lgeri n e expressed in terms of sums, differenes, produts, quotients, or rtionl powers of polynomils trnsendentl not lgeri ex. trigonometri, exponentil, logrithmi Trigonometry Rdins nd Degrees π rdins 180 rdin to degree: multiply y 180 π degree to rdin: multiply y π 180 Length of Cirulr Ar s r where s is the length of the r r rdius of the irle rdin mesure of the entrl ngle sutended y the r 1 Are of Cirulr Setor A r where is the rdin mesure of entrl ngle r rdius of the irle 6
Trigonometri Funtions A. of n ute ngle : sin s os se tn ot B. of ny ngle P (, ) r sin s r r os se r tn ot x + y r C. of rel numer vlue of trig funtion of rel numer x vlue of trig funtion t n ngle of x rdins Fundmentl Identities s 1 sin se 1 os ot 1 tn tn sin os ot os sin sin + os 1 1+ tn se 1+ ot s Speil Vlues of the Trigonometri Funtions (rdins) π 6 π 4 π (degrees) 0 1 45 60 sin os 1 tn 1 7
Referene Angle: π ; ( os,sin ) ; ( ) π + ; os, sin π Grphs of the Trigonometri Funtions ysin x y s x yos x y se x y tn x y ot x 8
Other Formuls 1. formuls for negtives sin( u) sinu os( u) osu tn( u) tnu s( u) s u se( u) seu ot( u) otu. ddition nd sutrtion formuls sin( u± v) sinuosv± osusinv os( u± v) osuosv sinusinv tnu± tnv tn( u± v) 1 tnutnv. doule-ngle formuls sinu sinuosu os u os u sin u os u 1 1 sin tnu tnu 1 tn u 4. hlf-ngle formuls 1 osu sin u 1+ osu os u u 9