Formula List for College Algebra Sullivan 10 th ed. DO NOT WRITE ON THIS COPY.

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1 Forula List for College Algera Sulliva 10 th ed. DO NOT WRITE ON THIS COPY. Itercepts: Lear how to fid the x ad y itercepts. Syetry: Lear how test for syetry with respect to the x-axis, y-axis ad origi. Liear Equatio Forulas: Stadard or Geeral For: Ax + By = C Slope forula: y y y 1 1 also 1 1 f x f x f f a f x h f x x x x x x a h Slope y-itercept for: y x or Liear Fuctio: f ( x) x Poit Slope for: ( ) y y x x or y x x y Systes of Liear Equatios: Icosistet the syste has NO SOLUTIONS (Cotradictio) Depedet the syste has INFINITELY OR MANY SOLUTIONS (Idetity) Cosistet the syste has at least ONE SOLUTION (Coditioal) Idepedet the syste has differet lies (ay have oe solutio or oe). Quadratic Equatio: A equatio is a equatio of the for : Square Root Method: Lear square root ethod. Quadratic Forula: ax x c 0, a, ad c are real uers ad a 0. 4 The solutios of the equatio ax x c 0, where a 0, are x The Discriiat: For a.. c. ax x c 0, a 0: If ac there are two uequal real solutios 4 0,. If ac there are two equal or oe real solutios 4 0, " " " ". If ac there is o real solutio 4 0,. ac a Zero-Factor priciple: a 0 if ad oly if a 0 or 0. Factorizatio Forulas: The Differece of Two Squares The Su of Two Squares The Differece of Two Cues A B A B A B A B prie ( )( ) 3 3 A B ( A B)( A AB B ) MAC115 College Algera Forula List Sulliva 10 th ed. Page 1 of 6

2 3 3 The Su of Two Cues A B ( A B)( A AB B ) Trioial Squares The Square of a Bioial A AB B ( A B)( A B) ( A B) A AB B ( A B)( A B) ( A B) Cue of a Bioial: x y x 3x y 3x y y x y x 3x y 3x y y Differece quotiet of a fuctio f at x is give y: f x h f x, where h 0. h The Algera of Fuctios: Su : f g x f x g x Differece : f g x f x g x f f x g g x Pr oduct : f g x f x g x Quotiet : x where g x 0 Quadratic Fuctio: A quadratic fuctio is oe i the for: MAC115 College Algera Forula List Sulliva 10 th ed. Page of 6 f x ax x c where a,, ad c are costats ad a is ot equal zero. Quadratic Equatio i Vertex For: The vertex for of the equatio ax x c 0, where a 0, is: v v y a x x y or f x a x h k, where x, y ( h, k ) is called the vertex. v v Vertex of a paraola: xv is the AXIS of Syetry ad yv f. a a 4ac I other words the vertex is : h,k x v, y v, f,. a a a 4a Distace Forula: The distace etwee poits (, ), give y: d x x y y 1 1 Midpoit Forula: The idpoit M x, y (, ), x y ad x y i the coordiate plae is give y: 1 1 x x1 y y1 M x, y, x y ad x y i the coordiate plae is 1 1 of a lie seget with edpoits

3 Modelig ad Regressio Aalysis: Scatterplot: Go to Y= ad press eter o STATPLOT #1 to tur ON. STAT, select 1. EDIT (eter x values i list 1 ad y values i list ) WINDOW (set viewig widow) or press Zoo #9 GRAPH Fid the Best Graphical Model or Regressio Lie / Curve: STAT CALC #4 for Liear Modelig, #5 for Quadratic Modelig, etc. ad press eter oce o the scree or press CALCULATE To paste your aswer oto Y= ad graph lie o scatterplot: Go to Y1 = ad ake sure is lak VARS select #5, arrow to EQ, select #1 (pastes eq. i Y1) GRAPH (graphs plot ad lie) CALC #1 (evaluates for a iput) The Equatio of a Circle: The stadard for of the Equatio of a Circle: x h y k r. The equatio of a circle with ceter (0, 0) ad radius r is give y: x y r The geeral for of the equatio of a Circle: For A, B, C, D, ad E real uers, A = B, A ad B ot zero, the geeral for of the equatio of the circle is give y: Ax By Cx Dy E 0. Other textooks ay have the geeral for as: x y ax y c 0. Vertical Lie Test: Kow what is ad how to do a vertical lie test. Polyoial Fuctio: 1 1 f ( x ) a x a 1 x... a1 x a0 where is deg ree of the polyoial. Power Fuctio: f ( x ) a x where a is a real uer, a 0, ad 0 is a it eger. MAC115 College Algera Forula List Sulliva 10 th ed. Page 3 of 6

4 Ratioal Fuctio: A ratioal fuctio is oe of the for f x P x Q x 0 where P x ad Q x are polyoials ad Q x. Vertical Asyptotes: If Q a 0, ut P a 0, the the graph of the ratioal fuctio P x Q x f x has a vertical asyptote at x a. Horizotal Asyptotes: P x Suppose f x is a ratioal fuctio where the deg ree of Q x P x is ad the deg ree of Q x is, ( ). a ) If, ( the deg ree of the uerator is less tha the deg ree of the deo i ator ) the the graph of f has a horizotal asyptote at y 0. ) If a, the the graph of f has a horizotal asyptote at y, where a is the lead coefficiet of P x ad is the lead coefficiet of Q x. c ) If, the the graph of f does ot have a horizotal asyptote. d ) If 1 ( the deg ree of the uerator is oe ore tha the deg ree of the deo i ator ), the the lie y ax is a olique asyptote, which is the quotiet foud u si g log divisio. e ) If the deg ree of the uerator is two or ore tha the deg ree of the deo i ator ), the there are o horizotal or olique asyptotes. Note : A ratioal fuctio will ever have oth a horizotal asyptote ad a olique asyptote. Copositio of Fuctios: Let f x ad g x repreet two fuctios. The copositio of f ad g,,., writte f g x is defied as f g x f g x Here g x ust e i the doai of f x. If it is ot, the f g x will e udefied. Oe-to-oe Fuctios: The iverse of a fuctio f is also a fuctio if ad oly if f is oe-to-oe. The graph of a oe-to-oe fuctio f ad the graph of its iverse fuctio are syetric with respect to the lie y = x. f 1 MAC115 College Algera Forula List Sulliva 10 th ed. Page 4 of 6

5 Iverse Fuctios: 1 Suppose the iverse of f is a fuctio, deoted y f. The 1 f y x if ad oly if f x y. Copositio of a Fuctio ad its Iverse: If a fuctio f x has a iverse f x the 1,, : 1 f f x x for every x i the doai of f, ad f f x x for every x i the doai of f 1 1 Expoets: a 1. a a a. a, a 0 a 3. a a 4. a a a a 1 5., 0 6. a, a 0 a a a 7., a 0, 0 8. a a a 1, a a a, is a it eger. 11. a a a a Expoetial Fuctio: 1 1 x f x a, where, a ad x are real uers, 0, 1 ad a 0 0 The ase the growth factor ad ecause f 0 a a, a is called the iitial value. The uer e is defied y the expressio: ad as e li 1. Expoetial Forulas: Siple Iterest Forula: I P r t Copoud Iterest: A P 1 r r t Copoud Iterest with Copoud ties per year: A P1, P pricipal, r aual rate, uer of copoudigs per year, t uer of years, A aout after t years. Copoud Iterest Cotiuously: A P e rt. MAC115 College Algera Forula List Sulliva 10 th ed. Page 5 of 6

6 Expoetial Equality: x y If, the x y where 0 ad 1. Logariths ad Expoets: Coversio Equatios If 0 ad x 0, the y y y log x if ad oly if x. y l x if ad oly if e x. Useful Logarith Properties: 1 1 log 1, ecause l e 1, ecause e e. 0 0 log 1 0, ecause 1 l 1 0, ecause e 1. x x x x x x log x, ecause l e x, ecause e e. log x l x x for x e x for x, 0, 0. Other Properties of Logariths: If x, y ad 0, the If x ad y 0, the a. log x y log x log y a. l x y l x l y x x. log log x log y. l l x l y y y k k c. log x k log x c. l x k l x Properties of Natural Logariths: If x ad y 0, the x k a. l x y l x l y. l l x l y c. l x k l x y The Natural log ad x e : x l x l e x, for all x ad e x, for x 0. Chage the ase of a logarith: log10 a l a log a log l 10 Asolute Value: Defiitio of Asolute Value: x x, if x 0 x, if x 0 Asolute Value Equatios ad Iequalities: a. ax c c 0 is equivalet to : ax c or ax c MAC115 College Algera Forula List Sulliva 10 th ed. Page 6 of 6. ax c c 0 is equivalet to : c ax c c. ax c c 0 is equivalet to : ax c or ax c