Generl Formuls Summry Informtion nd Formule MTH109 College Algebr Temperture: F = 9 5 C + 32 nd C = 5 ( 9 F 32 ) F = degrees Fhrenheit C = degrees Celsius Simple Interest: I = Pr t I = Interest erned (chrged) P = Principl (originl deposit, originl lon) r = nnul interest rte (APR) t = time in yers Compound Interest: A = P 1 + r nt n n = compounding periods per yer (number of times interest is figured) Time/Rte/Distnce: d = rt d = distnce trveled r = trvel rte (remember to combine rtes correctly if in moving medium) t = time of trvel Flling Body: s = gt 2 + v 0 t + s 0 s = distnce bove ground (note tht s is positive upwrds, t ground level s = 0) g = ccelertion of grvity ( g 16 ft. or g 4.9 m ) ( sec. ) 2 ( sec. ) 2 t = time flling v 0 = initil velocity s 0 = initil height Formuls for Geometric Objects Squre: A = s 2 nd P = 4s A = re P = perimeter s = one side Rectngle: A = lw nd P = 2l + 2w l = length w = width Circle: A = π r 2 nd C = 2π r C = circumference (perimeter) r = rdius Tringle: A = 1 bh nd P = + b + c 2 b = bse h = height, b, c = sides Right Tringle: 2 + b 2 = c 2, b = legs of the tringle c = hypotenuse Trpezoid: A = 1 2 h ( + b ), b = prllel sides Cube: V = s 3 nd SA = 6s 2 V = volume SA = surfce re Rectngulr Solid: V = lwh nd SA = 2 lw + hl + hw Circulr Cylinder: SA = 2π rh + 2π r 2 V = π r 2 h nd Sphere: V = 4 3 π r3 nd SA = 4π r 2
Chpter P: Prerequisites Properties of opertions on the Rel Numbers Closure: Opertion performed on two rel numbers results in rel number. Commuttive: Numbers dd in ny order. Numbers multiply in ny order. Associtive: Addition cn be grouped in ny order. So cn multipliction. Distributive: Multipliction distributes over ddition ( x + y) = x + y Frctions Multiplictive Identity: x y =1 x y = x y = x y Reducing (cnceling): x y = x y =1 x y = x y Adding frctions: d + b d = + b d Multiplying frctions: b x y = x by Simplifying complex frctions: b c d = b d c or b c bd bd d d Absolute Vlues Absolute Vlue: the distnce from zero = if 0 { if < 0 Properties: 1. 0 2. = 3. b = b 4. b = b = b c bd bd = d bc Distnce AB between points A, B on the number line (coordintes, b): AB = b Properties of Exponents Definition: x n mens x used s fctor (multiplied) n number of times (x is rel number, n is nturl number). Zero power: x 0 =1 Negtive exponent mens reciprocl: x n = 1 x n, 1 x n = x n 1 = x n, If the bses re the sme, multiply: x n x m = x n +m divide: exponentil to power: Power opertes before subtrcts: Power distributes over multipliction: x n x m = x n m ( x n ) m = x n m x y 1 = y x x 2 = ( x x) nd ( x) 2 = ( x) ( x) = x 2 ( xy) n = x n y n
Rtionl Exponents nd Rdicls Equivlent forms: m x n = x n m = x n 1m = x 1 m is n nth root of b if: n n = b or = b or = b 1 n Note: Rtionl exponents hve exctly the sme properties s integer exponents Polynomils Monomil in vribles x nd y: x n y m ( is constnt, n nd m non-negtive integers) Coefficient of the monomil: Degree of the monomil: n + m Degree of polynomil: Degree of the highest degree monomil Polynomil in x (generl) n x n + n 1 n n 1 +!+ 1 x + 0 where n 0, degree = n, leding coefficient is n, nd 0 is the constnt term Add (subtrct) polynomils: Combine like terms Multiple polynomils (generl): Multiply every term in first by every term in second. Multiply binomils: FOIL Specil products where u nd v cn be (nd will be) ny lgebric expression: Sum nd Difference u + v Squre of Binomil ( u v) = u 2 v 2 ( u + v) = u 2 + 2uv + v 2 ( u v) = u 2 2uv + v 2 ( u + v) = u 3 + 3u 2 v + 3uv 2 + v 3 ( u v) = u 3 3u 2 v + 3uv 2 v 3 Binomil Cubed Fctoring Formuls: Difference of Squres u 2 v 2 = u + v Perfect Squres u 2 + 2uv + v 2 = u + v Cubes u 3 + v 3 = u + v Grphs Digrm: Stndrd Crtesin Plne Point (x, y) x-coordinte is distnce from the y-xis y-coordinte is distnce from the x-xis Distnce Formul: Distnce between points ( x 1, y 1 ) nd ( x 2, y 2 ): ( u v) 2 u 2 2uv + v 2 = ( u v) 2 u 2 uv + v 2 u 2 + uv + v 2 d = ( x 2 x 1 ) 2 + ( y 2 y 1 ) 2 u 3 v 3 = u v nd Midpoint of line segment joining points x 1, y 1 ( x 2, y 2 ): x Midpoint coordintes: ( x m, y m ) = 1 + x 2, y 1 + y 2 2 2 (Note tht this is the verge of x, y coordintes) n Qudrnt II x-xis (negtive) Qudrnt III Origin y-xis (positive) y-xis (negtive). (x, y) Qudrnt I Qudrnt IV x-xis (positive)
Chpter 1: Equtions nd Inequlities Tests for symmetry: Grph of n eqution is symmetric with respect to the x-xis Grphicl: If whenever ( x, y) on the grph, ( x, y) is on the grph Algebric: Replcing y with y results in n equivlent eqution y-xis Grphicl: If whenever ( x, y) on the grph, ( x, y) is on the grph Algebric: Replcing x with x results in n equivlent eqution Origin Grphicl: If whenever ( x, y) on the grph, ( x, y) is on the grph Algebric: Replcing both x with x nd y with y results in n equivlent eqution Circle: Set of points ll the sme distnce from center point. Stndrd form: Center is h,k nd rdius is r. ( x h) 2 + y k 2 = r 2 Center t origin: x 2 + y 2 = r 2 Absolute Vlue Equtions u = v where u nd v re lgebric expressions Two possibilities: u = v (vlue of u inside bsolute vlue ws positive) u = v (vlue of u inside bsolute vlue ws negtive) Qudrtic Equtions Zero Product Principle: If b = 0, then = 0 or b = 0 If d > 0 nd x 2 = d Roots: x = ± d. Note there re two roots. Fctored Form: x 2 d = x + d ( x d ) = 0 Qudrtic Eqution - Stndrd Form: x 2 + bx + c = 0 Completing the Squre x + b 2 = b2 4c 2 4 2 Qudrtic Formul x = b ± b2 4c 2 Discriminnt b 2 4c > o there re two distinct rel solutions = 0 one rel root, multiplicity of two < 0 two complex solutions, conjugte pir Complex Numbers Define: i = 1, therefore: i 2 = 1 For x, write i x (or x i, but wtch til of rdicl!) for x > 0 Complex number: + bi where nd b re rel numbers nd i = 1 is the rel prt, bi is the imginry prt Imginry number: Any number in the form bi where b is rel number ( = 0)
Equlity: + bi = c + di if nd only if = c nd b = d. Addition ( + bi) + ( c + di) = ( + c) + ( b + d)i (dd the rel prts, dd the imginry prts) Subtrction: ( + bi) ( c + di) = ( c) + ( b d)i Multipliction: ( + bi) ( c + di) = ( c bd) + ( d + bc)i (by FOIL nd simplify) Complex conjugte of + bi is bi Product of complex nd its conjugte is rel: ( + bi) ( bi) = 2 + b 2 Division Multiply by 1 using complex conjugte of denomintor: + bi c + di = + bi c + di c di c di = ( c + bd) + ( bc d )i c 2 + d 2 Inequlities - Properties: Trnsitive If < b nd b < c, then < c Addition If < b nd c < d, then + c < b + d Adding constnt If < b, then + c < b + c Multiplying constnt For c > 0 if < b then c < bc For c < 0 if < b then c > bc Solving Inequlities with Absolute Vlue of Liner Algebric Expression u u < : Solve two inequlities for the vrible u < { u < All vlues of u re between - nd. < u <, solve for the vrible u, u > : Solve two inequlities for the vrible u > { u > u < or u >, solve for the vrible Note: open intervl for <, closed intervl for Solving Polynomil Inequlities n x n + n 1 n n 1 +!+ 1 x + 0 < 0 1. Find ll roots of polynomil n x n + n 1 n n 1 +!+ 1 x + 0 = 0 2. The roots re criticl vlues determining intervls to test for solutions 3. Choose representtive vlue to test in ech intervl between criticl vlues 4. If test vlue mkes inequlity true, then ll vlues in tht intervl re solutions. If test vlue is flse, ll vlues in tht intervl between criticl vlues re not solutions.
Chpter 2: Functions nd Their Grphs SOLUTION to n eqution (or other sttement), the roots of n eqution (function), the zeros of n eqution (function): ny nd ll vlues tht will mke the eqution TRUE. GRAPH: picture of ll solutions to sttement. Solutions re plotted on n pproprite set of xes. Function: Function f from set A to set B, is reltion which ssigns one x in set A, the domin, to exctly one y in set B, the rnge. 1. Ech element in A must be mtched to n element in B. 2. Not ll elements in B my be mtched with n element in A. 3. Two or more elements in A my be mtched to n element in B. 4. No element in A is mtched to more thn one element in B. Verticl Line Test: A grph represents function if no verticl line intersects the grph t more thn one point (no points is oky). Imge: Element y (dependent vrible) is the imge of x (independent vrible) Domin: set of ll inputs to function, set of ll independent vribles, set of from vlues in mpping. The x vlues. Rnge: set of ll outputs from function, set of ll dependent vribles, set of to vlues in mpping. This is the set of ll imges of the domin. The y vlues. Common limits on Domin: Division by zero (expression in denomintor cnnot be zero) Even roots of negtive vlues of the rdicnd (for rel numbers) Implied domin - function is not defined there. (e.g. rectngle width less thn zero) Liner Equtions Generl Form: Ax + By + C = 0 Slope-Intercept Form: y = mx + b where the slope is m nd the y-intercept is the point 0,b Point-Slope Form: y y 1 = m( x x 1 ) Two-Point Form: y y 1 = y 2 y 1 ( x x 1 ) x 2 x 1 Verticl Line: x = Horizontl Line: y = b Slope: "... is rise over run." Chnge in the y coordinte for given chnge in the x coordinte m = y 2 y 1 x 2 x 1 Prllel Lines: Two distinct, non-verticl lines re prllel if nd only if their slopes re equl. m 1 = m 2 Perpendiculr Lines: Two distinct, non-verticl lines re perpendiculr if nd only if their slopes re negtive reciprocls of ech other. (The product of slopes is -1.) m 2 = 1 m 1 or m 1 m 2 = 1 Function nme Input vlue Domin element f(x) = y Output vlue Rnge element
Grphs of Functions Zeros of function: All vlues of x where f ( x) = 0 Incresing Function: x 1 < x 2 implies f ( x 1 ) < f ( x 2 ) for the given intervl Decresing Function: x 1 < x 2 implies f ( x 1 ) > f ( x 2 ) for the given intervl Constnt Function: f ( x 1 ) = f ( x 2 ) for ny x 1 nd x 2 in the given intervl Reltive Minimum: Vlue in n intervl ( x 1, x 2 ) where f ll other x ( x 1, x 2 ). Reltive minimum is point (, f ( ) ) Reltive Mximum: Vlue in n intervl ( x 1, x 2 ) where f ll other x ( x 1, x 2 ). Reltive minimum is point (, f ( ) ) Averge Rte of Chnge of Function is the slope of the secnt line between the two function points: Chnge in f x < f ( x) for > f ( x) for for given chnge in the x, or m sec = f ( x 2 ) f ( x 1 ) = f ( x) (Like x tken to even power.) (Like x tken to odd power.) Even Function: f x Odd Function: f ( x) = f x Liner Function: f ( x) = mx + b Slope = m, y-intercept ( f ( x) = 0) = b or the point (0, b) x-intercept = f ( 0) = b m or the point b m, 0 Grph is incresing for m > 0, decresing for m < 0 Squring Function: f ( x) = x 2 Incresing on ( 0, ) Even function Domin: set of ll rel numbers. Rnge: 0, Odd function [ ) Grph: Intercept t 0,0 Decresing on (,0) Incresing on ( 0, ) Symmetric with respect to y-xis Reltive minimum t 0,0 = x 3 Cubic Function: f x Odd function Domin: set of ll rel numbers. Rnge: set of ll rel numbers. Grph: Intercept t 0,0 Incresing on, Symmetric with respect to origin Squre Root Function: f x [ ) [ ) Domin: 0, Rnge: 0, Grph: Intercept t 0,0 = x Reciprocl Function: f ( x) = 1 x x 2 x 1 Domin: (,0) ( 0, ), or x 0 Rnge: (,0) ( 0, ), or f x Grph: No intercepts Decresing on (,0) nd ( 0, ) Symmetric with respect to origin Absolute Vlue: f ( x) = x Even function Domin: set of ll rel numbers. Rnge: 0, [ ) Grph: Intercept t 0,0 Decresing on (,0) Incresing on ( 0, ) Symmetric with respect to y-xis Reltive minimum t ( 0,0) Gretest Integer: f x integer less thn or equl to x 0 = [[x]] = gretest
Trnsformtions of Grphs Grph of given function y = f ( x), nd where c is positive rel number Verticl nd Horizontl Shifts of grph Verticl shift c units up: h( x) = f ( x) + c Verticl shift c units down: h( x) = f ( x) c Horizontl shift c units right: h( x) = f ( x c) Horizontl shift c units left: h( x) = f ( x + c) Reflections of grphs Reflection in x-xis: h( x) = f ( x) Reflection in y-xis: h( x) = f ( x) Nonrigid Trnsformtions Verticl stretch h( x) = c f ( x) where c > 1 Verticl shrink h( x) = c f ( x) where 0 < c < 1 Horizontl stretch h( x) = f ( c x) where 0 < c < 1 Horizontl shrink h( x) = f ( c x) where c > 1 Arithmetic of Functions Functions f nd g hve overlpping domins Sum: ( f + g) ( x) = f ( x) + g( x) Difference: ( f g) ( x) = f ( x) g( x) Product: ( fg) ( x) = f ( x) g( x) Quotient: f ( x) = f ( x ) g g( x) ( x ) 0 Composition of Functions Composition of f with g ( f! g) ( x) = f ( g( x) ) Domin of ( f! g) All x in domin of g such tht g x domin of f. Inverse Functions Nottion: Function is f, Inverse of function f is written f 1 nd red "f inverse" is in the Grph: f 1 is the reflection of grph of f cross the line y = x. Not ll functions hve inverses Horizontl Line Test: f hs n inverse if nd only if no horizontl line intersects the grph of f t more thn one point. One-to-One Functions: Ech vlue of the dependent vrible corresponds to exctly one vlue of the independent vrible. It is function (verticl line test), nd in reverse - ny one vlue of f ( x) is given by only one x (horizontl line test). One-to-one functions hve inverses. Finding the Inverse of Function 1. Verify tht f ( x) is one-to-one. 2. Use y in plce of f ( x). 3. Swp vribles in the eqution: Put y where x ws nd x where y ws. 4. Solve for y. 5. The y is now f 1 ( x)
Chpter 3: Polynomil functions Polynomil function with degree n: f ( x) = n x n + n 1 n n 1 +!+ 1 x + 0 n is nonnegtive integer, nd n, n 1,!, 1 + 0 re rel numbers with n 0 = x 2 + bx + c Qudrtic Function: Polynomil function with degree 2 or f x Grph prbol Grph symmetric to xis of symmetry (or just xis) Axis intersects prbol t the vertex > 0, prbol opens up, vertex is minimum t x = b 2 < 0, prbol opens down, vertex mximum t x = b 2 Vertex Point b 2, f b 2 Stndrd Form of Qudrtic Functions Polynomil Function f ( x) = ( x h) 2 + k Axis is verticl line: x = h Vertex is point ( h,k) = n x n + n 1 n n 1 +!+ 1 x + 0 hs degree n f x If n is odd, left nd right tils of grph go in opposite directions: n > 0 left til to flls without bound, right til rises without bound. n < 0 left til to rises without bound, right til flls without bound. There must be t lest one rel zero (root). Grph must cross x-xis. If n is even, left nd right tils of grph go in the sme directions: n > 0 left til to rises without bound, right til rises without bound. n < 0 left til to flls without bound, right til flls without bound. Turning points: Grph hs t most n-1 turning points where the grph chnges from incresing to decresing or vice vers. These turning points re lso reltive mxim or reltive minim. Zeros: f ( x) hs t most n rel zeros. Rel Zeros of polynomil function f ( x): These ll men the sme thing Zero: A rel number vlue such tht f ( ) = 0 Solution: x = is solution if f ( x) = 0 Fctor: Quntity ( x ) is fctor of the polynomil f ( x) x-intercept Point,0 Repeted Zeros is n x-intercept of f ( x) hs binomil fctors ( x ) k, for k > 1, then f ( x) hs If polynomil function f x repeted zero, x = of multiplicity k. If k is odd, the grph crosses the x-xis t x =. If k is even, the grph touches (but does not cross) the x-xis t x =.
Intermedite Vlue Theorem Let nd b be rel numbers such tht < b nd f is polynomil function with f ( ) f ( b). In the intervl [,b], f tkes on every vlue between f ( ) nd f ( b). Note especilly if there is sign chnge between f ( ) nd f ( b), f pssed through zero nd there is root in [,b]. Division of polynomils Dividend = f ( x), Divisor = d( x), Quotient = q( x), ) d(x) Reminder r( x) (nd wtch your like terms!) f ( x) = d( x) q( x) + r( x) Synthetic Division of x 3 + bx 2 + cx + d by ( x h): h 1+ b c d 2x h h b c d j k R Quotient: q x Synthetic division by h: 3+ h b c d h 4x = x 2 + jx + k Reminder: r( x) = R Reminder is vlue of polynomil t h If reminder is 0, ( x h) is fctor of the polynomil If reminder is 0, h is root nd ( h,0) is the x-intercept Fundmentl Theorem of Algebr Every polynomil f(x) of degree n > 0 hs exctly n zeros (roots) mong the complex numbers (including multiplicity). Liner Fctoriztion Theorem: Every polynomil f(x) of degree n > 0 hs exctly n liner fctors. = n ( x c 1 )( x c 2 )! ( x c n ) f x where c 1, c 2,!, c n re complex numbers Reminder Theorem: After synthetic division by h, R = f ( h), i.e. the reminder is the vlue of the polynomil t x = h. Doing synthetic division by h is often esier thn evluting f ( h) for higher degree polynomils. Fctor Theorem: Polynomil f(x) hs fctor ( x h) iff f ( h) = 0... nd h is root, solution, zero, etc. of the polynomil f(x). Polynomil f(x) hs zero t x = h iff x h Rtionl Zero Theorem: for f x coefficients re integers: is fctor of f(x) = n x n + n 1 n n 1 +!+ 1 x + 0 where ll The only possible rtionl roots will hve the form p q where p is ± ll fctors of 0, the constnt term q is ± ll fctors of n, the leding coefficient Conjugte Zeros Theorem: If polynomil hs complex root, + bi, the complex conjugte, bi is lso root., i.e. complex roots pper in pirs. etc. q(x) f(x) r(x)
Chpter 4: Rtionl Functions nd Conics [Conics omitted] Intercepts - Generl y-intercept(s): evlute f ( 0) =? x-intercept(s): set f ( x) = 0, solve for x =? Asymptote: stright line which curve of grph cn pproch but never touch. 1. The line x = is verticl symptote of the grph of f if f ( x) or f ( x) s x from the right or from the left. 2. The line y = b is horizontl symptote of the grph of f if f x b s x or x. = N ( x ) D( x) Rtionl function: f x Where N ( x) nd D( x) hve no common fctors 1. Hs verticl symptote t ll roots (zeros) of D( x). 2. Hs one or no horizontl symptote depending on the degrees of N ( x) nd D( x): If the degree of N ( x) < degree of D( x), then f hs horizontl symptote on the line y = 0, the x-xis. If the degree of N ( x) = degree of D( x), then f hs horizontl symptote on the line y = n b m (y = rtio of leding coefficients). 3. If the degree of N ( x) is exctly one lrger thn the degree of D( x), then f hs slnt (oblique) symptote long the line determined by q( x), the first degree polynomil found by diving the numertor by the denomintor: N ( x) D( x) = q( x) + r( x) 4. If there is common fctor ( x h), between the numertor nd denomintor of rtionl function, the grph hs hole in where x = h. Horizontl Asymptotes of Rtionl Functions Using Limit Ide Rtionl function is rtio of polynomils: R( x) = n x n + n 1 x n 1 + n 2 x n 2 +... b m x m + b m 1 x m 1 + b m 2 x m 2 +... Degree of numertor = n nd degree of denomintor = m. 1. Fctor x n from ech term in numertor nd x m from ech term in denomintor 2. Cncel x n with x m. 3. Tke the limit s x goes to infinity of the tht expression. 4. Result is the eqution of the symptote: Horizontl Asymptotes of Rtionl Functions If n - m > 1, there is no symptote. If n - m = 1, the eqution will be liner eqution in x n oblique symptote. If n = m, it will be the horizontl line y = n b m If n < m, it will be the x xis, the line y = 0.
Chpter 5: Exponentil nd Logrithmic Functions Exponentil function: f ( x) = x ( > 0, 1, x is rel) Note: Exponent is vrible x Note: When bse 0 < < 1, n equivlent form exists with bse 1 (which is greter thn 1) nd n exponent of -x. Therefore, we usully use bse > 1. Properties: f ( x) = x For > 1: Properties: f ( x) = x For > 1: Horizontl symptote to y-xis on left o Horizontl symptote to y-xis on right Hs y-intercept ( 0,1 ) Hs y-intercept ( 0,1 ) Is strictly incresing Is strictly decresing Generl Properties: Domin is ll rel numbers. Rnge is y > 0. Function is one-to-one If u = v, then u = v If u = b u, (, b > 0), then either u = 0 or = b If u = v then u = v If u = b u (where, b > 0) then either u = 0 or = b. Nturl Exponent Function f ( x) = e x, where e = lim m or e 2.718281828459045 m m Models using the Exponentil Functions: Compound interest (specified compounding periods) formul: A = P 1 + r nt n A = Amount vilble P = Amount invested r = nnul interest rte n = number of times compounded nnully t = time in yers Compound interest with continuous compounding: A = Pe rt Exponentil growth Q( t) = q 0 e kt q 0 is quntity t t = 0 k is the growth constnt Exponentil decy Q t = q 0 e kt Logrithms Inverse of exponentil: If f x Equivlence: log x = y y = x Inverse of exponentil Common Logrithm: Nturl Logrithm: Logrithmic Identities: log 1 = 0 If f x = x, then f 1 ( x) = log x = x, then f 1 ( x) = log x log x = log 10 x = n where n is 10 n = x ln x = log e x = n where n is e n = x log x = x log =1 log x = x If log u = log v then u = v If log u = log b u, then u = 1, or = b.
Properties of logrithms: 1. log xy = log x + log y 2. log x y = log x log y 3. log x n = nlog x Chnge of bse: log b x = log x log b Solving Exponentil/Logrithmic Equtions 1. Use one-to-one property: If u = v, then u = v If log u = log v then u = v 2. Use equivlent forms: log x = y y = x 3. Use inverse opertions: = x, then f 1 ( x) = log x nd log x = x = log x, then g 1 ( x) = x nd log x = x f x g x Chpter 9: Systems of Equtions Solve by grphing: Grph equtions. Find intersection point for pproximte solution(s). Solve by Substitution: Solve one eqution for one vrible in terms of the other Substitute solved expression for first vrible Solve resulting eqution in one vrible Bck substitute to find the other vrible Solve by Elimintion: Multiply one or both equtions to obtin one pir of coefficients which differ only in sign Add the equtions to eliminte tht vrible nd solve the resulting eqution in the other vrible Bck substitute to find the first vrible Distnce = Trvel Rte Time Brek-even Point: Revenues = Costs Equilibrium Point: Supply = Demnd System of two liner equtions in two vribles l 1 : y = m 1 x + b 1 nd l 2 : y = m 2 x + b 2 Consistent System when m 1 m 2 : Exctly one solution Grphs of l 1 nd l 2 intersect t exctly one point Inconsistent Systems when m 1 = m 2 : Infinitely mny solutions if lines coincide (re identicl) No solutions if lines re prllel nd do not coincide Gussin Elimintion: System of liner equtions is mnipulted until it is in tringulr form where the only non-zero coefficient for the first vrible occurs in the first row, the only non-zero coefficients for the second vrible occur in the first nd second equtions, nd so on. The system cn now be solved by bck substitution.