1 PRECALCULUS READINESS DIAGNOSTIC TEST PRACTICE

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1 PRECALCULUS READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the smples, work the problems, then check your nswers t the end of ech topic. If you don t get the nswer given, check your work nd look for mistkes. If you hve trouble, sk mth tech or someone else who understnds the topic. TOPIC : ELEMENTARY OPERATIONS with NUMERICAL nd ALGEBRAIC FRACTIONS A. Simplifying frctions (by reducing): exmple: (note tht you must be ble to find common fctor in this cse 9 in both the top nd bottom in order to reduce frction). exmple: b 4b 4b 4 b 4b (common fctor: ) Problems -8: Reduce: 4x+7y y b 5+c x 6xy 5by x 8. 4(x+)(x ) (x )(x ) B. Equivlent frctions (equivlent rtios): exmple: is equivlent to how mny eighths? 4 4 8, exmple: 6 5 5b, 6 b 6 6b 5 b 5 5b exmple: x+ x+ 4( x+), x+ 4 x+ x+8 x+ 4 x+ 4x+4 exmple: x x+ ( x+) ( x ), x ( x ) ( x ) x+ ( x ) ( x+) x x+ ( x+) ( x ) Problems 9-: Complete: b ( +b) ( b) 0. x 7 7y. x 6 6 x x+. x+ ( x ) ( x+) C. Finding the lowest common denomintor: (LCD) by finding the lest common multiple (LCM) of ll denomintors: exmple: 5 6 nd 8. First find LCM of 6 nd 5: LCM 5 0, so , nd exmple: 4 nd 6 : 4 6 LCM, so 9 4, nd 6 exmple: ( x+) nd x 6( x+) : ( x + ) ( x + ) 6( x +) x + LCM ( x +) ( x + ), so ( x+) ( x+) ( x+) x+ nd x 6 x+ 4( x+) 6( x+) ( x+), x( x+) 6( x+) ( x+) Problems 4-8: Find equivlent frctions with the lowest common denomintor: 4. nd 9 7. x nd 4 x 5. x nd 5 8. x 5( x ) x 4 6. nd x+ 7x y nd 0( x ) D. Adding nd subtrcting frctions: If denomintors re the sme, combine the numertors: x exmple: x x x x y y y y If denomintors re different, find equivlent frctions with common denomintors: exmple: exmple: x + ( x+) x+ ( x ) ( x+) + ( x ) ( x ) ( x+) ( x+6+x x ) ( x+) 4x+5 ( x ) ( x+) Problems 9-6: Find the sum or difference s indicted (reduce if possible): x x b. b b+ b+ x. + x x x 6. ( x ) ( x ) + x x 4. x x 5. x x x+ x x+ ( x ) ( x ) ( x )

2 E. Multiplying frctions: Multiply the top numbers, multiply the bottom numbers, reduce if possible: exmple: exmple: 7. b 7 8. ( x+) x 4 ( x+) ( x+) ( x ) x x ( x ) ( x+) ( x ) x+6 x ( +b) 9. ( x+4) 5y 5y x 6 x y ( x y) 5 p ( 5 p ) ( +b) F. Dividing frctions: Mke compound frction nd then multiply the top nd bottom (of the lrger frction) by the LCD of both: exmple: c b bd b d b d bc bd exmple: 7 c d c d x 5x 5x exmple: x y y x y y x y 5x 0. b. 4. x+7 x 9 x. b 4 5 4xy 4y Answers: x 5. 5b x+y y b 5 +c 7. 4( x+) 8. x xy. x (cn t reduce) ( x + ) ( ). + b. 4. 6, , 5x x x 6. x( x +) ( x +), ( x +) 7., 4 x x x( x ) 8. 0( x ) ( x ), x ( x ) ( y ) 0( x ) ( x ) b b x +x x 4 x b 4 8. y x 4 ( x+) ( x ) ( 5 p) 9. +b x y 0. 9 b. x +7 x b TOPIC : OPERATIONS with EXPONENTS nd RADICALS A. Definitions of powers nd roots: Problems -0: Find the vlue: ( 4) 4. x B. Lws of integer exponents: I. b c b+c II. b c III. IV. b c V. b ( b ) c bc ( b) c c b c c c b c VI. 0 (if 0) VII. b b Problems -0: Find x:. 4 x x. 4 x 5. 4 x. 4 x 6. 8 x

3 7. x 9. c x c 4 8. b0 b x 0. y x b 5 y Problems -4: Find the vlue:. 7x 0 8. x 4 x. 4 x 9. c+ x c. 4 8x 40. x x 4. 6x x + x 6. ( ) 4. x x 7. x c+ x c Problems 44-47: Write two given wys: Given No negtive powers No frction 44. d 4 d x y 46. bc b c 47. x y z x 5 y 6 z C. Lws of rtionl exponents, rdicls: Assume ll rdicls re rel numbers: I. If r is positive integer, p is n integer, nd 0, then p r r p r p, which is rel number. (Also true if r is positive odd integer nd < 0). r r r II. b b, or b r b r III. IV. r r b r b rs r s r, or ( b ) r r b r s r, or rs s Problems 48-5: Write s rdicl: x x x r ( r ) Problems 54-57: Write s frctionl power: s Problems 58-6: Find x: x x x 6. x 8 4 x Problems 6-64: Write with positive exponents: 6. 9x 6 y b D. Simplifiction of rdicls: exmple: 6 4 exmple: exmple: exmple: 8 Problems 65-78: Simplify (ssume ll rdicls re rel numbers): x x x + x + x x x E. Rtionliztion of denomintors: 5 exmple: exmple: exmple: Problems 79-87: Simplify: b Answers:

4 x if x 0; x if x < y x c 8. x 9. x 6 4 x x 4. x 9 4. x d 8 d y 9 x 6 y 9x b 8b 47. y 9 z x y 9 z x x x x 5. x 5. x x y 64. b x x 7. x 7. 4 x x 74. if 0; if < b b TOPIC : LINEAR EQUATIONS nd INEQUALITIES A. Solving liner equtions: Add or subtrct the sme vlue on ech side of the eqution, or multiply or divide ech side by the sme vlue, with the gol of plcing the vrible lone on one side. If there re one or more frctions, it my be desirble to eliminte them by multiplying both sides by the common denomintor. If the eqution is proportion, you my wish to cross-multiply. Problems -5: Solve:. x 9 9. x 4 x +. 6x 5 0. x x+. x x x x 4. 4 x. x x 9. x+ 6. x x 5 x x 5 x x 6 x 5. x x+8 8. x x+ 6 7 B. Solving pir of liner equtions: The solution consists of n ordered pir, n infinite number of ordered pirs, or no solution. Problems 6-: Solve for the common solution(s) by substitution or liner combintions: 6. x + y 7 0. x y 5 x y 8 x + 5y 7. x + y 5. 4x y x y 4x + y 8. x y 9. x + y x 8 x + y 9. x y. x y y x 5 6x 9 y C. Anlytic geometry of one liner eqution: The grph of y mx + b is line with slope m nd y-intercept b. To drw the grph, find one point on it (such s (0, b)) nd then use the slope to find nother point. Drw the line joining the two points.

5 exmple: y x + 5 hs slope nd y-intercept 5. To grph the line, locte (0, 5). From tht point, go down (top of slope frction), nd over (right) (bottom of frction) to find second point. Drw the line joining the points. Problems 4-8: Find slope nd y-intercept, nd sketch grph: 4. y x x y 5. y x 8. x y + 6. y 4x 8 To find n eqution of non-verticl line, it is necessry to know its slope nd one of its points. Write the slope of the line through (x, y) nd the known point, then write n eqution which sys tht this slope equls the known slope. exmple: Find n eqution of the line through (-4, ) nd (-, 0). Slope 0 4+ Using (-, 0) nd (x, y), Slope y 0 ; cross multiply, x+ get y x +, or y x Problems 9-: Find n eqution of line: 9. Through (-, ) nd (-, -4) 0. Through (0, -) nd (-, -5). Through (, -) nd (5, -). Through (8, 0), with slope. Through (0, -5), with slope A verticl line hs no slope, nd its eqution cn be written so it looks like x k (where k is number). A horizontl line hs zero slope, nd its eqution looks like y k. exmple: Grph on the sme grph: x nd + y. The first eqution is x 4 The second is y 4 5 D. Anlytic geometry of two liner equtions: Two distinct lines in plne re either prllel or intersecting. They re prllel if nd only if they hve the sme slope, nd hence the equtions of the lines hve no common solutions. If the lines hve unequl slopes, they intersect in one point nd their equtions hve exctly one common solution. (They re perpendiculr if their slopes re negtive reciprocls, or one is horizontl nd the other is verticl.) If one eqution is multiple of the other, ech eqution hs the sme grph, nd every solution of one eqution is solution of the other. Problems 6-4: For ech pir of equtions in problems 6 to, tell whether the lines re prllel, perpendiculr, intersecting but not perpendiculr, or the sme line: 6. Problem Problem 0 7. Problem 7 4. Problem 8. Problem 8 4. Problem 9. Problem 9 4. Problem E. Liner inequlities: exmple: One vrible grph: solve nd grph on number line: x 7 (This is n bbrevition for {x: x 7 }) Subtrct, get x 6 Divide by, x Grph: Problems 44-50: Solve nd grph on number line: 44. x > x < x < x > x 46. x x > < x exmple: Two vrible grph: grph solution on number plne: x y > (This is n bbrevition for {( x, y): x y > }) Subtrct x, multiply by, get y < x. Grph y x, but drw dotted line, nd shde the side where y < x : - Problems 4-5: Grph nd write eqution for 4. The line through (-, 4) nd (-, ) 5. The horizontl line through (4, -) Problems 5-56: Grph on number plne: 5. y < 54. x < y + 5. y > x 55. x + y < 5. y x x y >

6 F. Absolute vlue equtions nd inequlities: exmple: x Since the bsolute vlue of both nd is, x cn be either or. Write these two equtions nd solve ech: x x x or x 5 x x 5 Grph: Problems 57-6: Solve nd grph on number line: 57. x 60. x x 6. x x exmple: x < The bsolute vlue of n number between nd (exclusive) is less thn. Write this inequlity nd solve: < x <. 6 Subtrct Multiply by, get 5 > x >. (Note tht this sys x > nd x < 5). Grph: exmple: x +. The bsolute vlue is greter thn or equl to for ny number or. So, x + x + x or x 4 x x Grph: Problems 6-66: Solve nd grph on number line: 6. x < 65. x + 6. < x x < x + < Answers: (9, -) 7. (, 4) 8. (8, 5) 9. (-4, -9) 0. ( 8 9, 9 ). ( 4, 0). no solution. (, ), where is ny number; infinite number of solutions 4. m, b m, b 6. m, b 4 7. m, b 8. m, b 9. y 5 x 0. y x. y. y x + 8. y x 5 4. x 5. y intersecting, not intersecting, not 9. intersecting, not 40. intersecting, not 4. intersecting, not 4. prllel 4. sme line 44. x > x < x x > x > x < x >

7 x ± (on number line) 58. no solution 59. x, x 6. x, 6. < x < x > or x < < x < x or x < x <0 0 0 TOPIC 4: POLYNOMIALS nd POLYNOMIAL EQUATIONS A. Solving qudrtic equtions by fctoring: If b 0, then 0 or b 0 exmple: If ( x) ( x + ) 0 then ( x) 0 or ( x + ) 0 nd thus x or x Note: there must be zero on one side of the eqution to solve by the fctoring method. exmple: 6x x Rewrite: 6x x 0 Fctor: x( x ) 0 So x 0 or x Thus, x 0 or x. 0. Problems -: Solve by fctoring:. x( x ) 0 7. x x 6 0. x x 0 8. x x. x x 9. 6x x + 4. x(x + 4) 0 0. x + x 6 5. ( x + ) x 6. ( x +) ( x ) 0. x x x 6x B. Monomil fctors: The distributive property sys b + c ( b + c) exmple: x x x(x ) exmple: 4x y + 6xy xy(x + ) Problems -7: Fctor:. + b 6. x y y x 4. b + b 7. 6x y 9x 4 y 5. 4xy +0x C. Fctoring: ( x ) ( x + b) + ( x ) ( x + c): The distributive property sys jm + jn j( m + n). Compre this eqution with the following: ( x +) ( x + ) + ( x +) ( x 4) x + ( x + ) + ( x 4) Note tht j x +, m (x + ), nd n (x 4), nd we get ( x +) ( x + 6x x 4) ( x +) x + 7x + 5 Problems 8-0: Find P which completes the eqution: 8. (x )(x ) (x )(x + ) (x )( P ) ( x ) + ( x 4) ( x ) 9. x + 4 ( x ) ( P ) 0. ( x ) ( x +) x x ( x + ) ( P ) D. The qudrtic formul: If qudrtic eqution looks like x + bx + c 0, then the roots (solutions) cn be found by using the qudrtic formul: x b± b 4c exmple: x + x 0,, b, nd c x ± 4 ( ) ± 4+ ± ± or 6 exmple: x x 0,, b, c ± +4 x ± nd 5 Problems -4: Solve: So there re two roots:. x x 6 0. x x 0. x + x 4. x x 4 0 E. Qudrtic inequlities: exmple: Solve x x < 6. First mke one side zero: x x 6 < 0. Fctor: ( x ) ( x + ) < 0. If ( x ) 0 or ( x + ) 0, then x or x.

8 These two numbers ( nd ) split the rel numbers into three sets (visulize the number line): x ( x ) ( x + ) ( x ) ( x + ) solution? x < < x < x > negtive negtive positive negtive positive positive positive negtive positive no yes no Therefore, if ( x ) ( x + )< 0, then < x < Note tht this solution mens tht x > nd x < Problems 5-9: Solve, nd grph on number line: 5. x x 6 > 0 8. x > x 6. x + x < 0 9. x + x > 0 7. x x < F. Completing the squre: x + bx will be the squre of binomil when c is dded, if c is found s follows: find hlf the coefficient of x, nd squre it this is c. Thus c b b 4, nd x + bx + c x + bx + b 4 x + b 8 exmple: x + 5x 5 4 Hlf of 5 is 5, nd 5, which must be dded to complete the squre: x + 5x x + 5 If the coefficient of x is not, fctor so it is. exmple: x x x x 6, nd ( 6) 6, so Hlf of is ( x x + 6) ( x 6), nd ( x x + 6) x x + 6. Thus, 6 (or must be dded to x x to complete the squre. Problems 0-: Complete the squre, nd tell wht must be dded: 0. x 0x. x x. x + x. x + 8x G. Grphing qudrtic functions: Problems 4-40: Sketch the grph: 4. y x 8. y (x +) 5. y x 9. y (x ) 6. y x y ( x + )(x ) 7. y x ) Answers:. 0,. 0,. 0, 4. 0, 4 5., 6., 7., 8., 9., 0.,..,. + b 4. b + b 5. x y + 5x 6. xy x y 7. x y(y x) 8. x x 9. x + x 6 0. x 4x + 4.,. ± , 4 5. x < or x > 6. < x < 0 7. no solution, no grph 8. 0 < x < 9. x < or x > 0. (x 5), dd 5, dd 4. x ( x 4), dd 9 6. (x + ), dd (,-) - -

9 TOPIC 5: FUNCTIONS 9 A. Wht functions re nd how to write them: The re of squre depends on the side length s, nd given s, we cn find the re A for tht vlue of s. The side nd re cn be thought of s n ordered pir: (s, A). For exmple, (5, 5) is n ordered pir. Think of function s set of ordered pirs with one restriction: no two different ordered pirs my hve the sme first element. Thus {(s, A) : A is the re of the squre with side length s} is function consisting of n infinite set of ordered pirs. A function cn lso be thought of s rule: for exmple, A s is the rule for finding the re of squre, given side. The re depends on the given side nd we sy the re is function of the side. ' A f (s)' is red ' A is function of s', or ' A f of s'. There re mny functions of s, the one here is s. We write this f (s) s nd cn trnslte: the function of s is s. Sometimes we write A(s) s. This sys the re is function of s, nd specificlly, it is s. In some reltions, s x + y 5, y is not function of x, since both (, 4) nd (, -4) mke the reltion true. Problems -7: Tell whether or not ech set of ordered pirs is function: {(, ) ( 0, ) }. {( 0, 5) } {(, ) (,0) }.,., 4. {(x, y) : y x nd x is ny rel number} 5. {(x, y) : x y nd y is ny rel number} 6. {(x, y) : y nd x is ny rel number} 7. {(x, y) : x 4 nd y is ny rel number} Problems 8-: Is y function of x? 8. y 4x + x 0. ( x + y) 6 9. y x. y x B. Function vlues nd substitution: If A(s) s, A(), red A of, mens replce every s in A(s) s with, nd find the re when s is. When we do this, we find A() 9. exmple: g( x) is given: y g(x) πx exmple: g() π 9π exmple: g(7) π 7 49π exmple: g() π exmple: g(x + h) π(x + h) πx + πxh + πh. Given y f ( x) x ; complete these ordered pirs: (, ), (0, ), (, ), (, 0), (, -), ( x, ) Problems -7: Given f (x) x x+ Find:. f 6. f ( ) 4. f ( ) 7. f ( x ) 5. f ( 0) C. Composition of functions: exmple: If f (x) x, nd g( x) x, f g( x) every x in f (x) x with g( x) giving f ( g( x) ) ( g( x ), which equls (x ) x 6x + 9. exmple: g f ( ( )) g( ) ) g( 4) 4 4 is red f of g of x, nd mens replce Problems 8-6: Use f nd g s bove: 8. g( f( x) ). f( x) g( x) 9. f( g ) 4. f( x) g x) 0. g( g( x) ) 5. g( x ). f( x)+ g( x) 6. ( g( x ). f( x) g( x) exmple: If k( x) x 4x, for wht x is k( x) 0? If k( x) 0, then x 4x 0 nd since x 4x x( x 4) 0, x cn be either 0 or 4. (These vlues of x: 0 nd 4, re clled zeros of the function, becuse ech mkes the function zero.) Problems 7-0: Find x so: 7. k( x) 4 9. x is zero of x( x +) 8. k( x) 5 0. x is zero of (x x ) D. Grphing functions: An esy wy to tell whether reltion between two vribles is function or not is by grphing it: if verticl line cn be drwn which hs two or more

10 0 points in common with the grph, the reltion is not function. If no verticl line touches the grph more thn once, then it is function. exmple: x y hs this grph: Not function (the verticl line hits it more thn once). Problems -9: Tell whether or not ech of the following is function: Since y f( x), the vlues of y re the vlues of the function, which correspond to specific vlues of x. The heights of the grph bove (or below) the x-xis re the vlues of y nd so lso of the function. Thus for this grph, f is the height (vlue) of the function t x nd vlue is : At x, the vlue (height) of f ( x)is zero; in other words, f ( ) 0. Note tht f > f ( ), since > 0, nd tht f ( 0) < f( ), since f ( ) nd f ( 0) <. Problems 40-44: For this grph, tell whether the sttement is true or flse: 40. g( ) g( 0) 4. g( )> g 4. g( 0) g 44. g< g( 0) < g( 4) 4. g> g( ) To grph y f( x), determine the degree of f( x) if it is polynomil. If it is liner (first degree) the grph is line, nd you merely plot two points (select ny x nd find the corresponding y) nd drw their line. If f( x)is qudrtic (second degree), its grph is prbol, opening upwrd if the coefficient of x is positive, downwrd if negtive. To plot ny grph, it cn be helpful to find the following: ) The y-intercept (find f ( 0) to locte y-xis crossing) b) The x-intercept (find x for which f ( x) 0 x-xis crossing) c) Wht hppens to y when x is very lrge (positive) or very smll negtive? d) Wht hppens to y when x is very close to number which mkes the bottom of frction zero? e) Find x in terms of y, nd find wht hppens to x s y pproches number which mkes the bottom of frction zero. (d, e, nd sometimes c bove will help find verticl nd horizontl symptotes.) exmple: y x nlysis: qudrtic function (due to x ) y-intercept: f ( 0) x-intercepts: y x 0 so x ± exmple: y x+ ) y-intercepts: f ( 0) b) x-intercept: none, since there is no solution to y 0 x+ c) Lrge x: negtive y pproches zero; very negtive x mkes y positive nd going to zero. (So y 0, the x-xis, is n symptote line.) d) The bottom of the frction, x +, is zero if x. As x moves to from the left, y gets very lrge positive, nd if x pproches from bove, y becomes very negtive. (The line x is n symptote.)

11 Grph: e) To solve for x, multiply by x + nd divide by y to get x + y or x y y y Note tht y close to zero results in very positive or negtive x nd mens y 0 is n symptote, which we lredy found bove in prt c. Problems 45-54: Sketch the grph: 45. f ( x) x 50. y x 46. y 4 x 5. f ( x) 47. y x 5. y x 48. y x 5. y x 49. y x + x y x x Answers:. yes. no. yes 4. yes 5. no 6. yes 7. no 8. yes 9. no ( intersecting lines) 0. no ( prllel lines). yes. 7, -,, 4,, x none (undefined) 7. x x 8. x x 6. x + x. x x. x x + 4. x x 5. x 6. x 6x , 5 9., 0 0.,. yes. no. yes 4. no 5. yes 6. yes 7. yes 8. no 9. yes 40. F 4. T 4. T 4. T 44. T TOPIC 6: TRIGONOMETRY A. Trig functions in right tringles: The sine rtio for n cute ngle of right tringle is defined to be the length of the opposite leg to the length of the hypotenuse. Thus the sine rtio for ngle B, bbrevited sin B, is b c. c B A b C The reciprocl of the sine rtio is the cosecnt (csc), so csc B c b. The other four trig rtios (ll functions) re cosine cos djcent leg hypotenuse secnt sec cos hypotenuse djcent leg tngent tn cotngent cot ctn opposite leg djcent leg djcent leg opposite leg Problems -8: For this right tringle, give the following rtios: x. tn x. sin x 6 0 cos x θ. cos θ 8

12 4. sinθ cosθ 5. cos x, which mens (cos x) 6. sin x 8. sinθ cosθ 7. cosx sinθ B. Circulr trig definitions: Given circle with rdius r, centered on (0, 0). Drw the rdius connecting line from the vertex to ny point on the circle, mking n ngle θ with the positive x-xis (θ my be ny rel number, positive mesure is counterclockwise). The coordintes (x, y) of point P together with rdius r re used to define the functions: sinθ y r, cosθ x r, tnθ y x, nd the reciprocl functions s before. (Note tht for 0<θ < π, these definitions gree P( x, y) with the right tringle definitions. Also note tht sinθ, cosθ, nd tnθ cn be ny rel number.) Problems 9-: For the point (-, 4) on the bove circle, give: 9. x y r 0. tnθ. cosθ. cotθ sinθ Note tht for ny given vlue of trig function, (in its rnge), there re infinitely mny vlues of θ. Problems -4: Find two positive nd two negtive vlues θ for which:. sinθ 4. tnθ tn45 Problems 5-6: Given sin θ 5 nd π < θ < π, then: 5. tnθ 6. cosθ C. Pythgoren reltions (identities): + b c (or x + y r ) bove, cn be divided by c (or r ) to give c + b c c c, or sin A + cos A, (or sin θ + cos θ ), clled n identity becuse it is true for ll vlues of A for which it is defined. y r x θ Problems 7-8: Get similr identity by dividing + b c by: 7. b 8. D. Similr tringles: If ABC ~ DEF, nd if tn A 4, then tn D 4 lso, since EF : DF BC : AC Find DC, given DB 5 nd sin E.4 E. Rdins nd degrees: For ngle θ, there is point P on the circle, nd n rc from A counter-clockwise to P. The length of the rc is θ 60 C θ 60 πr, nd the rtio of the length of rc to rdius is π θ, where θ is 80 the number of degrees (nd the rtio hs no units). This is the rdin mesure ssocited with point P. So P cn be locted two wys: by giving the centrl ngle θ in degrees, or in number of rdii to be wrpped round the circle from point A (the rdin mesure). Converting: rdins π 80 degrees or degrees 80 π rdins. π exmple: (rdins) π π exmple: π 7 π (rdins) 80 (which mens tht it would tke little over 7 rdii to wrp round the circle from A to 40.) Problems 0-: Find the rdin mesure for centrl ngle of: Problems 4-6: Find the degree mesure, which corresponds to rdin mesure of: 4. π π 6 Problems 7-9: Find the following vlues by sketching the circle, centrl ngle, nd verticl segment from point P to the x-xis. (Rdin

13 mesure if no units re given.) Use no tbles or clcultor. 7. cos 5π 9. sin( 5 ) 6 8. tn( 5 ) Problems 0-: Sketch to evlute without tble or clcultor: 0. sec80. sinπ. cot( π ). cos π F. Trigonometric equtions: exmple: Solve, given tht 0 θ < π : tn θ tnθ 0. Fctoring, we get tnθ(tnθ ) 0, which mens tht tnθ 0 or tnθ. Thus θ 0 (degrees or rdins) plus ny multiple of 80 (or π ), which is n 80 (or n π ), or θ 45 ( π 4 rdins), or 45 + n 80 (or π 4 + n π ). Thus, θ cn be 0, π 4, π, or 5π 4, which ll check in the originl eqution. Problems 4-9: Solve, for 0 θ π : 4. sinθ cosθ 7. sinθ cotθ 5. sin θ + cosθ 8. cosθ tnθ 6. sin θ cos θ 9. sinθ tnθ secθ G. Grphs of trig functions: By finding vlues of sin x when x is multiple of π, we cn get quick sketch of y sin x. The sine is periodic (it repets every π, its period). sin x never exceeds one, so the mplitude of sin x is, nd we get this grph: To grph y sinx, we note tht for given vlue of x, sy x, the vlue of y is found on the grph of y sin x three times s fr from the y-xis s. Thus ll points of the grph. y sinx re found by moving ech point of y sin x grph to its previous distnce from the y- xis, showing the new grph repets times in the period of y sin x, so the period of y sin x, so the period of y sinx is π. Problems 40-46: Sketch ech grph nd find its period nd mplitude: 40. y cos x 44. y 4sin x 4. y cosx 45. y sin x 4. y tn x 46. y + cos x 4. y tn x H. Identities: exmple: Find formul for cosa, given cos A + B Substitute A for B: cos A cos A + A cos A sin A cos A cosb sin Asin B. cos A cos A sin Asin A Problems 47-49: Use sin x + cos x nd the bove to show: 47. cosa cos A 48. cosa sin A 49. cos x + cos x 50. Given cos A, sina tn A, nd seca cosa sin A + cos A, show tht tn x + sec x. 5. Given sin A sin Acos A, show 8sin x cos x 4sin x. Answers: , 4, π, π, π, 7π 4. 5, 5, 45, tn A + sec A 8. + cot A csc A π 5. π 4. π. 7π ( 540 π)

14 π 4, 5π 4 5. π, π, π 6. 0 θ < π 7. π, 5π 8. π 9. π, π 40. π 4. - π - π P π A P π A π 4 π π π - π π - π π P π no A P π no A P π A 4 P π A P π A 47. cosa cos A sin A cos A ( cos A) cos A 48. cosa cos A sin A ( sin A) sin A sin A 49. From problem 47, cosx cos x. Add, divide by. 50. tn A + sin A cos A + sin A +cos A cos A cos A sec A 5. Multiply given by 4, then let A x TOPIC 7: LOGARITHMIC nd EXPONENTIAL FUNCTIONS A. Logrithms nd exponents: Exponentil form: 8 Logrithmic form: log 8 Both of the equtions bove sy the sme thing. log 8 is red log bse two of eight equls three nd trnsltes the power of which gives 8 is. Problems -4: Write the following informtion in both exponentil nd logrithmic forms:. The power of, which gives 9 is.. The power of x, which gives x is.. 0 to the power is 00 is the power of 69 which gives. 4. exponent rules: (ll quntities rel) b c b+c b c b c ( b ) c bc ( b) c c b c c c b b c 0 (if 0) b b p p r r p r (think of p r s power root ) log rules: (bse ny positive rel number except ) log b log + logb log log logb b log b blog log b b (log b) b log b log c b log c (bse chnge rule) Problems 5-5: Use the exponent nd log rules to find the vlue of x: x x 6. x 9. log 7 x 7. x log 6 x 8. log 0 x log log 0. log x 5 5. log 4 log x. log 6. log x log x log 6 x 7. log (7 4 ) x 8. log( x 6) log( 6 x) 9. log 4 64 x. 7 x x x. log 8 log log 7 8 x 7 5. x 4. log 4 0 x log 4 0 Problems 6-: Find the vlue: log log log 6 6. log Find log, given: log log 0.0 Problems -4: Given log 04 0, find:. log log 04 Problems 5-46: Solve for x in terms of y nd z: 5. x y z 6. 9 y z x

15 7. x y 9. x y 8. x y 40. log x log y 4. log x log y logz 4. log x log y 4. log x log y + log z 44. log x + log y logz 45. log 7 y; log 7 z x log 46. y log 9; x log B. Inverse functions nd grphing: If y f( x)nd y g( x) re inverse functions, then n ordered pir (, b) stisfies y f( x) if nd only if the ordered pir (b, ) stisfies y g( x). In other words, f nd g re inverses of ech other mens f ( ) b if nd only if g( b) To find the inverse of function y f( x) ) Interchnge x nd y ) Solve this eqution for y in terms of x, so y g( x) ) Then if g is function, f nd g re inverses of ech other. The effect on the grph of y f( x) when x nd y re switched, is to reflect the grph over the 45 line (bisecting qudrnts I nd III). This reflected grph represents y g( x). exmple: Find the inverse of f ( x) x or y x ) Switch x nd y: x y (note y nd x 0) l f 5 ) Solve for y : x y, so y x + ( x 0 is still true) ) Thus g(x) x + (with x 0) is the inverse function, nd hs this grph: Note tht the f nd g grphs re reflections of ech other in the 45 line, nd tht the ordered pir (,5) stisfies g nd (5,) stisfies f. exmple: Find the inverse of y f (x) x nd grph both functions on one grph: ) Switch: x y ) Solve: log x y g(x), the inverse to get the grph of g(x) log x, reflect the f grph over the 45 line: Problems 47-48: Find the inverse function nd sketch the grphs of both: 47. f ( x) x 48. f (x) log ( x) (note tht x must be positive, which mens x must be negtive) Problems 49-56: Sketch the grph: 49. y x 4 5. y 4 x 50. y 4 x 54. y log 4 x 5. y 4 x 55. y 4 x 5. y log 4 x 56. y log 4 (x ) l f f l l g l g Answers:. 9, log 9. x x,log x x. 0 00,log , log ny rel number > 0 nd log 4 log 5. (ny bse; if log bse, x log ) y + z 6. z y 7. y 8. log y 9. y log 40. y y y

16 4. y z 4. y 4. yz z y z y 46. y 47. g(x) x g(x) x g f f g (,) TOPIC 8: MATHEMATICAL MODELING WORD PROBLEMS Word Problems:. of 6 of 4 of number is. Wht is the number?. On the number line, points P nd Q nd units prt. Q hs coordinte x. Wht re the possible coordintes of P?. Wht is the number, which when multiplied by, gives 46? 4. If you squre certin number, you get 9. Wht is the number? 5. Wht is the power of 6 tht gives 6? 6. Point X is on ech of two given intersecting lines. How mny such points X re there? 7. Point Y is on ech of two given circles. How mny such points Y? 8. Point Z is on ech of given circle nd given ellipse. How mny such Z? 9. Point R is on the coordinte plne so its distnce from given point A is less thn 4. Show in sketch where R could be. Problems 0-: A 0. If the length of chord AB is x nd length of CB is 6, wht is AC?. If AC y nd CB z, how long is AB (in terms of y nd z)?. This squre is cut into two smller squres nd two non-squre rectngles s shown. C O B Before being cut, the lrge squre hd re ( + b). The two smller squres hve res nd b. Find the totl re of the two nonsqure rectngles. Do the res of the 4 prts dd up to the re of the originl squre?. Find x nd y: 4. When constructing n equilterl tringle with n re tht is 00 times the re of given equilterl tringle, wht length should be used for side? Problems 5-6: x nd y re numbers, nd two x s equl three y s: 5. Which of x or y must be lrger? 6. Wht is the rtio of x to y? Problems 7-: A plne hs certin speed in still ir. In still ir, it goes 50 miles in hours: 7. Wht is its (still ir) speed? 8. How long does it tke to fly 000 miles? 9. How fr does the plne go in x hours? 0. If the plne flies ginst 50 mph hedwind, wht is its ground speed?. If it hs fuel for 7.5 hours of flying time, how fr cn it go ginst this hedwind? Problems -5: Georgie nd Porgie bke pies. Georgie cn complete 0 pies n hour.. How mny cn he mke in one minute?. How mny cn he mke in 0 minutes? 4. How mny cn he mke in x minutes? 5. How long does he tke to mke 00 pies?

17 7 Problems 6-8: Porgie cn finish 45 pies n hour: 6. How mny cn she mke in one minute? 7. How mny cn she mke in 0 minutes? 8. How mny cn she mke in x minutes? Problems 9-: If they work together, how mny pies cn they produce in: 9. minute. 80 minutes 0. x minutes. hours Problems -4: A nurse needs to mix some lcohol solutions, given s percent by weight of lcohol in wter. Thus in % solution, % of the weight would be lcohol. She mixes x grms of % solution, y grms of 0% solution, nd 0 grms of pure wter to get totl of 40 grms of solution which is 8% lcohol:. In terms of x, how mny grms of lcohol re in the % solution? 4. The y grms of 0% solution would include how mny grms of lcohol? 5. How mny grms of solution re in the finl mix (the 8% solution)? 6. Write n expression in terms of x nd y for the totl number of grms in the 8% solution contributed by the three ingredients (the %, 0%, nd wter). 7. Use your lst two nswers to write totl grms eqution. 8. How mny grms of lcohol re in the 8%. 9. Write n expression in terms of x nd y for the totl number of grms of lcohol in the finl solution. 40. Use the lst two nswers to write totl grms of lcohol eqution. 4. How mny grms of ech solution re needed? 4. Hlf the squre of number is 8. Wht is the number? 4. If the squre of twice number is 8, wht is the number? 44. Given positive number x. The squre of positive number y is t lest 4 times x. How smll cn y be? 45. Twice the squre of hlf of number is x. Wht is the number? Problems 46-48: Hlf of x is the sme s onethird of y. 46. Which of x nd y is the lrger? 47. Write the rtio x: y s the rtio of two integers. 48. How mny x s equl 0 y s? Problems 49-50: A gthering hs twice s mny women s men. If W is the number of women nd M is the number of men 49. Which is correct: MW or MW W 50. Write the rtio s the rtio of two M +W integers. Problems 5-5: If A is incresed by 5%, it equls B. 5. Which is lrger, B or the originl A? 5. B is wht percent of A? 5. A is wht percent of B? Problems 54-56: If C is decresed by 40%, it equls D. 54. Which is lrger, D or the originl C? 55. C is wht percent of D? 56. D is wht percent of C? Problems 57-58: The length of rectngle is incresed by 5% nd its width is decresed by 40%. 57. Its new re is wht percent of its old re? 58. By wht percent hs the old re incresed or decresed? Problems 59-6: Your wge is incresed by 0%, then the new mount is cut by 0% (of the new mount). 59. Will this result in wge, which is higher thn, lower thn, or the sme s the originl wge? 60. Wht percent of the originl wge is this finl wge? 6. If the bove steps were reversed (0% cut followed by 0% increse), the finl wge would be wht percent of the originl wge? Problems 6-75: Write n eqution for ech of the following sttement bout rel numbers nd tell whether it is true or flse: 6. The product of the squres of two numbers is the squre of the product of the two numbers. 6. The squre of the sum of two numbers is the sum of the squres of the two numbers 64. The squre of the squre root of number is the squre root of the squre of the number. 65. The squre root of the sum of the squres of two numbers is the sum of the two numbers. 66. The sum of the bsolute vles of two numbers is the bsolute vlue of the sum of the two numbers.

18 67. The product of the bsolute vlues of two numbers is the bsolute vlue of the product of the two numbers. 68. The negtive of the bsolute vlue of number is the bsolute vlue of the negtive of the number. 69. The cube root of the squre of number is the cube of the squre root of the number. 70. The reciprocl of the negtive of number is the negtive of the reciprocl of the number. 7. The reciprocl of the sum of two numbers is the sum of the reciprocls of the two numbers The reciprocl of the product of two numbers is the product of the reciprocls of the two numbers. 7. The reciprocl of the quotient of two numbers is the quotient of the reciprocls of the two numbers. 74. The multiple of x which gives xy is the multiple of y which gives yx. 75. The power of, which gives xy is the power of, which gives yx. Answers:. 44. x +, x ,, or 8. 0,,,, or 4 9. Inside circle of rdius 4 centered on A 0. x 6. y + z. b, yes: ( + b) + b + b. x 40 7 ; y times the originl side 5. x mph hours x miles mph. 000 miles x min x x x 4..y x + y x + y x +.y 40..0x +.y. 4. x 80 7 ; y , , x 45. x 46. y 47. : M W B 5. 5% 5. 80% 54. C % % % 58. 5% decrese 59. lower % 6. sme (96%) 6. b (b) T 6. ( + b) + b F T (if ll expressions re rel) b + b F b + b F 67. b b T 68. F 69. F 70. T 7. +b + b F 7. b b T 7. b b T 74. y x F 75. xy yx T

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