PRECALC ALG & TRIG
PRECALC ALG & TRIG MAC 1147
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Review Unit III Name one positive and one negative angle coterminal with 3 Give the general formula for all angles coterminal with 2 7 3 Find the reference angle 0 for 6 Find sinz 27139 cos 3 Suppose that a machine contains a Wheel of diameter 3ft rotating at 10 revolutions per second Find the linear speed of the Wheel in feet per min 4 Gwen cost csct gt 0 9 Find cott cos 90 t Evaluate cot 270 4sin180 tan7r 4sin 500s2 7139 III Given the function y 1 25in 3x 3 Find the amplitude period and phase shift Find all asymptotes ofthe graph ofy cot 2x Given tanu and 37 lt u lt 27139 Find sin2u Simplify cot6tan6 COSZ 2 tan2 2 Find sin 75 using the halfangle formula VII Evaluate Without a calculator si 11 1 l sincos 1 l 1 107139 cos cos 3 VIII Silfl Sin97r a Solve the equation on the interval 027139 b Find all solutions of the equation 00526 0056 2 1 Evaluate tan cos 1 Given triangle with a 2 b 3 c 439 nd the angle 0 Find the area ofa triangle if a 6 b 9 and c 5 Find the area ofa triangle with a 4 b 6 and y 2 120 Given triangle with c 3 0 2 105 and y 45 nd 13 XI XII REVIEW UNIT II MAC1147 FORMULAS TO MEMORIZE Equation of a circle with radius r and center h k x h2 y k2 r2 Distance on the plane between points x1 y1 and x2y2 d x2 x12 y2 y1 2 Midpoint Formula M 2 2 The slope ofthe line through xly1 and x2y2 m g H Ax x2 q Equation ofthe line with slope m through the point x1y1 y y1 mx xl Equation ofthe line with x and y intercepts a and b l a 7i 0 b at 0 a Average rate of change of f on the interval 0 x is m x i C x c Vertex ofthe parabola ax2 bx C is h k where h g b k a Properties of EXponents a gt 0 a i l b gt 0 b i l A a y l axayaxy axy ax a aX x a0l ay a A abx axbx b b Identities amg x x gt 0 loga ax x x is any real number 1 Given points P 3 5 and Q3 2 a Find dP b Find the midpoint of the segment joining the two points 0 Determine whether the point 69 79 is on the line passing through the points P and 2 Find the equation of the circle with center 5 3 passing through the point 1 0 3 Find the centerradius equation of a circle if the endpoints of the diameter are l 2 and 9 6 4 Complete the square to nd center and radius of the circle 3x2 3y2 l2x 6y l 0 5 Write the equation of the line passing through point 6 3 and parallel to the line through 1 2 and 4 Give the nal answer in a standard form 6 Write the equation of the line through the point 2 perpendicular to the line 3x 2 y 6 Give the nal answer in the slopeintercept form What is the y intercept 4x 3yl3 7 Solve 5x2y l 8 When moving against the wind a bicyclist rides 50 mi in 3 hours and with the wind he rides 80 mi in 4 hours Find the bicyclist s speed and the wind s speed 9 If fx 4x2 3x 2 nd and simplify the difference quotient 10 A piecewise defined function is given x l if x S 2 fx 1 if 2ltxltl x2 l if x Z 1 Find f2 f0 f0 11 Sketch the graph of the piecewise defined function x if x lt 2 fx 2 if 2SxS0 lx 3l l if xgt0 a Give the largest open intervals on which the function is increasing decreasing or constant b Give the intervals on which f x S 0 12 Which of the following relations define y as a function of x a x 2y35 b y2 23x cyxx29 d y3 4x l0 l3 Classify the following functions as even or odd or neither even nor odd 1 x2 1 a fx x3 2x2 b fx 2 c fx 3 d fx x xix 5 x 14 Find the average rate of change of fx I on the interval 3 1 x 15 Explain how gx x l4 2 can be obtained by a translation of the graph of f x x4 l6 Analyze and graph the function f x 3x2 6x 5 a Give the vertex aXis xintercepts yintercept domain and range b Graph the parabola 17 The height 3 of an object after t seconds is given by s l6t2 128t 50 Find the maximum height of the object and the time it takes the object to reach this height 3 2 2 2 3 18 Find all zeros of gx x 4xx 5x 6 x 4 and their mult1p11c1t1es Write the multiplicity in parentheses next to the zero Which of the zeros are the x intercepts 19 Make a sketch ofthe graph ofthe function y x 1 3x 42 Determine the function s behavior when lxl becomes in nitely large 20 Analyze the rational functions a Find the vertical and horizontal or oblique asymptotes if 2 JCS 5x2 xZ x 6 x 2 2 x 2 x 10x 25x 3x 2 x 2x2 4x3 x2 6x8x139 Find the domain symmetry holes asymptotes intercepts and points where the graph crosses its horizontal or oblique asymptote If any of the above does not eXist write None b Sketch the graph of the rational function f x 21 Solve the inequalities Give the answers in interval notation a 5x 3x2 22 b 6x3x 12 12x2x 1lt0 1 x2 x 236 x2 gt l x5x 22x 5 c y3lty3 0 y3 y 3 d 22 Use synthetic division to find the quotient and remainder when fx 2xquot5 4x 5 is divided by x 3 23 Use synthetic division to determine f 2 if fx 3x5 4x4 2x2 7x 13 24 Use the factor theorem to determine if x 4 is a factor of 4x4 33 2x2 5x8 25 Find all zeros of fx x5 5x4 5x3 25x2 36x 180 if 5 is one zero of 26 Use the intermediate value theorem for polynomials to determine if fx 2x3 3x2 5x 4 has azero on each interval 3 2 2 1 10 01 12 i z 27 Name two functions f and g such that o gx 5x2 3x 4 28 Let i2 and gx 1 x Find x a f gx and their domains g b 34 c fog0 d o gx g ofx and their domains 29 Which ofthe following functions f g and h are onetoone Vx 5 l 2 x x x h x g H 30 Graphf l if fx 1 31 If fx is onetoone nd f 1 x and its domain and range if x 5 a x x2 32 Sketch the graph of the function f x 2 6 Find its domain range and asymptote Is function increasing or decreasing 33 Solve l6 2 8 34 Find the domain range asymptote intercepts and sketch the graph of the function y2 210g2 xl 35 Write the statements below in equivalent exponential form b fxx2 2x x21 c fxx22xx20 l 10 1253 b ln2x 10 a g5 C glooo 36 Write the statements below in equivalent logarithmic form 4 1 7x a e0 1 b 102 4 0 l 16 d 21 2 y 37 Evaluate a log25 5 b logz c log100 d In 83 e logL 81 L1 Real Numbers Algebra Review Sets of Numbers Natural numbers Whole numbers Integers Rational numbers x x 2 where a and b are integers and b 2 0 Note 1 Any integer m can be written as thus all integers are rational numbers 2 Rational numbers include repeating and terminating decimals for example O333O 0521621 Irrational numbers Includes any real number that is not a rational number for example Real numbers The rational and irrational numbers together Operations on the Set of Real Numbers Addition a b Multiplication ab Properties of Real Numbers Commutative Properties Associative Properties Identity Properties There is a unique number 0 called additive identity such that There is a unique number 1 called multiplicative identity such that Inverse Properties There is a unique number b additive inverse or negative ofb such that Subtraction a b For b 2 0 there is a unique number 1 b multiplicative inverse or reciprocal ofb such that Division I 72 O ab Distributive Property Order of Operations 1 Work separately above and below each fraction bar 2 Work within parentheses or square brackets by working inner to outer 3 Simplify all powers and roots in order left to right 4 Multiply and divide in order from left to right 5 Negate add or subtract in order from left to right Example Evaluate the expression 6 42205 232 186331 Cancellation Property 6 i O acbc gt ac E Example 2x 3 gt 15 g ZeroProduct Property If ab 0 then Example 7x O gt ZeroQuotient Property gzo ltgtaO providedthatb 0 Note is undefined Example i0 gt 2 x O gt x Arithmetic of Fractions b Qd O amp6 ifb 0d 0 ampln ifb 0c 0d 0 ampq Q WIS WIS Example Use the Least Common Multiple to add the fractions 5 4 18 15 Real Number Line Order on the Number Line If two numbers a and b are given only one of the statements below is true a lt b a is to the left of b a b a is at the same location as b a gt b a is to the right of b 5 Absolute Value Absolute value of a number a is the distance on a number line from O to a Notation used la The absolute value of a real number a is defined by the rules 04 a if aZO a if altO Note la is always nonnegative Properties of Absolute Value aid 2 190 b at 16420 iallal labllal lbl lbl la bl S a b the triangle inequality Example Write without absolute value signs l15l l3 7r7r 3 12 2x Distance Between Two Numbers on a Number Line dab b al m Since I a and then dab dba Example Find the distance between 7 and 3 Example Write each statement as an inequality involving absolute value The distance between x and 3 is no more than 2 The distance between x and 5 is at least 6 Exponents If a is a real number and n is a natural number then a aaaa a J n factors 4 8gt2 43 82 l a0 if a 72 O Negative Exponents If a i O and n is natural then n l 1quot a 2 H a a Laws of Exponents a ifa 0 9 a ifb 0 aquot b b Example Simplify using the Laws of Exponents 4 6 43 32x 2 2x3 3 Square Roots l A number I is called a square root of a if b2 a I There are two square roots of a positive number a one is positive and the other is negative The notation xZ is used for the positive root which is called the principal square root of a Thus all square roots of a are ixg Example Find all square roots of 9 Find the principal square root of 9 For any real number a J lal W Jlt 7gt2 Decimal and Scienti c Notation Example Write the number in scientific notation One Micron Millions of Meter 000003937quot 2 Example Write the number in decimal form Interior temperature of Sun 13 X 107C 2 REVIEW UNIT III MAC1147 FORMULAS TO MEMORIZE Identities alog x x gt 0 loga ax x x is any real number Properties of logarithms x gt 0 y gt 0 a gt 0 a 1 andp 2 0 a 10gaxy 10ga x 10ga y b 10a log x 10ga y y c loga x rloga x d log p x iloga x a p e loga a 1 f loga l 0 log b x Changeof Base Theorem loga x abx gt 0 a 2 l b 2 1 log a V1 Compound Interest Formula A P l 1 n Continuous Compounding A Pe Exponential Growth or Decay At A061 Length of an Arc 9 in radians s H9 1 Area ofa Sector 9 in radians A Erz s 9 L1near Speed 0 Angular Speed a 7 Relation between D and a u m2 9 in radians Trigonometric Identities tant9 Slug cott9cfsg tant9cott9l cost9 sm 9 l l csct9 sect9 tant9 s1n 9 cos 9 cot t9 sin2t9cos2t9l tan2t9lsec2t9 lcot2t9csc2t9 sint927msint9 cost927mcost9 tant97mtant9 sint9i7r sint9 cost9i7r cost9 cos t9cost9 sin t9 sint9 tan t9 tant9 Sum and Difference Formulas cos a cos acos sinasin cosa cosacos sinasin a gt a sinacos cosasin sin sinacos cosasin tanatan tanlta tana w l tanatan ltanatan Doubleangle Formulas cos 26 cos2 6 sin2 6 cos 26 2cos2 6 1 cos 26 l 2sin2 6 sin 26 2sin6cos6 tan26 l tan 6 Halfangle Formulas a l cosa a lcosa a l cosa s1n i cos i tan i 2 2 2 2 2 lcosa a sin a a 1 cos a tan tan 2 2 1 cos a 2 sm a General Formulas for the Solutions of the Trigonometric Equations k 0ili2 sin6a lalSl 6arcsina27rk and 67r arcsina27rk cos6 a lalSl 6arccosa27rk and 627r arccos a27rk OR 6iarccosa27rk tan6a 6arctana7rk The Law of Sines sm w a b c The Law of Cosines c2 a2 b2 2ab cos y Area ofa Triangle A bh l A ab s1n y A ss as bs c Heron sFormula s ltabcgt n quot n n g The Binomial Formula x y anjyj where 10 J J JnJ l Solve the exponential equations a 32xz 99 b 83x 125 c 51 4 1 use natural logarithm d e 114 6 2 Solve the logarithmic equations a log2 x 32 4 b log3 2x l log3 x l 2 c lnxlnx 2ln3 dlog3x 62 3 Simplify where it is possible Give all restrictions on the variables 1 9 a ogs log5 25 4 Given log52x l evaluate b e31 c loglohz dlog42xz e 51 flnx2y2 5 Wnte as alnganllnrn ufa smgle quantity 510g 27 310g x210gy a Wntetheslngleluganthm asasurn andurdlfferenee log6 6 f y z 7 se ChangerufrBaseTheurEm tn rewnte eaelr luganllnrn wth anewbase b Slmphfywhere ltlspusslble 210g22b hlogmxb10 210g36216b6 a 30000 r fur 27 tn merease tn SKELEIEIEI 314 eumpuunded a uanerl h euntlnuuuslw 9 urrnula N0 Nngarnnm Where lsln huurs 1U gven by At 20002 Flnd une amuunt quhe substaneepresenta er lnu days h and me halfrllfe ufthe substance 11 Let S ZED a and the angle quhe smallestpusslble nnsuve rneasure eutenmnal ml 0 h and llnereferenee angle dln degrees andln radlans lz and the area ufa seetur ufa nrele mta39eepted by aeenual angle uf15U ln aerrele uf radlus 5 em and the angular weed quhe pulley ln radlans per mmute 14 If478 ls un Lheta mlnal slde ufan angle 0 ln standard nusrunn nd sln a eusa lane 15 FlndLhe exaetvalues a see 1 b lanelzu e slnmeeusw d ese732 l l u u u u u 16 H V n r y4i sln3x n 17 Graph yeus2xen 18 and the eduauun fur the slnusuldal gaph 19 Find the asymptotes for a y tan2x 7239 b y cscx J c y arctanxg d y sec391 x 20 Find the exact values a sin391 J b cos391 c arctanxg d cot391 e csc391 l 21 Evaluate a sincos391 b cossin 1 c cos sin391 3 d s1n 2tan391 5 e tan arccos lt gt lt gt lt gt l t 4 22 Find the exact value a sin391tan b cos391cos227 c cot391cot15 j d tan391cos0 23 Express in terms ofu sincos391 u l S u S1 24 Simplify a cot2 9 ll cos2 9 b cos 9 sint92 c csc2 6 sin 29 25 Verify 1 cos 2a a sin cos b csc2a 2 tan cot sinza I I c w csc2 a sec2 at d sec2 9 csc2 t9 sec2 9 csc2 9 sm acosa 26 Let sin 3 with s in quadrant III and let cost with t in quadrant 11 Find a cos s t b sin s t c cos 2s d tan 2t 27 Evaluate sinarccos arcsin D 2 7 28 Let sina cosa and 3 lt a lt 27239 Use the halfangle formula to nd a tan b cos c sin 29 Find the exact value a sinl65 sin15 b cosl coss l2 12 30 Find all solutions asin2t9l b2cost9 l ccsct9 2dtan2t9 l0 31 Solve in the interval 027239 cos 2x 2 3cos x 32 Given the equation sin 26 J2cost9 a nd all solutions b list those that are in the interval 0 27239 33 A fire ghter at the top of a tower 500 ft tall spots a re The angle of depression to the re is 29 How far is the re from the base of the tower 34 A regular octagon is inscribed in a circle with a radius of 158 cm Find the perimeterP of the octagon 35 An airplane pilot measures the angle of depression to two ships to his right to be 30 and 45 If the pilot is ying at an elevation of 6 miles nd the distance between the two ships Round up your answer to one decimal place 36 Solve the following triangles a a12 30 y40 b a135 b56 y25 c a5 b6 c10 37 A parallelogram has sides of length 20 cm and 50 cm If one ofthe angles has measure 55 approximate the length of each diagonal 38 Find the angle a if a 5 b 4J2 0 1 Do not use a calculator 39 Approximate the area of the triangle a 60 a10cm c20cm b a 403 629 b 56 ft c a 200 b 200 c 100 40 The Pentagon is the largest of ce building in the world in terms of ground area The perimeter of the building has the shape of a regular pentagon with each side of length 921 feet Find the area enclosed by the perimeter of the building 5 41 Expand by using the Binomial Theorem x l x n n n n n 42 Express in terms of n and simplify 0 l 2 3 4 l 43 Find the coef cient of x14 in the binomial expansion of x 2d Review Unit 111 Name one positive and one negative angle coterminal with 3 Give the general formula for all angles coterrninal with 2 7 3 Find the reference angle a for 6 Find sin 2 COS Suppose that a machine contains a wheel of diameter 3ft rotating at 10 revolutions per second Find the linear speed of the wheel in feet per min 4 Given cost csct gt 0 Find cot t cos90 t Evaluate cot 270 4sin180 tan 4sin j 5008272quot 111 Given the function y 2 1 2 sin 3x Find the amplitude period and phase shift Find all asymptotes 0f the graph of y cot2x Simplify 1 tan2 6 1 cot2 6 Given tanu and 37 lt u lt 27 Find sin 2u u cos 2 u tan 2 VI Find sin 75 using the halfangle formula VII Evaluate without a calculator sin 14 sincos1 1 1 COS COS 3 VIII 1 97 sm sm 8 Evaluate tan cos 1 a Solve the equation on the interval 027r b Find all solutions of the equation cos26 cos6 1 A tunnel for a new highway is to be cut through a mountain that is 260 feet high At a distance 200 feet from the base of the mountain the angle of elevation is 36 From a distance of 150 feet on the other side the angle of elevation is 47 Approximate the length of the tunnel to the nearest foot XI REVIEW UNIT 1 MAC 1147 FORMULAS TO MEMORIZE x if x Z 0 Ix x 2 0 x 1f x lt 0 xy2 x22xyy2 xy2xz2xyy2 xZy2xyxy xSy3xyx2xyy2 x3y3xyx2xyy2 xy3 x3 3x2y3xy2 y3 xy3x33x2y3xy2y3 Jx Zlx bib2 4ac The Quadratic Formula x 2a 1 Write Without absolute value signs and simplify 2x l x2 1 a b 7z 5 3 7z c 3lZ 4rr ll ifrgt3 xl 2 Let x l and y 3 Evaluate 2x y1642 x 1 2 20 mot y3 ly xl xyl x y 3 LetA n 16i06 List the elements that belong to each set a Natural numbers b Integers c Rational numbers d Irrational numbers e Real numbers 1 Perform the indicated operations and simplify 71 xl xl l l a b x l x l x32x2 x34x24x 5 Factor completely over the real numbers a 4x2x 23 6x3x 22 2x2x 22 b 4x4 x2 16x3 4x c x4 x2 12 d 32x5 4x2 6 Reduce the fraction to the lowest terms Find the domain of the fraction 12 y2 4y 5 6 y2 y 2 7 Simplify each rational expression Write the domain of the variable 4x2 4 x4x 6x25x 6 8x327 a b xZ xl x22xl 3x2 2x 4x3 6x29x 8 Simplify each compound fraction Find the domains of the fractions in a b 3 l 3 7i i 2 x l x Zx l xZ a 1 x6 b x x1l c 2 x 1 772 1 7 x x x 9 Simplify each eXpression Give your answer using only positive eXponents 27 3 72 73 72 72 6 7 71 7 a w b u c xly xzy x y My2 10 Simplify Rationalize the denominator where it is necessary 4x10y12 x2 xy 2 2 2 a 5 b c d 3 x 4 x 164x523y 0 56 Nixg N yd y 11 Simplify each radical eXpression where it is possible a 41618 b JE J32 c16x l6 2xlx l 3xl4x 4 d d2 132 12 Evaluate 1 a 9gji 16 l7 20 b 2562 256 8 l3 Perform the indicated operations a a3 a l Za3 a2 1 b x3 2x 3x2 x 1 4 2 c M d 318 8 e 2x 33 2xl l4 Solve the equations 4 l x a b xx 3 x x 3 3x l l 3x 15 Evaluate a l 64 b V 125wl 80 c 13944 l6 Perform the indicated operations Give the answers in standard form 2 i2 3 2i 3 a 4 3i 43i b 2 i 6 2i c d z 17 Find m of the equations a x4 10 b y12 16 c 2x2 x 5 0 d x2 3x3 0 18 What must be the value ofk for the equation 6x2 3x k 0 to have one repeated real solution 19 Solve the quadratic equation by completing the square 4x2 4x 35 0 2 0 Solve the equation 2 21 Find all four solutions of the equation 41f1 3x 2 1 0 22 In the morning Mary drove to the of ce at average speed of 45 mph Her average speed on return trip home was 30 mph The return trip in the afternoon took 10 minutes longer because of heavy traf c How far away from home is her of ce 23 The width ofa rectangle is 2 feet less than its length Find the length of the rectangle if its diagonal is 10 ft 24 Jack can paint his apartment in 12 hours His wife Cheryl requires 20 hours to do the same job How long would it take them to complete the job if they worked together 25 An arrow is shot upward from a platform 40 feet high with the initial velocity of 224 ftsec Its height h in ft after t seconds is given by h 16t2 224t40 At what time will the arrow be 824 ft above the ground 26 Solve the equations a l3x 430 b Vx210x113x c 5x4 2J 1 d xl 3xli 100 l 27 Solve for r and completely simplify S L m 28 Solve the inequalities Give answers in interval notation a 2x5S43x b 3lt1 2x5ss 29 Solve the absolute value equations a 5x100 b x32x2 3x210 4x 3 c 1 2x 3x 5 d 3 039 l l l 0 3x4 30 Solve the absolute value inequalities Give answers in interval notation a 2l2 xISO b 2l2 xIZO 8 296 3 1 1 c 4 gt0 d 3 xllt 3 4 2 l l 2 31 Write the statement below as absolute value inequality and solve it for k Give your answer in interval notation k is no more than 13 units away from 5 Review Final Part II Solve the equations 210g3 x 10g3x 4 10g3 2 4x 2 3x5 Find the angle which lies in 027r and coterrninal with Find the reference angle a for 6 2 113 2 Find the length of the arc of the circle with r 9 ft subtended by the central angle 6 60 The car is moving at 60 kmhr The diameter of its wheels is l in Find the number of revolutions per minute of the rotating wheels If sin 2 3 and 7 lt 6 lt 37 find tan and cosg Find the amplitude range period and phase shift for y1 2sin 27zx Find the asymptotes the graph of y tan x Find arcsin sin Solve the equation on the interval 027r 200s2 6 3sin6 3 Find all solutions of the equation A woman on a 40 m tower sees a man at an angle of depression of 15 How far is the man from the base of the tower Shawn Johnson wants to measure the distance across the Big Muddy River If 7 105 a30 b 2350 ft find a Findcifa3 cmb5 cmand7120 Find the angle a and the area of the triangle with a 13 m b 5 m and c 2 10m Find the area of the triangle if a 13ft 6 10 ft and 30 AnwersrsziBN Unit m In 33 1H 20 3116 1 a t 1112 h Ins aha4 Wm z 2 717 Mg 13 a 87 3 21 b 2 no c 2x2 g 12 efcannutbes1mpl1 ed 4 1 5 log 32f a 1llog xilog y7210g z x 2 2 7 a i as lm a g a at265yezrs t263years 3 huurs 111 a 22539 1 2711 67 yams 1 13 TZZMA 65 days 11 2130 b asu sl 1z 13 sq cm 13 80 rad1ansmm l4 51n57 BESS lsltagt on 16 A3 15 4 3 l7 71 71 71 71 18 Zsm x ur Zcus x y z 4 y z 4 13 a x gm n 111112 b x rm n 111112 ltgtyn andyn ltagty 20 a b c d e 21 a g b g c i d i e 2 2 3 13 7 22 a g b 67 c 7 d g 23 1 112 24 a 1 b 1 sin219 c 2cot19 26 4 b c d 3J7 27 2J5 2 284a 2JE 7 b 72JE G 7 26 15 5 14 14 2 J3 J2 29 a b 4 2 30 a 197m n0r1r2 b i2 27m n0i1i2 c 27m57 27m n0i1i2 d19i7m n0r1r2 31 0251 3 3 32 a 1nn127m3 27m n0i1i2 b 2 4 4 4 2 4 2 33 x500tan61 m902 feet 34 13967 cm 35 44 mi 36 a a110 b64 c82 b c88 16 a139 c y131 a22 27 37 42 cm and 64 cm 38 a 45 39 a A 50x3 w 87 sq cm b Am111 sq ft 0 255968 40 A1459379 sq ft 6 15 6 1 41x x66x415x220 2 4 6 x x x x 421 n quotn 1 21 quotn 1 2 3 43 3 4 39 16 Review Unit 11 Find k so that the distance between 3 2 and 2k is J5 Find the center and radius of the circle 2x2 2y2 12x4y14 O Is y3 x 1 2 0 an equation which defines y as a function of x If yes is the function even or odd Given fx 4x2 3x 2 fah fa h Find Find an equation of the line passing through the point 35 and perpendicular to 2x 7 y 8 Give the final answer in standard form 111 Determine whether the function f x 2x2 6x 4 has the minimum or maximum value and find it Find the domain all asymptotes and holes if any x3 x2 2x R x x2 2x 3 Determine the interval on which the graph lies above the xaXis Given fx 2 1 V3 x 4 Find f1x if it exists VI Describe the transformations that need to be performed on the graph of f x in order to obtain the graph of gx Find the domain range asymptote and intercepts of the function gx a fx 2x and gx 23x 1 VII b fx10g2x and gx210g2 x 4 2 VIII Write as a single logarithm 10g3 210g3 310g3 m n Solve the equation 2x1 51 x Solve the equation 10gx110gx 2 1 L1 Real Numbers Algebra Review Sets of Numbers Natural numbers Whole numbers Integers Rational numbers xl x 2 Where a and b are integers and b i 0 Note 1 Any integer in can be written as thus all integers are rational numbers 2 Rational numbers include repeating and terminating decimals for example 03330 0521521 Irrational numbers Includes any real number that is not a rational number for example Real numbers The rational and irrational numbers together Operations on the Set of Real Numbers Addition a b Multiplication ab Properties of Real Numbers Commutative Properties Associative Properties Identim Properties There is a unique number 0 called additive identity such that There is a unique number 1 called multiplicative identity such that Inverse Properties There is a unique number b additive inverse or negative of I such that Subtraction a b For b i 0 there is a unique number 1 b multiplicative inverse or reciprocal of I such that Division b i 0 ab Distributive Property Order of Operations 1 2 3 4 5 Work separately above and below each fraction bar Work within parentheses or square brackets by working inner to outer Simplify all powers and roots in order left to right Multiply and divide in order from left to right Negate add or subtract in order from left to right Example Evaluate the expression 6 42205 232 186331 Cancellation Property 0 0 aczbc 3 ac Example 2x23 3 15 i ZeroProduct Property If ab 0 then Example 7x 0 3 ZeroQuotient Property 20 3a0 providedthatb 0 Note 3 is undefined Miler E0 2 2 x 0 3 x Arithmetic of Fractions ifb 0d 0 b d 33 ifb 0d 0 b d E g ifb 0c 0d 0 3 Example Use the Least Common Multiple to add the fractions 5 4 18 15 Real Number Line Order on the Number Line If two numbers a and b are given only one of the statements below is true a lt b a is to the left of b a b a is at the same location as b a gt b a is to the right of b 5 Absolute Value Absolute value of a number a is the distance on a number line from 0 to a Notation used lal The absolute value of a real number a is de ned by the rules a if a Z 0 lal a 1f a lt 0 Note lal is always nonnegative Properties of Absolute Value M b 0 a Ialzo lallal labllallbl la bl S lal lbl the triangle inequality Example Write without absolute value signs 15 3 7r7r 3 12 2x Distance Between Two Numbers on a Number Line dab b al m Since I a and then dab dba Example Find the distance between 7 and 3 Example Write each statement as an inequality involving absolute value The distance between x and 3 is no more than 2 The distance between x and 5 is at least 6 Exponents If a is a real number and n is a natural number then aquot aaaa n factors 4 8 43 82 I do ifa 0 Negative Exponents If a i 0 and n is natural then n 1 1quot a 2 quot a a Laws of Exponents amaquot 2 amquot amquot 2 amquot abquot aquotbquot a quotquot ifa 0 mych aquot b bquot Example Simplify using the Laws of Exponents 4 5 43 32 x72 2x3 3 2 Square Roots I A number I is called a square root of a if b2 a I There are m square roots of a positive number a one is positive and the other is negative The notation J is used for the positive root which is called the principal square root of a Thus all square roots of a are 3 Example Find all square roots of 9 Find the principal square root of 9 For any real number a 2 a lal x9 2 72 Decimal and Scienti c Notation Example Write the number in scienti c notation One Micron Millions of Meter 000003937quot 2 Example Write the number in decimal form Interior temperature of Sun 13 X107C Review Final Part I 3 3 4 Simplify 4 Write without absolute value signs 7r 2 72 4 y7 Sim lif 5 p y V64x3 1x4 Sim lif p y 2x 2 Fred and Wilma start from the same point and travel on a straight road Fred travels at 30 mph while Wilma travels at 50 mph If Wilma starts 3 hr after Fred find the distance they travel before Wilma catches up with Fred Solve V7x 26 1 Solve in the complex number system x4 x2 2 O Solve 4x 7 lt 21 Solve 4x 7 lt 21 x Solve 2 lt 3 x 3