College Algebra MTH 103
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Date Created: 09/19/15
Section 32 Polynomial Functions and Their Graphs Defmition of a Polynomial Function Let n be a nonnegative integer and let an aH a2 a1 do be real numbers with an at 0 The function de ned by fx anxquot anilx39 1 azx2 alx1 610 is called a polynomial function of degree n The number an the coefficient of the variable to the highest power is called the leading coefficient Polynomial functions of degree 2 or higher have graphs that are smooth and continuous Example 1 Find the degree and the leading coefficient of each following function fX3 gx3 2x1 hx x4x6 The Leading Coef cient Test As x increases or decreases without bound the graph of the polynomial function eventually rises or falls The behavior of a graph to the far left or the far right is called end behavior The end behaviors are determined by the degree n and the leading coefficient an 1 For 71 odd Opposite behavior at each end an gt 0 a lt 0 2 For 71 even Same behavior at each end a gt 0 a lt 0 y y Example 2 Use the Leading Coefficient Test to determine the end behavior of the graph of fX X4 3x32 Zeros 0f Polynomial Functions Zeros the values of x for which fX 0 ie roots or solutions of fX 0 Each real root of the polynomial appears as an x intercept of the graph Example 3 Find all zeroes of fX A 21K 4X 8 Example 4 Find all zeroes of fX X4 4x Multiplicities 0f Zeros In factoring the equation for the polynomial function f if the same factor X r occurs k times but not k1 times we call r a zero with multiplicity k ie X 0quot appears Example 5 Find the zeros and each multiplicity of fX X 2 X 33 If r is a zero of even multiplicity then the graph touches the xaxis and turn around at r If r is a zero of odd multiplicity then the graph crosses the xaxis at r Graphs tend to atten out at zeros with multiplicity greater than one Even Multiplicity Odd Multiplicity y y Final Exam Review HOMEWORK Review all in class review problems review problems below ALL exams quizzes and homework problems ALL Story Problem Applications from sec 13 15 17 26 31 45 Solving equations algebraically linear quadratic rational radical literal polynomial exponential log absolute value etc and terms and de nitions 1 Solving Equations i Linear ax b 0 a 7i 0 Example 1 mc 23 5x 4x1 illnswer x 7 t i it 2 ii quadratic ax bx C I 0 Solve by factoring QE Quadratic Formula If ax2 bx c 0 where b i V b2 4ac a 0then x 2 a Complete the square Watch common errors 1 Watch negatives care llly If b is negative then Z is positive You cannot cancel over i You must factor rst or break up fraction You must have 0 on one side to use the zero property Example 2 Solve 3x2 7 2x 4 4 Page 1 Final Exam Review iii absolute value The absolute value of a number a denoted lal is the distance from 0 0n the real number line a aZO a a alt0 If x 2 then x i2 In general x athen xiaifagt0 Strategy for Solving Absolute Value Equations Step 1 Isolate the absolute value Step 2 rewrite something a where a 2 0 as something a or something a Note Ifa lt 0 then something has no solution Step 3 Solve each resulting equation Step 4 Check your answer Example 3 Solve 2 5x ll 3 4 iv radical Example 4 Solve x 1 5 x x 1 V 5 x Must isolate raalicalfirst x l2 5 x T Square both sides NEVER square each term x1xl 5 x x22x15 x lIUSTFOIL x23x 40 Get00n one side x4x l 0 Factor x 4 0 or x 1 0 Use zero property to set each factor0 x 4 orx 1 Solve Page 2 Final Exam Review MANDATORY Check in ORIGINAL EQUATION x 1 xS x x 4 4 15 4 x1 1 151 41 1 1JZ 4 13 1 12 11 So answer is only VVVVV v rational Solving Rational Equations Step 1 Factor the denominators of all rational expressions completely Step 2 IDENTIFY ANY RESTRICTIONS Step 3 Identify the LCD of all of the expressions in the equation Step 4 Multiply both sides of the equation by the LCD Watch Remember if there is more than one term in the numerator or a negative you must distribute Step 5 Solve the resulting equation Step 6 CHECK EACH Potential Solution Example 5 Solve 2 x 2 1 7 a1 ebraicall xZ Sx 3 2x1 x 3 g y Mae 4 Page 3 Final Exam Review W literal Example 6 Solve M 4y 1 for u 5x 2u 20xy5x 3b 7a8y2 Answer u Strategy Clear Fractions by mult by LCD Get all terms with the variable you want on one side and terms 1 without that variable on the other Factor out the variable you want Then divide by the coef cient viz exponential Example 7 Solve 23 quot 5e2 7 ln57 ln2 4ln3 N 39071 1113 2 3 fAnswer x viiz logarithmetz39c Example 8 Solve log7 2x 1 log7 x 3 log7 2x 12 Magi 15 Page 4 Final Exam Review Example 9 Solve 2 logx 1 10g3 x ix polynomial Example 10 20x4 4x3 45xZ 9x 0 7777777777 asomething2 bsomething c 0 Let u something Example 11 a x392 8x391 200 bx 8x 200 c e 8e 200 d x6 8x3 200 e xm 8x13 200 f 5 85 200 All ofthese are ofthe form it2 8u 20 0 Answer a 503 1 100 cln10 d Wy e Jami gm Page 5 H Inequalities Final Exam Review Properties of Inequalities Property of Inequality Multiplication Pro erty of Inequality If a lt b then c gt 0 c is positive c lt 0 c is negative If altbthen If altbthen ac gt bc a c lt b c ac lt bc a b a c lt b c a b g gt Z c Adding or subtracting does not affect the gt or lt sign Multiplying 0r dividing by a positive number does not affect the gt or lt sign When multiplying 0r dividing by a negative number reverse the gt or lt sign Watch out for common mistakes Whenever you multiply 0r divide by a negative you must ip the inequality Remember to ip the inequality in the same step you mult or divide Watch that you read the inequality correctly if the variable is on the right side of the equation 1 linear Example 12 Solve 2 3x 1 S 5 1391 Compound 39 J Procedure Solving a Compound Inequality with and or or Step1 Solve and graph each inequality separately Step 2 Analyze and vs or If the inequalities are joined by the word and nd the intersection of the two solution sets If the inequalities are joined by the word or nd the union of the two solution sets Step 3 Express the solution set in interval notation Page 6 Final Exam Review 1 iii absolute value 39 J Strategy Solving a Absolute value inequality Step1 Solve for the absolute value Step 2 rewrite the absolute value as 2 inequalities lt or S becomes an gt or 2 becomes or something S becomes something S and something 2 something lt becomes something lt and something gt something 2 becomes something 2 or something S something gt becomes something gt or something lt Step 3 Solve and graph each inequality separately Step 4 Analyze and vs or If the inequalities are joined by the word and nd the intersection of the two solution sets If the inequalities are joined by the word or nd the union of the two solution sets Step 5 Express the solution set in interval notation i Watch out for common mistakes Make sure you both ip the inequality and change the sign when rewriting i Make sure you maintain the equal sign fOIS or 2 Example 13 Solve 31 2 Z 5 Example 14 Solve 11 2 lt3 Page 7 Final Exam Review iv polynomiaVrationalr 39 1 Express as fX gt 0 or fX lt 0 Get 0 on right side 2 Find values that make the numerator amp the denominator 0 These are the Boundary Points Numerator 0 zeros If original inequality has Then solid dot If original inequality has Then open dot Den 0 restrictions Always open 3 Plot the boundary points on a number line to obtain Intervals 4 Test Value within each interval and evaluate fc for each value If fc gt 0 then fX is for interval If fc lt 0 then fX is 7 for interval 5 Write solution using interval notation Check solution on calculator x 5x S2 x2 x2 Example 15 Solve Page 8 Final Exam Review HI Graphs i xint where graph crosses x axis Substitute y0 into the equation anal solving for x ii yint where graph crosses x axis Substitute x0 into the equation anal solving for y iii asymptotes Rational polynomials Finding Vertical and Hori7ontal As mptotes Vertical Horizontal x n n71 If fx p 3 15 a reduced ratlonal Given the rational anction f x W q x bmx39quot bmilxm39l b0 maion and a is a zero 0f 10 then where n degree of numerator and m degree of denominator the then the Horizontal Asymptote of the graph of f is Vertical Asymptote of the graph of f y 0 if n lt m IS y if n m x a X zeros of the denominator b NoHAifngtm Log equations y 10gb x vertical at x0 If y alogb cx h k then at x transform original asymptote c Exponential equations y b horizontal at y0 If y abmh k ten at yk transform original asymptote Examples 16 Find the horizontal anal vertical asymptotes of 3x2 1 x1 a x b x 5x2 9x 2 4x2 9 cy210g75 3x1 aly 45 32 herizyf9 here1 iVert x 15 x 2i ivertx 32 x 32 ivert x 531 Page 9 iv domain Final Exam Review The domain of the relation is the set of rst components inputs often corresponds with the x value The domain of a function that does not model data or verbal conditions is the largest set of real number for which the value of fX is a real number You must exclude the following from the domain a real numbers that cause division by zero b real numbers that result in a square root of a negative Number c real numbers that result in a log of a negative Number Tip Use a number line when using interval notation Example 1 7 find the domain of v 5 2x f x x3 5x2 6x 10g4 x x g x225 WiEf fif1a93 91 v range I The range of the relation is the set of second components outputs often corresponds with the y value vi relative extrema Relative MaximumMinimum The p0ints at which a lnction changes its increasing or decreasing behavior vii intervals of incdec Increasing Functions Decreasing Functions Constant Functions A lnction is increasing on an open interval 1 if for any X1 and X2 in the interval where X1 ltX2 then fX1 lt A lnction is decreasing on an open interval 1 if for any X1 and X2 in the interval where X1 ltX2 then fX1 gt A lnction is constant on an open interval 1 if for any X1 and X2 in the interval where X1 ltX2 then fX1 Page 10 Final Exam Review viiz evenodd Even Functions Odd Functions f x f x f x fx Symmetric over x axis Symmetric over origin Examples Examples x2 1 x3 fx x4 4x2 x Note If fx 2 f x and fx 2 f x then fx is neither odd nor even ix one to one x Sketch graph Steps for Graphing a Polynomial Function f x anx aHxH azx2 a1x a0 1 Use Leading Coef cient Test to determine End Behavior Find the x intercepts Let fX 0 Check for evenodd multiplicities N 9 Find the y intercept Let X 0 Use symmetry if appropriate to graph Symmetric t0 yaXis if fX fX Symmetric t0 origin if fX fX 4 Use of turning points to check graph accuracy UI Page 11 Final Exam Review xi Transformations of graphs Basic strategy y af bx c k 1 Identify the base graph 2 Find the key points of the base graph 3 Transform the key points H horizontal shi 7 Add c to x values Watch sign There is a negative in the form so it is the opposite of what it looks like S stretch Mult y by a for vert stretch Mult x by lb for horizontal stretch R re ect Mult y by 1 for re ect over X axis alt0 Mult x by 1 for re ect over y aXis blt0 V vertical shi add k to y values 4 Plot the new key points 5 Connect the dots Verify endpoints arrows open or closed etc 6 If you know the speci c equation check it on your calculator Example 18 If2 1 is on the graph off whatpoirit is on the graph of y 5f x 4 3 7 3 H s R V 3 Answer 2 l gt6 l gt6 5 gt 6 5 gt 6 2 x4 y5 xri y3 If2 1 is on the graph off whatpoirit is on the graph of y 3fx 4 5 7 3 H s R V 3 fAriswer 2 1 gt 2 1 gt 2 3 gt 23 gt 2 2 i 4 y3 yel yrs 3 xii basic shape anal properties of basic graphs quadratic cubic abs value sqrt reciprocal exponential log etc Page 12 Final Exam Review HI polynomials i degree ii end behavior End Behavior of a Polynomial what happens to the graph of the lnction to the far left x gt 00 and far right x gt 00 and Leading Coef cient an Test Degree 11 is Even Degree 11 is Even Degree 11 is Odd Degree 11 is Odd angt0 anlt0 angt0 anlt0 iii turning points I Turning Points A polynomial of degree n has at most n 1 turning points iv Multiplicity If the same factor x7 r occurs k times then the zero r is called a zero with multiplicity k Even Multiplicity gt Graph touches XaXiS and turns around Odd Multiplicity gt Graph crosses XaXis If kgt1 graph attens out v Intervals of incdec vi evaluate vii factor completely viii division algorithm The Division Algorithm If x and dx are polynomials where dx7 0 and deg dxlt deg fx qx doc f 96 V06 f x V06 fx dx 6100 V06 0 doc 6100 doc Page 13 Final Exam Review ix remainder thrn The Remainder Theorem If the polynomial fX is divided by X 7 c then the remainder is fc x factor thrn The Factor Theorem Let fX be a polynomial If fc 0 then X 7 c is a factor offX If 7 c is a factor of fX then fc 0 xi long division and synthetic division xii rational root thrn The Rational Zero Theorem IffX auxquot amxr 1 alx a0 has integer coefficients then 3 q reduced to lowest terms is a rational zero of fX where p is a factor of the constant term a0 and q is a factor of the leading coefficient an Possible Rational Zeros of fX W i factors of an leading term xiii Linear Factorization Thm Linear Factorization Theorem IffX auxquot an1Xquot391 anzXquot392 a1X a0 where n 2 1 and an iquot 0 then fX anX 7 c1X7 c2 X 7 c Properties of Polynomial Functions fx auxquot an1xquot391 amzxn392 alx a0 1 If a polynomial equation is of degree n then counting multiple roots multiplicities separately the equation has n roots zeros 2 If a bi is a zero of a polynomial equation b at 0 then the complex imaginary number a 7 bi is also a zero In other words complex imaginary zeros occur in conjugate pairs Page 14 Final Exam Review Exanqde19hvenfny2x sx118x 20x536x 45xi4y212x state the end behavior list all possible rational zeros find all of the zeros of f and state the multiplicity of each Describe the graph at each zero touch cross nicely cross and atten etc State the max turning points F ind f2 Final the remainder whenfis divided by x1 Degree Sketch the graph t bampampQamp Answer factors as xx 13x 3 x222x1 a x gt oofx gtoo x gtoofx gtoo b take out gcffirst then use RRT W i1 i 23 i 4i6i12i i c amp d x0 mult 1 crosses nicely x mult 3 crosses and attens x3 mult 1 crosses nicely x 2 mult 2 touches and bends back x 12 mult 1 crosses nicely e7 160 g32 h8 NM Page 15 Final Exam Review Example 20 Given the graph of f The graph of a polynomial lnction f x is shown below Use the graph to answer the following questions Is the degree of the polynomial lnction f x evenodd Is the leading coefficient positive negative The number of real zeros of f x is Label with letters A B on the graph all turning points of the polynomial f x Is the multiplicity k of the zero a0 odd even Is the multiplicity k of the zero b0 equal tol greater than 1 State the quotEnd behaviorquot of fx F ind f 4 Isfafunction S f 1 1 7 Find the value of the relation f 15 State the intervals where f is increasing decreasing constant In State the relative extrema Give answer in the form a minmax of 7 at H wavvsaveeege Answer a odd b positive c 4 eamp f X5 odd kl X2 k even kgtl Xl k odd kl X6 k odd k l g x gt oofx gt oo x gt oofx gt 00 h 1 1 yes 1 No does notpassHLT k 62 1 inc oo 4U 2 0U44 00 dec 4 2U0 44 m max of at 4 a min ofO at 2 a max of at 00 min of12 75 at 44 Page 16 Final Exam Review Example 21 Given fx 3x 15x 23 x 64 2x 7final degree zeros anal multiplicity of each describe the graph at each zero touch cross nicely or cross anal atten etc enal behavior max turning points 33983 Answer a 13 b amp c x1 mult 5 croses analflattens x 2 mult 3 crosses anf attens x 6 mult 4 touches anal turns back x 72 crosses nicely d x gt oofx gt 00 x gt oo fx gt oo e 12 Example 2239 For each function below final a enal behavior b intervals of incdecconstant c domain al range j equations of any asymptotes g x int h y int i if evenoalal or neither Sketch the graph Show transformation work i fx 3x12 12 1 gx32m5 iAnswer ax gt oofx gt oo x gtoofxoo axgtoofxgtoo xawfxgt5 b inc oo 1 alec 1 00 c 00 00 al 00 2 b alec oo 00 c 00 00 al 5 00 e none e x 1 19 none g 1 0 3 0 n 0 9 z neither y 5 g lnzl21 0 0260 3 n l h 0 1 i neither 39 5 111 y 4log2 x 3 iv fx x32 1 Answer a x gt oofx gtoo x gt239fx gt oo a x gt oofx gt 1 x gtoofx gt 1 i b alec 00 00 c 00 2 al 00 oo binc 00 3 alec 3 00 c 00 3 U 3 oo 1 e none j x2 g 0 4log2 30 1796 al 1 00 e none x 3 y 1 n 2 100 n 360 z neither 1 y 5 g 3 6 o 3 0 n 0 49 1 neither Page 17 Final Exam Review I V functions i de nition Given a relation in X and y we say y is a function of x if for each element x in the domain there in exactly one element value of y in the range ii Horizontal line test iii Vertical line test iv 1 v Function notation I The notation fx is read fofx or the value of the lnction f at x Vi Evaluate functions Find fsomething iffgiven by a equation iff given by a graph Difference quotient W for h at 0 Example 23 Find the difkrence quotient if 4 a fx 3x25x 1 b fxi3 x Page 18 Final Exam Review vii Combine functions sumdi rence mult and quot Combination of Functions X and Given two Jnctions f and g 1 Sum f gx fxgx 2 Difference f gx fx gx 3 Product fgx f x gx 4 Quotient x E g gx gx 0 Note For X and of f and g the domain of the combination must be common to both f and g domain of f 0 domain of g For we must also exclude where gx 0 viii Composition of functions Composition of Functions The composition of the lnction f with g is denoted by f o g and is de ned by the equation f 0 mo fgx fgx Meansfofthefunction g ofx The domain of the composite function f o g is the set of all x such that x is in the domain of g gx is in the domain of f Think beginningend or insideoutside ix Domain of combination or composition Page 19 Final Exam Review Example 24 If f x i1 and gx 3 evaluate each of the following Be sure to simplify your answer x x and to write the domain in interval notation a f gx b f gx C ngX e fogXx V Lines 139 Slope Definition The slope m of the line through the distinct points 9 y1and x2 y2 is changez39ny or rise or Q or yZ y1 or yl y2 where xZ x1 or xl xzi0 changez39nx run Ax xz x1 xl x2 Page 20 Final Exam Review 1391 equation Point Slope Form Slope Intercept Form The pointslope form of the equation of a non The slopeintercept form of the equation of a non vertical line with slope m that passes through vertical line with slope m and yintercept 0b is Xuyl s y y1mx x1 ymxb vertical line with an unde ned slope iv Horizontal y b v Vertical x a v1 Average rate of change If the graph of a lnction is not a straight line the average rate of change between any two points is the slope of the line containing the two points This line is called a secant line Let xl f x1 and x2 f x2 be distinct points on the graph of a lnction f The average rate of change from q to x2 is f 962 f 99 x2 x1 Example 24 Find the equation of a line through the point 1 3 and perpendicular to the line whose equation is 3x 2 y 5 0 Write the equation in slopeintercept form Page 21 Final Exam Review Example 27 Final the equation of the lone whose x int is 12 anal y int is 6 Put your answer in slope intercept form WI Quadratics The Standard Form of a Quadratic Function is f x ax h2 k a 7i 0 If a gt 0 then the parabola opens up If a lt 0 then the parabola opens down Graph is a parabola with vertex at h k Graph is symmetric to line x h The Vertex ofa parabola whose equation is in the form fx ax2 bx c is 2i f a a Example 28 Final the x int y int vertex line of symmetry anal graph of a fx 2x324 b fx 2x2 3x5 Answer xint 3 J10 3 0 Answer xint 52 0 1 0 y int 0 14 y int 0 5 vertex 3 4 vertex axis symmetry x 3 axis symmetry x Page 22 Final Exam Review V711 Equation of Circles distance and midpoint i distance formula Distance Formula The distance d between the points X1 yl and X2 yz I the rectangular coordinate system is d x2 x12 y2 y12 ii Midpoint The coordinates of the midpoint of a line segment with endpoints X1 yl and X2 yz are Jami MHZ 2 2 iii circles Definition of a Circle A circle is the set of all points in a plane that are equidistant same distance from a xed point called the center This xed distance from the center of the circle is called the radius Standard Form of the Equation of a Circle General From of the Equation of a Circle x h2y k2r2 x2y2DxEyFO with center h k and radius r D E and F are real numbers Example 29 Find the equation of the circle whose endpoints of the diameter is 2 5 and 6 9 Elami E fEQEEfE25 Example 30 Find the center radius domain and range of the circle Put the equation in standard form x2y2 6x4y 120 Answer center 3 2 radius r 5 Equation in standardform x 32 y 22 25 3 Domain 2 8 Range 7 3 Page 23 Final Exam Review IX Inverses The lnctions f and g are inverse functions if fgx x for every x in the domain ofg AND gfx x for every x in the domain of f The inverse of the function f is denoted by f 1 Read as finverse domain of f range of g range of f domain of g ff 1x x f 1fx x The graphs of f and g are symmetric to the line y X If the point a b is on the graph off then the point b a is on the graph of f 1 i Definition determine if 2 functions are inverses ii Graphs Example 31 a Sketch the graph of the inverse of f b What type of finction is f What is the equation the 7 What is 1 What is the equation of the asymptote on f 1 Answer a swtich coordinates of x amp y on all key points and redraw b exponential y 5 c logarithmetic x 5 iii nding equation of inverse Steps for finding the eg nation for the inverse of a function f 1 Replace x with y 2 Interchange x and y 3 Solve for y If y is not a lnction of x then f does not have an inverse 4 Replacey with f 1x 5 Check Does fg gf x Page 24 Final Exam Review Example 32 Find the inverse of y 210g3x 4 1 State the domain and range ofboth thefunction and the inverse e wriif oifif Example 33 Find the inverse of f x 2x x 3 4 State the domain and range of both the function and the inverse Example 34 Veri v the above functions are inverses by computing the compositions 2 14x 1 Answer foam 133 Z 214x13x 2 Z 28x3x 2 2 2 3m 4 314x 43x 2 312x 12x8 11 1413x 442x1 3x 48x4 11x f39 fx 32 32x1 23x 4 6x3 6x8 iv find f 1 given equation offor graph off Page 25 Final Exam Review X Applications Sales Tax cost of merchandisetax rate Commission dollars in salesrate Simple Interest I Prt Distance rate time alrt Geometry Common formulas forArea perimeter anal volume see chart on page 123 of text Compound Interest A P1 f Continuous Interest A Pe Projectile motion h 16t2 vet s0 where h inft t in sec v0 initial velocity s0 initial height 1 linear simple interest mixture alrt cost etc Example 35 Suppose you plan to borrow 6000 from 2 lenders to pay for your tuition next year One lender charges 10 simple interest and the other charges 4 simple interest How much did you borrow from each lender if you paid a total of 39183 in interest after 1 year Answer Equation 01x 0046000 x 39183 253050from the 10 lender and 346950from the 4 lenaler Example 36 Suppose you invested 5000 into 2 accounts a CD paying 6 simple interest and a stock paying 8 simple interest If you receive 70640 in interest after 2 years how much was invested in each account Answer a Letx amt in CD Then 006x2 5000790 082 7064 CD 2340 Stock 2660 Page 26 Final Exam Review Example 37 How many liters of 40 antifreeze should be added to 4 L of a 10 antifreeze solution to produce a 35 antifreeze solution Answer a Equation Letx amt of 40 antifreeze soln 040x 0104 035 4x Answer 20 L of 40 solution Example 38 How many liters of25 bleach should be added to a 10 bleach solution to produce 600 ml a 5 bleach solution Answer Equation Letx amt of25 bleach soln 0 025x 600 x010 005 600 Answer 400 ml of25 anal 200 ml of 0 Example 40 Two trains are 190mi apart and travel toward each other on the same road They meet in 2 hours One travels 4 mph faster than the other What is the average speed of each train Set up as an equation in one variable and solve Answer Drt table TOTAL DISTANCE SUAI Letx be the rate of the first train Equation 2x 2x4190 I train 455 mph 2quot train 495 mph Example 4 39 The cost of a monthly texting plan is 12 per month plus 01 0 per text a Final aformulafor the cost C as afunction ofthe number oftext messages x b How much will it cost you to sendreceive 50 text messages 2500 text messages c How many text messages would you need to sendreceive to make it more economical to switch over to the unlimited plan with a fixed rate of 65 per month iAnswer a C 1201x b 17 262 c gt 530 Page 27 Final Exam Review ii rational Example 42 A motorist travels 80 mi while driving in a bad rainstorm In sunny weather the motorist drives 20mph faster and covers 120 mi in the same amount of time Find the speed of the motorist in the rainstorm and in sunny weather Answer Drt table SAME TIME Letx be the rate in rainstorm 120 x 20 In rainstorm 40 mph Sunny Day 60 mph 80 Equation 7 x iii Quadratic Example 43 A model rocket is launched straight upwards from the top of a 100 balcony The initial velocity is 160 s The height of the rocket in feet ht at any time t in seconds is given by ht 16t2 160t 100 Find the following a Find the coordinates ofthe vertex of ht 16t2 160t 100 Show your work b 160 xcoordmate 7 2a 2 16 ycoordinate h 23 h5 1652 160 5 100 500 ANSWER 5 500 a b How long does it take the rocket to reach its maximum height Include units d How long does it take for the rocket to hit the ground Include units Show your work Hits ground when h 0 0 l6t2 160t100 i Solve with QE 160i1602 4 16 100 ANSWER 1059 sec Round answer to 2 dec1ma1 places Page 28 Final Exam Review Example 44 The width ofa rectangle is 5 in less than 3 times the length The area is 28 ml Find the dimensions Equation 28 l3l 5 3 Answer l4 in w7in iv exponential simple interest compound interest exp growthdecay half life etc Example 45 39 Suppose you invest 2000 in an account that pays 15 interest compounded quarterly a how much will you have in the account after 1 0 years b how much will you have in the account after 45 years c How long will it take the investment to double d How long will it take the investment to triple e How long will it take to reach 1000000 Answer a 872076 b 150969412 c 47years d 746years e 422years Example 46 Suppose you invest 2000 in an account that pays 15 interest compounded continuously a how much will you have in the account after 10years b how much will you have in the account after 45 years c How long will it take the investment to double d How long will it take the investment to triple e How long will it take to reach 1000000 Answer a 896338 b 170811753 c 462years d 732years e 4143 years Example 4739 Suppose the population of a town was 6500 in 1970 and 7000 in 1990 a Find a model for the population t years after 1950 b What will the population be in 2025 c When will the population double d How long until the population reaches 10000 Answer a A 6500e0 003705 b 7969people c187years d116years Page 29 Final Exam Review Example 48 Suppose the half life of the nuclear waste dumped into the ocean is 30 years a Find a model representing the amount of radioactive matter present at time tin years b What ofthe original amount will be left after 10years c Suppose you expect to live another 70 years What percent of the original amount will be left at the end ofyour lifetime When will you have 70 of the original amount left When will 60 of the original amount have decayed If it is considered safe to live in the area when there is only 15 of the original amount left how long until residents can move back to their home bee Answer model1140 e39039023105 b 7937 c 1985 d 154years e 397years j 82 years Make sure you can do all examples from the section 45 lecture notes and exam 4 review V711 Complex s i definition a bi where i si l and i2 1 ii simplifving complex s v operations on complex s vi complex zeros Example 49 Simplifv the following a 25 4 b 2 302 c d e reciprocal of 5i 1 Answer a 10 b 5421 c i d i357 Watch common mistakes You must simplify each expression to complex form rst and then multiply 4 9 6 NEVER square each term ALWAYS FOIL Page 30 Final Exam Review XI logarithms See exam 4 review sheet anal see 45 notes i properties ii alef conversion to exp form Example 50 convert to exponential form y 21n3x 41nx to log y 3 Lima quot1395g5 y335 397139 iii Change of base formula Example 51 Convert the following to ln log anal log 4 0 anal approximate as a decimal a 10g2 3 b 10g31n5 Answer a 3 1 g3 1 g43158469 b mans logans w 043317 ln2 10g2 10g42 ln3 10g3 10g43 iv Expanal expressions 2x3 J W7 V Example 52 Expand 1n Answer ln231nx121ny 71nw 451nz Page 31
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