the exam is next Thursday the 20th from 8:00 am to 10:00 am
QCC MA-119 Final Review Questions. HSI QCC MA-119 Spring 2021 Final Review Questions Page 1 of 12 Graphs and Functions 1. Use the graph of the function, ? = ?(?), to answer the question. a) Estimate ?(1) Answer: ?(1) ≈ 5 (find a point on the graph that has ?-coordinate = 1). b) Estimate the solution(s) to ?(?) = 9 Answer: ? ≈ 1.75 (find all points on the graph that have ?-coordinate=9; in this case, there's just one such point). c) Estimate the coordinates of all intercepts. Answer: ?-intercept is approximately(−8.5,0). (All ?-intercepts have ?-coordinate= 0.) There’s only one ?-intercept in this graph; ?-intercept is approximately(0,3). (All ?- intercepts have ?-coordinate= 0. There’s only one y-intercept in this graph.) d) Estimate the domain and range of the function Answer: Domain: (−∞, ∞). (The arrows indicate that the graph extends, without breaks, indefinitely to the left and indefinitely to the right.) Range: (−∞, ∞). (The arrows indicate the graph extends, without breaks, indefinitely up and indefinitely down.) e) Estimate the minimum or maximum ?-value Answer: no minimum ?-value and no maximum ?-value. (The arrows indicate that the graph extends indefinitely up and indefinitely down.) f) Estimate the ?-value at which the minimum or maximum is achieved? Answer: no such ?-values because of the answer to part e). g) Estimate the values of ? for which ? < 0="" answer:="" (−∞,="" 8.5)="" for="" these="" -values="" the="" graph="" is="" below="" the="" -axis="" so="" for="" these="" -="" values,=""><0. see="" how="" this="" answer="" relates="" to="" part="" c)="" h)="" when="" 1=""><>< 2,="" is="" the="" function="" increasing="" or="" decreasing?="" answer:="" increasing.="" move="" to="" the="" right="" within="" the="" stated="" interval="" of="" -values.="" the="" graph="" is="" going="" upwards="" for="" those="" -values.="" 2.="" use="" the="" graph="" of="" the="" function,="" =="" (?),="" to="" answer="" the="" question.="" a)="" estimate="" (−11)="" answer:="" =="" (−11)="" ≈="" 7="" b)="" estimate="" the="" solution(s)="" to="" (?)="−4" answer:="" ≈="" −13="" and="" ≈="" −6="" c)="" estimate="" the="" coordinates="" of="" all="" intercepts.="" hsi="" qcc="" ma-119="" spring="" 2021="" final="" review="" questions="" page="" 2="" of="" 12="" answer:="" -intercepts:="" (−14,0),="" (0,0),="" (9,0);="" -intercept:="" (0,0)="" d)="" estimate="" the="" domain="" and="" range="" of="" the="" function="" answer:="" domain="" (−∞,="" ∞)="" range="" [−7.75,="" ∞)="" e)="" estimate="" the="" minimum="" or="" maximum="" -value="" answer:="" minimum="" -value="" ≈="" −7.75;="" no="" maximum="" -value="" f)="" estimate="" the="" x-value="" at="" which="" the="" minimum="" or="" maximum="" is="" achieved?="" answer:="" for="" minimum:="" ≈="" −10="" (the="" -coordinate="" of="" the="" point="" with="" the="" minimum="" -="" value);="" for="" maximum:="" no="" such="" -value="" g)="" estimate="" the="" values="" of="" for="" which="">< 0="" answer:="" (−14,0)="" ∪="" (0,9)="" (the="" symbol="" ∪="" stands="" for="" union="" or="" “together="" with”.="" the="" endpoints="" are="" not="" included="" because="" there="" we="" have="" =="" 0="" not="">< 0="" h)="" when="" 4=""><>< 5, is the function increasing or decreasing? answer: decreasing 3. compute ?(4)−?(1) 4−1 for the given function. a) ?(?) = ?2 − 3? + 1 answer: 2 b) ?(?) = √? answer: 1 3 linear functions 4. suppose that a manufacturer wishes to use a linear function to model how the demand for its product affects the product’s price. the unit price function has the form ?(?) = ?? + ? where ? is the number of units of the product in demand. suppose that when 2400 units are demanded then the unit price is $250 and when 3000 units are demanded then the unit price is $300. number of units ? price per unit ? 2400 250 3000 300 a) determine the equation of the function ?(?) = ?? + ? b) according to this model, what is the unit price ?(?) if the demand is ? = 3600? hsi qcc ma-119 spring 2021 final review questions page 3 of 12 answer: a) ?(?) = 1 12 ? + 50 b) $350 5. a linear function ? passes through (−1,5) and (2, −4). find the function value ?(−2). answer: ?(−2) = 8 6. markos currently has 200 songs in his music collection. every month, he adds 15 new songs. write a formula for the number of songs, ?, in his collection as a function of time, ?, the number of months. how many songs will he own in a year? answer: ?(?) = 15? + 200; 380 songs 7. a town’s population has been growing linearly. in 2004 the population was 6,200. by 2009 the population had grown to 8,100. assume this trend continues. a) write a linear equation to model the population growth. answer: ?(?) = 380? + 6200, where ? is the number of years since 2004. b) predict the population in 2013. answer: 9,620 c) identify the year in which the population will reach 15,000. answer: sometime around the year 2027 quadratic equations (factoring and quadratic formula) 8. factor completely. 40?4 + 22?3 − 6?2 answer: 2?2(4? + 3)(5? − 1) 9. solve the quadratic equations a) 4?2 + ? = 3 answer: ? = −1 or ? = − 3 4 b) (? + 1)(? − 3) = 2 answer: ? = 1 ± √6 c) ?2 + 7? + 12 = 0 answer: ? = −3 or ? = −4 d) 2?2 + 1 = ? answer: ? = 1±?√? 4 e) 2(? + 3)2 + 8 = 0 answer: ? = −3 ± 2? f) 1 3 ?2 + ? = − 1 2 answer: ? = −3±√3 2 g) 4?2 − ? − 3 = 0 answer: ? = 1 or ? = − 3 4 h) (? − 1 3 ) 2 − 4 9 = 0 answer: ? = 1 or ? = − 1 3 hsi qcc ma-119 spring 2021 final review questions page 4 of 12 quadratic functions 10. graph the following quadratic functions. a) ?(?) = −?2 − 6? − 5 b) ℎ(?) = −(? − 3)(? + 2) c) ?(?) = 2?2 − 3? − 2 d) ?(?) = −2?2 − 4? + 5 e) ?(?) = (2? − 1)(? + 3) f) ?(?) = 2?2 + 8 before graphing the parabolas, state which open upwards and which open downwards. for each parabola include a) the coordinates of all intercepts b) the coordinates of the vertex c) the equation and graph of the axis of symmetry d) the domain and range in interval notation e) the coordinates of an additional point on the graph f) the maximum or minimum range value g) the domain value at which the max or min is reached h) ?-interval(s) at which the function is negative i) ?-interval(s) at which the function is positive application of quadratic equations/functions 11. the length of a rectangle exceeds twice its width by 3 inches. if the area is 10 square inches, find the rectangle's dimensions. round to the nearest tenth of an inch. answer: approximately 1.6 in by 6.2 in. hsi qcc ma-119 spring 2021 final review questions page 5 of 12 12. the hypotenuse of a right triangle is 6 feet long. one leg is 2 feet shorter than the other. find the lengths of the legs. round to the nearest tenth of a foot. answer: approximately 5.1 ft and 3.1 ft. 13. a retailer estimates that, by charging x dollars each for a particular phone case, she can sell 40 − ? units each week. the function ?(?) = ?(40 − ?), models the weekly revenue, ?(?), received when the selling price is ?. a) interpret (0, ?(0)) and (1, ?(1)) in context. answer: when the charge is $0, the revenue is $0; when the charge is $1, the revenue is $39 b) find the selling price that will give the maximum revenue, and then find the amount of the maximum revenue. answer: the maximum revenue is $400 when the charge is $20 c) for which selling price(s) will the weekly revenue equal $300? answer: $10 and $30 14. a ball is thrown vertically upward, at the rate of 120 ft/sec, from a rooftop with a height of 50 feet. the quadratic function ℎ(?) = −16?2 + 120? + 50 models the height of the ball, from the ground, at time ?. a) interpret (0, ℎ(0)) and (1, ℎ(2)) in context. answer: 0 seconds after the ball is thrown the ball is 50 feet from the ground; 1 second after the ball is thrown the ball is 154 feet from the ground. b) find how long it will take the ball to reach its maximum height, and then find the maximum height. answer: the maximum height of 275 feet is reached in 3.75 seconds c) at which time(s) does the ball reach 200 feet above ground? answer: 1.58 and 5.92 seconds d) when does the ball hit the ground? answer: after 7.896 seconds rational and radical functions 15. find the domain of the following functions. a) ?(?) = ? ?−3 − 2 answer: (−∞, 3) ∪ (3, ∞) hsi qcc ma-119 spring 2021 final review questions page 6 of 12 b) ?(?) = 1 − √3? − 6 answer: [2, ∞) 16. given functions ?(?) = √? − 3, find: a) ?(7) − 2 answer: 0 b) the domain of the function ? which is define as ?(?) = ?(?) − 5,="" is="" the="" function="" increasing="" or="" decreasing?="" answer:="" decreasing="" 3.="" compute="" (4)−?(1)="" 4−1="" for="" the="" given="" function.="" a)="" (?)="?2" −="" 3?="" +="" 1="" answer:="" 2="" b)="" (?)="√?" answer:="" 1="" 3="" linear="" functions="" 4.="" suppose="" that="" a="" manufacturer="" wishes="" to="" use="" a="" linear="" function="" to="" model="" how="" the="" demand="" for="" its="" product="" affects="" the="" product’s="" price.="" the="" unit="" price="" function="" has="" the="" form="" (?)="??" +="" where="" is="" the="" number="" of="" units="" of="" the="" product="" in="" demand.="" suppose="" that="" when="" 2400="" units="" are="" demanded="" then="" the="" unit="" price="" is="" $250="" and="" when="" 3000="" units="" are="" demanded="" then="" the="" unit="" price="" is="" $300.="" number="" of="" units="" price="" per="" unit="" 2400="" 250="" 3000="" 300="" a)="" determine="" the="" equation="" of="" the="" function="" (?)="??" +="" b)="" according="" to="" this="" model,="" what="" is="" the="" unit="" price="" (?)="" if="" the="" demand="" is="" =="" 3600?="" hsi="" qcc="" ma-119="" spring="" 2021="" final="" review="" questions="" page="" 3="" of="" 12="" answer:="" a)="" (?)="1" 12="" +="" 50="" b)="" $350="" 5.="" a="" linear="" function="" passes="" through="" (−1,5)="" and="" (2,="" −4).="" find="" the="" function="" value="" (−2).="" answer:="" (−2)="8" 6.="" markos="" currently="" has="" 200="" songs="" in="" his="" music="" collection.="" every="" month,="" he="" adds="" 15="" new="" songs.="" write="" a="" formula="" for="" the="" number="" of="" songs,="" ,="" in="" his="" collection="" as="" a="" function="" of="" time,="" ,="" the="" number="" of="" months.="" how="" many="" songs="" will="" he="" own="" in="" a="" year?="" answer:="" (?)="15?" +="" 200;="" 380="" songs="" 7.="" a="" town’s="" population="" has="" been="" growing="" linearly.="" in="" 2004="" the="" population="" was="" 6,200.="" by="" 2009="" the="" population="" had="" grown="" to="" 8,100.="" assume="" this="" trend="" continues.="" a)="" write="" a="" linear="" equation="" to="" model="" the="" population="" growth.="" answer:="" (?)="380?" +="" 6200,="" where="" is="" the="" number="" of="" years="" since="" 2004.="" b)="" predict="" the="" population="" in="" 2013.="" answer:="" 9,620="" c)="" identify="" the="" year="" in="" which="" the="" population="" will="" reach="" 15,000.="" answer:="" sometime="" around="" the="" year="" 2027="" quadratic="" equations="" (factoring="" and="" quadratic="" formula)="" 8.="" factor="" completely.="" 40?4="" +="" 22?3="" −="" 6?2="" answer:="" 2?2(4?="" +="" 3)(5?="" −="" 1)="" 9.="" solve="" the="" quadratic="" equations="" a)="" 4?2="" +="" =="" 3="" answer:="" =="" −1="" or="" =="" −="" 3="" 4="" b)="" (?="" +="" 1)(?="" −="" 3)="2" answer:="" =="" 1="" ±="" √6="" c)="" 2="" +="" 7?="" +="" 12="0" answer:="" =="" −3="" or="" =="" −4="" d)="" 2?2="" +="" 1="?" answer:="" =="" 1±?√?="" 4="" e)="" 2(?="" +="" 3)2="" +="" 8="0" answer:="" =="" −3="" ±="" 2?="" f)="" 1="" 3="" 2="" +="" =="" −="" 1="" 2="" answer:="" =="" −3±√3="" 2="" g)="" 4?2="" −="" −="" 3="0" answer:="" =="" 1="" or="" =="" −="" 3="" 4="" h)="" (?="" −="" 1="" 3="" )="" 2="" −="" 4="" 9="0" answer:="" =="" 1="" or="" =="" −="" 1="" 3="" hsi="" qcc="" ma-119="" spring="" 2021="" final="" review="" questions="" page="" 4="" of="" 12="" quadratic="" functions="" 10.="" graph="" the="" following="" quadratic="" functions.="" a)="" (?)="−?2" −="" 6?="" −="" 5="" b)="" ℎ(?)="−(?" −="" 3)(?="" +="" 2)="" c)="" (?)="2?2" −="" 3?="" −="" 2="" d)="" (?)="−2?2" −="" 4?="" +="" 5="" e)="" (?)="(2?" −="" 1)(?="" +="" 3)="" f)="" (?)="2?2" +="" 8="" before="" graphing="" the="" parabolas,="" state="" which="" open="" upwards="" and="" which="" open="" downwards.="" for="" each="" parabola="" include="" a)="" the="" coordinates="" of="" all="" intercepts="" b)="" the="" coordinates="" of="" the="" vertex="" c)="" the="" equation="" and="" graph="" of="" the="" axis="" of="" symmetry="" d)="" the="" domain="" and="" range="" in="" interval="" notation="" e)="" the="" coordinates="" of="" an="" additional="" point="" on="" the="" graph="" f)="" the="" maximum="" or="" minimum="" range="" value="" g)="" the="" domain="" value="" at="" which="" the="" max="" or="" min="" is="" reached="" h)="" -interval(s)="" at="" which="" the="" function="" is="" negative="" i)="" -interval(s)="" at="" which="" the="" function="" is="" positive="" application="" of="" quadratic="" equations/functions="" 11.="" the="" length="" of="" a="" rectangle="" exceeds="" twice="" its="" width="" by="" 3="" inches.="" if="" the="" area="" is="" 10="" square="" inches,="" find="" the="" rectangle's="" dimensions.="" round="" to="" the="" nearest="" tenth="" of="" an="" inch.="" answer:="" approximately="" 1.6="" in="" by="" 6.2="" in.="" hsi="" qcc="" ma-119="" spring="" 2021="" final="" review="" questions="" page="" 5="" of="" 12="" 12.="" the="" hypotenuse="" of="" a="" right="" triangle="" is="" 6="" feet="" long.="" one="" leg="" is="" 2="" feet="" shorter="" than="" the="" other.="" find="" the="" lengths="" of="" the="" legs.="" round="" to="" the="" nearest="" tenth="" of="" a="" foot.="" answer:="" approximately="" 5.1="" ft="" and="" 3.1="" ft.="" 13.="" a="" retailer="" estimates="" that,="" by="" charging="" x="" dollars="" each="" for="" a="" particular="" phone="" case,="" she="" can="" sell="" 40="" −="" units="" each="" week.="" the="" function="" (?)="?(40" −="" ),="" models="" the="" weekly="" revenue,="" (?),="" received="" when="" the="" selling="" price="" is="" .="" a)="" interpret="" (0,="" (0))="" and="" (1,="" (1))="" in="" context.="" answer:="" when="" the="" charge="" is="" $0,="" the="" revenue="" is="" $0;="" when="" the="" charge="" is="" $1,="" the="" revenue="" is="" $39="" b)="" find="" the="" selling="" price="" that="" will="" give="" the="" maximum="" revenue,="" and="" then="" find="" the="" amount="" of="" the="" maximum="" revenue.="" answer:="" the="" maximum="" revenue="" is="" $400="" when="" the="" charge="" is="" $20="" c)="" for="" which="" selling="" price(s)="" will="" the="" weekly="" revenue="" equal="" $300?="" answer:="" $10="" and="" $30="" 14.="" a="" ball="" is="" thrown="" vertically="" upward,="" at="" the="" rate="" of="" 120="" ft/sec,="" from="" a="" rooftop="" with="" a="" height="" of="" 50="" feet.="" the="" quadratic="" function="" ℎ(?)="−16?2" +="" 120?="" +="" 50="" models="" the="" height="" of="" the="" ball,="" from="" the="" ground,="" at="" time="" .="" a)="" interpret="" (0,="" ℎ(0))="" and="" (1,="" ℎ(2))="" in="" context.="" answer:="" 0="" seconds="" after="" the="" ball="" is="" thrown="" the="" ball="" is="" 50="" feet="" from="" the="" ground;="" 1="" second="" after="" the="" ball="" is="" thrown="" the="" ball="" is="" 154="" feet="" from="" the="" ground.="" b)="" find="" how="" long="" it="" will="" take="" the="" ball="" to="" reach="" its="" maximum="" height,="" and="" then="" find="" the="" maximum="" height.="" answer:="" the="" maximum="" height="" of="" 275="" feet="" is="" reached="" in="" 3.75="" seconds="" c)="" at="" which="" time(s)="" does="" the="" ball="" reach="" 200="" feet="" above="" ground?="" answer:="" 1.58="" and="" 5.92="" seconds="" d)="" when="" does="" the="" ball="" hit="" the="" ground?="" answer:="" after="" 7.896="" seconds="" rational="" and="" radical="" functions="" 15.="" find="" the="" domain="" of="" the="" following="" functions.="" a)="" (?)="?" −3="" −="" 2="" answer:="" (−∞,="" 3)="" ∪="" (3,="" ∞)="" hsi="" qcc="" ma-119="" spring="" 2021="" final="" review="" questions="" page="" 6="" of="" 12="" b)="" (?)="1" −="" √3?="" −="" 6="" answer:="" [2,="" ∞)="" 16.="" given="" functions="" (?)="√?" −="" 3,="" find:="" a)="" (7)="" −="" 2="" answer:="" 0="" b)="" the="" domain="" of="" the="" function="" which="" is="" define="" as="" (?)="?(?)">