M122 College Algebra Review for Final Exam

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1 M1 College Algebra Review for Final Eam Revised Fall 015 for College Algebra Beecher All answers should include our work (this could be a written eplanation of the result, a graph with the relevant feature pointed out or an algebraic solution). On the Final Eam, less work = fewer points!! On final NO WORK = NO POINTS Determine whether or not the relationship shown is a function. Eplain wh or wh not. 1) Januar Weight (lbs) Does the table define weight as a function of the da in Januar? Wh or wh not? ) Suppose a cost-benefit model is given b 1.5p C(p) =, where C is the cost in 0 - p thousands of dollars for removing p percent of a given pollutant. Find C(85) to the nearest dollar and interpret it. 3) Suppose the sales of a particular brand of appliance are modeled b the linear function S() = , where S() represents the number of sales in ear, with = 0 corresponding to 198. Find the number of sales in 199. ) The population of a small U.S. town can be modeled b P = -37t + 13,500, where t is the number of ears since Find and interpret all intercepts of the graph of this function. 5) A boat is moving awa from shore after 1 hour it is 13. km from shore, after hours it is.7 km from shore. What is the average rate of change of the distance from shore? Find the zero of f. ) f() = + 8 Solve the formula for the specified variable. 7) F = 9 C + 3 for C 5 8) A gas station sells 80 gallons of regular unleaded gasoline on a da when the charge $1.35 per gallon, whereas the sell 3917 gallons on a da that the charge $1.0 per gallon. Find a linear function that epresses gallons sold as a function of price. (Use the ordered pairs (1.35, 80) and (1.0, 3917) 9) An average score of 90 for 5 eams is needed for a final grade of A. Johnʹs first eam grades are 79, 89, 97, and 95. Determine the grade needed on the fifth eam to get an A in the course. Use the table to determine whether the data set represented is eactl linear, approimatel linear, or nonlinear. ) ) A shoe compan will make a new tpe of shoe. The fied cost for the production will be $,000. The variable cost will be $37 per pair of shoes. The shoes will sell for $3 for each pair. Find a cost function C() for the total cost of producing pairs of shoes and a revenue function R() for the total revenue generated. What is the profit if 00 pairs are sold? 1) At Allied Electronics, production has begun on the X-15 Computer Chip. The total revenue function is given b R() = and the total cost function is given b C() = + 1, where represents the number of boes of computer chips produced. Find P(). 13) Midtown Deliver Service delivers packages which cost $1.70 per package to deliver. The fied cost to run the deliver truck is $80 per da. If the compan charges $.70 per package, how man packages must be delivered dail to break even? 1

2 1) Find the market equilibrium point for the following suppl and demand functions and eplain what it means. D(p) = 790-1p; S(p) = p 15) Suppose that the sales of a particular brand of appliance satisf the relationship S() = , where S() represents the number of sales in ear, with = 0 corresponding to For what ears will sales be between 70 and 3? Determine the intervals on which the function is increasing, decreasing, and constant. 3) ) John owns a hot dog stand. He has found that his profit is given b the equation P = , where is the number of hot dogs sold. How man hot dogs must he sell to earn the most profit? Find the coordinates of the verte and determine whether the graph opens up or down. 17) = ( + 8) ) = -( - 3) - Scale the aes and sketch the graph listing the verte and all intercepts. 19) = Find the -intercepts. 0) = Use the quadratic formula to solve the equation and give the solution in simplest radical form (eact answers). 1) 7m + m + = 0 ) The demand equation for a certain product is P = , where is the number of units sold per week and P is the price in dollars at which one is sold. The weekl revenue R is given b R = P. What number of units sold produces a weekl revenue of $33,000? Write the equation of the graph after the indicated transformation(s). ) The graph of = is shifted units to the left. This graph is then verticall stretched b a factor of 5 and reflected across the -ais. Finall, the graph is shifted 8 units downward. Solve the equation algebraicall. 5) - 3 = - 3 ) A stud in a small town showed that the percent of residents who have college degrees can be modeled b the function P = , where is the number of ears since According to the model, what percent of residents had a college degree in 1995? Round to the nearest whole number. Graph the function. f() = 7) - for > 0-1 for 0

3 8) Given the graph of f() = 3, draw the graph of g() = -3( - 3) Answer the question ) How can the graph of = be obtained from the graph of = 3 30) How can the graph of f() = be obtained from the graph of =? 3) For f() = - 3 and g() = + 3 f Find () and its domain. g 35) The cost of manufacturing clocks is given b C() = Also, it is known that in t hours the number of clocks that can be produced is given b = 8t, where 1 t 1. Epress C as a function of t. Determine whether or not the function is one -to-one and eplain wh or wh not. 3) This chart shows the number of hits for five Little League baseball teams. Team Hits Hawks 8 Lions 1 Eagles 55 Bears 55 Dolphins 0 37) f() = + Find the requested value. 31) A married coupleʹs federal income ta in 00 could be determined using the following function where is the taable income up to $9,50: f() = 0., if (-00), if 00 < (-500), if 500 < 950 What would the couples income ta be if their taable income was $35,000. For problems 7-51 find the new function and the domain. 1 3) f() = -7 and g() = - 8 Find (f g)() and its domain. Find the inverse of the function. 38) f() = 3-3 Use algebraic and graphical or numerical methods to solve the inequalit. 39) + 3 0) The cost C of producing t units is given b C(t) = t + 9t, and the revenue R generated from selling t units is given b R(t) = 5t + t. For what values of t will there be a profit? 1) A coin is tossed upward from a balcon ft high (ho) with an initial velocit (vo) of 1 ft/sec, according to the formula h(t) = -1t + vot + ho, where t is time in seconds. During what interval of time will the coin be at a height of at least 70 ft? 7 33) Given f() = - and g() = 5 7, Find f (g()) and its domain. 3

4 ) The number of books in a small librar increases according to the function B = 300e0.05t, where t is measured in ears after opening. a) How man books did the librar have when the opened? b) How man books will the librar have ears after opening? Graph the function showing all asmptotes and intercepts. 3) f() = (+1) - ) f() = log ( + ) 5) The logarithmic function f() = ln models the number of visitors (in millions) to U.S. museums from 190 to 1990, where is the number of ears since Use this function to estimate the number of visitors in the ear Provide an appropriate response. 55) For the function: f() = log? a) What is the domain of the function? b) What is the range of the function? Solve the equation algebraicall. 5) 9 = 89 (Round to two decimal places.) Provide an appropriate response. 5) Eplain how the graph of = 1 3 (-) + 5 can 57) 50e-0.7 = 15(Round to three decimal places.) be obtained from the graph of = 3. 58) log = log 5 + log ( - ) ) Eplain how the graph of = -5log (+3) - 1 can be obtained from the graph of =log. Write the logarithmic equation in eponential form. 7) log 1 = -3 8) ln = - Write the eponential equation in logarithmic form. 9) 3 = 1 50) e = Find the value of the logarithm without using a calculator. 51) log 5 5 Use a change of base formula to evaluate the given logarithm. Approimate to three decimal places. 5) log (0.5) Find the inverse of the function. 53) f() = ) log (3 - ) - log ( - 5) = 0) 5 ln( - ) = Rewrite the epression as the sum and/or difference of logarithms, without using eponents. Simplif if possible. 1) log Rewrite as a single logarithm. ) (log t t - log t s) + 3 log t u 3) Assume the cost of a car is $1,000. With continuous compounding in effect, the cost of the car will increase according to the equation C = 1,000ert, where r is the annual inflation rate and t is the number of ears. Find the number of ears it would take to double the cost of the car at an annual inflation rate of.1%. Round the answer to the nearest hundredth.

5 ) In the formula A(t) = A 0 ekt, A is the amount of radioactive material remaining from an initial amount A0 at a given time t, and k is a negative constant determined b the nature of the material. A certain radioactive isotope decas at a rate of 0.175% annuall. Determine the half-life of this isotope, to the nearest ear. 5) A bacteria culture starts with,000 bacteria, and the number doubles ever 0 minutes. a) Find the eponential growth function that models the number of bacteria after t minutes. b) Find the number of bacteria after one hour. c) After how man minutes will there be 50,000 bacteria? ) A radioactive substance has a half-life of das. Suppose a sample of this substance has a mass of 300 mg. a) Find an eponential deca function that models the amount of the sample remaining after t das. Determine a. The interval where the graph is increasing b. The interval where the graph is decreasing c. The intercepts d. The local maima e. The local minima 71) = (-0., -.) (-1.7, -38.8) (0.1, -.1 Determine the minimum degree of the polnomial and state whether the leading coefficient is positive or negative. 7) b) Find the mass remaining after one ear. c) How long will it take for the sample to deca to a mass of 00 mg? 7) Find the amount of mone in an account after 9 ears if $3500 is deposited at % annual interest compounded monthl. 8) How long would it take $9000 to grow to $5,000 at 7% compounded continuousl? Round our answer to the nearest tenth of a ear. 9) Find the annual interest rate if $3500 is deposited in an account that compounds semi-annuall and after 1.5 ears the future value is $000 70) The number of students infected with the flu on a college campus after t das is modeled b 30 the function P(t) =. What is the e-0.3t maimum number of infected students possible? 5

6 73) Suppose that the population of a certain cit can be approimated b: P() = where is time in ears since 190. Using the graph of P() below,, estimate during what time period the population of the cit was increasing (.5, 087) Use the graph of the polnomial function f() to solve f() = 0. 78) ) S() = , 0 is an approimation to the number of salmon swimming upstream to spawn, where represents the water temperature in degrees Celsius. Using the graph below, find the temperature that produces the maimum number of salmon. Number of salmon (1, 59) Temperature in Celsius ) If the price for a product is given b p = 900 -, where is the number of units sold, then the revenue is given b R = p = How man units must be sold to give zero revenue? 80) Suppose a business can sell gadgets for p = dollars apiece, and it costs the business c() = dollars to produce the gadgets. Determine the production level required to maimize profit. 81) The Cool Compan determines that the suppl function for its basic air conditioning unit is S(p) = p3 and that its demand function is D(p) = 50-0.p, where p is the price. Determine the price for which the suppl equals the demand. Graph the function showing all asmptotes and intercepts. 8) f() = Solve the polnomial equation b using the root method. 75) = 0 Solve the polnomial equation b factoring. 7) 3-0 = 0 77) = 0 For the given rational function f, find all values of for which f() has the indicated value. 83) f() = - - ; f() = 3 Use analtical methods to solve the equation. 0 8) - = 1 + +

7 85) In the following formula, is the minimum number of hours of studing required to attain 0.38 a test score of : =. How man hours of stud are needed to score 90? 8) The percent of concentration of a certain drug in the bloodstream hr after the drug is administered is given b K() = + 5. At what time is the concentration a maimum? 87) Suppose that a cost-benefit model is given b f() = where f() is the cost in thousands of dollars of removing percent of a given pollutant. What is the vertical asmptote of the graph of this function? What does this suggest about the possibilit of removing all of the pollutant? Eplain our reasoning. Solve the inequalit. 88) ( + 9)( + 5)( - 8) > 0 89) > 0 91) An open-top bo is to be made b cutting small identical squares from each corner of a 1-b-1-in. sheet of tin and bending up the sides. If each corner square is inches on a side, the volume of the bo (in in.3) is given b: V() = Using the sketch of the graph of V() below, estimate what values of result in a bo with a volume greater than in (0.5, ) (, ) ) The monthl sales volume (in thousands of dollars) is related to monthl advertising ependitures (in thousands of dollars) according to the equation = 0 +. Spending how much mone on advertising will result in sales of at least $75,000 per month? Use the graph of f to solve the inequalit. 90) f() <

8 Answer Ke Testname: M1 FINAL REVIEW F15TST 1) Yes. For each input value (da) there is eactl one output value (weight) ) $8500; It will cost $8500 to remove 85% of the pollutant. 3) 500 ) (0, 13500) : The population of the town was 13,500 in (3.8, 0): If the population continues to decrease at the same rate, the population will be zero after 35 ears (30). 5).5 km/hr ) - 7) C = 5 (F - 3) 9 8) G(p) = -18,00 + 9,01 9) 90 ) Nonlinear 11) C() =, ; R() = 3; $15,00 1) P() = ) 1 packages 1) When the price is $.7, the amount demanded equals the amount supplied which is ) Between 199 and ) 33 hot dogs 17) (-8, -3); concave up 18) (3, -) Concave down 19) verte: (-1.5, -139,8375) intercepts: (3., 0) (-.5, 0) (0, -175) ) (, 0) and (-3, 0) 1) -5 ± 11 7 ) 00 or 1,500 3) Increasing on (-, 0) and (3, 5); Decreasing on (1, 3); Constant on (-5, -) ) = -5( + ) - 8 5) 3 ) 1% 8

9 Answer Ke Testname: M1 FINAL REVIEW F15TST 7) ) - 9) Reflect it across the -ais. Shrink it verticall b a factor of 0.. Shift it verticall 5 upward. 30) Reflect it across the -ais. Shrink it verticall b a factor of 0.5. Shift it verticall units downward. 31) $,50 3) 33) -7 ; domain: [7,8) (8, ) ; domain: { 0, 5 } 3) -3-3 ; { +3 } 35) C(t) = t - t 3) No, the output value of 55 is repeated with different input values. 37) No, its graph does not pass the horizontal line test. 38) f-1() = ) ) t > 8 1) 0 sec t sec ) 300; 80 9

10 Answer Ke Testname: M1 FINAL REVIEW F15TST 3) ) ) The graph is reflected over the -ais, shifted units to the right and 5 units up. ) The graph is shifted 3 units to the left, stretched b a factor of 5, reflected across the -ais and shifted 1 unit down. 7) -3 = 1 8) = e- 9) log 1 = 3 50) ln = 51) 1 5) ) f-1() = -log ( - 3) 5) 8. million 55) a) (0, ) b) (-, ) 5) ) ) {0} 59) = 0) e ) 8 log + 9 log - log 3 tu3 ) log t s 3) ) 39 r

11 Answer Ke Testname: M1 FINAL REVIEW F15TST 5) a) n(t) = 000e t b) 8,87 bacteria c) 93 minutes ) a) m(t) = 300e t b) 9 mg. c) 8 das 7) $ ) 3.0 ears 9) 9.1% 70) 30 71) a. (-7, -.) (-., ) b. (-, -.7) (-.,.) c. (0, -5), (-.5, 0), (1.5, 0) d. -. e , -.1 7) Minimum degree is 5. Leading coefficient is negative. 73) Between 190 and ) 1 C 75) 3 7) 0,, - 77) -, 1, -1 78) -3, -1, 1, 3 79) 0, 30 80) 11,50 gadgets 81) $1.8 8) ) 5 8) 1, -1 85) 3. hr 8) 5 hr 87) Answers will var. Possible answer: The vertical asmptote is = 0. This suggests that as approaches 0 percent, the cost becomes infinitel large. It is therefore not possible to remove all of the pollutant. 88) -9 < < -5 or > 8 89) (-,-1) (1, ) 90) < - or - < < 91) 0.5 in. in. 9) At least $30,000 11