Pre-Calculus Midterm Eam REVIEW Januar 0 Name: Date: Teacher: Period: Your midterm eamination will consist of: 0 multiple-choice questions (including true/false & matching) these will be completed on the Scantron. short answer questions these will be completed in the test booklet. Show our work. essa question in which ou will be asked to eplain a concept or describe our work. The eam will cover material from Chapters,, and and section 7.6. NOTE: School polic mandates a penalt for cheating on an eam to be a grade of ZERO for that eam. The term cheating includes "intent to cheat." NO CELL PHONES. All cell phones must be kept out of sight. If a cell phone is seen during an eam, ou will receive a grade of ZERO. All calculators ma be checked for inclusion of etraneous material. No papers should be placed in calculators. No information should be written on the front/back of calculators. The program portion of the graphing calculator will be checked. An information entered there can be considered intent of cheating. Before the eamination, clear our calculator of an formulas, notes or an such items, which could be perceived as "useful" or providing unfair advantage. The best solution is to RESET and clear the memor completel. The following pages provide a comprehensive review of the materials to be studied for this eam. We will take a few das of class time to review for this eam. Please feel free to stop in on our own time for further assistance. Good Luck! Mr. Dominguez, Ms. Eisen, Mrs. Pogach, Mr. Hman and Mr. Stella
PRE-CALCULUS JANUARY 0 Review for MIDTERM EXAM. Match the graph with the correct function. a) f() = ( ) + b) f() = ( + ) + c) f() = + d) f() = ( + ) + 6. The verte of the graph of = ( - ) + is: a) (, ) b) (, 7) c) (, ). Determine whether or not each equation defines as a function of. 7. If = + 0, find the ais of smmetr and the coordinates of the verte. a. = b. =. Determine whether each function is even, odd, or neither. a) f() = b) f() = 8 8. Graph the following piecewise function.., f() =,. Evaluate h(7) if h() =, 7, 7 9. Find the domain (f g)() if f() = and g() =. Find the equation of the ais of smmetr of the graph of: = 8 7 0. Between what two successive integers does a real zero of P() = lie? Livingston High School Page
PRE-CALCULUS JANUARY 0 Review for MIDTERM EXAM. Find the domain of f() = 6. State the possible rational roots for: + 6 = 0. State each root and its multiplicit: ( 9)( + )( ) = 0 7. On a certain route, a train line carries 7000 passengers per month each paing $90. A market surve indicates that for each $ decrease in the ticket price, the airline will gain 60 passengers. Epress the monthl revenue for the route, R, as a function of the ticket price,.. Solve: ( ) ( + ) > 0 8. Find the remainder using snthetic division: ( ) ( ). Find the remainder when: + + 7 is divided b ( ) 9. Solve: + + 6 = 0. If i is a root of the equation = 0, find the other roots. 0. A rectangular dog park is to be fenced off into two adjacent rectangular dog runs, one for small dogs and one for big dogs. 600 feet of fencing is to be used. Find the dimensions that maimized the enclosed area and find the maimum area. Livingston High School Page
PRE-CALCULUS JANUARY 0 Review for MIDTERM EXAM. Solve completel: = 0 6. Graph & discuss multiplicit of the roots: a) = ( )( + )( ) b) c) ( ) d) = ( ) e) = ( + ) ( + ). Find the horizontal asmptote of the graph of: 9 f ( ) 7. Sketch: = ( + ). Find the vertical asmptotes, horizontal asmptotes, and the -intercepts of the graph of: f ( ) 8. Graph the function given b the equation f() = and its inverse f - on the same coordinate aes.. The zero(s) of ( ) 6 f is (are): 9. Sketch: = ( ). Given that the polnomial equation 8 + 7 0 = 0 has as a root, find the other two roots. 0. Graph the function: f() = + Livingston High School Page
PRE-CALCULUS JANUARY 0 Review for MIDTERM EXAM. Let f() = + and g() = + Find: a) (f + g)() b) (f g)() State in simplest form. 6. Use the rational root theorem to solve: 6 0. Let f() = 7 and g() = + Find: a) (f g)() b) (g f)() State in simplest form.. Find the maimum or minimum value of the function f() = +. State whether this value is a maimum or minimum value. Match each inequalit in Column with the graph of its solution set in Column. Column Column 7. ( )( + ) 0 8. ( + )( ) > 0 9. ( )( + ) > 0 0. ( + )( ) < 0. ( + )( ) 0. ( )( + ) < 0 A. (-,-) U (, ) B. (-,) C. (-,) D. (-,-) U (, ) E. [-,] F. (-,-] U [, ). Find P( ) if P() = + 7 + + + 9. Divide. 8 9. Simplif: a) i 00 b) 6 c) d) ( i) + (6 + i) e) ( i) (6 + i) f) ( i) (6 + i) g) i 6 i. Which one of the following equations has two imaginar conjugate roots? a) + = 0 b) = 0 c) + + = 0 d) + = 0 e) = 0. True or False: The rational function eactl zero. ( 6)( 7) ( ) 6 f has Livingston High School Page
PRE-CALCULUS JANUARY 0 Review for MIDTERM EXAM 6. True or False: A fourth-degree equation with real coefficients will alwas have real roots.. What is the domain of f ( )? a) set of all real numbers b) set of all real numbers greater than c) set of all real numbers ecept d) set of all real numbers ecept e) N.O.T. 7. How do ou know that a cubic equation with real coefficients cannot have roots,, and i?. Graph the equation f() = + ; determine the domain and range. State in interval notation. 8. The graph of a third degree polnomial function is given below. Sketch the -ais so that the function has real zeros, one with a multiplicit of.. The graph of f() = is transformed to the congruent graph of g shown below. Write a formula for g. 9. What is the range of the function g() = 6 +. Use the graph of = to find a formula for the function = f(). a) f() = ( ) + b) f() = ( ) + c) f() = ( + ) + d) f() = ( + ) e) N.O.T. 0. The snthetic division shown below illustrates the division of 0 + b d() with a remainder of r, where a) d() = + and r = b) d() = and r = c) d() = + and r = d) d() = and r = - -0 6 9 - -. Using the graphs of g and h below, find: a) (g + h)() b) (g h)( ) c) (g + h)(0) d) (g h)() Livingston High School Page
PRE-CALCULUS JANUARY 0 Review for MIDTERM EXAM 6. Give the new coordinates if the point (8,-7) is reflected across: a) the -ais b) the -ais 6. Find the accumulated value of an investment of $,000 over a period of 6 ears at an interest rate of.% compounded: a) monthl c) the origin b) continuousl d) the line = 7. Find the value for each letter in this snthetic division process: - 8 0 0 A B 8 - - C D 6. a) Rewrite = log 9 in eponential form. b) Rewrite 6 in logarithmic form. 8. Complete the graph for < < so that it will be smmetric about the -ais. 6. Eplain. a) log b (de ) b) ln 7 9. Complete the graph for < < so that it will be smmetric about the -ais. 60. Complete the graph for < < so that it will be smmetric about the origin. 6. When our sink overflows, ou call a plumber to snake the pipes. His fee varies linearl with the amount of time that he has to work. If he works for 0 minutes, the fee is $. If he works for an hour, his fee is $. a) Define the variables, write the ordered pairs, find the slope, and find the particular equation of this function epressing dollars earned in terms of minutes worked. b) What will his fee be for 0 minutes of work? c) If the fee is $, how long did he work? d) What is the cost intercept and what is its real world meaning? e) What are the units of the slope? What is its real world meaning? Livingston High School Page 6
PRE-CALCULUS JANUARY 0 Review for MIDTERM EXAM 6. Determine the equation of the curve drawn below. 68. Using our graphing calculator, eamine the following graphs. In each case, state if the graph is even, odd, or neither. Eplain what it means for a graph to be even, odd, or neither. a) = + + b) = 6 + c) = + 66. Eamine the graph of = using our graphing calculator. a) Cop the graph from our displa. b) Does the graph open up or down? c) Does the graph have a maimum or minimum? d) Use the CALC button to estimate the coordinates of the verte. e) Estimate the -intercepts to the nearest hundredth (using CALC). f) Using the formula, find the equation of the ais of smmetr. g) Algebraicall determine the verte. h) Using the quadratic formula, find the eact roots. 69. Using our graphing calculator, eamine the graph of: f() = ( ) ( ) ( + ) a) What are the real zeros of this function? b) Is there a double root? How do ou know? c) What is the range of this function? d) Determine the intervals (of ) where f() < 0. e) How would the graph of f() differ from the graph of f()? 70. Using our graphing calculator, eamine the graph of: f() = +. a) Sketch the graph. b) Determine the inverse function of f(). c) Sketch the inverse function of f(). d) How do the two graphs compare? e) How do ou know that the function has an inverse function? 67. Using our graphing calculator, eamine the graph of: = + + 7 a) How man real roots do ou see? b) Estimate the real roots to the nearest hundredth. 7. How does the graph of f() = + compare to the parent graph of f() =? Discuss horizontal and vertical translations, stretches, or shrinks as the appl. Livingston High School Page 7
PRE-CALCULUS JANUARY 0 Review for MIDTERM EXAM 7. Craig Browning bakes cookies for the elementar school cookie sale. His chocolate chip cookies sell for $.00 a dozen, and his oatmeal brownie cookies sell for $.0 a dozen. He will bake up to 0 dozen chocolate chip cookies, and up to 0 dozen oatmeal brownie cookies, but no more than 0 dozen cookies total. Also, the number of oatmeal brownie cookies will be no more than three times the number of chocolate chip cookies. How man of each kind should Craig make in order for the elementar school to make the most mone? 7. The revenue equation of a compan, in terms of the price of their product is: R = p + 00p + 000. Find the price that will ield the maimum revenue and determine the maimum revenue. If represents the number of dozen of chocolate chip cookies and represents the number of dozen of oatmeal brownie cookies, graph the region to be maimized. a) Determine the revenue function. b) Shade the appropriate region on the graph that satisfies the constraints. Label all boundar lines and corner points. c) How man of each kind should Craig make in order for the elementar school to make the most mone? 7. Write the epression as a single logarithm whose coefficient is. 7 ln - ln + ln z 7. Solve: 8 7. Solve: 6. =.8 You ma round to the nearest /00 th. 76. Solve: log ( 7) = 77. a) Solve: log ( + 6) log = log ( ) b) Solve: ln = Livingston High School Page 8
PRE-CALCULUS JANUARY 0 Review for MIDTERM EXAM 78. Use the following graph of f() to answer the questions below: 9 8 7 9 b) 6 8 7 6-9 -8-7 -6 - - - - - 6 7 8 9 - - - - - -6-7 -8-9 - - - - - - c) 9 8 7 Sketch the following: a) f() b) ( ) 6-9 -8-7 -6 - - - - - 6 7 8 9 - - - - - c) f(-) -6-7 -8-9 79. If ( ) [ ] determine the transformation of h() if d) a) ( ) [ ] a) vertical stretch b) ( ) [ ] b) vertical shrink c) ( ) [ ] c) horizontal stretch d) horizontal shrink e) vertical shift up f) vertical shift down - - - - - - - - 80. Determine the domain and range of the following graphs: a) - - - - - 8. Ever da Rhonda needs a dietar supplement of mg of vitamin A, mg of vitamin B, and 00 mg of vitamin C. Either of two brands of vitamin pills can be used, Brand X at 6 or Brand Y at 8. Brand X supplies mg of vitamin A, mg of vitamin B and mg of vitamin C. Brand Y supplies mg of vitamin A, mg of vitamin B and 0 mg of vitamin C. Write and graph a sstem of constraints to find how man of each pill Rhonda should take to minimize cost. - - - Livingston High School Page 9
PRE-CALCULUS JANUARY 0 Review for MIDTERM EXAM The following essa questions are comparable in difficult to those on the midterm eam. The are not, however, the same questions. Create a polnomial function of the sith degree whose zeros have multiplicities of,, and, respectivel. Eplain our process. Eplain how ou determine horizontal and vertical asmptotes in rational function graphs. What is the discriminant and what role does it pla in determining the nature of the roots in a quadratic equation. How does the nature of the roots affect the graph of the quadratic function? Hannah solved the equation: + 6 6 80 = 0 and determined =, =, =, or =. How can ou tell that she is incorrect? Create a quadratic function and eplain what is meant b verticall stretching and shrinking the graph of this function. Over three ears, would it be better to invest our mone at % compounded weekl or.9% compounded continuousl? How is smmetr about the -ais, -ais, and the origin defined? How do ou determine end behavior, domain and range of a polnomial function? Livingston High School Page 0
PRE-CALCULUS JANUARY 0 Review for MIDTERM EXAM ANSWERS. D.,, 9 9. A) B)) C) 9 D) 6.. X = 6. a 7. =, (, ) 8. 9. 7,, c Pos.Real NegReal 0 0 0. < <. 7. Root i i 0 0 Multiplict Imag. < or 0<< or >.. i,, 6.,,,,, 6 7. Sum =, product = 8. 9.,,, 0. < <.,, i. Y = 0. V.A.: =, = H.A.: = -int.:,.., 0 6. a) b) c) d) e) f()=(-)*(+)*(-)^ f()=(-)/(+) f()=/(*(^-)) f()=*(-)^ f()=-(+)^*(+) 7. 8. 9. 0.. a) 9 b). a) 8 b) 6 0 f()=*(+)^- f()=- f()=(+)/ line of reflection f()=-(-)^ f()=abs(-)+. Minimum value is 8 Livingston High School Page
PRE-CALCULUS JANUARY 0 Review for MIDTERM EXAM. a) (, - ) b) 7 - =. a) - f) 6 8i b) 9i g) i 7 0 c) d) 9 i e) - 6i 6. 7. E 8. A 9. D 0. B. F. C. -9. D. True 6. False 7. Because if i is a root then + i must also be a root, since comple roots come in conjugate pairs. Then there would be roots not three. The equation there fore cannot be a cubic. 8. 0 - -0-0 - -0-9. Range: { : -} 0. b. b. D : { : - } R : { : - }. g(). a. a) b) - c) d) - 6. False 7. A = -6 B = - C = D = - 8. 9. 60. 6-6 ---- - - 6 - - - - -6 - - - - - - - - - - - - - - - - - - - - - - - - 6. a) $869.60 b) $8,699.9 6. a) b) 6. a) b) ln +.ln( +)-7ln(-) 6. a) Let = no. of minutes worked Let = no. of dollars earned (0, ) and (60, ) Slope is 0 b) $8 c) minutes d) $0 service charge e) $ per minutes intervals or 0 per minute 6. 9 66. a) b) up c) minimum point d) (0.8, -.) e) (-0.8, 0) & (.78, 0) f) 7 g), 8 7 h) 67. a) one b) = -0.9 f()=^-- Livingston High School Page
- - - - - - - - - - - - - - - 9 8 7 6 9 8 7 6 - - 6 7 - PRE-CALCULUS JANUARY 0 Review for MIDTERM EXAM 68. a) even b) neither c) odd An even function is smmetric with respect to the -ais; whenever (, ) is a point on the graph, so is (-, ); f() = f(-), for all in the domain. An odd function is smmetric with respect to the origin; whenever (, ) is a point on the graph, so is (-, -); f(-) = -f() for all in the domain. Neither means that the function is neither even nor odd. 69. a) -,, b) Yes at = there is a turning point or relative minimum point. c) d) e) if() would flip or reflect the graph of f() over the -ais. 70. a) b) c) f ( ) f()=^+ f()=(-)^(/) 7. The graph of f() verticall stretches the graph of f() and translates or shifts the graph horizontall units to the left and verticall unit down. 7. a) P(,). b) 70 Constraints: 0 0 0 0 0 Corner Points: 0,0.,7. 0,0 0,0 c) ½ dozen of choc. chip 7 ½ dozen of oatmeal brownie 7. price: $0 Ma. rev.: $,00 7. 60 0 (.,7.) 0 0 0 0 (0,0) 7. =7 76. =7. 77a) = b) = 78.a) (0,0) 0 0 0 0 0 (0,0) b) c) 79.a) d b) a c) f 80. a) D: ( ) ( ) R: ( ) ( ) b) D: ( ) R: [ ] c) D: ( ) R: ( ] d) D: [ ] R: [ ] d) The two graphs are reflections over the line =. e) f() passes the horizontal line test Livingston High School Page
PRE-CALCULUS JANUARY 0 Review for MIDTERM EXAM 8. Brand X (mg) Brand Y (mg) Min. Vit. A B C 0 00 Cost 6 8.. (0,).. (,). 0. (,.) -0. 0.... (,0) -0. Constraints: 0 0 Corner Points: 0,,,0.,0 c) Brand X pill and Brand Y pills Livingston High School Page