MATH 115 MIDTERM EXAM NAME: _______________________________ May-July, 2016 Instructor: B. Stec I have completed this assignment myself, working independently and not consulting anyone except the instructor. INSTRUCTIONS The exam is worth 100 points. There are 15 problems. This exam is open book and open notes. This means that you may refer to your textbook, notes, and online classroom materials, but you must work independently and may not consult anyone (and confirm this with your submission). You may take as much time as you wish, provided you turn in your exam no later than Sunday, June 12. Show work/explanation where indicated. Answers without any work may earn little, if any, credit. You may type or write your work in your copy of the exam, or if you prefer, create a document containing your work. Scanned work is acceptable also. In your document, be sure to include your name and the assertion of independence of work. If you have any questions, please contact me by e-mail. 1. (6 pts) Divide the polynomials (reduce to the simplest form) 2 x 2 3x 4 . Show work. x 1 A) Check by direct substitution of two selected x values that the resulting polynomial expression is correct B) Define the domain and range f ( x) Graph the function. 2 x 2 3x so that the formula represent a function? x 1 2. (6pts) The formulas below represent the relations (graphs) (A) (B) (C) (D) (E) (x-2)2 +(y-3)2=3 y=-2x+3 y=3 x=4 x=y2 (a) Which of these graphs represents a function? Answer(s): ____________ (b) Which of these graphs represents a one-to-one function? Answer(s): ____________ (c) Which of these graphs represents an even function? Answer(s): ____________ 3. (4 pts) Which of the following is TRUE about the line through the points (-2, 5) and (2, -3)? Answer. _______ (no explanation required) A. B. C. D. The slope is undefined. The slope is negative. The slope is 0. The slope is positive. 4. (4 pts) Convert to a logarithmic equation: 12x = 43. A. log 43 = 12 B. log12 43 = C. log 12 = 43 D. log12 = 43 5. (4 pts) Convert a base of the equation 12x = 43 to 10 (Reminder; the equation should look like 10power=43) (no explanation required) Answer. ______ 6. (11 pts) Bob is a salesperson who must decide between two monthly income options. Option A: Salary of $1000 per month, plus 12% of monthly sales (in dollars) or Option B: Salary of $2000 per month, plus 8% of monthly sales (in dollars) (a) State a linear function f (x) that represents the Option A total income, given monthly sales of x dollars. (b) State a linear function g(x) that represents the Option B total income, given monthly sales of x dollars. (c) Laura (Bob's wife) wants to determine the following: For what amount of monthly sales is Option A the better choice rather than Option B? To answer the question, should you solve the inequality f (x) g(x) or the inequality f (x) g(x) ? Write the appropriate inequality, show how to solve it, and state your answer to the question in a sentence. 7. (10 pts) Let f (x) = 2x2 - 6x + 4 and g(x) = 3x + 6 (a) Find the composite function ( )() and simplify the results. Show work. Please remember the composition function ( )() is defined as ( )() = ((()) (b) Find ( )(2) . Show work. 8. (8 pts) Let f ( x) 2x 2 4 . (no explanations required) x2 3 (a) State the y-intercept. (b) State the x-intercept(s). (c) State the vertical asymptote(s). (d) State the horizontal asymptote. 9. (10 pts) Let ( ) = 6 7 4 3 (a) Graph the function (b) Find f 1, the inverse function of f. Show work. (c) What is the domain of f ? (d) What is the domain of the inverse function? 10. (5 pts) The graph below represents a rational function y = f (x) . (no explanations required) (a) State the vertical asymptote. (b) State the horizontal asymptote. (c) The graph represents which of the following functions? Choice: _____ A. () = B. () = C. () = D. () = 5 2 2 9 5 3 3 5 5 3 11. (5 pts) The graph below represents a polynomial function y = f (x) . (no explanations required) (a) Is the degree of the polynomial odd or even? (b) State the interval(s) on which the function is increasing. (c) State the value of f (2). 12. (7 pts) Solve the equation. Check all proposed solutions. Show work in solving, and state your final conclusion. x5 4 2 0 x 4 2 x 10 x 8 13. (5 pts) Find the real number solution to the equation. (Tricky; if helpful graph it.) Check all proposed solutions by checking the domains of both sides. Show work in solving, and state your final conclusion. 19 3 = 5 14. (5 pts) Let f (x) = e x - 2 - 3. Answers can be stated without additional work/explanation. (a) Which describes how the graph of f can be obtained from the graph of y = ex ? Choice: ________ A. Shift the graph of y = ex up by 1 unit. B. Shift the graph of y = ex to the left by 2 units and up by 3 units. C. Shift the graph of y = ex to the right by 2 units and down by 3 units. D. Shrink the graph of y = ex horizontally by a factor of 2 and shift up by 3 units. (b) What is the x and y-intercept? State the approximation to 2 decimal places (i.e., the nearest hundredth). 15. (10 pts) In one event, the competitor at the Olympic Games, throws the javelin to the height h in feet, above the ground in t seconds. The throw is modeled by the equation h(t) = -16t2 + 22:08t + 6. (a) Using algebraic techniques we have learned, find the maximum height predicted by the quadratic model and find the time when it occurred. (b) What is the hammer's total time in the air? How the answer would change if you assume that the javelin was thrown not from a height 0 but from a height of 7 feet. . . . . 2 2 3, + 4 1. (6 pts) Dlwde the palynonuala (reduce to the Slnlplest fowl) I +: x 2 r xW xii-25c . Show work. \"5' +4 -6 #5.. j; 6? P65 "-7 2&4? +3 M1 A) Check by direct substitution of two selected x values that the resulting polynomial PUD= 2x~5 4: .9... express 1011 1 ><+ 1 Xao FCC) 1-" O+O+Y on ---LI ._ ._ 3.. '2: *S'l'cl =5\" 1 7H m .._. 20) 50M \"' (0+1 = 2514; 2-; 2 2. POD: QED5 +_'L 074'! 12533... 2 +2_ 2 _ B) Dene the domain and range f(x) = 2x 13x x + Gm?h$' __ Qt\") H19Qx2\"5>DCD : O so that the formula represent a mction? 7x17: 0' 58'\" 45(13wa _= - Drew Se, 120.? ALL ["b'mg/w) BM (Fm/\"DUH/w) . _ . -=.. __= .__.=__-______ [Eu-.1\". __=___.=_____=_ IIIII-I. . : I -__-=_ A. __=_=_____=.- In A.\" H H v.\" H H H H H _ 2. (6pts) The formulas below represent the relations (graphs) (a) 1ET'IJhieh of these graphs represents a function? Answer(s): E) at (b) Which of these graphs represents a one-to-one function? Answer(s): E? E (c) 1Which of these graphs represents an even function? Answer): A} I: 3. (4 pts) Which of the following is TRUE about the line through the points (2. 5) and (2, -3)? Answer. (no explanation required) A. The slope is undened. The slope is negative. . The slope is 0. D. The slope is positive. 4. (4 pts) Convert to a logarithmic equation: 12Jr = 43. (no explanation requiIed) Answer. A. log: 43 = 12 B. log\" 4-3 = x C. log. 12 = 43 log\" I = 43 5. (4 pts) Convert a base of the equation I? = 43 to ID (Reminder; the equatinn ehnnld look like lm=43) lotH?) a fat?" M333 [DI-4:33 L413 6. (1] pts) Bob is a salesperson 1who must decide between two monthly income options. Option A: Salary of $1000 per month, plus 12% of monthly sales (in dollars) or Option B: Salary of $2000 per month, plus 3% of monthly sales (in dollars) (a) State a linear mconfc) that represents the Option A total income, given monthly sales ofx dollars. %C% = 'D'l2x+lwo (b) State a linear inaction gr} that represents the Option B total income, given monthly sales of 3: dollars. $99 = 9-0834 +2300 (-2) Laura (Bob's wife) wants to determine the following: For what amount of monthly sales is Option A the better choice rather than Option B? To answer the question, should you solve the inequality f (x) 2 3(1) or the inequality f (x) 5 g(x) '1" Write the appropriate inequality, show how to solve it, and state your answer to the question in a 5mm fe} 33M) 1)th\"le 2" WWW\" b'llxr 0-be 7'? 20'5\"\" le'lpt B )900 a 129 X o-'b'f y :3, 2500:) So, option it would be a better option when amount of sales is greater than or equal to $25000 1(1opts)Letf(x)=2x25x+4and 31x3=3x+5 (3) Find the composite metioo {f 0 3X1} and simplify the results. Show work- Please remember the composition mction {f o g){x] is dened as [f o g)[x) = {f{g{x]) EW *- ECM3 = @5166 HQ +7 :: TxL-TBGX *5643X $591 :1. 9x?" +15%! '1'\" (b) Find {f o y) {2) _ Show work. #9923 = aezsw 3636 1*") q ur . I- 2x14 Jig3 3. (3 pts) Let f(x) = . (11c: explanatiuns required) {a} State HR: yiutercept. Q 0 ) % (b) State the Iiutercepgl @Ti/D) ) 66149 (a) State the vertical asmptuteil: E f' .; .Ji ((1) State the hrizmltal asymptute. 3:1 10. (5 pts} The gmph below represents a Iatienal flmetieny =f(:r) _ [m eqmtimls mquired} {a} State the vertical asymptote. X '2 3 {b} State the herizemal Mute. F . 5' {e} The graph imam which ef the fellewihg metiens'? Cheiee: A. fix) = 5:\": 1' 11. (5 pta) The graph belew represents a pelynemial metieny =f[x) _ [In eqanat'mns required} {aJIsmedegIee efthepeljmumialaddor a? M {b} State the EMS] en whjeh the :metien is increasing. L L: , D {e} State the 1value eff(2]. D 12. (\"f puts) Solve the equation. Check all proposed solutions. Shaw work in solng, and state your nal conclusion x+5 4 + m 2x3+10x+s= XI'F'E + '2. :0 JEFF" X +KH 7.. 1 0 295. m \"L @m) 13. (5 pts) Find the real number solution to the equation. (Tricky; if helpful graph it.) Check all proposed solutions by checking the domains of both sides. Show work in solving. and state Your nal conclusion. v193xe5 (W): er- 5) 14. (5 pts) Letx) = ex2 - 3. Answers can he stated without additional workfexplanation. (a) 1Which describes how the graph of f can be obtained from the graph of y = e' '3 Choice: C A. Shift the graph of y = 3" up by 1 unit. B. Shift the graph of y = a' to the left by 2 units and up by 3 units. Shift the graph of y = e' to the right by 2 units and down by 3 units. D. Shrink the graph of y = e\" horizontally by a factor of 2 and shift up by 3 units. (13) What is the x and yintercept'? State the approximation to 2 decimal places (i.e., the nearest hundredth). vii-2 15. {10 pts) In one event, the competitor at the Olympic Games, throws the jas-'elin to the height 1".- in feet, above the ground in 1' seconds. The throw is modeled by the equation hm = 4er + 22:081'+ 5. (a) Using algebraic techniques we have learned: find the maximum height predicted by the quadratic model and find the time when it occurred. _ __ .. . 3 ... '- time? It" 2% : viii) _ 060' We ew: M): 460- 5691+ 22. DEJ'EED-tg :: '3'5176 (b) What is the hammer's total time in the air? How the answer would change if you assume that the javelin was thrown not from a height 0 but from a height of T feet. 0 --I 65'} 22-09%, +6 :s o -t:: -0*2325' , \"5'25; SQ'W Hm = l-GlZSY-CO'Zg) ":- FEHE 35 AU) =7 1 r: 45 Jails-22.08% r 6 14+1+22-D ] = 0 1: 0 "W68 9 (0:9 Afar: #61255 39) Hm Mm : illQFFD'OLIM F: [-56567