Re - do all handouts and do the review from the book. Remember to SHOW ALL STEPS. You must be able to solve analytically and then verify with a graph.

Transcription

1 Math Review Chapter 3 Name Re - do all handouts and do the review from the book. Remember to SHOW ALL STEPS. You must be able to solve analticall and then verif with a graph. Find the rational zeros of the polnomial. List an irrational zeros correct to two decimal places. 1) f() = Match the polnomial function graph to the appropriate zeros and multiplicities. ) Do not use the calculator. Eplain our reasoning A) -1 (multiplicit 3), 5 (multiplicit 3) B) -1 (multiplicit 3), 5 (multiplicit ) C) -1 (multiplicit ), 5 (multiplicit 3) D) -1 (multiplicit ), 5 (multiplicit ) Use regression to solve the problem. Round numbers to the nearest hundredth. 3) The first column shows the independent variable (). The second column shows the dependent variable () Solve the problem. Find the quadratic regression equation. (Use our calculator: STAT, CALC, choose the quadratic model) (This is similar to the linear regression done in section.) 4) A flare fired from the bottom of a gorge is visible onl when the flare is above the rim. If it is fired with an initial velocit of 19 ft/sec, and the gorge is 560 ft deep, during what interval can the flare be seen? (h = -16t + vot + ho.) 1

2 5) The price p dollars and the quantit sold of a certain product obe the demand equation = -15p + 450, 0 p 30. (a) Epress the revenue R as a function of. (b) What quantit maimizes the revenue? (c) What price should the compan charge to maimize revenue? 6) The number of mosquitoes M(), in millions, in a certain area depends on the June rainfall, in inches: M() = What rainfall produces the maimum number of mosquitoes? Solve algebraicall. Then check the work with the calculator. Show our graph. 7) The concentration C of a certain drug in a patient's bloodstream is given b 30t t (a) Find the horizontal asmptote of C(t). (b) Using a graphing utilit, determine the time at which the concentration is highest. 8) A piece of rectangular sheet metal is 8 inches wide. It is to be made into a rain gutter b turning up equal edges to form parallel sides. Let represent the length of each of the parallel sides. For what value of will the area of the cross section be a maimum (and thus maimize the amount of water that the gutter will hold)? If necessar, round to decimal places. 9) A developer wants to enclose a rectangular grass lot that borders a cit street for parking. If the developer has 30 feet of fencing and does not fence the side along the street, what is the largest area that can be enclosed? 10) A ball is thrown verticall upward with an initial velocit of 19 feet per second. The distance in feet of the ball from the ground after t seconds is s = 19t - 16t. For what interval of time is the ball more than 43 above the ground? Write the product in standard form. 11) ( + 6i)(3 + 3i) 1) (8 + 9i)(8-9i)

3 Graph the function without the calculator. What is the degree? What is the end behavior? What are the zeros? Specif the multiplicit of each zero and indicate whether the graph crosses, bounces or sits and crosses the -ais. 13) f() = -(-1)( - ) 14) f = List the - and -intercepts ) f() = - 6 Make up a rational function that has the given graph. 16) Give the equation of the specified asmptote(s). 17) Horizontal asmptote: h() = ) Horizontal asmptote: g() = ) Oblique asmptote: f() = ) Vertical asmptote(s): f() =

4 Use the Factor Theorem to determine whether - c is a factor of f. 1) f() = ; c = 3 Find the zeros of the function. ) f = (do this algebraicall, b factoring!!!) For the polnomial, list each real zero and its multiplicit. Determine whether the graph crosses, bounces or sits and crosses the -ais at each -intercept. 3) f() = ( + 1 )4 ( - 6)3 Write two cubic functions with the given zeros. 4) -5,, -6 Find the - and -intercepts of the graph. 5) = -( + 3) + 8 Using the Remainder Theorem, find the remainder when f() is divided b g(). 6) f() = ; g() = + 1 State the domain of the rational function. 7) f() = For the given function, find all asmptotes of the tpe indicated (if there are an) 8) f() = , horizontal 9) f() = + + 4, slant ) f() = - 6-4, vertical 4

5 Give the domain of the function. 31) Find the domain of R() = Solve the inequalit, then graph its solution. Use interval notation. ( - 1)(3 - ) 3) 0 ( - ) 33) Do analticall, then verif with a graph. Find the verte and ais of smmetr of the graph of the function. 34) f() = Match the given graph with its polnomial function. 35) A) f() = B) f() = C) f() = D) f() = ) Eplain how ou decide. Do not graph the given functions given as answers with our calculator. A) f() = B) f() = C) f() = D) f() =

6 Match the correct function to a given graph. 37) Select the function given that matches the graph. You should be able to do this without the calculator!!!!! Show how ou make the selection A) f() = + 1 B) f() = + 1 C) f() = + D) f() = + Use the intermediate value theorem to show that the polnomial has a real zero in the interval [a, b]. 38) a = -1 and b = 0 f() = Analze the graph of the rational function for the given step. 39) Find the vertical asmptote(s) and/or hole(s) for R() = Solve the inequalit. - 40) ( + 9) < 0 41) (b + 6)(b + 1)(b - 1) < 0 Do analticall. Then verif with a graph. 6

7 Determine the quadratic function whose graph is given. 4) (0, 1) (-, -3) For the polnomial, one zero is given. Find all others. 43) P() = ; -i 44) P() = ; -i 45) P() = ; 1 + i Answer the question. 46) For the polnomial f() = ( - )3( - 3)( - 4) (a) Find the - and -intercepts of the graph of f. (b) Determine whether the graph crosses or touches the -ais at each -intercept. (c) End behavior: find the power function that the graph of f resembles for large values of. (d) Use a graphing utilit to graph the function. Determine the number of turning points on the graph. (e) Approimate the local minima rounded to two decimal places. (f) Put all the information together, and connect the points with a smooth, continuous curve to obtain the graph of f. 47) Which of the following polnomial functions might have the graph shown in the illustration below? A) f() = ( - )( - 1) B) f() = ( - )( - 1) C) f() = ( - )( - 1) D) f() = ( - )( - 1) 7

8 Find the real solutions of the equation. 48) = 0 49) Solve the inequalit Construct a rational epression with the given characteristics. 50) The graph of R() crosses the -ais at -1, touches the -ais at -4, has vertical asmptotes at = - and = 3, and has one horizontal asmptote at = -. Write the sum or difference in the standard form a + bi. 51) ( - 4i) + (4 + i) 5) (5 + 6i) - (-7 + i) Determine the intervals where the function is positive. 53) f() = (+5) ( - 4) ( - 8) Form a polnomial whose zeros and degrees are given. 54) Zeros: -3, multiplicit ; 1, multiplicit 1; 5, multiplicit 3; degree = 6 Graph the quadratic function b determining whether the graph opens up or down and b finding its verte, ais of smmetr, -intercept, and -intercepts, if an. 55) f() = ) Find the maimum or minimum value of the function. Specif whether it's a maimum or a minimum. f() = Find the comple zeros of the polnomial P. 57) P() = Use transformations of the graph of = 4 to sketch the graph of the function. 58) g() = - ( + )4-3 8

9 List the potential rational zeros of the polnomial function. Do not find the zeros. 59) f() = Find all real zeros of the given polnomial function P, and use the real zeros to factor P. 60) P() = Write the epression in the form bi, where b is a real number. 61) -16 Form a polnomial f() with real coefficients having the given degree and zeros. 6) Find a third-degree polnomial function with real coefficients and with zeros 1 and 3 + i. A) f() = z3 + 7z + 16z + 10 B) f() = z3-7z + 4z - 10 C) f() = z3-5z + 4z + 10 D) f() = z3-7z + 16z - 10 Find the requested function. 63) Find the cubic function with the given table of values f Solve the equation. 64) = 0 Information is given about a comple polnomial f() whose coefficients are real numbers. Find the remaining zeros of f. 65) Degree 6; zeros: -, + i, -3 - i, 0 Construct a polnomial with the given properties. 66) The graph of the polnomial crosses the -ais at - and 3, touches the -ais at 5, crosses the -ais at -5 and is below the -ais between - and 3. Determine the values that cause the function to be (a) zero, (b) undefined, (c) positive, and (d) negative ) f = + 1 9

10 Answer Ke Testname: REVCHAP3 1) -3, -, ) C 3) = ) 5 < t < 7 5) (a) R() = dollars, (b) = 5 units (c) p = $15 6) 8.5 in. 7) (a) = 0 (b) t = 7 8) 9) 1,800 ft 10) { v 3 sec < < 9 sec} 11) i 1) ) )

11 Answer Ke Testname: REVCHAP3 15) No -intercepts, -intercept: 0, 1 ; ) R() = ( + )( - 3) 17) = ) None 19) = ) None 1) Yes ) 0, 5, and ) - 1, multiplicit 4, touches -ais; 6, multiplicit 3, sits and crosses -ais 4) f = ) (-1, 0), (-5, 0), (0, -10) 6) 16 7) (-«, «) 8) = 1 9) = ) =, = - 31) { 6, 8} 3) (-«, 1] or [3, «) 33) (-«, -4] or [-1, «) ) (-3, -37); = -3 35) B 36) D 37) D 38) f(a) = -3 and f(b) = 9 39) vertical asmptote: = -6; hole at 7, ) (-«, -9) U (-9, ) 41) (-«, -6) or (-1, 1)

12 Answer Ke Testname: REVCHAP3 4) = ) i, 3, -3 44) i, 7, -7 45) 1 - i, -5 46) (a) The -intercepts are, 3, and 4. The -intercept is 88. (b) The graph crosses the -ais at and 4, and touches it at 3. (c) The graph resembles f() = 6 for large values of. (d) The graph has 3 turning points. (e) (.57, -0.05), (3.77, -0.76) (f) ) D 48) { 1, 10, - 10 } 49) -«, -5/ U 5/3, 4 50) R() = -( + 1)( + 4), -, 3 ( + )( - 3) 51) 6 - i 5) 1 + 5i 53) (-«, -5)(-5, 4) U (8, «) 54) P() = ( + 3)( - 1)( - 5)3 55) ) maimum; 18 1

13 Answer Ke Testname: REVCHAP3 57) The comple zeros are -1,, 58) 5 10 i, and - 10 i ) ± 1 5, ± 3 5, ± 1, ± 3 60) real zeros of P: - 5, -1, ; P() = ( + 5)( + 1)( - ) 61) 4i 6) D 63) f = ) - 1 ± 7 i 65) - i, -3 + i 66) P() = ( + )( - 3)( - 5) 67) (a) 6, (b) 7, -1/, (c) -1/, 6 U 7, «, (d) -«, -1/ U 6, 7 13