Math 0 Final Review Factor completely ) 6 ) 6 0 ) a b a y by ) n n ) 0 y 6) y 6 7) 6 8 y 6 yz 8) 9y 0y 9) ( ) ( ) 0) Factor out from 6 0 ) Given P ( ) 6 a) Using the Factor Theorem determine if is a factor of P. ( ) b) Using the Factor Theorem determine if is a factor of P. ( ) ) Divide using synthetic division: 6 ) Let A= (, 7) and B= (9,) be points in the y -plane. a) Find the distance between them. b) Find the equation of the line joining A and B. c) Find the midpoint of the segment joining A and B. ) Given the equation of the line y 0. a) Determine the slope. b) Determine the -intercept. c) Determine the y-intercept. d) Find an equation of the line through (,7) parallel to the given line. e) Graph the given line y 0 and the line found in part d) on the same set of aes. ) Find the equation of the line perpendicular to y through the point (, ). 6) Given,, =7, and 8 determine a) i b) k i k c) j j
Solve for the indicated variable y y 7) m, for 8) y y for y 9) A P Pr t, for P 0) log b A for Perform the indicated operations and reduce if possible. ) 8 0 8 ) 70 9 68 7 ) 6 ) 6 ) 8 0 6 9 6 Perform the indicated operations and simplify, assume all variables are positive. 6) abc abc 7) 8 7 7 8) 7 8 9 9) 0) y z 6 60 yz 9 7 ) Rewrite the radicals into fractional eponential form: a) b) 7 ) Rewrite into radical form: / a) b) 6 /7
Perform the indicated operations and leave your answer in simplest form. Assume all variables are unrestricted ecept by the natural domains. ) ( yz) ( z) ) 6 y y 7 ) n n 6) / / / 7) j j j 0 8) ( ) ( ) ()( ) 9) 0 y y 0y 0) 9 9 6 ) Answer in a bi form ) 8i i Answer in a bi form ) i i Answer in bi 0 7 a form ) i ) 6 6) 7 7) 9 8) 8 9) Find each value without the aid of a calculator 0) log 6 ) log 8 ) log6 ) log 7 7 ) log e 6 6) 8 / ln )
7) Eplain why ln is undefined. 8) Write the epression as sums or differences of simpler logarithmic epressions. Epress your answer in such a way that no logarithm of products, quotients, or power appears:. log b b y 7 z 9) Write the epression as one logarithm: log ( ) log y logz Solve each equation, if possible. 60) y 7 9 6) 6y y 9 6) 6) 6 6) 6) 8 98 6 0 66) 9 67) 8 8 68) 0 8 0 69) 0 70) 6 7) ( ) 7) 7 7) 6 8 7) 8 7) e e 0 76) ln( ) 9 77) log 6 log 7
Solve each inequality. Give the result in interval notation and graph it. 78) 79) 7 6 80) 7 8) 6 9 8) 0 8) ( ) 9 or 7 8) 6 8) 7 86) 0 8 ( ) ( ) 0 Solve each of the following system of equations, if possible. If a system is inconsistent or if the equations are dependent, so indicate. 87) 7y y 8 88) y y 89) 6 y 0 y 90) y 60y9 9) Solve each of the following system using Gauss-Jordan Elimination. yz yz yz 9) Given f, determine a) f b) f ( t ) c) f( ) 9) Given g ( ), find a) g (0) b) g( ) c) g() g() 9) Given f ( ), find f ( h) f ( ) h
9) Given f ( ) and g ( ) find a) f g() b) g f ( ) f c) () g d) f g() t e) ( ) fg f) g f ( ) 96) Given g ( ) 0 7 a) Using the method of completing the square rewrite into standard form: b) Determine the ais of symmetry c) Determine the verte d) Determine the domain e) Determine the range f) Graph g ( ) a ( h) k 97) A company s cost function is given by C ( ),000,000 where represents the quantity produced in thousands and C represents the company s cost in thousands of dollars. a) What is the company s cost of producing,000 units? b) If the company has allocated $70 million dollars, how many units can they produce? 98) Given f( ) a) Is this function one to one? Eplain your answer. b) Determine the inverse of this function and if possible epress your answer as f ( ). 99) For the following three problems use the graph of y f ( ). Assume each grid mark represents one unit. a) Estimate f ( ). b) For which values of (if any) does f ( )? c) Give the -intercepts. d) Give its domain. e) Give its range. - - - - - 00) Given the relation y a) Is it a function? b) Give its domain c) Give its range d) Find its inverse e) Determine whether or not the inverse is a function
For problems 0-08 determine: a) Domain b) Range c) -intercepts, if any d) y-intercepts, if any e) asymptotes, if any f) verte, if the graph is a parabola g) Graph 0) f( ) 0) f( ) 8 0) h ( ) 7 0) f ( ) 8 0) f( ) 06) f( ) 07) f( ) 08) f( ) log ( ) 09) Graph y. Find an eact value for the radius and also a decimal approimation for the radius, to the nearest tenth. 0) Graph y 6 ) Graph y y 0 6 ) For the equation below, find the verte. State which way the parabola opens. y 0y 60 ) Estimate the solution to the following system of equations by graphing. Determine if your estimate is the y actual solution. y ) Graph each inequality on a set of coordinate aes: a) 6y 0 b) y 8 6
) Find the equation of the boundary line or lines. Then give the inequality whose graph is shown. Graph the solution set of each system. y 6) y 7) y0 y6 0 8) Solve the following linear programming problem. Maimize P y subject to the following constraints: 0 y 0 y y y 9 if 9) Given f ( ) e if, find if f d) Domain e) Range f) Graph a) f b) f c) y 0) Given, graph the system of inequalities. Make sure to label the corners points where y the two curves intersect that lie adjacent to the feasibility region. 7
) The volume of a cylindrical tank varies jointly with the height of the tank and the square of the radius of its circular base. The volume is cubic feet when h feet and r foot. Find the height when the volume is 8 cubic feet and r feet. ) Use the discriminant to determine what type of solutions eist for the equation 0 0 ) An initial deposit of $,00 earns % interest, compounded semiannually. How much will be in the account after years? Applications: Solve each of the following by identifying unknowns, setting up equations, solving the equation(s) and answering in a sentence which includes quantifiers. ) The dimensions of a rectangle are inches by 9 inches. When both dimensions are increased by the same amount, the area of the new rectangle increased by 0 in. Find the dimensions of the new rectangle. ) 00 liters of a 0% alcohol solution is required for a chemistry eperiment. Determine the amount of pure alcohol that needs to be added to a % alcohol solution to get the required concentration. 6) A retired couple invested part of $,000 at % interest and the rest at 6.%. If their annual income from these investments is $8, how much was invested at each rate? 7) The owner of a home decorating shop wants to mi dried rose petals selling for $6 per pound, dried lavender selling for $ per pound, and buckwheat hulls selling for $ per pound to get 0 pounds of miture that would sell for $.0 per pound. She wants to use twice as many pounds of rose petals as lavender. How many pounds of each should she use? 8) You have the following scores on three eams this semester: 88, 8, and 9. There is still the final eam to be taken. In order to receive an A in the course, you must have at least a 90 average. If the weight of the final eam is twice that of a regular eam, determine the lowest score you can get on the final eam worth 00 points to receive an A in the course. 9) Helen can spend at most $.00 that she got from her grandparents. She goes to the candy store and wants to buy bubble gum that cost $0.0 each and gummy bears that cost $0. each. a) Write a linear inequality that describes Helen s options for buying candy. b) Can Helen buy 8 bubble gums and gummy bears? c) Can Helen buy 9 bubble gums and gummy bear? 0) A light plane flew for 76 km directly into a strong wind and then flew back with the same wind. If the total flying time was hours and the speed of the plane was 0 km/hr in still air, determine the speed of the wind. ) Dan and his brother want to meet their friends at the beach at :00 p.m. At 0:00 a.m., their father informs them they can go only after the garden has been weeded. It takes Dan hours to do the weeding by himself, while his brother who is younger requires 6 hours to complete the job himself. If they work together, will they finish in time to meet up with their friends at :00 p.m.? 8
Estimate the irrational numbers in the left column without a calculator to the nearest two integers. Use this estimate to bound the more complicated epression. ) 9 ) 9 ) log 8 ) log 8 Estimate with a calculator to three decimal places 6) 7 7) log 00 8) 0. e ) 9 ) ANSWERS n n ) ya b ) ) y y 6) y y y y y y 7) 6 y z z z 8) y 9) 7 0 0) ) a) P( ) 0 so ( ) not a factor b) P( ) 0 so ( ) is a factor ) 08 ) a) b) y 9 c), ) a) b) (,0) c) (0, ) d) y e) - - - - - - - - - -6-7 8 ) m y y y 6) a) 9 b) 8 c) 7 7) m 8) y 9) A P 0) rt A b ) 6 9
) 07 ) 6 ) ) 6) a b c c 7) 8) 9) 0) y yz 8 z ) a) / b) 7 / ) a) b) 6 7 ) 8 8 y z ) 0 ) 7 y n n 6) 7) 8) 7 78 8 9) y 0) y ) i ) i ) i ) ) 6 6 6) 7) 8) 9) 0) ) log8 log log log log 8 ) ) 0 ) ) e 6) 7) f ( ) ln( ) is the inverse function of g ( ) e 0. So the domain of f equals the range of g D R (0, ). f ( ) is not defined then because is not in the domain of f. f g 8) log ( ) 7 log y log z 9) b b b log z y 60) {,} 0
6), 6),,, 9 6),0, 6) 6) { } 66) 67) { } 68) { 6} 69) { 8, } 70) { 7, 7} 7) 7) 7 i If directions said Find all real solutions answer is t 7) Use conversion: b A0 logb A t. So ln 8 Remark: log68 ln 6 7) { } 7) {ln()} 76) 6 8 log68 log68 9 e 77) 78) 7, 7 79), 80) 8) 8), 8,, 9 86) 7,, 8),, 8 9 7 8), 8), 87) Dependent ( y, ) y 7 7 88),,, 89), 90) Inconsistent 9),, 9) a) 7 b) 6t c) 9) a) b) 9 c) 9) h
9) a) 7 b) 0 c) 6 7 d) t 6t 6 e) 0 9 8 f) 0 96) a) g b) c) (, 7) d) (, ) ( ) ( ) 7 e) 7, f) g) 0, 7 h), 0,, 0 0-9 -8-7 -6 - - - - 9 - - -9-7 - - 6 7 8 9 0 97) Make sure to take into account that,000 units means, and Cost of $,000 means C a) $8 million b) 9,000 units - -9-7 -6-7 -8 98) a) Yes, because graph passes horizontal line test. Or Yes because the definition of a one to one function is a function that satisfies f ( a) f( b) implies a b. So if we assume f ( a) f( b) then a b so a b. Our function is one to one because it satisfies the definition. b) f ( ) 99) a). b)., 0,. c), 0 (, 0) d) (, ) e) 0, 00) a) Yes since each value of determines a unique value for y. b) (, ) c) (, ) d) f ( ) e) Yes since the original was one to one. 0) a) (, ) b) (, ) c),0 d) (0, ) e) none f) NA g) -8-7 -6 - - - - - - 6 7 - - -7 slope y-intercept - line 0) a) (, ) b) 76, g) c) 7,0 d) (0, 8) e) none f) (, 76) - 9 - - -9-7 - - - -9-7 -6-7 -8 7 9 right down 76 parabola
0) a) 7, b) 0, c) 7,0 g) d) (0, 7) e) none f) NA 9-8 -7-6 - - - - - - - left 7 half parabola 0) a) (, ) b) (, ) c) 0,0 g) d) (0, 0) e) none f) NA -7-6 - - - - - - - - -7-9 - - - left down 8 cubic 0) a),, b),, c),0 g) - -9-7 - - - d) (0, ) - e), y -7 f) NA -0 shift right up of reciprocal function 0 7 7 9 06) a) (, ) b),0 c), 0 g) d) (0, ) e) none f) NA - - - 7 9 shift absolute value function right, reflected about -ais. - - - - 07) a) (, ) b), c) none g) d) (0, ) -6 - - - - - - e) y - f) NA shift up of eponential growth function 08) a), b) (, ) c),0 g) - - 7 9 d) none e) - f) NA shift right of logarthmic growth curve 09) Circle with center (0, 0) and radius r.7 - - - - - - 0) Circle with center (, ) and radius r 6 - - - - - 6 7 - - - -7
) Circle with center (, 7) and radius r 9 ) Verte, opens to the right - - 7 9 7 9 - - ) Intersection of two curves is., but actual solution. ) a) b) y..9 which is false. So estimate is not the ) 6) 7)
8) P at8, 9) a. b. e or e, c. d. e., f. see graph - (-,e^-) - - - (-,-) (,) (,e) 0) ) h feet ) Two unequal irrational solutions ) $,8. ) in by in ) 0 liters would be needed of pure alcohol 6) $0,000 at % and $,000 at 6.% 7) 6 lbs of rose petals, lbs of lavendar and lb of buckwheat hulls 8) The lowest score on the final you can get is 9. 9) a) Let be the number of bubble gum pieces she can buy and y be the number of gummi bear pieces she can buy. 0 0.00.y b) No c) Yes 0) The speed of the wind is 70 mph. ) Yes, they will finish in time because they have hours and it will only take them hours and minutes. ) 9 ) 9 7 ) log 8 ) 6 log 8 7 6).6 7).86 8).66