MATH 110: FINAL EXAM REVIEW


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1 MATH 0: FINAL EXAM REVIEW Can you solve linear equations algebraically and check your answer on a graphing calculator? (.) () y y= y + = ( ) ( ) ( ) ( ) y+ 7 7 y = 9 (d) ( ) ( ) 6 = + + Can you set up and solve word problems involving linear equations? (.6) () America West Airlines has three models of Boeing aircraft in their fleet. The contains more seats than the The has 6 less seats than twice the number of seats in the Find the number of seats for each of the aircraft is the total number of seats for the three models is 7. A premedical student at a local college was complaining that she had just paid $86. for her algebra book, including ta. Find the price of the book before taes if the ta rate at this college is 9%. Find the angles of a triangle whose two base angles are equal and whose third angle is 0 less than three times the base angle. (d) The length of a rectangular playing field is meter less than twice its width. If 0 meters of fencing goes around the field, find the dimensions of the field. Can you create tables and use tables to solve problems? (.7) () Mark Cook is the general manager of the Acme Used Car lot. At the beginning of the month, he has 80 cars on the lot and his sales teams has been selling the cars at the rate of per day. Write an algebraic representation for the number of cars on the lot, y, in terms of the numbers of days that have passed,. (Assume no new cars are delivered. Complete the table for the number of cars remaining on the lot given the number of days that have elapsed. Days Elapsed Cars Remaining If Mark does not replace any of the sold inventory, when will his lot be empty of cars? Can you graph a linear equation by hand and on a graphing calculator? (.) () Sketch the graph of y= by hand. Check the graph using a graphing calculator. Revised April 0
2 Can you find the domain and the range of a relation? Can you determine if a relation is a function? (.) () Find the domain and the range of this relation. Is this relation a function? Why? (7) Find the domain and the range of this relation. Is this relation a function? Why?  0 y 7 (6) Is this relation a function? Why? = y Revised April 0 8) Find the domain and range of f ( ) = Can you evaluate a function using function notation and eplain what the result means? (.) (9) Let f ( ) =. Find f (6) and eplain what it represents. Find f ( ) and eplain what it represents. h t = 6t + 80t represent the height h of a rocket launched from the ground with a (0) Let ( ) velocity of 80 feet per second. Find h ( ) and eplain what it represents. Find h ( ) and eplain what it represents. Can you use the graph of a function to estimate either values or f ( ) values? (.) () Estimate f ( ) Estimate f (.) Estimate when f ( ) = (d) Estimate when f ( ) = 0
3 Can you graph a linear function by hand and on a graphing calculator? (.) () Sketch the graph of ( ) f = by hand. Check the graph using a graphing calculator. () Sketch the graph of y= 8 by hand by finding the intercept and the yintercept. Then write y= 8 using function notation. Can you calculate the slope of a line given information about the line? (.) () Calculate the slope of the line through the points (,) and (6, ). Calculate the slope of the line with the given graph. () Find both the slope and the yintercept of the line given the equation of the line, y= 8. Can you set up and find the slope given a word problem? (.) (6) Upon takeoff, A Delta Airlines jet climbs to miles as it passes over miles of land below it. Find the slope of its climb. Sketch this figure before you find the slope. Can you graph a linear function by hand using only its slope and yintercept? (.) (7) Sketch the graph of y 6 + = by hand using only the slope, m, and yintercept, ( ) Check the graph using a graphing calculator. Can you determine whether a pair of lines are parallel, perpendicular, or neither? (.) (8) Determine whether the pair of lines is parallel, perpendicular or neither. Eplain y= + y= 6 0, b. Revised April 0
4 Can you use the equation of a linear model to make predictions and answer questions? (.) (9) The yearly cost of tuition and required fees for attending a public twoyear college fulltime can be estimated by the linear function f ( ) = , where is the number of years after 990 and f ( ) is the total cost. Find and interpret the slope of this equation. Find and interpret the yintercept of this equation. What is the yearly cost of tuition and fees in 008? Round to the nearest cent. Can you find the equation of a line satisfying given conditions? (.) (0) Find the slopeintercept form of the equation of the line with slope and through the point,. ( ) () Find an equation of the line through the points (, ) and (, ). Write the equation in standard form. () Find an equation of the line through the point (, 6) Write the equation in function form. and perpendicular to the line + y= 8. Can you solve linear equations algebraically and check it on a graphing calculator using either the intersect method or the zero method? (.) () r r 8 r + = Round your answer to two decimal places. 0 Can you solve a linear inequality in one variable and graph the solution set and write it in interval notation? (.) () Solve ( + ) (+ ). Write the result in both set builder notation and interval notation and graph the result on the number line. Can you solve a compound linear inequality in one variable and graph the solution set and write it in interval notation? (.) () Solve < < 7 Can you set up and solve a word problem using a linear inequality? (.) (6) A clerk must use the elevator to move boes of paper. The elevator s weight limit is 00 pounds. If each bo of paper weighs 66 pounds and the clerk weigh 7 pounds, use an inequality to find the maimum number of boes she can move on the elevator at one time. (7) A car rental company offers two subcompact rental plans. Plan A charges $ per day for unlimited mileage, and Plan B chares $ per day plus cents per miles. Use an inequality to find the number of daily miles for which Plan A is more economical than Plan B. Revised April 0
5 Can you solve a system of linear equations graphically by using a hand drawn graph? (.) (8) Solve each system of equations by using a hand drawn graph and state whether the system is independent, dependent, or inconsistent. y= y= y= 6 y= 0 Can you solve a system of linear equations algebraically by the method of substitution or elimination and check your answer on a graphing calculator using the intersect method? (.) (9) Solve the system using substitution. Check your result using a graphing calculator. y= y= (0) Solve the system using elimination. Check your result using a graphing calculator. y= + y= y= + 6y= 0 Can you set up and solve a word problem using a system of linear equations? (.) () A school sold tickets for a play. Student tickets cost 0 cents each while nonstudent tickets cost $.0. If the total receipts were $8.0, find how many tickets of each type were sold. () Two cars leave Chicago, one traveling east and the other west. After hours they are 9 miles apart. If one car is traveling 7mph faster than the other, find the speed of each. () A lab has available a 0% alcohol solution and a 60% alcohol solution. Find how many liters of each solution he should mi to make 0 liters of a 0% alcohol solution. Can you use properties of eponents to simplify epressions with integer eponents? (.l &.) () Simplify ( ab ) ( a b). () Simplify Can you evaluate a polynomial function using function notation? (.) y 8 y (6) Find the indicated output for f = + ( ) 8. a) f ( ) b) f Revised April 0
6 Can you add, subtract and multiply polynomials? (. &.) (7) Perform the following operations and simplify: ( 7 + 8) ( + + ) y ( ) ( ) (d) ( + )( ) Can you factor a polynomial with four terms by grouping and check the result by multiplying your factors? (. &.7) (8) Factor completely and check the result by epanding the factors y Can you factor trinomials and check the result by multiplying your factors? (.6 &.7) (9) Factor completely and check the result by epanding the factors. 0 y+ y 6 9 Can you factor higher degree polynomials? (.7) (0) Factor completely. 9 y + y+ 8 (d) 8 Can you solve quadratic or cubic equations by factoring and check your answer on a graphing calculator? (.8) () Solve each equation by factoring and check your answer on a graphing calculator. = ( ) = 0 + = (d) + 6= Can you set up and solve word problems that can be modeled by polynomial equations? (.8) () The shorter leg of a right triangle is centimeters less that the other leg. Find the length of the two legs if the hypotenuse is centimeters. () The floor of a shed has an area of 90 square feet. The floor is in the shape of a rectangle whose length is feet less than twice the width. Find the length and the width of the floor of the shed. Revised April 0 6
7 () After t seconds, the height h( t ) of a model rocket launched from the ground into the air is h t = 6t + 80 t. Find how ling it takes the rocket to reach a height of 96 feet. given by ( ) Can you evaluate a rational function using function notation? (6.) () Given f ( ) = + 8, Find f ( ) Find f ( ) Can you find the domain of a rational function algebraically? (6.) (6) Algebraically find the domain of each function. Use interval notation. + f ( ) = f ( ) = + 8 Can you simplify rational functions? (6.) (7) Simplify each rational function Can you multiply or divide rational epressions? (6.) (8) Perform the following operations and simplify the result Can you add or subtract rational epressions? (6.) (9) Perform the following operations and simplify the result Can you simplify a comple fraction? (6.) (0) Revised April 0 7
8 Can you solve rational equations algebraically and check your answer on a graphing calculator? (6.) () Solve each equation. Verify your result using a graphing calculator. + = = = Can you set up and solve word problems that can be modeled by rational equations? (6.6) () The sum of a number and twice its reciprocal is. Find the number(s). () Three postal workers can sort a stack of mail in 0 minutes, 0 minutes, and 60 minutes, respectively. Find how long it takes them to sort the mail if all three work together. () The speed of a boat in still water is mph. If the boat travels miles upstream in the same time that it takes to travel 90 miles downstream, find the speed of the current. Can you simplify radicals without using the calculator? (7.l) () 9 9 (d) 9 6 Can you convert epressions in eponential form to radical form and simplify? (7.) (6) Convert to radical form and simplify without using a calculator. 6 ( 8) ( 6) (d) 9 Can you use properties of eponents to simplify epressions that contain rational eponents? (7.) (7) Use the properties of eponents to simplify each epression. Write with positive eponents. 7 y ( ) 6 9 a a a Revised April 0 8
9 Can you use the product and quotient property for radicals to simplify radical epressions? (7.) (8) Simplify each radical epression. Assume that all variables represent nonnegative quantities. 6 0 y 6 6 y z (d) y 9 (e) 7 Can you add or subtract radical epressions by combining like radicals? (7.) (9) Add or subtract as indicated. 9y+ 8 8y Can you multiply radical epressions with the same inde? (7.) (60) Multiply. Assume that all variables represent nonnegative quantities. ( 7 )( ) 7( 7) ( )( + ) (d) ( 6 y+ ) Can you rationalize denominators and simplify quotients of radical epressions? (7.) (6) Simplify, leaving no radicals in denominators, no radicals of fractions. Assume that all variables represent nonnegative quantities. a b 6 + Can you solve equations containing radicals? (7.6) (6) Solve each square root equation. = + = + = (d) + = (e) = Revised April 0 9
10 Can you set up and solve word problems involving radical equations? (7.6) (6) A spotlight is mounted on the eaves of a house feet above the ground. A flower bed runs between the house and the sidewalk, so the closest the ladder can be placed to the house is feet. How long a ladder is needed so an electrician can reach the place where the light is mounted? Can you solve quadratic equations of the form answer on a graphing calculator? (8.l) = k using the square root property and check your (6) Solve the quadratic equation and check your answer on a graphing calculator. = 0 ( ) = 7 Can you solve quadratic equations by completing the square and check your answer on a graphing calculator? (8.) (6) Solve by completing the square and check your answer on a graphing calculator. 6 0= 0 + = 0 Can you solve quadratic equations by using the quadratic formula and check your answer on a graphing calculator? (8.) (66) Use the quadratic formula to solve 7 0 =. Check your answer on the graphing calculator. (67) Use the quadratic formula to solve ( ) 6+ =. Check your answer on the graphing calculator. (68) Use the quadratic formula to solve 9+ 6=. Provide the eact solutions as well as estimations rounded to three decimal places. Can you solve quadratic equations using any method of your choice? (8.) (69) + = 0 8= = + 9 (d) = Can you set up and solve word problems involving quadratic equations? (8. & 8.) (70) The hypotenuse of an isosceles right triangle is 6 centimeters longer than either of its legs. Find the eact length of the legs. (7) An entry in a poster contest must be rectangular and have an area of 00 square inches. In addition its length must be 0 inches longer than its width. Find the dimensions each entry must have. Revised April 0 0
11 (7) Mark s workout consists of jogging for miles and then riding his bike for miles at a speed miles per hour faster than he jogs. If his total workout time is hour, find his jogging speed and his biking speed. Can you algebraically find the and yintercepts, verte, ais of symmetry and use the information to graph a quadratic function? (8.6) (7) For f ( ) = 8; find the and yintercepts, verte, and ais of symmetry. Use this information to graph the function by hand. Can you estimate the and yintercepts, verte, ais of symmetry, and or, f ( ), given the other variable, by reading the graph of a quadratic function? (8.6) (7) Use the graph of the quadratic function to estimate the indicated value. a) Estimate the verte. b) Estimate the intercepts. c) Estimate the yintercept. d) Estimate the ais of symmetry. e) Estimate f ( ). f) Estimate when f () =. Can you estimate the and yintercepts and the maimum or minimum value of a quadratic function by using the value, zero and maimum or minimum programs on a graphing calculator? (8.6) (7) Estimate the and yintercepts and the maimum or minimum value of graphing calculator. Round to the nearest hundredth. f ( ) = using a Can you set up and solve application problems dealing with a maimum or minimum value of a quadratic function? (8.6) (76) A company s cost C in dollars of manufacturing bicycles is given by the function C( ) = , 000. Find the number of bicycles that must be manufactured to minimize the cost. Find the minimum cost. (77) The length and width of a rectangle must have a sum of 0. Find the dimensions of the rectangle that will have maimum area. Revised April 0
12 Answers (with occasional eplanations): 0 () y= 7 y= 9 All real numbers (d) No solution or () The has seats. The has seats. The has 90 seats. The price of the book before taes is $ The angles of the triangle are 8, 8, and 0. (d) The width is 0 meters and the length is 7 meters. () y= 80 Days Elapsed Cars Remaining In days, the car lot will be empty of cars. (Just set 80 = 0 and solve for.) () y y = 6 () The domain is {, 0,, } and the range is {,, 7 } Yes, it is a function because each is paired with only one y. (6) No, as each 0 is paired with two different y values, not one y value. (7) The domain is [ 0, ) and the range is (, ) No, it is not a function as a vertical line can intersect the graph in two points, not one point. This means the same is paired with two different y values. (8) The domain is (, ) and the range is (, ). Revised April 0
13 (9) f (6) = is the value of y when = 6. f ( ) = is the value of y when =. (0) h ( ) = 96 The rocket is 96 feet above the ground in seconds. h = 0 ( ) Thus the rocket hits the ground in seconds. () f ( ) f (.) 0. and. (d) 0. () f ( ) = 0 0 () The intercept is (, 0 ) 8 The yintercept is 0, or 0, 8 f ( ) = () 7 m= 9 m= Revised April 0
14 () y= 8 y= + 8 y 8 = + 8 y= 8 m= and the yintercept is 0, or 0, (6) (7) rise up miles m= = = run right miles y= + Thus m= The yintercept is (0,) rise m= = run rise = so down from 0, run = so now go right ( ) (8) m = and m = as y = while y = + = or can say the two slopes are opposite reciprocals of each other. Thus these two lines are perpendicular. (9) m =.6 The cost of tuition and fee are increasing at a rate of $.60 per year since , The yintercept is ( ) This means that the cost of tuition and fees in 990 was $ f(8) = 8.68 Thus is 008, the total cost of tuition and fees is $,8.68. (0) y y = m( ) y+ =( ) y= + 6 Revised April 0
15 () 7+ 6y= 6 () f ( ) = () r 8.9 () Set Builder Notation { / 8/} Interval Notation [ 8 /, ) () Set Builder Notation { / < < } Interval Notation (, ) (6) The maimum number of boes that you can move is 0 boes. (Note: Need whole boes.) (Hint: Solve for where = number of boes, getting 0. ) (7) Plan A is more economical than Plan B if you drive more than miles. (8) (, ); independent system as the lines intersect in one point. No solution; inconsistent system as when you graph them, the lines are parallel. (9) (, ) (0) 7 6, Infinite number of solutions as the solution set = {(, y) / y= } dependent system as the two lines when graphed overlay one another. () You sold 8 student tickets and 0 nonstudent tickets. () One car travels at 8 mph while the other car travels at 6 mph. () You should mi 0 liters of the 0% solution with 0 liters of the 60% solution. () 8a b 9 7 Revised April 0
16 () y 7 (6) f ( ) = 6 8 f = (7) (d) 0y+ y + (8) ( )( 7) ( 7+ y)( 7 y) (9) ( )( + ) ( + )( ) (0) ( + )( ) ( )( + ) () = 7 or = = 0, =, = y ( ) y+ ( ) (d) ( )( + 0+ ) = or = (d) = or = 6 () The two legs of the right triangle are cm and 9 cm. () The width of the shed is 7 feet and the length is feet. () It will take seconds and seconds (ascending and descending) for the rocket to reach 96 ft. 7 8 (6) Domain is, U, as f = 7. () f ( ) = ( ) Domain is ( ), U,, U as and. (7) + + (8) + 8 Revised April 0 6
17 (9) + ( )() ( )( + ) (0) + () = or = 0 = = (Please note that as it causes division by 0.) () The numbers are and. () It will take all three postal workers only 0 minutes to sort the mail. () The speed of the current is 6 mph. () Not a real number 7 (d) (6) 8 (d) 7 (7) 9y a (8) y y z z (d) 9 y (9) 6 y + (e) (60) 6 0 (d) 6y+ 60 y+ (6) 6ab b 0 Revised April 0 7
18 (6) = No Real Solution = 6 (d) = (Please note that 0 as your equation does not check out when = 0.) (e) = 9 (6) The ladder must be feet long. (Hint: Use the Pythagorean Theorem.) (6) = or = = or = + (Don t cancel the s as the st term,, can t be divided by ) (6) = ± = or = + (66) (67) (68) = or = = or = = or = or.089 (69) =, =, =, or = = 8 or = 6 =, or = + (d) = 8 Careful as it doesn t check out in your equation. (70) Each leg of the isosceles right triangle is 6+ 6 centimeters. (7) The width is 0+ 0 inches and the length is 0+ 0 inches. (7) Mark s jogging speed is 6 mph and his biking speed is 0 mph. Revised April 0 8
19 (7) intercepts:,0 and,0 ( ) ( ) yintercepts: 0, 8 ( ) verte:, 9 ( ) Ais of symmetry: = (7) The verte is (, ) The intercepts are (, 0 ) and ( 0,0) The yintercept is ( 0,0 ) (d) The ais of symmetry is = f = (e) ( ) (f) 0.8 and. (7) intercepts: ( 0.8, 0 ) and (., 0) yintercept: ( 0,9 ) Minimum value: None f.. Maimum value: ( ) (76) The minimum number of bicycles is 00. The minimum cost is $,000 (77) The length is and the width is Revised April 0 9
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