Math 0- Functions and Equations Multiple Choice Questions 8 simplifies to: A. 9 B. 0 C. 90 ( )( ) simplifies to: A. B. C. 8 A. 9 B. C. simplifies to: The area of the shaded region below is: 0 0 A. B. 0 C. 0 0 0 0
When written as a mied radical, equals A. B. 8 C. 9 The following problem involves the subtraction of radicals. In which step was the first error made? Step Question 0 8 A. Step B. Step C. Step Step A student is working on a problem, and correctly enters it into her calculator. Her calculator entry is shown to the right. Which of the following accurately represents the problem she is attempting to evaluate? A. B. C. 9 9 9 9 8 Correct to the nearest tenth, the value of is A. 0. B.. C. -. -.
9 A rectangular room measures m by m. In one corner of the room is a door, and in the furthest opposite corner is a thermostat. Which of the following most closely represents the shortest walking distance between the door and the thermostat? A. B. 0 C. 8 0 8 0 When simplified the value of is A. B. C. A student is asked to rationalize the denominator in the epression. The value of k k. Her answer is A. B. C. 8 When written as an entire radical, is equal to A. 8 B. C. 98 9
Jeff is working on questions involving radicals. He is asked to respond to whether the statement in the question is true or false. His answers are in the chart below. Question Jeff s Answer (true/false) 9 9 8 true true 90 9 0 false 9 true How many questions did Jeff answer correctly? A. one B. two C. three four A student is working on a problem, and correctly enters it into her calculator. Her calculator entry is shown to the right. Which of the following accurately represents the problem she is attempting to evaluate? A. B. C. The dimensions of a rectangle are 8 m by m, shown in the diagram on the right. The length of the diagonal can be calculated using the Pythagorean thereom; a b c. The eact value of the diagonal, shown by the dashed line is A. B. C. 8 m m When epanded and simplified as a mied radical, the value of is A. B. C. 8
Sylvia is rationalizing the denominator for the epression below. Step. Her four step solution is shown 9 9 Sylvia s first error was made in A. Step B. Step C. Step Step ( ) 8 The rational epression ( )( ) A. one restriction on. B. two restrictions on. C. three restrictions on. four restrictions on. has 9 Which of the following is the correct lowest common denominator for the rational addition problem? A. ( )( ) B. ( ) ( ) C. ( )( )( ) ( )( ) 0 When simplified, A. B. C., simplifies to
A student is working on the rational equation,. After multiplying both sides of this equation ( ), the equation that will need to be solved is A.. B.. C... Peter drives during the day travelling 00 km for t hours. On the same journey, Peter s father drives at night also travelling 00 km for hour longer and 0 km/h slower than Peter. Which of the following equations finds the velocity, v, that Peter drove? A. 00 00 v v 0 B. v v 0 00 00 00 00 C. v 0 v v 0 v 00 00 In solving the equation 9, a student should conclude that A. There are no non-permissible values. B. There are two non-permissible values. C. There are three non-permissible values. There are four non-permissible values. The area and side width of a rectangle are given. In order to find the side length of this triangle, a student could A. Simplify the epression to determine a side length of B. Simplify the epression to determine a side length of C. Simplify the epression Simplify the epression + - to determine a side length of to determine a side length of -
When simplified, the epression q A. p B. p q C. q q pq p 8 pq equals p p q,, 0 The sum of A. B. C. 8 ( ) ( ) 8 0 ( ) ( ) The simplified epression A. B. C., 0, is equals 8 The restrictions on the rational epression ( ) are A. B. C.,0,0 only only
9 A student is completing the addition problem. Which of the following is the student s lowest common denominator? A. ( )( ) B. ( ) C. ( ) ( ) 0 The epression A. B. C.,, simplifies to Rachel is working on the radical equation. Which of the following should be her first step? A. Multiply all the terms on the left side by ( ). B. Multiply all the terms on the right side by ( ). C. Multiply all the terms on both sides by ( ). Multiply all the terms on both sides by ( ). Two cars are travelling from Airville to Broadtown. Car A travels at 0 km/h for t hours. Car B travels at 00 km/h for hours longer. Which of the following equations could be used to solve for the distance between Airville and Broadtown? d d A. 00 0 d d B. 0 00 d d C. 00 0 d d 0 00 In solving the equation ( )( ) ( )( ), a student should conclude that A. There are no non-permissible values. B. There is one non-permissible value. C. There are two non-permissible values. There are three non-permissible values.
The area of a rectangle is 0. If it has a side length of then the width can be represented by the epression A. B. C. When simplified, the epression A. B. C. c ab c ab ab c ab c a b 8c c ab,,, 0 a c b equals When divided and simplified, 0m A. 9n B. m n C. m n 8mn mn 8m m n n, mn, 0, equals The simplified epression equals A. B. C.
8 A student entered two functions in their calculator; Y and Y. The result is shown in the graph to the right. The intersection points of the two graphs are the solution to the equality; A. 0 B. C. ; f g 0 8 9 0 9 In solving the equality is; A. k B. k, C. k, No solution k k k 0 The graph to the right is of the function f (). According to the graph, A. f () has no solutions B. f () has one solution C. f () has two solutions f () has three solutions k, the result f 0 0 8 A quadratic equation that could be derived in the attempt to solve is; A. 0 B. 0 C. 0 0 A student graphs the function f(), which is shown to the right. The largest solution to the equation f( ) appears to be; A. B. C. f 0 9 8 0 8 8
A student is attempting to solve the quadratic equation. Her steps are shown in the table below; Step ( ) ( ) Step Step Step Solution The student made her first error in; A. Step B. Step C. Step Step The reciprocal function pairs must be on f()? A. (0, ) B. (0, -) C. (, 0) (-, 0) f ( ) has a vertical asymptote at. Which of the following ordered Randy travels the 0 km from Airdrie to Edmonton driving 0 km/h for hours. His parents strongly suggest he slow down by v km/h, even though it will take him t hours longer to make the trip, or he will lose his driving privileges. Which of the following relations shows the time, t, it will now take Randy to make the trip relative to the speed, v with which he slows down? A. t 0 0 v B. t 0 0 v C. t 0 0 v t 0 0 v
The factored form of f () is represented by the function f ( ) k( ) ( ), k R. Which of the following correctly describes the graph of the function? f ( ) A. f ( ) will have three asymptotes at k,, B. f ( ) will have three asymptotes at k,, C. f ( ) will have two asymptotes at, f ( ) will have two asymptotes at, The rational function is given by m 8 n. When m 0, the value of n must be; A. B. 0 C. 0 0 8 The function f ( ) is shown in the graph to the right. If f () is a quadratic function, which of the following statements would be true about the graph of f ()? A. f () opens up with no -intercepts B. f () opens down with no -intercepts C. f () opens up with one -intercept f () opens down with one -intercept f. 0. 0 8 9 0. 9 A cubic function, f(), has three unique zeros. The reciprocal function A. At least one, and not more than si invariant points with f() B. At least two, and not more than si invariant points with f() C. At least one, and not more than four invariant points with f() At least two, and not more than four invariant points with f() f( ) must have; 9
0 Geoff must travel 00 km. He correctly determines that travelling 0 km/h for hours will cover that distance, but if he decreases his speed by v km/h, his time will increase by t hours. A function that correctly determines the change, v, in speed, relative to the time t in hours is 00 (0 v)( t ). When written as v in terms of t, this function is; A. v 0 00 t, B. v 0 00 t, C. v 0 00 t, v 0 00 t, The function f ( ) is shown in the graph to the right. There are no other asymptotes. If f () is a quadratic function, which of the following statements would be true about the graph of f ()? A. f () opens up with no -intercepts B. f () opens down with no -intercepts C. f () opens up with one -intercept f () opens down with one -intercept The function shown to the right is of f(). How many ordered pairs will f() share with? f ( ) A. B. C. f b f 0 9 8 0, a 0 The function y = f() is such that f () 0 when. Which of the following is true regarding the function y f ()? A. f () 0 when B. f () 0 when C. f () 0 when f () 0 when
The graph of y f () is shown to the right. Which of the following could be said about y f ()? A. y f () is a complete reflection of y f () around the y- ais. B. y f () is a complete reflection of y f () around the - ais. C. y f () is a partial reflection of y f () around the y- ais. y f () is a partial reflection of y f () around the -ais f In graphing the function g ( ), it is discovered that g () has the domain f ( ) R. Which of the following could be said about the graph of f ()? A. f () has no y-intercepts B. f () has no -intercepts C. f () cannot be a function f () cannot share any ordered pairs with g (). Numerical Response Epressed as a mied radical, 0 would have a coefficient of. In the radical epression, the inde number is. Epressed as an entire radical would have a radicand of. Correct to the nearest tenth, the value of 0 is. If 000 k, then the value of k must be. If 88 k, then the value of k must be. The epression 0 is simplified yielding k ( ). The value of k must be. 8 The only non-permissible value in the epression is.
9 The chart below provides four factorable epressions. Epression # Epression # Epression # Epression # When two of those epressions are divided and simplified, the result is. This is true because epression is divided by epression. 0 Correct to the nearest hundredth, the solution to the equation 9, 0 is. The equation 9 has two solutions. The only positive solution to this equation, correct to the nearest tenth, is. When simplified, the value of is. The solution for is. The largest non-permissible value in the epression 8 0 is. n The epression n an ( ), n, simplifies to. The values of a and b are (place your n b n value for a in the first available bo, and your value for b in the net available bo). The rational epression 9 k, k, simplifies to because k equals. The perimeter of the isosceles triangle shown in the diagram on the right could be written m. The value of m must be units. ( ) / /(+) 8 Correct to the nearest tenth, the solution to the radical equation 8 0 is.
9 The graph shown to the right is of f (). The largest solution to the epression f () appears to be. 0 A student is determining the solution to the equation graphically. The student correctly determines two solutions. If one solution is 0., correct to the nearest tenth, the other solution must be. The radical equation k 0, k 0, has one solution of 8 if k equals. The largest restriction on in the solution to the equation is. The graph of the reciprocal function is shown to the right. f ( ) The largest -intercept of f () must be. 0 f 0 9 8 8 0 8 8 f 0 8 k The rational function f ( ), where k R has asymptotes at a. The value of a must be. The rational function shown in the graph to the right has two vertical asymptotes. The function could be represented by the equation f( ). The value of k must be. ( )( k ) f 0
Written Response Determine the eact values of a and b in simplified mied radical form. b a Epress each of the below in simplest form. All denominators should be rationalized and all eponents shown as positive. a. b. c. d. 8
e. 0 80 f. g. 9 h. 8 8 A student is given three problems to solve. She correctly answers each question. Her answers are shown in the table below. Problem Problem # Problem # Problem # Simplify 8 Epand and simplify Simplify by rationalizing the denominator. Solution 0 Determine how this student found her answers by showing all steps in each problem solution. 8 The following question is based on the diagram shown on the right. A (0,) a. Find the length of side AC using Pythagorean theorem ( a b c ), and represent your answer in simplest mied radical form. b. Using your answer from (a), epress the perimeter of the triangle in simplest mied radical form. -0 - B (-,0) C (,0) -
Complete each of the following; a. Multiply by its conjugate, and simplify your solution. b. An estimated value of is 0. Eplain how a student can conclude that 0 is a correct estimation of. c. Write the epression 8 in simplest mied radical form. d. Rationalize the denominator for the epression e. Write 0 as an entire radical.. Complete each of the following; a. Multiply by its conjugate, and simplify your solution. b. Write the epression in simplest mied radical form. 0 c. Divide, and epress your simplified answer with a rationalized denominator. 00 d. Multiply and epress your answer in its simplest form. Simplify each of the following and state all restrictions. 8y a. y b. c. a a a 0 t t t t t t a a a 9a 0 d.
8 Solve each of the following rational equations and state all restrictions. a. b. 9 A student is trying to solve the following problem. Dividing 0 by a number gives the same result as dividing by less than the same number. a. Set up an equation using the variable that will solve the problem stated. b. Solve your problem from (a) for and state the restrictions. 0 Simplify each of the following and state all restrictions. b b a. c ac b. c. 9 9 8 d. Solve each of the following rational equations and state any restrictions. a a a. b. In a regular polygon with n sides, the measure of each angle, a, is given by the 0 formula a 80. Susan reasons that a rectangle is a regular polygon n 0 with sides, and so a 80 giving each angle a measure of 90. a. The diagram to the right is a regular polygon. What is the measure of each interior angle? b. If each interior angle of a regular polygon is, how many sides does it have?
Solve each of the following equations/inequalities algebraically. No marks will be given for solutions generated by graphical methods only. For complete marks, all restrictions must be stated. 8 a. b. c. d. e. f. A student attempts to solve the equation. Their solution is shown below; 9 ( ) a. At this point in their solution, the student has made eactly two errors. Recopy their steps shown above, and circle eactly where their errors have occurred. What are the restrictions to the solution of this problem? b. The student correctly does the problem in their second attempt, and determines a simplified relation 0 89 0. Solve this problem by factoring. c. Show that your answers from c correctly solve the problem.. Each of the following equations has been correctly solved. Show, by algebraic methods, that the answers given must be the only solutions to each equation. Category Equation Solution Rational Radical Absolute Value 8,, No real solution
. A student is taking a quilting class. Her project will be a quilt, made of 0 squares sewn together. The original design calls for the quilt to be 0 squares wide. a. What is the length of the quilt according to the original design? b. The student decides to increase the length by l squares, at the same time reducing the width by w squares. She 0 derives the formula l, but later discovers w 0 some errors in her relation. State the proper relation between l and w. 0 c. The correct graph of the relation from question (b) is shown on the right, with an ordered pair labelled on the 0 w, graph. Show that the ordered pair results in a different quilt than the original design, but all 0 squares will still be used, and eplain your answer. l w y 0 9 8 0 8 9 0 0. Rita trained for a marathon. After running the marathon, she looked back at her training program, and realized that she had run a total of 00 km spread over the 0 training days she used. Rita ran the same distance each day. a. How far did Rita run each day? b. Rita decides to write an article about her training methods. As part of her article, she would like to include a formula readers can use if they choose to use more than 0 days to run 00 km. She lets the variable t represent the increase in number of training days, and the variable d represent the corresponding decrease in km run each day. She 00 generates the incorrect equation d 8. 0 t Correct Rita s equation, and write your final epression as change in distance d relative to change in running days, t. c. After generating the correct equation (with your help), Rita correctly publishes the graph shown on the right. Eplain how a runner can make effective use of this graph to assist them in their training program, and use specific information from the graph in your detailed eplanation. d t 0.9.8....... 0.9 0.8 0. 0. 0. 0. 0. 0. 0. 0 8 0 8 0 0 t 0 8. Sketch m( ) using the graph to your right as a guide. m 0
Decrease in number of cards in each row 9. A student buys a memory card game for her younger brother. There are 8 cards in the game. When the game is played, the cards are arranged face down on a table, in rows, with an equal number of cards in each row. The goal of the game is to turn over matching cards, two at a time, in an attempt to match cards. There are pairs of matching cards. a. The first time the game is played, the cards are arranged in rows as the rules of the game suggest. How many cards are in each row? b. The Pure Math 0 student correctly identifies that she can represent different configurations of the game by increasing the number of rows, r, and decreasing the number of cards in each row, c, based on the suggestion in the rules as eplained in (a). She determines an equation for the decrease in the number of cards in each row, c, relative to the increase in the number of rows, r, 8 as c, but has made an error in her epression. Determine the correct epression she r is trying to model. c. After generating the correct equation (with your help), the Pure Math 0 student correctly determines the graph on the right, which shows c as a function of r. According to this graph, if she increases the number of rows by, so she now has rows, and she must decrease the number of cards in each row by. Eplain why this will not work as a configuration for this game. c r 0 8 0 8 d. While the graph to the right is correct, it will only assist in certain configurations of the game. By choosing an ordered pair from this graph, eplain how it will yield a possible configuration of the 8 cards used to play this game. 0 8 9 0 r Increase in rows