Mathematics Algebra II Unit 07: Rational Functions, Equations & Inequalities

Transcription

1 Mathematics Algebra II Unit 07: Rational Functions, Equations & Inequalities

2 1 Consider the rational function 4 Rhonda is looking at a table of data representing a translation of the A. Completely factor both the numerator and the denominator. Simplify the expression. B. What is the equation of the vertical asymptote? C. At what x value is the function discontinuous? Identify the type of discontinuity. function. She notices that when x is 2, the table says error for the y value. She notices that as the x values approach negative infinity, the y values seem to approach 3; and as the x values approach positive infinity, the y values again seem to approach 3. What is the equation of the translation of the parent function she is observing in the table? 2 For which rational function is the domain equal to all real numbers and the range is y > 0.25? F G H J 3 If the x in a rational parent function is replaced with x + 8, how is the graph changed? A B C D The graph is shifted up 8 units. The graph is shifted down 8 units. The graph is shifted right 8 units. The graph is shifted left 8 units. Page 2 GO ON

3 5 With the stopper in place and the faucet running, a bathtub can be filled in 1 minutes. With the faucet off and the stopper removed, the tub will empty in 20 minutes. If the faucet is running and the stopper is not in place, how long will it take to fill the tub? 7 Which graph could be the graph of an inverse variation relationship? A B C The cost per person to rent a chartered bus varies inversely to the number of people who ride the bus. If 40 people pay $9.50 each to ride the chartered bus, what is the cost per person if only 25 people go? D F $15.20 G $19.00 H $ J $ Page 3 GO ON

4 8 Which graph represents the solution of the following rational inequality? F 10 A motorboat goes 3 miles upstream on a river whose current is running at 3 miles per hour. The trip up and back takes 5 hours. What is the speed of the boat (assuming that it maintains a constant speed relative to the water)? F 9 mph G H G H J 12 mph 15 mph 18 mph J 11 The graph below shows the solution set for the inequality 9 Carl can do a particular job in 4 hours. It takes Mike.5 hours to do the same job. Which equation shows how long it will take the boys to complete the job if they work together? A B C 4x +.5x = 1 D Which ordered pair is not in the solution set of the inequality? A ( 2, 0) B (0, 2) C (0, 1) D (2, 0) Page 4 GO ON

5 12 A. Explain how the number line below can 15 State the domain of the rational function be used to solve. A {x x є R; x 3} B {x x є R; x ±3} C {x x є R; x 5} D all real numbers B. Solve the inequality. 13 Find a solution to the following equation, if one exists. Show the work that leads to your answer. 14 Solve by graphing: Page 5 GO ON

6 1 The graph of f(x) = below. is shown 17 What is a reasonable domain and range for this scenario? Water is added to a 30 milliliter solution that is 50% acid. The equation represents the concentration of acid, A, in the mixture as, x, amount of water added. If the graph is translated 3 units to the left and 3 units up, what is a reasonable domain and range for the new function? F Domain: {x is the set of all real numbers, x 3} Range: {y is the set of all real numbers, y 3} G Domain: {x is the set of all real numbers, x 5} Range: {y is the set of all real numbers, y 0} 18 The Johnson family is preparing to vacation at a famous amusement park. The park offers a family ticket discount package for an unlimited number of rides and will charge an entry fee for each person in the family. The equation below represents the cost per person as a function of the number of people in the family. H J Domain: {x is the set of all real numbers, x 3} Range: {y is the set of all real numbers, y 2} Domain: {x is the set of all real numbers, x 2} Range: {y is the set of all real numbers, y 3} Which of the following statements is TRUE? F The family ticket package costs $120, and the entry fee is $7.50 per person. G The family ticket package costs $7.50, and the entry fee is $120 per person. H J If people use the family package, the cost per person would be $9.50. If the cost per person is $19.50, then there are 12 people in the family. Page GO ON

7 19 An automobile's velocity starting from a complete stop is where v is measured in feet per second. What happens to the auto's velocity as time increases? A B C D The velocity continues to increase. The velocity begins to decrease. The velocity approaches 140 feet per second. The velocity approaches 28 feet per second. 20 A football team has a record of 9 wins and 12 losses. How many consecutive games must they win in order to raise their winning record to 50%? Page 7 STOP

8 Test Key Mathematics Algebra II Unit 07: Rational Functions, Equations & Inequalities ## Item # Correct Answer Primary SE Secondary SE Obj/Cat 1 A R8CS A. B. A vertical asymptote is present at x = - 2. A2.10(A) C. A hole is present at x = 2. 2 A CS G A2.10(A) 3 A CS D A2.10(A) 4 A CS A2.10(B) 5 M0AII01313CS 80 A2.10(G) LOC20857 F A2.10(G) 7 A CS B A2.10(G) 8 LOC20858 J A2.10(F) [R] 9 A220133CS D A2.10(F) [R] 10 A220859CS H A2.10(F) [R] 11 A22012CS D A2.10(E) 12 A CS A. Since x = -3 makes the quotient 0 and x = 7 makes the quotient undefined, these divide the number line into 3 test intervals. Pick a number from each interval and plug that number into the inequality to see which intervals make the inequality true OR these are the points on the x-axis where the value of y changes. A2.10(E) B. x < -3 or x > A CS No solution. A2.10(D) 14 A CS 0 to 3 A2.10(D) 15 A220130CS D A2.10(C) 1 A220128CS J A2.10(C) 17 A CS Domain: x > 0; Range: because there will always be some acid in the mixture. A2.10(C)

9 18 A220134CS F A2.10(B) 19 A220131CS D A2.10(B) 20 M0AII01314CS 3 A2.10(F) [R]

10 Scoring Rubrics 14 The graphs cross at (, 4). The equation will equal 4 when x =. 3 The response shows full understanding of the essential mathematics applicable to the task and a sound approach toward solution that includes logical reasoning and appropriate conclusions. Computation and procedures used are generally accurate, but the response may contain minor computational or procedural flaws that do not detract from evidence of full understanding. 2 The response shows a satisfactory understanding of the essential mathematics applicable to the task, but reasoning may not be completely clear, and there may be minor flaws in computation and/or use of procedures as a result of carelessness or non-essential misunderstandings. The flaws do not detract from evidence of satisfactory understanding. A score of 2 may also be earned if the response is partially correct but some aspect of the task is omitted. 1 The response indicates limited understanding of the essential mathematics applicable to the task. While an effort is made to address the task, omissions and/or errors related to insufficient mathematical knowledge or incorrect application of skills or procedures bring into question that student's ability to deal successfully with tasks of this type. 0 The response indicates no understanding of the essential mathematics applicable to the task, or there is no response.