Chapter 9 Prerequisite Skills

Size: px
Start display at page:

Download "Chapter 9 Prerequisite Skills"

Transcription

1 Name: Date: Chapter 9 Prerequisite Skills BLM 9. Consider the function f() 3. a) Show that 3 is a factor of f(). If f() ( 3)g(), what is g()?. Factor each epression fully. a) 30g 4g 6fg 8g c) 6 5 d) 5 e) 6a a 3 f) 0.0 g) 0 y Consider the sketch of the function h() a) What are the zeros of the function? Epress h() in factored form. c) What are the roots of the equation ? 4. Identify all non-permissible s for each epression. 4a a) 3bc 5 c) d) 3 5 t 6t 9 6 e) f) (3 )( 5) ( a )( a )( a 3) 5. Consider P() a) Use the Factor Theorem to find a factor of P(). Completely factor P(). c) What is the solution of P() 0? d) Sketch the graph of P(). Eplain the significance of the -intercepts. 6. a) Arrange the equation 4 into the form a b c 0, where a, b, and c are integers. State the solution for this equation. Rewrite the function you developed in part a) in the form y ( h) k, where h and k are real numbers. c) Graph the function. Eplain the relationship of the -intercepts and coordinates of the verte to the functions in parts a) and. 7. Solve each equation algebraically. Epress answers as eact s and identify non-permissible s, where necessary. a) c) d) e) f) 4 8. Complete the square. Leave your answers in the form a( h) k 0, where a, h, and k are real numbers. a) c) 3 6 d) Solve each equation by graphing the corresponding function. a) Simplify each rational epression. State any non-permissible s for the variables. 0k 55k 75 a) 3 0k 0k 50. Simplify each product or quotient. Identify all non-permissible s. 4z 5 z 4 a) z 3z 0 4z Copyright 0, McGraw-Hill Ryerson Limited, ISBN:

2 Name: Date: Section 9. Etra Practice BLM 9. Match each function with its graph. 4 4 a) y 3 y 3 4 c) y 3 d) 4 y 3 A D B C 5. Graph the function y using a table of s. Analyse your graph and use a table to summarize the following characteristics: non-permissible (s) behaviour near non-permissible (s) end behaviour domain range equation of vertical asymptote equation of horizontal asymptote 3. Sketch and graph each function. Identify the domain and range, intercepts, and asymptotes. 3 a) y y 6 5 c) y 4 d) y 8 4. Graph each function using technology, and identify any asymptotes and intercepts. 5 a) y 4 3 y Copyright 0, McGraw-Hill Ryerson Limited, ISBN:

3 Name: Date: BLM 9 5. Write the equation of each function in the a form y k. h a) c) d) a 6. The rational function y k passes 5 through points (6, 7) and (4, ). a) Determine the of a and k. Graph the function. 7. Sketch the graph of y and y on the same set of aes. 6 9 Describe how one is a transformation of the other. 8. Use a table of s and a graph to analyse the function y. Then, complete the 7 table. Characteristic Non-permissible Behaviour near non-permissible End behaviour Domain Range Equation of vertical asymptote Equation of horizontal asymptote y 7 9. The distance between two cities is 35 km. a) Write an epression that you can use to calculate the time, t, in hours, that it takes to travel distance, d, in kilometres, at an average speed of s km/h. Use your formula from part a) to determine how long it will take to travel from one city to the other at an average speed of 65 km/h. c) If the trip from one city to the other took 5 h, determine the average speed, s, in kilometres per hour. Copyright 0, McGraw-Hill Ryerson Limited, ISBN:

4 Name: Date: Section 9. Etra Practice BLM 9 3. Eplain the behaviour at each non-permissible in the graph 5 6 of the rational function y Analyse each function and predict the location of any vertical asymptotes, points of discontinuity, and intercepts. Then, graph the function to verify your predictions. 5 7 a) y y c) 5 3 y d) y 3 6. Complete the table and compare the behaviour of the two functions near any nonpermissible s. Characteristic y y Eplain how the equation of the rational function y can be analysed to 3 determine whether the graph of the function has an asymptote or a point of discontinuity. 3. Complete the table for the given rational function. Characteristic Non-permissible (s) Feature ehibited at each non-permissible Behaviour near each non-permissible Domain Range ( 3)( ) y ( 5)( 3) 4. Create a table of s for each function for s near its non-permissible (s). Eplain how your table shows whether a point of discontinuity or an asymptote occurs in each case. 5 4 a) 5 4 y y 6 8 Non-permissible (s) Feature ehibited at each nonpermissible Behaviour near each nonpermissible 7. Without using technology, match the equation of each rational function with the most appropriate graph. Eplain your reasoning. 4 4 a) y y c) y 4 A Copyright 0, McGraw-Hill Ryerson Limited, ISBN:

5 Name: Date: BLM 9 3 B C c) 8. Write the equation for each graphed rational function. a) d) 9. Write the equation of a possible rational function that has an asymptote at, has a point of discontinuity at.5, and passes through (6, 3). Copyright 0, McGraw-Hill Ryerson Limited, ISBN:

6 Name: Date: Section 9.3 Etra Practice BLM 9 4. Solve each equation algebraically. a) c) d) Solve algebraically. Check your solutions. 3 a) 3 9 c) d) 3 3. a) Determine the roots of the rational equation 5 60 algebraically. 5 Graph the rational function y 6 and determine the -intercepts. c) Eplain the connection between the roots of the equation and the -intercepts of the graph of the function. 4. Solve each of the following equations by graphing each side of the equation as a separate function. 6 a) c) 3 5. Solve by rearranging as a single function and then graphing. a) Solve each equation algebraically. Give your answers to the nearest hundredth. a) c) Determine the approimate solution(s) to each rational equation graphically, to the nearest hundredth. 3 a) Solve the equation algebraically. 8 n n 9 n 3 9. It takes James 9 h longer to construct a fence than it takes Carmen. If they work together, they can construct the fence in 0 h. How long would it take each of them, working alone, to construct the fence? Copyright 0, McGraw-Hill Ryerson Limited, ISBN:

7 Name: Date: Chapter 9 Study Guide BLM 9 5 This study guide is based on questions from the Chapter 9 Practice Test in the student resource. Question I can Help Needed Refer to # eplain the behaviour of the graph of a rational function for s of the variable near a non-permissible some none # match a function to its graph some none #3 eplain the behaviour of the graph of a rational function some for s of the variable near a non-permissible none #4 describe the relationship between the roots of a rational some equation and the -intercepts of the graph of the none corresponding rational function #5 write equations in equivalent forms some #6 determine if the graph of a rational function will have an asymptote or a point of discontinuity for a nonpermissible #7 determine the solution of a rational equation algebraically and graphically #8 analyse the graph of a rational function to identify characteristics graph a rational function with linear epressions in the numerator and denominator #9 determine, graphically, an approimate solution of a rational equation none some none some none some none some none some none #0 sketch a graph with or without technology some none eplain the behaviour of the graph of a rational function some for s of the variable near a non-permissible none # analyse a rational function to identify characteristics some none # match a set of rational functions to their graphs, and some eplain the reasoning none #3 analyse the graphs of a set of rational functions to some compare characteristics none #4 solve a rational equation algebraically and graphically some none #5 solve problems involving rational equations some none #6 solve problems involving rational equations some none 9. Eample 9. Eample Eample 9.3 Eample 9.3 Eample 9. Eample 9.3 Eample 9. Eample 9. Eample Eample 9. Eample 9. Eamples, 9. Eample 3 9. Eample 3 9. Eample 9.3 Eamples, 3 9. Eample 5 9. Eample 5 Copyright 0, McGraw-Hill Ryerson Limited, ISBN:

8 Name: Date: Chapter 9 Test Multiple Choice For # to #8, choose the best answer. k. The -intercept of y is 0.5. What is the of k? A.0 B.5 C.5 D 3.0. Consider the function g ( ). Which statement is false? A g() has two vertical asymptotes. B g() is not defined when 0. C g() has one zero. D g() is a rational function. 3. Consider the functions f(), g(), ( ) and h ( ) f. Which statement is true? g( ) A f(), g(), and h() have the same domain. B The zero of f() is the vertical asymptote of h(). C The non-permissible of h() is the zero of g(). D h() is equivalent to y Consider the following graph of the function f ( ) r. What is the of r? A 3 B C D 3 5. Which of the following is true of the 3 rational function y 6? BLM 9 6 A It has a zero at. B Its range is {yy R}. 6 9 C It is equivalent to y. D It has a vertical asymptote at The graph of which function has a point of discontinuity at? A y B y C y D y 7. Which function has a domain of {, R} and a range of {yy 3, y R}? A y B y C y D y 8. How many roots does the equation 8 have? 6 4 A 0 B C D 3 Short Answer 9. a) Sketch the graph of the function y. 4 Identify the domain, range, and asymptotes of the function. c) Eplain the behaviour of the function as the of becomes very large. 0. a) Sketch the graph of the function y. 5 State the s of the -intercept and y-intercept. c) Solve 0 algebraically. 5 d) How is your answer to part c) related to your answers to parts a) and? Copyright 0, McGraw-Hill Ryerson Limited, ISBN:

9 Name: Date: BLM 9 6. Select the graph that matches the given function. a) y 5 y 3 ( 3) ( 5) c) y ( 5) A B C 3 3. a) Solve the equation 7 algebraically. Check your answer to part a) graphically. Etended Response 3. a) Graph the functions f( ) and 3 g ( ). Use a table to compare the characteristics of the two graphs. Write g() as a transformation of f(): g() f( a) b. c) Describe the transformation of f() to g(). 4. a) Describe two methods you could use to solve the equation 3 5 ( ) graphically. 4 Use one of the methods from part a) to solve the equation. 5. A rectangle has an area of 6000 cm. a) Write an equation to represent length, l, as a function of the width, w, for this rectangle. Write an equation to represent the change in length, as a function of width, w, when the width is increased by cm. c) Determine the width, w, of the rectangle if the change in length is 0 cm. 6. An emergency patrol boat is patrolling a river. The river has a 5 km/h current. The patrol boat travels 0 km upriver and 0 km back. The total time, t, in hours, for the round trip is given by the function 0v t, where v is the speed of the boat v 5 in kilometres per hour. a) State the domain and range for this function. Sketch the graph over the domain determined in part a). c) Determine the speed of the boat if the round trip took.5 h. Copyright 0, McGraw-Hill Ryerson Limited, ISBN:

10 Chapter 9 BLM Answers BLM 9 Prerequisite Skills. a) g() 5 4. a) g(5 g) g(3f 4g) c) ( 5)( ) d) ( )( 5) e) (a 3)(3a ) f) ( 0.)( 0.) g) 5(y 3)(y 3) 3. a),, and 3 h() ( )( )( 3) c),, and a) b 0, c 0 c), d) t 3 5 e), f) a, a, and a a) P() () 3 5() () 8 0, therefore, ( ) is a factor. P() ( )( 4)( ) c),, 4 d) 7. a) 7 3, c) d) e), 0, 4 3 f) 6,, 8. a) ( 5) 0 ( 4) 3 0 c) 3( ) 0 d) ( ) a) 4, 5,, BLM 9 7 The -intercepts are the solution to P() a) 4 0; 6 and y ( ) 6 c) 0. a),, 3 3. a), z 4, 5 k 3 5, k,3 k 3 3 3,,0,5 BLM 9 Section 9. Etra Practice. a) B C c) D d) A. The -intercepts are the solution to the equation. The -coordinate of the verte is the that makes ( h) 0 and the y-coordinate is equal to k. Copyright 0, McGraw-Hill Ryerson Limited, ISBN:

11 BLM 9 7 Characteristic 5 y Non-permissible Behaviour near nonpermissible As approaches, y becomes very large. End behaviour As becomes very large, y approaches 0. Domain {, R} Range {yy 0, y R} Equation of vertical asymptote Equation of horizontal y 0 asymptote 3. a) d) domain: {, R}; range: {yy 8, y R}; intercepts: (0, 8.5), (.5, 0); asymptotes:, y 8 4. a) domain: {, R}; range: {yy 0, y R}; intercept: (0, 3); asymptotes:, y 0 asymptotes:, y ; intercepts: (.5, 0), (0, 5) domain: { 0, R}; range: {yy 6, y R}; intercept: 3,0 ; asymptotes: 0, y 6 c) asymptotes:, y 4; intercepts: (0,.5), (0.75, 0) a) y y c) y d) 5 6. a) a 3, k 4 y 4 domain: { 4, R}; range: {yy, y R}; intercepts: (0, 0.75), (.5, 0); asymptotes: 4, y Copyright 0, McGraw-Hill Ryerson Limited, ISBN:

12 BLM a) t d s 35 t 5.4, so 5.4 hours or 5 h and 4 min 65 c) 70. km/h The graph of y is the graph of y 6 9 translated 3 units left. 8. y undefined Characteristic y 7 Non-permissible 7 Behaviour near nonpermissible End behaviour As approaches 7, y becomes very large. As becomes very large, y approaches. Domain { 7, R} Range {yy, y R} Equation of vertical asymptote Equation of horizontal asymptote 7 y BLM 9 3 Section 9. Etra Practice. point of discontinuity at (3, 0 ) vertical asymptote: 7. You can factor the denominator: y ( )( ). Since the factor ( ) appears in the numerator and denominator, the graph will have a point of discontinuity at (, ). The factor ( ) appears in the denominator only, so there will be an asymptote at. 3. Characteristic ( 3)( ) y ( 5)( 3) Non-permissible (s) 5 and 3 Feature ehibited at each non-permissible Behaviour near each non-permissible asymptote at 5; point of discontinuity at (3,.5) As approaches 5, y becomes very large. As approaches 3, y approaches.5. Domain { 3, 5, R} Range 4. a) y undefined {yy, As approaches, y approaches 3. 5, y R} Copyright 0, McGraw-Hill Ryerson Limited, ISBN:

13 BLM 9 7 y undefined y undefined As approaches, y approaches 4.5, and as approaches 4, y becomes very large, approaching negative infinity or positive infinity. 5. a) vertical asymptote: ; point of discontinuity at (5, 5 3 ); -intercept: (0, 0); y-intercept: (0, 0) vertical asymptote: 3; point of discontinuity at (3, ); 6 4 -intercept: (4, 0); y-intercept: (0, ) 3 c) no vertical asymptote; point of discontinuity at (, 3); -intercept: (4, 0); y-intercept: (0, 4) d) no vertical asymptote; point of discontinuity at (3, 7); -intercept: (0.5, 0); y-intercept: (0, ) 6. Characteristic Non-permissible (s) Feature ehibited at each nonpermissible Behaviour near each nonpermissible y y 3 3 point of discontinuity As approaches 3, y approaches asymptote As approaches 3, y becomes very large. 7. a) C; Eample: In factored form, the rational function has two non-permissible s in the denominator, which do not appear in the numerator. Therefore, the graph with two asymptotes is the most appropriate choice. B; Eample: In factored form, the rational function has one non-permissible that appears in both the numerator and denominator, and another nonpermissible that is only in the denominator. Therefore, the graph with one asymptote and one point of discontinuity is the most appropriate choice. c) A; Eample: In factored form, one non-permissible appears in the numerator and denominator. Therefore, the graph has a point of discontinuity, but no asymptote. Copyright 0, McGraw-Hill Ryerson Limited, ISBN:

14 BLM or y or y 4 y 4 or y 4 c) y or y 5 d) y a) y 9. Eample: y ( 5) ( )( 5) BLM 9 4 Section 9.3 Etra Practice 3. a) 5 and 6 c) 0.5 and 5 c) 4 d) 4. a) 0 and 4 7 and c) 0 and 3 d) 3 and 3. a) 5 and c) The of the function is 0 when the of is or 5. The -intercepts of the graph of the function are the same as the roots of the corresponding equation. 4. a) 0 and a) 0 8 and 6 Copyright 0, McGraw-Hill Ryerson Limited, ISBN:

15 BLM y BLM 9 6 Chapter 9 Test. D. B 3. C 4. D 5. C 6. A 7. C 8. B 9. a) 3 6. a) 0.76 and and.79 c) 0.53 and a) 0.63 domain: {, R}; range: {yy 0,, y R}; 4 vertical asymptote: ; horizontal asymptote: y 0 c) As becomes very large, y approaches a) -intercept 3; y-intercept 0.6 c) 3 d) The root of the equation is the same as the -intercept.. a) B A c) C. a) 0.85 and The solution n 3 is a non-permissible, so there is no solution. 9. Carmen: 36 h; James: 45 h Copyright 0, McGraw-Hill Ryerson Limited, ISBN:

16 BLM a) g() = f ( ) + 3 g ( ) 3 c) a vertical translation 3 units up and a horizontal translation unit right 4. a) Eample: 3 5 Method : Graph y ( ), and 4 determine the -intercepts of the graph. 3 5 Method : Graph y and y ( ), 4 and determine the -coordinates of the points where the two graphs intersect. Characteristic Nonpermissible Behaviour near nonpermissible End behaviour Domain Range Equation of vertical asymptote Equation of horizontal asymptote f( ) 0 As approaches 0, y becomes very large. As becomes very large, y approaches 0. { 0, R} {y y 0, y R} 0 y 0 y 3 3 g ( ) As approaches, y becomes very large. As becomes very large, y approaches 3. {, R} {y y 3, y R} and a) l w change in l w w 6000 w w c) w 4 cm 6. a) domain: {vv > 5, v R}; range: {tt > 0, t R} c) v 5 km/h Copyright 0, McGraw-Hill Ryerson Limited, ISBN:

Chapter 9 BLM Answers

Chapter 9 BLM Answers Chapter 9 BLM Answers BLM 9 Prerequisite Skills. a) x 5x4 x x x x x x 5x x 5x 5x 4x 4x 0 g(x) x 5x 4. a) g(5 g) g(f 4g) c) (x 5)(x ) d) (x )(x 5) e) (a )(a ) f) (x 0.)(x 0.) g) 5(xy )(xy ). a),, and h(x)

More information

Section 5.1 Extra Practice

Section 5.1 Extra Practice Name: Date: Section.1 Etra Practice BLM 1. Epress each radical as a simplified mied radical. 0 98, 0 6 y, 0, y 0. Epress each mied radical as an equivalent entire radical. 1 9, 0 y 7 y, 0, y 0. Order each

More information

RADICAL AND RATIONAL FUNCTIONS REVIEW

RADICAL AND RATIONAL FUNCTIONS REVIEW RADICAL AND RATIONAL FUNCTIONS REVIEW Name: Block: Date: Total = % 2 202 Page of 4 Unit 2 . Sketch the graph of the following functions. State the domain and range. y = 2 x + 3 Domain: Range: 2. Identify

More information

Chapter 8 Prerequisite Skills

Chapter 8 Prerequisite Skills Chapter 8 Prerequisite Skills BLM 8. How are 9 and 7 the same? How are they different?. Between which two consecutive whole numbers does the value of each root fall? Which number is it closer to? a) 8

More information

Chapter. Part 1: Consider the function

Chapter. Part 1: Consider the function Chapter 9 9.2 Analysing rational Functions Pages 446 456 Part 1: Consider the function a) What value of x is important to consider when analysing this function? b) Now look at the graph of this function

More information

Exam 2 Review F15 O Brien. Exam 2 Review:

Exam 2 Review F15 O Brien. Exam 2 Review: Eam Review:.. Directions: Completely rework Eam and then work the following problems with your book notes and homework closed. You may have your graphing calculator and some blank paper. The idea is to

More information

PACKET Unit 4 Honors ICM Functions and Limits 1

PACKET Unit 4 Honors ICM Functions and Limits 1 PACKET Unit 4 Honors ICM Functions and Limits 1 Day 1 Homework For each of the rational functions find: a. domain b. -intercept(s) c. y-intercept Graph #8 and #10 with at least 5 EXACT points. 1. f 6.

More information

Composition of and the Transformation of Functions

Composition of and the Transformation of Functions 1 3 Specific Outcome Demonstrate an understanding of operations on, and compositions of, functions. Demonstrate an understanding of the effects of horizontal and vertical translations on the graphs of

More information

Five-Minute Check (over Lesson 8 3) CCSS Then/Now New Vocabulary Key Concept: Vertical and Horizontal Asymptotes Example 1: Graph with No Horizontal

Five-Minute Check (over Lesson 8 3) CCSS Then/Now New Vocabulary Key Concept: Vertical and Horizontal Asymptotes Example 1: Graph with No Horizontal Five-Minute Check (over Lesson 8 3) CCSS Then/Now New Vocabulary Key Concept: Vertical and Horizontal Asymptotes Example 1: Graph with No Horizontal Asymptote Example 2: Real-World Example: Use Graphs

More information

7.4 RECIPROCAL FUNCTIONS

7.4 RECIPROCAL FUNCTIONS 7.4 RECIPROCAL FUNCTIONS x VOCABULARY Word Know It Well Have Heard It or Seen It No Clue RECIPROCAL FUNCTION ASYMPTOTE VERTICAL ASYMPTOTE HORIZONTAL ASYMPTOTE RECIPROCAL a mathematical expression or function

More information

UNIT 3. Rational Functions Limits at Infinity (Horizontal and Slant Asymptotes) Infinite Limits (Vertical Asymptotes) Graphing Rational Functions

UNIT 3. Rational Functions Limits at Infinity (Horizontal and Slant Asymptotes) Infinite Limits (Vertical Asymptotes) Graphing Rational Functions UNIT 3 Rational Functions Limits at Infinity (Horizontal and Slant Asymptotes) Infinite Limits (Vertical Asymptotes) Graphing Rational Functions Recall From Unit Rational Functions f() is a rational function

More information

For problems 1 4, evaluate each expression, if possible. Write answers as integers or simplified fractions

For problems 1 4, evaluate each expression, if possible. Write answers as integers or simplified fractions / MATH 05 TEST REVIEW SHEET TO THE STUDENT: This Review Sheet gives you an outline of the topics covered on Test as well as practice problems. Answers are at the end of the Review Sheet. I. EXPRESSIONS

More information

The region enclosed by the curve of f and the x-axis is rotated 360 about the x-axis. Find the volume of the solid formed.

The region enclosed by the curve of f and the x-axis is rotated 360 about the x-axis. Find the volume of the solid formed. Section A ln. Let g() =, for > 0. ln Use the quotient rule to show that g ( ). 3 (b) The graph of g has a maimum point at A. Find the -coordinate of A. (Total 7 marks) 6. Let h() =. Find h (0). cos 3.

More information

UNIT #9 ROOTS AND IRRATIONAL NUMBERS REVIEW QUESTIONS

UNIT #9 ROOTS AND IRRATIONAL NUMBERS REVIEW QUESTIONS Answer Key Name: Date: UNIT #9 ROOTS AND IRRATIONAL NUMBERS REVIEW QUESTIONS Part I Questions. Which of the following is the value of 6? () 6 () 4 () (4). The epression is equivalent to 6 6 6 6 () () 6

More information

UNIT 3. Recall From Unit 2 Rational Functions

UNIT 3. Recall From Unit 2 Rational Functions UNIT 3 Recall From Unit Rational Functions f() is a rational function if where p() and q() are and. Rational functions often approach for values of. Rational Functions are not graphs There various types

More information

4.5 Rational functions.

4.5 Rational functions. 4.5 Rational functions. We have studied graphs of polynomials and we understand the graphical significance of the zeros of the polynomial and their multiplicities. Now we are ready to etend these eplorations

More information

Section 10.1 Extra Practice

Section 10.1 Extra Practice Section 0. Extra Practice BLM 0. For each pair of functions, determine h(x) f (x) g(x). a) f (x) x 4 g(x) f (x) x 7 g(x) 5x c) f (x) x 3x g(x) x x 5 d) f (x) (x 4) g(x) 7x. Consider the functions f (x)

More information

Lesson 7.1 Polynomial Degree and Finite Differences

Lesson 7.1 Polynomial Degree and Finite Differences Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 1 b. 0.2 1. 2 3.2 3 c. 20 16 2 20 2. Determine which of the epressions are polynomials. For each polynomial,

More information

8.2 Graphing More Complicated Rational Functions

8.2 Graphing More Complicated Rational Functions 1 Locker LESSON 8. Graphing More Complicated Rational Functions PAGE 33 Name Class Date 8. Graphing More Complicated Rational Functions Essential Question: What features of the graph of a rational function

More information

UNIT THREE FUNCTIONS MATH 611B 15 HOURS

UNIT THREE FUNCTIONS MATH 611B 15 HOURS UNIT THREE FUNCTIONS MATH 611B 15 HOURS Revised Jan 9, 03 51 SCO: By the end of Grade 1, students will be epected to: C1 gather data, plot the data using appropriate scales, and demonstrate an understanding

More information

Lesson 4.1 Exercises, pages

Lesson 4.1 Exercises, pages Lesson 4.1 Eercises, pages 57 61 When approimating answers, round to the nearest tenth. A 4. Identify the y-intercept of the graph of each quadratic function. a) y = - 1 + 5-1 b) y = 3-14 + 5 Use mental

More information

CHAPTER 8 Quadratic Equations, Functions, and Inequalities

CHAPTER 8 Quadratic Equations, Functions, and Inequalities CHAPTER Quadratic Equations, Functions, and Inequalities Section. Solving Quadratic Equations: Factoring and Special Forms..................... 7 Section. Completing the Square................... 9 Section.

More information

Name Date Period. Pre-Calculus Midterm Review Packet (Chapters 1, 2, 3)

Name Date Period. Pre-Calculus Midterm Review Packet (Chapters 1, 2, 3) Name Date Period Sections and Scoring Pre-Calculus Midterm Review Packet (Chapters,, ) Your midterm eam will test your knowledge of the topics we have studied in the first half of the school year There

More information

5.2 Solving Quadratic Equations by Factoring

5.2 Solving Quadratic Equations by Factoring Name. Solving Quadratic Equations b Factoring MATHPOWER TM, Ontario Edition, pp. 78 8 To solve a quadratic equation b factoring, a) write the equation in the form a + b + c = b) factor a + b + c c) use

More information

Honors Algebra 2 Chapter 9 Page 1

Honors Algebra 2 Chapter 9 Page 1 Introduction to Rational Functions Work Together How many pounds of peanuts do you think and average person consumed last year? Us the table at the right. What was the average peanut consumption per person

More information

Section 3.3 Limits Involving Infinity - Asymptotes

Section 3.3 Limits Involving Infinity - Asymptotes 76 Section. Limits Involving Infinity - Asymptotes We begin our discussion with analyzing its as increases or decreases without bound. We will then eplore functions that have its at infinity. Let s consider

More information

Introduction to Rational Functions

Introduction to Rational Functions Introduction to Rational Functions The net class of functions that we will investigate is the rational functions. We will eplore the following ideas: Definition of rational function. The basic (untransformed)

More information

Analyzing Rational Functions

Analyzing Rational Functions Analyzing Rational Functions These notes are intended as a summary of section 2.3 (p. 105 112) in your workbook. You should also read the section for more complete explanations and additional examples.

More information

G r a d e 1 1 P r e - C a l c u l u s M a t h e m a t i c s ( 3 0 S ) Final Practice Exam Answer Key

G r a d e 1 1 P r e - C a l c u l u s M a t h e m a t i c s ( 3 0 S ) Final Practice Exam Answer Key G r a d e P r e - C a l c u l u s M a t h e m a t i c s ( 3 0 S ) Final Practice Eam Answer Key G r a d e P r e - C a l c u l u s M a t h e m a t i c s Final Practice Eam Answer Key Name: Student Number:

More information

NAME DATE PERIOD. Study Guide and Intervention. Solving Quadratic Equations by Graphing. 2a = -

NAME DATE PERIOD. Study Guide and Intervention. Solving Quadratic Equations by Graphing. 2a = - NAME DATE PERID - Study Guide and Intervention Solving Quadratic Equations by Graphing Solve Quadratic Equations Quadratic Equation A quadratic equation has the form a + b + c = 0, where a 0. Roots of

More information

Introduction. A rational function is a quotient of polynomial functions. It can be written in the form

Introduction. A rational function is a quotient of polynomial functions. It can be written in the form RATIONAL FUNCTIONS Introduction A rational function is a quotient of polynomial functions. It can be written in the form where N(x) and D(x) are polynomials and D(x) is not the zero polynomial. 2 In general,

More information

Maths A Level Summer Assignment & Transition Work

Maths A Level Summer Assignment & Transition Work Maths A Level Summer Assignment & Transition Work The summer assignment element should take no longer than hours to complete. Your summer assignment for each course must be submitted in the relevant first

More information

Pre-Calculus Module 4

Pre-Calculus Module 4 Pre-Calculus Module 4 4 th Nine Weeks Table of Contents Precalculus Module 4 Unit 9 Rational Functions Rational Functions with Removable Discontinuities (1 5) End Behavior of Rational Functions (6) Rational

More information

9.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED LESSON

9.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED LESSON CONDENSED LESSON 9.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations solve

More information

Equations and Inequalities

Equations and Inequalities Equations and Inequalities Figure 1 CHAPTER OUTLINE 1 The Rectangular Coordinate Systems and Graphs Linear Equations in One Variable Models and Applications Comple Numbers Quadratic Equations 6 Other Types

More information

SOLVING QUADRATICS. Copyright - Kramzil Pty Ltd trading as Academic Teacher Resources

SOLVING QUADRATICS. Copyright - Kramzil Pty Ltd trading as Academic Teacher Resources SOLVING QUADRATICS Copyright - Kramzil Pty Ltd trading as Academic Teacher Resources SOLVING QUADRATICS General Form: y a b c Where a, b and c are constants To solve a quadratic equation, the equation

More information

Calculus I Practice Test Problems for Chapter 2 Page 1 of 7

Calculus I Practice Test Problems for Chapter 2 Page 1 of 7 Calculus I Practice Test Problems for Chapter Page of 7 This is a set of practice test problems for Chapter This is in no way an inclusive set of problems there can be other types of problems on the actual

More information

Math 20-1 Functions and Equations Multiple Choice Questions

Math 20-1 Functions and Equations Multiple Choice Questions 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

More information

#1, 2, 3ad, 4, 5acd, 6, 7, 8, 9bcd, 10, 11, 12a, 13, 15, 16 #1-5

#1, 2, 3ad, 4, 5acd, 6, 7, 8, 9bcd, 10, 11, 12a, 13, 15, 16 #1-5 MHF4U Unit 3 Rational Functions Section Pages Questions Prereq Skills 146-147 #1, 2, 3bf, 4ac, 6, 7ace, 8cdef, 9bf, 10abe 3.1 153-155 #1ab, 2, 3, 5ad, 6ac, 7cdf, 8, 9, 14* 3.2 164-167 #1ac, 2, 3ab, 4ab,

More information

1 DL3. Infinite Limits and Limits at Infinity

1 DL3. Infinite Limits and Limits at Infinity Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 78 Mark Sparks 01 Infinite Limits and Limits at Infinity In our graphical analysis of its, we have already seen both an infinite

More information

(a) Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3]. (2)

(a) Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3]. (2) . f() = 4 cosec 4 +, where is in radians. (a) Show that there is a root α of f () = 0 in the interval [.,.3]. Show that the equation f() = 0 can be written in the form = + sin 4 Use the iterative formula

More information

GUIDED NOTES 5.6 RATIONAL FUNCTIONS

GUIDED NOTES 5.6 RATIONAL FUNCTIONS GUIDED NOTES 5.6 RATIONAL FUNCTIONS LEARNING OBJECTIVES In this section, you will: Use arrow notation. Solve applied problems involving rational functions. Find the domains of rational functions. Identify

More information

SEE and DISCUSS the pictures on pages in your text. Key picture:

SEE and DISCUSS the pictures on pages in your text. Key picture: Math 6 Notes 1.1 A PREVIEW OF CALCULUS There are main problems in calculus: 1. Finding a tangent line to a curve though a point on the curve.. Finding the area under a curve on some interval. SEE and DISCUSS

More information

Name: Class: Date: A. 70 B. 62 C. 38 D. 46

Name: Class: Date: A. 70 B. 62 C. 38 D. 46 Class: Date: Test 2 REVIEW Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Divide: (4x 2 49y 2 ) (2x 7y) A. 2x 7y B. 2x 7y C. 2x 7y D. 2x 7y 2. What is

More information

Calculus Summer Packet

Calculus Summer Packet Calculus Summer Packet Congratulations on reaching this level of mathematics in high school. I know some or all of you are bummed out about having to do a summer math packet; but keep this in mind: we

More information

Baruch College MTH 1030 Sample Final B Form 0809 PAGE 1

Baruch College MTH 1030 Sample Final B Form 0809 PAGE 1 Baruch College MTH 00 Sample Final B Form 0809 PAGE MTH 00 SAMPLE FINAL B BARUCH COLLEGE DEPARTMENT OF MATHEMATICS SPRING 00 PART I (NO PARTIAL CREDIT, NO CALCULATORS ALLOWED). ON THE FINAL EXAM, THERE

More information

This problem set is a good representation of some of the key skills you should have when entering this course.

This problem set is a good representation of some of the key skills you should have when entering this course. Math 4 Review of Previous Material: This problem set is a good representation of some of the key skills you should have when entering this course. Based on the course work leading up to Math 4, you should

More information

Study Guide and Intervention. The Quadratic Formula and the Discriminant. Quadratic Formula. Replace a with 1, b with -5, and c with -14.

Study Guide and Intervention. The Quadratic Formula and the Discriminant. Quadratic Formula. Replace a with 1, b with -5, and c with -14. Study Guide and Intervention Quadratic Formula The Quadratic Formula can be used to solve any quadratic equation once it is written in the form a 2 + b + c = 0. Quadratic Formula The solutions of a 2 +

More information

Math 155 Intermediate Algebra Practice Exam on Chapters 6 & 8 Name

Math 155 Intermediate Algebra Practice Exam on Chapters 6 & 8 Name Math 155 Intermediate Algebra Practice Eam on Chapters 6 & 8 Name Find the function value, provided it eists. 1) f() = 7-8 8 ; f(-3) Simplify by removing a factor equal to 1. ) 5 + 0 + 3) m - m - m Find

More information

Radical and Rational Functions

Radical and Rational Functions Radical and Rational Functions 5 015 College Board. All rights reserved. Unit Overview In this unit, you will etend your study of functions to radical, rational, and inverse functions. You will graph radical

More information

Lesson 8.2 Exercises, pages

Lesson 8.2 Exercises, pages Lesson 8. Eercises, pages 38 A Students should verif the solutions to all equations.. Which values of are not roots of each equation? a) ƒ - 3 ƒ = 7 = 5 or =- Use mental math. 5: L.S. 7 R.S. 7 : L.S. 7

More information

SYSTEMS OF THREE EQUATIONS

SYSTEMS OF THREE EQUATIONS SYSTEMS OF THREE EQUATIONS 11.2.1 11.2.4 This section begins with students using technology to eplore graphing in three dimensions. By using strategies that they used for graphing in two dimensions, students

More information

Calculus 1 (AP, Honors, Academic) Summer Assignment 2018

Calculus 1 (AP, Honors, Academic) Summer Assignment 2018 Calculus (AP, Honors, Academic) Summer Assignment 08 The summer assignments for Calculus will reinforce some necessary Algebra and Precalculus skills. In order to be successful in Calculus, you must have

More information

f ( x ) = x Determine the implied domain of the given function. Express your answer in interval notation.

f ( x ) = x Determine the implied domain of the given function. Express your answer in interval notation. Test Review Section.. Given the following function: f ( ) = + 5 - Determine the implied domain of the given function. Epress your answer in interval notation.. Find the verte of the following quadratic

More information

Chapter XX: 1: Functions. XXXXXXXXXXXXXXX <CT>Chapter 1: Data representation</ct> 1.1 Mappings

Chapter XX: 1: Functions. XXXXXXXXXXXXXXX <CT>Chapter 1: Data representation</ct> 1.1 Mappings 978--08-8-8 Cambridge IGCSE and O Level Additional Mathematics Practice Book Ecerpt Chapter XX: : Functions XXXXXXXXXXXXXXX Chapter : Data representation This section will show you how to: understand

More information

Vocabulary. Term Page Definition Clarifying Example. combined variation. constant of variation. continuous function.

Vocabulary. Term Page Definition Clarifying Example. combined variation. constant of variation. continuous function. CHAPTER Vocabular The table contains important vocabular terms from Chapter. As ou work through the chapter, fill in the page number, definition, and a clarifing eample. combined variation Term Page Definition

More information

MATH 111 Departmental Midterm Exam Review Exam date: Tuesday, March 1 st. Exam will cover sections and will be NON-CALCULATOR EXAM.

MATH 111 Departmental Midterm Exam Review Exam date: Tuesday, March 1 st. Exam will cover sections and will be NON-CALCULATOR EXAM. MATH Departmental Midterm Eam Review Eam date: Tuesday, March st Eam will cover sections -9 + - and will be NON-CALCULATOR EXAM Terms to know: quadratic function, ais of symmetry, verte, minimum/maimum

More information

Advanced Calculus Summer Packet

Advanced Calculus Summer Packet Advanced Calculus Summer Packet PLEASE DO ALL YOUR WORK ON LOOSELEAF PAPER! KEEP ALL WORK ORGANIZED This packet includes a sampling of problems that students entering Advanced Calculus should be able to

More information

Suggested Problems for Math 122

Suggested Problems for Math 122 Suggested Problems for Math 22 Note: This file will grow as the semester evolves and more sections are added. CCA = Contemporary College Algebra, SIA = Shaum s Intermediate Algebra SIA(.) Rational Epressions

More information

Math RE - Calculus I Functions Page 1 of 10. Topics of Functions used in Calculus

Math RE - Calculus I Functions Page 1 of 10. Topics of Functions used in Calculus Math 0-03-RE - Calculus I Functions Page of 0 Definition of a function f() : Topics of Functions used in Calculus A function = f() is a relation between variables and such that for ever value onl one value.

More information

A: Super-Basic Algebra Skills. A1. True or false. If false, change what is underlined to make the statement true. a.

A: Super-Basic Algebra Skills. A1. True or false. If false, change what is underlined to make the statement true. a. A: Super-Basic Algebra Skills A1. True or false. If false, change what is underlined to make the statement true. 1 T F 1 b. T F c. ( + ) = + 9 T F 1 1 T F e. ( + 1) = 16( + ) T F f. 5 T F g. If ( + )(

More information

Algebra I Practice Questions ? 1. Which is equivalent to (A) (B) (C) (D) 2. Which is equivalent to 6 8? (A) 4 3

Algebra I Practice Questions ? 1. Which is equivalent to (A) (B) (C) (D) 2. Which is equivalent to 6 8? (A) 4 3 1. Which is equivalent to 64 100? 10 50 8 10 8 100. Which is equivalent to 6 8? 4 8 1 4. Which is equivalent to 7 6? 4 4 4. Which is equivalent to 4? 8 6 Page 1 of 0 11 Practice Questions 6 1 5. Which

More information

Unit 11 - Solving Quadratic Functions PART ONE

Unit 11 - Solving Quadratic Functions PART ONE Unit 11 - Solving Quadratic Functions PART ONE PREREQUISITE SKILLS: students should be able to add, subtract and multiply polynomials students should be able to factor polynomials students should be able

More information

MA Review Worksheet for Exam 1, Summer 2016

MA Review Worksheet for Exam 1, Summer 2016 MA 15800 Review Worksheet for Eam 1, Summer 016 1) Choose which of the following relations represent functions and give the domain and/or range. a) {(,5), (,5), ( 1,5)} b) equation: y 4 1 y 4 (5, ) 1 (

More information

AP CALCULUS AB & BC ~ er Work and List of Topical Understandings~

AP CALCULUS AB & BC ~ er Work and List of Topical Understandings~ AP CALCULUS AB & BC ~er Work and List of Topical Understandings~ As instructors of AP Calculus, we have etremely high epectations of students taking our courses. As stated in the district program planning

More information

Math Pre-Calc 20 Final Review

Math Pre-Calc 20 Final Review Math Pre-Calc 0 Final Review Chp Sequences and Series #. Write the first 4 terms of each sequence: t = d = - t n = n #. Find the value of the term indicated:,, 9,, t 7 7,, 9,, t 5 #. Find the number of

More information

Honors Calculus Summer Preparation 2018

Honors Calculus Summer Preparation 2018 Honors Calculus Summer Preparation 08 Name: ARCHBISHOP CURLEY HIGH SCHOOL Honors Calculus Summer Preparation 08 Honors Calculus Summer Work and List of Topical Understandings In order to be a successful

More information

Lesson 9.1 Using the Distance Formula

Lesson 9.1 Using the Distance Formula Lesson. Using the Distance Formula. Find the eact distance between each pair of points. a. (0, 0) and (, ) b. (0, 0) and (7, ) c. (, 8) and (, ) d. (, ) and (, 7) e. (, 7) and (8, ) f. (8, ) and (, 0)

More information

Summer Review Packet for Students Entering AP Calculus BC. Complex Fractions

Summer Review Packet for Students Entering AP Calculus BC. Complex Fractions Summer Review Packet for Students Entering AP Calculus BC Comple Fractions When simplifying comple fractions, multiply by a fraction equal to 1 which has a numerator and denominator composed of the common

More information

6.1 Polynomial Functions

6.1 Polynomial Functions 6.1 Polynomial Functions Definition. A polynomial function is any function p(x) of the form p(x) = p n x n + p n 1 x n 1 + + p 2 x 2 + p 1 x + p 0 where all of the exponents are non-negative integers and

More information

Unit 7 Review - Radicals and Rational Expressions

Unit 7 Review - Radicals and Rational Expressions Name: Class: Date: D: A Unit 7 Review - Radicals and Rational Epressions. Simplify 8 4. b. 8 6 True or False:.. For any integer a:;:. 0, a n = an'. For any integer n > 0 and any positive real number a,

More information

MAC1105-College Algebra

MAC1105-College Algebra MAC1105-College Algebra Chapter -Polynomial Division & Rational Functions. Polynomial Division;The Remainder and Factor Theorems I. Long Division of Polynomials A. For f ( ) 6 19 16, a zero of f ( ) occurs

More information

* A graphing calculator is highly recommended for this class!!!

* A graphing calculator is highly recommended for this class!!! AP Calculus AB Summer Packet 08-09 Ms. Febus - García marlene_febus@gwinnett.k.ga.us This packet includes a sampling of problems that students entering AP Calculus AB should be able to answer. The questions

More information

Lesson 7.1 Polynomial Degree and Finite Differences

Lesson 7.1 Polynomial Degree and Finite Differences Lesson 7.1 Polnomial Degree and Finite Differences 1. Identif the degree of each polnomial. a. 1 b. 0. 1. 3. 3 c. 0 16 0. Determine which of the epressions are polnomials. For each polnomial, state its

More information

AP Calculus AB - Mrs. Mora. Summer packet 2010

AP Calculus AB - Mrs. Mora. Summer packet 2010 AP Calculus AB - Mrs. Mora Summer packet 010 These eercises represent some of the more fundamental concepts needed upon entering AP Calculus AB This "packet is epected to be completed and brought to class

More information

SANDY CREEK HIGH SCHOOL

SANDY CREEK HIGH SCHOOL SANDY CREEK HIGH SCHOOL SUMMER REVIEW PACKET For students entering A.P. CALCULUS BC I epect everyone to check the Google classroom site and your school emails at least once every two weeks. You will also

More information

Precalculus Notes: Functions. Skill: Solve problems using the algebra of functions.

Precalculus Notes: Functions. Skill: Solve problems using the algebra of functions. Skill: Solve problems using the algebra of functions. Modeling a Function: Numerical (data table) Algebraic (equation) Graphical Using Numerical Values: Look for a common difference. If the first difference

More information

Algebra II Notes Rational Functions Unit Rational Functions. Math Background

Algebra II Notes Rational Functions Unit Rational Functions. Math Background Algebra II Notes Rational Functions Unit 6. 6.6 Rational Functions Math Background Previously, you Simplified linear, quadratic, radical and polynomial functions Performed arithmetic operations with linear,

More information

kx c The vertical asymptote of a reciprocal linear function has an equation of the form

kx c The vertical asymptote of a reciprocal linear function has an equation of the form Advanced Functions Page 1 of Reciprocal of a Linear Function Concepts Rational functions take the form andq ( ) 0. The reciprocal of a linear function has the form P( ) f ( ), where P () and Q () are both

More information

MAT12X Intermediate Algebra

MAT12X Intermediate Algebra MAT12X Intermediate Algebra Workshop 3 Rational Functions LEARNING CENTER Overview Workshop III Rational Functions General Form Domain and Vertical Asymptotes Range and Horizontal Asymptotes Inverse Variation

More information

Section 5.5 Complex Numbers

Section 5.5 Complex Numbers Name: Period: Section 5.5 Comple Numbers Objective(s): Perform operations with comple numbers. Essential Question: Tell whether the statement is true or false, and justify your answer. Every comple number

More information

Troy High School AP Calculus Summer Packet

Troy High School AP Calculus Summer Packet Troy High School AP Calculus Summer Packet As instructors of AP Calculus, we have etremely high epectations of students taking our courses. We epect a certain level of independence to be demonstrated by

More information

Summer AP Assignment Coversheet Falls Church High School

Summer AP Assignment Coversheet Falls Church High School Summer AP Assignment Coversheet Falls Church High School Course: AP Calculus AB Teacher Name/s: Veronica Moldoveanu, Ethan Batterman Assignment Title: AP Calculus AB Summer Packet Assignment Summary/Purpose:

More information

3.2 Logarithmic Functions and Their Graphs

3.2 Logarithmic Functions and Their Graphs 96 Chapter 3 Eponential and Logarithmic Functions 3.2 Logarithmic Functions and Their Graphs Logarithmic Functions In Section.6, you studied the concept of an inverse function. There, you learned that

More information

Polynomial Functions of Higher Degree

Polynomial Functions of Higher Degree SAMPLE CHAPTER. NOT FOR DISTRIBUTION. 4 Polynomial Functions of Higher Degree Polynomial functions of degree greater than 2 can be used to model data such as the annual temperature fluctuations in Daytona

More information

2. Find the value of y for which the line through A and B has the given slope m: A(-2, 3), B(4, y), 2 3

2. Find the value of y for which the line through A and B has the given slope m: A(-2, 3), B(4, y), 2 3 . Find an equation for the line that contains the points (, -) and (6, 9).. Find the value of y for which the line through A and B has the given slope m: A(-, ), B(4, y), m.. Find an equation for the line

More information

Full Name. Remember, lots of space, thus lots of pages!

Full Name. Remember, lots of space, thus lots of pages! Rising Pre-Calculus student Summer Packet for 016 (school year 016-17) Dear Advanced Algebra: Pre-Calculus student: To be successful in Advanced Algebra: Pre-Calculus, you must be proficient at solving

More information

LEARN ABOUT the Math

LEARN ABOUT the Math 1.5 Inverse Relations YOU WILL NEED graph paper graphing calculator GOAL Determine the equation of an inverse relation and the conditions for an inverse relation to be a function. LEARN ABOUT the Math

More information

MATH 115: Review for Chapter 5

MATH 115: Review for Chapter 5 MATH 5: Review for Chapter 5 Can you find the real zeros of a polynomial function and identify the behavior of the graph of the function at its zeros? For each polynomial function, identify the zeros of

More information

Rational Functions. A rational function is a function that is a ratio of 2 polynomials (in reduced form), e.g.

Rational Functions. A rational function is a function that is a ratio of 2 polynomials (in reduced form), e.g. Rational Functions A rational function is a function that is a ratio of polynomials (in reduced form), e.g. f() = p( ) q( ) where p() and q() are polynomials The function is defined when the denominator

More information

Algebra Final Exam Review Packet

Algebra Final Exam Review Packet Algebra 1 00 Final Eam Review Packet UNIT 1 EXPONENTS / RADICALS Eponents Degree of a monomial: Add the degrees of all the in the monomial together. o Eample - Find the degree of 5 7 yz Degree of a polynomial:

More information

3.5 Continuity of a Function One Sided Continuity Intermediate Value Theorem... 23

3.5 Continuity of a Function One Sided Continuity Intermediate Value Theorem... 23 Chapter 3 Limit and Continuity Contents 3. Definition of Limit 3 3.2 Basic Limit Theorems 8 3.3 One sided Limit 4 3.4 Infinite Limit, Limit at infinity and Asymptotes 5 3.4. Infinite Limit and Vertical

More information

(c) Find the gradient of the graph of f(x) at the point where x = 1. (2) The graph of f(x) has a local maximum point, M, and a local minimum point, N.

(c) Find the gradient of the graph of f(x) at the point where x = 1. (2) The graph of f(x) has a local maximum point, M, and a local minimum point, N. Calculus Review Packet 1. Consider the function f() = 3 3 2 24 + 30. Write down f(0). Find f (). Find the gradient of the graph of f() at the point where = 1. The graph of f() has a local maimum point,

More information

4. (6 points) Express the domain of the following function in interval notation:

4. (6 points) Express the domain of the following function in interval notation: Eam 1-A L. Ballou Name Math 131 Calculus I September 1, 016 NO Calculator Allowed BOX YOUR ANSWER! Show all work for full credit! 1. (4 points) Write an equation of a line with y-intercept 4 and -intercept

More information

(2.5) 1. Solve the following compound inequality and graph the solution set.

(2.5) 1. Solve the following compound inequality and graph the solution set. Intermediate Algebra Practice Final Math 0 (7 th ed.) (Ch. -) (.5). Solve the following compound inequalit and graph the solution set. 0 and and > or or (.7). Solve the following absolute value inequalities.

More information

Rational Numbers, Irrational Numbers, Whole Numbers, Integers, Natural Numbers

Rational Numbers, Irrational Numbers, Whole Numbers, Integers, Natural Numbers College Readiness Math Semester Eam Review Write the set using interval notation and graph the inequality on the number line. 1) { -6 < < 9} Use the order of operations to simplify the epression. 4 (8

More information

of multiplicity two. The sign of the polynomial is shown in the table below

of multiplicity two. The sign of the polynomial is shown in the table below 161 Precalculus 1 Review 5 Problem 1 Graph the polynomial function P( ) ( ) ( 1). Solution The polynomial is of degree 4 and therefore it is positive to the left of its smallest real root and to the right

More information

( ) ( 4) ( ) ( ) Final Exam: Lessons 1 11 Final Exam solutions ( )

( ) ( 4) ( ) ( ) Final Exam: Lessons 1 11 Final Exam solutions ( ) Show all of your work in order to receive full credit. Attach graph paper for your graphs.. Evaluate the following epressions. a) 6 4 6 6 4 8 4 6 6 6 87 9 b) ( 0) if ( ) ( ) ( ) 0 0 ( 8 0) ( 4 0) ( 4)

More information

Chapter 9. Rational Functions

Chapter 9. Rational Functions Chapter 9 Rational Functions Lesson 9-4 Rational Epressions Rational Epression A rational epression is in simplest form when its numerator and denominator are polnomials that have no common divisors. Eample

More information

Math 103 Intermediate Algebra Final Exam Review Practice Problems

Math 103 Intermediate Algebra Final Exam Review Practice Problems Math 10 Intermediate Algebra Final Eam Review Practice Problems The final eam covers Chapter, Chapter, Sections 4.1 4., Chapter 5, Sections 6.1-6.4, 6.6-6.7, Chapter 7, Chapter 8, and Chapter 9. The list

More information