MAT Polynomial and Rational Inequalities

Size: px
Start display at page:

Download "MAT Polynomial and Rational Inequalities"

Transcription

1 MAT Polynomial and Rational Inequalities A. Quadratic Inequalities Mar 15 10:35 AM B. Polynomial Inequalities Solving Polynomial Inequalities Given f (x) is a polynomial in standard form, 1. Write f in completely factored form. 2. Plot real zeroes on the x axis, noting their multiplicity. If the multiplicity is odd the function will change sign. If the multiplicity is even, there will be no change in sign. 3. Use the end behavior to determine the sign of f in the outermost intervals, then label the other intervals as f(x) < 0 or f(x) > 0 by analyzing the multiplicity of neighboring zeroes. 4. State the solution in interval notation. Mar 15 10:48 AM 1

2 C. Rational Inequalities D. Solving Function Inequalities Using Interval Tests E. Applications of Inequalities 2

3 311/8. Solve each quadratic inequality by locating the x intercept( s) ( if they 311/12. Solve each quadratic inequality by locating the x intercept( s) ( if they 3

4 312/22. Solve each quadratic inequality by locating the x intercept( s) ( if they 312/26. Solve each quadratic inequality by locating the x intercept( s) ( if they 4

5 312/28. Solve each quadratic inequality by locating the x intercept( s) ( if they 312/30. Solve each quadratic inequality by locating the x intercept( s) ( if they 5

6 312/34. Recall that for a square root expression to represent a real number, the radicand must be greater than or equal to zero. Applying this idea results in an inequality that can be solved using the skills from this section. Determine the domain of the following radical functions. 312/36. Recall that for a square root expression to represent a real number, the radicand must be greater than or equal to zero. Applying this idea results in an inequality that can be solved using the skills from this section. Determine the domain of the following radical functions. 6

7 312/42. Solve the inequality indicated using a number line and the behavior of the graph at each zero. Write all answers in interval notation. 312/48. Solve the inequality indicated using a number line and the behavior of the graph at each zero. Write all answers in interval notation. Mar 17 11:42 AM 7

8 312/70. Solve the inequality indicated using a number line and the behavior of the graph at each zero. Write all answers in interval notation. Mar 17 11:42 AM 312/72. Solve the inequality indicated using a number line and the behavior of the graph at each zero. Write all answers in interval notation. Mar 17 11:42 AM 8

9 314/95. Average speed for a round trip: Surprisingly, the average speed of a round trip is not the sum of the average speed in each direction divided by two. For a fixed distance D, consider rate r 1 in time t 1 for one direction, and rate r 2 in time t 2 for the other, giving r 1 = D/t 1 and r 2 = D/t 2. The average speed for the round trip is R = 2D / (t 1 + t 2 ). The distance from St. Louis, Missouri, to Springfield, Illinois, is approximately 80 mi. Suppose that Sione, due to the age of his vehicle, made the round trip with an average speed of 40 mph. a. Use the relationships stated to verify that r 2 = 20r 1 / (r 1 20). b. Discuss the meaning of the horizontal and vertical asymptotes in this context. c. Verify algebraically the speed returning would be greater than the speed going for 20 < r 1 < 40. In other words, solve the inequality 20r 1 / (r 1 20) > r 1 using the ideas from this section. Mar 17 11:42 AM Mar 18 9:14 AM 9

10 314/96. The distance from Boston, Massachusetts, to Hartford, Connecticut, is approximately 100 mi. Suppose that Stella, due to excellent driving conditions, made the round trip with an average speed of 60 mph. a. Use the relationships above to verify that r 1 = 30r 1 / (r 1 30). b. Discuss the meaning of the horizontal and vertical asymptotes in this context. c. Verify algebraically the speed returning would be greater than the speed going for 30 < r 1 < 60. In other words, solve the inequality 30r 1 / (r 1 30) > r 1 using the ideas from this section. Mar 17 11:42 AM Mar 18 10:24 AM 10

11 Mar 18 10:33 AM Mar 18 10:36 AM 11

To get horizontal and slant asymptotes algebraically we need to know about end behaviour for rational functions.

To get horizontal and slant asymptotes algebraically we need to know about end behaviour for rational functions. Concepts: Horizontal Asymptotes, Vertical Asymptotes, Slant (Oblique) Asymptotes, Transforming Reciprocal Function, Sketching Rational Functions, Solving Inequalities using Sign Charts. Rational Function

More information

Section 2.7 Notes Name: Date: Polynomial and Rational Inequalities

Section 2.7 Notes Name: Date: Polynomial and Rational Inequalities Section.7 Notes Name: Date: Precalculus Polynomial and Rational Inequalities At the beginning of this unit we solved quadratic inequalities by using an analysis of the graph of the parabola combined with

More information

11 /2 12 /2 13 /6 14 /14 15 /8 16 /8 17 /25 18 /2 19 /4 20 /8

11 /2 12 /2 13 /6 14 /14 15 /8 16 /8 17 /25 18 /2 19 /4 20 /8 MAC 1147 Exam #1a Answer Key Name: Answer Key ID# Summer 2012 HONOR CODE: On my honor, I have neither given nor received any aid on this examination. Signature: Instructions: Do all scratch work on the

More information

Precalculus. How to do with no calculator 1a)

Precalculus. How to do with no calculator 1a) Precalculus UNIT 2 Review NAME PERIOD This assessment covers many concepts which you must be able to understand without the use of your calculator to view the graph. Please complete the following table

More information

MAT116 Final Review Session Chapter 3: Polynomial and Rational Functions

MAT116 Final Review Session Chapter 3: Polynomial and Rational Functions MAT116 Final Review Session Chapter 3: Polynomial and Rational Functions Quadratic Function A quadratic function is defined by a quadratic or second-degree polynomial. Standard Form f x = ax 2 + bx + c,

More information

MAT 107 College Algebra Fall 2013 Name. Final Exam, Version X

MAT 107 College Algebra Fall 2013 Name. Final Exam, Version X MAT 107 College Algebra Fall 013 Name Final Exam, Version X EKU ID Instructor Part 1: No calculators are allowed on this section. Show all work on your paper. Circle your answer. Each question is worth

More information

PAP Geometry Summer Work- Show your work

PAP Geometry Summer Work- Show your work PRE- PAP Geometry Summer Work- Show your work Solve the equation. Check your solution. 1. 2. Solve the equation. 3. 4. 5. Describe the values of c for which the equation has no solution. Write the sentence

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

1. The graph of a quadratic function is shown. Each square is one unit.

1. The graph of a quadratic function is shown. Each square is one unit. 1. The graph of a quadratic function is shown. Each square is one unit. a. What is the vertex of the function? b. If the lead coefficient (the value of a) is 1, write the formula for the function in vertex

More information

CA-A1-B Interpret the dependent and independent variables in the context of functions.

CA-A1-B Interpret the dependent and independent variables in the context of functions. STEM Units of Study The STEM course features four units of instruction and a capstone project. This document clearly articulates a list of the key performance indicators that are included in the units.

More information

Relations and Functions (for Math 026 review)

Relations and Functions (for Math 026 review) Section 3.1 Relations and Functions (for Math 026 review) Objective 1: Understanding the s of Relations and Functions Relation A relation is a correspondence between two sets A and B such that each element

More information

Fall 09/MAT 140/Worksheet 1 Name: Show all your work. 1. (6pts) Simplify and write the answer so all exponents are positive:

Fall 09/MAT 140/Worksheet 1 Name: Show all your work. 1. (6pts) Simplify and write the answer so all exponents are positive: Fall 09/MAT 140/Worksheet 1 Name: Show all your work. 1. (6pts) Simplify and write the answer so all exponents are positive: a) (x 3 y 6 ) 3 x 4 y 5 = b) 4x 2 (3y) 2 (6x 3 y 4 ) 2 = 2. (2pts) Convert to

More information

CHAPTER 3: Quadratic Functions and Equations; Inequalities

CHAPTER 3: Quadratic Functions and Equations; Inequalities MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 3: Quadratic Functions and Equations; Inequalities 3.1 The Complex Numbers 3.2 Quadratic Equations, Functions, Zeros, and

More information

PreCalculus Notes. MAT 129 Chapter 5: Polynomial and Rational Functions. David J. Gisch. Department of Mathematics Des Moines Area Community College

PreCalculus Notes. MAT 129 Chapter 5: Polynomial and Rational Functions. David J. Gisch. Department of Mathematics Des Moines Area Community College PreCalculus Notes MAT 129 Chapter 5: Polynomial and Rational Functions David J. Gisch Department of Mathematics Des Moines Area Community College September 2, 2011 1 Chapter 5 Section 5.1: Polynomial Functions

More information

. State whether the triangle is an isosceles triangle, a right triangle, neither of these, or both.

. State whether the triangle is an isosceles triangle, a right triangle, neither of these, or both. PRECALCULUS MIDTERM PRACTICE TEST (2008-2009) Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the distance d(, ) between the points and.

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

MATH 1314 College Algebra Scott Travis Fall 2014 Review for Exam #2

MATH 1314 College Algebra Scott Travis Fall 2014 Review for Exam #2 MATH 1314 College Algebra Scott Travis Fall 2014 Review for Exam #2 There are eight sections from Chapters 4 and 5 included in the exam: 4.1, 4.3, 5.1 to 5.6. This review should help you prepare. For each

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) x 8. C) y = x + 3 2

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) x 8. C) y = x + 3 2 Precalculus Fall Final Exam Review Name Date Period MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Simplify the expression. Assume that the variables

More information

HONORS GEOMETRY Summer Skills Set

HONORS GEOMETRY Summer Skills Set HONORS GEOMETRY Summer Skills Set Algebra Concepts Adding and Subtracting Rational Numbers To add or subtract fractions with the same denominator, add or subtract the numerators and write the sum or difference

More information

9.5. Polynomial and Rational Inequalities. Objectives. Solve quadratic inequalities. Solve polynomial inequalities of degree 3 or greater.

9.5. Polynomial and Rational Inequalities. Objectives. Solve quadratic inequalities. Solve polynomial inequalities of degree 3 or greater. Chapter 9 Section 5 9.5 Polynomial and Rational Inequalities Objectives 1 3 Solve quadratic inequalities. Solve polynomial inequalities of degree 3 or greater. Solve rational inequalities. Objective 1

More information

NAME DATE PERIOD. Power and Radical Functions. New Vocabulary Fill in the blank with the correct term. positive integer.

NAME DATE PERIOD. Power and Radical Functions. New Vocabulary Fill in the blank with the correct term. positive integer. 2-1 Power and Radical Functions What You ll Learn Scan Lesson 2-1. Predict two things that you expect to learn based on the headings and Key Concept box. 1. 2. Lesson 2-1 Active Vocabulary extraneous solution

More information

Controlling the Population

Controlling the Population Lesson.1 Skills Practice Name Date Controlling the Population Adding and Subtracting Polynomials Vocabulary Match each definition with its corresponding term. 1. polynomial a. a polynomial with only 1

More information

2-7 Solving Quadratic Inequalities. ax 2 + bx + c > 0 (a 0)

2-7 Solving Quadratic Inequalities. ax 2 + bx + c > 0 (a 0) Quadratic Inequalities In One Variable LOOKS LIKE a quadratic equation but Doesn t have an equal sign (=) Has an inequality sign (>,

More information

Student: Date: Instructor: kumnit nong Course: MATH 105 by Nong https://xlitemprodpearsoncmgcom/api/v1/print/math Assignment: CH test review 1 Find the transformation form of the quadratic function graphed

More information

Cumulative Review. Name. 13) 2x = -4 13) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Cumulative Review. Name. 13) 2x = -4 13) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Cumulative Review Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Evaluate the algebraic expression for the given value or values of the variable(s).

More information

Part I: Multiple Choice Questions

Part I: Multiple Choice Questions Name: Part I: Multiple Choice Questions. What is the slope of the line y=3 A) 0 B) -3 ) C) 3 D) Undefined. What is the slope of the line perpendicular to the line x + 4y = 3 A) -/ B) / ) C) - D) 3. Find

More information

CHAPTER 4: Polynomial and Rational Functions

CHAPTER 4: Polynomial and Rational Functions MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial

More information

x 1 2 i 1 5 2i 11 9x 9 3x 3 1 y 2 3y 4 y 2 1 Poudre School District s College Algebra Course Review

x 1 2 i 1 5 2i 11 9x 9 3x 3 1 y 2 3y 4 y 2 1 Poudre School District s College Algebra Course Review . Perform the indicated operations and simplify the result. Leave the answer in factored form. 9x 9 x a. b. 9x 9 x x. Solve: x 7x x 0. a. x., b. x 0,., x,0,. x.,0,. Find the quotient and the remainder

More information

M30-1: Polynomial, Radical and Rational Functions, Graphs and Equations Exam /20

M30-1: Polynomial, Radical and Rational Functions, Graphs and Equations Exam /20 Class: Date: ID: A M30-1: Polynomial, Radical and Rational Functions, Graphs and Equations Exam /20 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which

More information

2-6 Nonlinear Inequalities

2-6 Nonlinear Inequalities 31. Find the domain of each expression. For the expression to be defined, x 2 3x 40 0. Let f (x) = x 2 3x 40. A factored form of f (x) is f (x) = (x 8)(x + 5). f (x) has real zeros at x = 8 and x = 5.

More information

Final Exam Review. Determine whether the relation represents a function. If it is a function, state the domain and range.

Final Exam Review. Determine whether the relation represents a function. If it is a function, state the domain and range. Final Exam Review Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find that value. 1) f(x) = x2-5 1) A) minimum; 0 B) minimum; -5 C) maximum;

More information

Section 5.1 Determine if a function is a polynomial function. State the degree of a polynomial function.

Section 5.1 Determine if a function is a polynomial function. State the degree of a polynomial function. Test Instructions Objectives Section 5.1 Section 5.1 Determine if a function is a polynomial function. State the degree of a polynomial function. Form a polynomial whose zeros and degree are given. Graph

More information

Bishop Kelley High School Summer Math Program Course: Honors Pre-Calculus

Bishop Kelley High School Summer Math Program Course: Honors Pre-Calculus 017 018 Summer Math Program Course: Honors Pre-Calculus NAME: DIRECTIONS: Show all work in the packet. Make sure you are aware of the calculator policy for this course. No matter when you have math, this

More information

Department of Mathematics, University of Wisconsin-Madison Math 114 Worksheet Sections (4.1),

Department of Mathematics, University of Wisconsin-Madison Math 114 Worksheet Sections (4.1), Department of Mathematics, University of Wisconsin-Madison Math 114 Worksheet Sections (4.1), 4.-4.6 1. Find the polynomial function with zeros: -1 (multiplicity ) and 1 (multiplicity ) whose graph passes

More information

Midterm. Multiple Choice Identify the choice that best completes the statement or answers the question.

Midterm. Multiple Choice Identify the choice that best completes the statement or answers the question. Midterm Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Factor completely. If the polynomial cannot be factored, say it is prime. 10x 2-95x + 225 2. Solve

More information

ALGEBRA 1B GOALS. 1. The student should be able to use mathematical properties to simplify algebraic expressions.

ALGEBRA 1B GOALS. 1. The student should be able to use mathematical properties to simplify algebraic expressions. GOALS 1. The student should be able to use mathematical properties to simplify algebraic expressions. 2. The student should be able to add, subtract, multiply, divide, and compare real numbers. 3. The

More information

Chapter 1- Polynomial Functions

Chapter 1- Polynomial Functions Chapter 1- Polynomial Functions WORKBOOK MHF4U W1 1.1 Power Functions MHF4U Jensen 1) Identify which of the following are polynomial functions: a) p x = cos x b) h x = 7x c) f x = 2x, d) y = 3x / 2x 0

More information

ICM ~ Unit 4 ~ Day 3. Horizontal Asymptotes, End Behavior

ICM ~ Unit 4 ~ Day 3. Horizontal Asymptotes, End Behavior ICM ~ Unit 4 ~ Day 3 Horizontal Asymptotes, End Behavior Warm Up ~ Day 3 1. Find the domain, then convert to fractional/rational eponent. f ( ) 7. Simplify completely: 3( + 5). 3. Find the domain, & y

More information

evaluate functions, expressed in function notation, given one or more elements in their domains

evaluate functions, expressed in function notation, given one or more elements in their domains Describing Linear Functions A.3 Linear functions, equations, and inequalities. The student writes and represents linear functions in multiple ways, with and without technology. The student demonstrates

More information

2012 Texas Essential Knowledge and Skills for Algebra II in Pearson Texas Algebra II

2012 Texas Essential Knowledge and Skills for Algebra II in Pearson Texas Algebra II 2012 Texas Essential Knowledge and Skills for Algebra II in Pearson Texas Algebra II The following table shows where each of the from the 2012 Texas Essential Knowledge and Skills (TEKS) for Algebra II

More information

Rational and Radical Functions. College Algebra

Rational and Radical Functions. College Algebra Rational and Radical Functions College Algebra Rational Function A rational function is a function that can be written as the quotient of two polynomial functions P(x) and Q(x) f x = P(x) Q(x) = a )x )

More information

Rational Functions 4.5

Rational Functions 4.5 Math 4 Pre-Calculus Name Date Rational Function Rational Functions 4.5 g ( ) A function is a rational function if f ( ), where g ( ) and ( ) h ( ) h are polynomials. Vertical asymptotes occur at -values

More information

A2.MidtermRev2015. Algebra 2 Midterm Exam Review Part 1: Multiple Choice (75pts)

A2.MidtermRev2015. Algebra 2 Midterm Exam Review Part 1: Multiple Choice (75pts) Name: UNIT 1 Algebra 2 Midterm Exam Review Part 1: Multiple Choice (75pts) Patterns & Expressions 1. Which of the following is the seventh term in the pattern below? 2. Which of the following is the eighth

More information

1) Solve the formula for the indicated variable. P = 2L + 2W for W. 2) Solve the formula for the variable y. 5 = 7x - 8y

1) Solve the formula for the indicated variable. P = 2L + 2W for W. 2) Solve the formula for the variable y. 5 = 7x - 8y Math120 Cumulative Review This is to help prepare you for the 40 question final exam. It is not all inclusive of the material covered in your course. Therefore items not on this review may appear on the

More information

Algebra II Assessment. Eligible Texas Essential Knowledge and Skills

Algebra II Assessment. Eligible Texas Essential Knowledge and Skills Algebra II Assessment Eligible Texas Essential Knowledge and Skills STAAR Algebra II Assessment Mathematical Process Standards These student expectations will not be listed under a separate reporting category.

More information

171S3.2 Quadratic Equations, Functions, Zeros, and Models September 30, Quadratic Equations, Functions, Zeros, and Models

171S3.2 Quadratic Equations, Functions, Zeros, and Models September 30, Quadratic Equations, Functions, Zeros, and Models MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 3: Quadratic Functions and Equations; Inequalities 3.1 The Complex Numbers 3.2 Quadratic Equations, Functions, Zeros, and

More information

CHAPTER 3: Quadratic Functions and Equations; Inequalities

CHAPTER 3: Quadratic Functions and Equations; Inequalities MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 3: Quadratic Functions and Equations; Inequalities 3.1 The Complex Numbers 3.2 Quadratic Equations, Functions, Zeros, and

More information

Horizontal and Vertical Asymptotes from section 2.6

Horizontal and Vertical Asymptotes from section 2.6 Horizontal and Vertical Asymptotes from section 2.6 Definition: In either of the cases f(x) = L or f(x) = L we say that the x x horizontal line y = L is a horizontal asymptote of the function f. Note:

More information

Unit 5 RATIONAL FUNCTIONS. A function with a variable in the denominator Parent function 1 x Graph is a hyperbola

Unit 5 RATIONAL FUNCTIONS. A function with a variable in the denominator Parent function 1 x Graph is a hyperbola Unit 5 RATIONAL FUNCTIONS A function with a variable in the denominator Parent function 1 x Graph is a hyperbola A direct variation is a relationship between two variables x and y that can be written in

More information

Math 120, Sample Final Fall 2015

Math 120, Sample Final Fall 2015 Math 10, Sample Final Fall 015 Disclaimer: This sample final is intended to help students prepare for the final exam The final exam will be similar in structure and type of problems, however the actual

More information

Sect Polynomial and Rational Inequalities

Sect Polynomial and Rational Inequalities 158 Sect 10.2 - Polynomial and Rational Inequalities Concept #1 Solving Inequalities Graphically Definition A Quadratic Inequality is an inequality that can be written in one of the following forms: ax

More information

# 1-11, 12(don't graph), 13, 14, 15, 17, 18 # 8abd, 13

# 1-11, 12(don't graph), 13, 14, 15, 17, 18 # 8abd, 13 MHF4U Unit 1 Polynomial Functions Section Pages Questions Prereq Skills 2 3 # 1ace, 2cde, 3bce, 4, 5, 6, 7, 8ace, 9, 10b, 11b, 12 & Factoring Practice 1.1 11 14 # 1, 2, 3, 4, 5, 7, 8, 9(in class) 1.2 26

More information

Topics from Algebra and Pre-Calculus. (Key contains solved problems)

Topics from Algebra and Pre-Calculus. (Key contains solved problems) Topics from Algebra and Pre-Calculus (Key contains solved problems) Note: The purpose of this packet is to give you a review of basic skills. You are asked not to use the calculator, except on p. (8) and

More information

Final Exam Review for DMAT 0310

Final Exam Review for DMAT 0310 Final Exam Review for DMAT 010 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Factor the polynomial completely. What is one of the factors? 1) x

More information

Algebra I. Course Outline

Algebra I. Course Outline Algebra I Course Outline I. The Language of Algebra A. Variables and Expressions B. Order of Operations C. Open Sentences D. Identity and Equality Properties E. The Distributive Property F. Commutative

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

Section 1.4 Solving Other Types of Equations

Section 1.4 Solving Other Types of Equations M141 - Chapter 1 Lecture Notes Page 1 of 27 Section 1.4 Solving Other Types of Equations Objectives: Given a radical equation, solve the equation and check the solution(s). Given an equation that can be

More information

CHAPTER 3: Quadratic Functions and Equations; Inequalities

CHAPTER 3: Quadratic Functions and Equations; Inequalities 171S MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 3: Quadratic Functions and Equations; Inequalities 3.1 The Complex Numbers 3.2 Quadratic Equations, Functions, Zeros,

More information

Unit 5 RATIONAL FUNCTIONS. A function with a variable in the denominator Parent function 1 x Graph is a hyperbola

Unit 5 RATIONAL FUNCTIONS. A function with a variable in the denominator Parent function 1 x Graph is a hyperbola Unit 5 RATIONAL FUNCTIONS A function with a variable in the denominator Parent function 1 x Graph is a hyperbola I will be following the Alg 2 book in this Unit Ch 5 Sections 1-5 Use the Practice Packet

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

STEM Course Master Rubric

STEM Course Master Rubric STEM Course Master Rubric This competency rubric places emphasis on the importance of mastery of pre-requisite skills and concepts to move to meet the expectations of the competencies and key performance

More information

Polynomial Degree Leading Coefficient. Sign of Leading Coefficient

Polynomial Degree Leading Coefficient. Sign of Leading Coefficient Chapter 1 PRE-TEST REVIEW Polynomial Functions MHF4U Jensen Section 1: 1.1 Power Functions 1) State the degree and the leading coefficient of each polynomial Polynomial Degree Leading Coefficient y = 2x

More information

What kind of number is? Write the number in scientific notation ,000

What kind of number is? Write the number in scientific notation ,000 Chapter 1: 1.1, 1.2, 1.3, 1.4 Chapter 2: 2.1, 2.2, 2.3, 2.4 Chapter 3: 3.1, 3.2, 3.3, 3.4 Chapter 4: 4.1, 4.2, 4.3, 4.5, 4.7 Chapter 5: 5.1, 5.2, 5.3, 5.4, 5.6, 5.7 Chapter 6: 6.1 1.1 What kind of number

More information

MATHEMATICS Trigonometry. Mathematics 30-1 Mathematics (10 12) /21 Alberta Education, Alberta, Canada (2008)

MATHEMATICS Trigonometry. Mathematics 30-1 Mathematics (10 12) /21 Alberta Education, Alberta, Canada (2008) MATHEMATICS 30-1 [C] Communication Trigonometry Develop trigonometric reasoning. 1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians. [CN, ME, R, V] 2. Develop

More information

ALGEBRA 2 FINAL EXAM REVIEW

ALGEBRA 2 FINAL EXAM REVIEW Class: Date: ALGEBRA 2 FINAL EXAM REVIEW Multiple Choice Identify the choice that best completes the statement or answers the question.. Classify 6x 5 + x + x 2 + by degree. quintic c. quartic cubic d.

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

Chapter 3: Polynomial and Rational Functions

Chapter 3: Polynomial and Rational Functions .1 Power and Polynomial Functions 155 Chapter : Polynomial and Rational Functions Section.1 Power Functions & Polynomial Functions... 155 Section. Quadratic Functions... 16 Section. Graphs of Polynomial

More information

Summer Assignment for AP Calculus AB

Summer Assignment for AP Calculus AB This assignment is a review of Pre-calculus and Algebraic concepts that you need to be familiar with in order to make a smooth transition into AP Calculus AB. It will be due when you return to school on

More information

2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY

2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY 2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY The following are topics that you will use in Geometry and should be retained throughout the summer. Please use this practice to review the topics you

More information

Basic Fraction and Integer Operations (No calculators please!)

Basic Fraction and Integer Operations (No calculators please!) P1 Summer Math Review Packet For Students entering Geometry The problems in this packet are designed to help you review topics from previous mathematics courses that are important to your success in Geometry.

More information

Complete your Parent Function Packet!!!!

Complete your Parent Function Packet!!!! PARENT FUNCTIONS Pre-Ap Algebra 2 Complete your Parent Function Packet!!!! There are two slides per Parent Function. The Parent Functions are numbered in the bottom right corner of each slide. The Function

More information

Intermediate Algebra Final Exam Review

Intermediate Algebra Final Exam Review Intermediate Algebra Final Exam Review Note to students: The final exam for MAT10, MAT 11 and MAT1 will consist of 30 multiple-choice questions and a few open-ended questions. The exam itself will cover

More information

Section 4.1: Polynomial Functions and Models

Section 4.1: Polynomial Functions and Models Section 4.1: Polynomial Functions and Models Learning Objectives: 1. Identify Polynomial Functions and Their Degree 2. Graph Polynomial Functions Using Transformations 3. Identify the Real Zeros of a Polynomial

More information

Algebra 1 Prince William County Schools Pacing Guide (Crosswalk)

Algebra 1 Prince William County Schools Pacing Guide (Crosswalk) Algebra 1 Prince William County Schools Pacing Guide 2017-2018 (Crosswalk) Teacher focus groups have assigned a given number of days to each unit based on their experiences and knowledge of the curriculum.

More information

SB CH 2 answers.notebook. November 05, Warm Up. Oct 8 10:36 AM. Oct 5 2:22 PM. Oct 8 9:22 AM. Oct 8 9:19 AM. Oct 8 9:26 AM.

SB CH 2 answers.notebook. November 05, Warm Up. Oct 8 10:36 AM. Oct 5 2:22 PM. Oct 8 9:22 AM. Oct 8 9:19 AM. Oct 8 9:26 AM. Warm Up Oct 8 10:36 AM Oct 5 2:22 PM Linear Function Qualities Oct 8 9:22 AM Oct 8 9:19 AM Quadratic Function Qualities Oct 8 9:26 AM Oct 8 9:25 AM 1 Oct 8 9:28 AM Oct 8 9:25 AM Given vertex (-1,4) and

More information

Math 1314 Lesson 1: Prerequisites. Example 1: Simplify and write the answer without using negative exponents:

Math 1314 Lesson 1: Prerequisites. Example 1: Simplify and write the answer without using negative exponents: Math 1314 Lesson 1: Prerequisites 1. Exponents 1 m n n n m Recall: x = x = x n x Example 1: Simplify and write the answer without using negative exponents: a. x 5 b. ( x) 5 Example : Write as a radical:

More information

Calculus First Semester Review Name: Section: Evaluate the function: (g o f )( 2) f (x + h) f (x) h. m(x + h) m(x)

Calculus First Semester Review Name: Section: Evaluate the function: (g o f )( 2) f (x + h) f (x) h. m(x + h) m(x) Evaluate the function: c. (g o f )(x + 2) d. ( f ( f (x)) 1. f x = 4x! 2 a. f( 2) b. f(x 1) c. f (x + h) f (x) h 4. g x = 3x! + 1 Find g!! (x) 5. p x = 4x! + 2 Find p!! (x) 2. m x = 3x! + 2x 1 m(x + h)

More information

Algebra II Unit #2 4.6 NOTES: Solving Quadratic Equations (More Methods) Block:

Algebra II Unit #2 4.6 NOTES: Solving Quadratic Equations (More Methods) Block: Algebra II Unit # Name: 4.6 NOTES: Solving Quadratic Equations (More Methods) Block: (A) Background Skills - Simplifying Radicals To simplify a radical that is not a perfect square: 50 8 300 7 7 98 (B)

More information

Polynomial Functions and Their Graphs

Polynomial Functions and Their Graphs Polynomial Functions and Their Graphs Definition of a Polynomial Function Let n be a nonnegative integer and let a n, a n- 1,, a 2, a 1, a 0, be real numbers with a n 0. The function defined by f (x) a

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Calculus I - Homework Chapter 2 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the graph is the graph of a function. 1) 1)

More information

2. If the values for f(x) can be made as close as we like to L by choosing arbitrarily large. lim

2. If the values for f(x) can be made as close as we like to L by choosing arbitrarily large. lim Limits at Infinity and Horizontal Asymptotes As we prepare to practice graphing functions, we should consider one last piece of information about a function that will be helpful in drawing its graph the

More information

2. If the discriminant of a quadratic equation is zero, then there (A) are 2 imaginary roots (B) is 1 rational root

2. If the discriminant of a quadratic equation is zero, then there (A) are 2 imaginary roots (B) is 1 rational root Academic Algebra II 1 st Semester Exam Mr. Pleacher Name I. Multiple Choice 1. Which is the solution of x 1 3x + 7? (A) x -4 (B) x 4 (C) x -4 (D) x 4. If the discriminant of a quadratic equation is zero,

More information

Lesson 2.1: Quadratic Functions

Lesson 2.1: Quadratic Functions Quadratic Functions: Lesson 2.1: Quadratic Functions Standard form (vertex form) of a quadratic function: Vertex: (h, k) Algebraically: *Use completing the square to convert a quadratic equation into standard

More information

Unit 3: HW3.5 Sum and Product

Unit 3: HW3.5 Sum and Product Unit 3: HW3.5 Sum and Product Without solving, find the sum and product of the roots of each equation. 1. x 2 8x + 7 = 0 2. 2x + 5 = x 2 3. -7x + 4 = -3x 2 4. -10x 2 = 5x - 2 5. 5x 2 2x 3 4 6. 1 3 x2 3x

More information

Algebra 1. Standard 1: Operations With Real Numbers Students simplify and compare expressions. They use rational exponents and simplify square roots.

Algebra 1. Standard 1: Operations With Real Numbers Students simplify and compare expressions. They use rational exponents and simplify square roots. Standard 1: Operations With Real Numbers Students simplify and compare expressions. They use rational exponents and simplify square roots. A1.1.1 Compare real number expressions. A1.1.2 Simplify square

More information

8-5. A rational inequality is an inequality that contains one or more rational expressions. x x 6. 3 by using a graph and a table.

8-5. A rational inequality is an inequality that contains one or more rational expressions. x x 6. 3 by using a graph and a table. A rational inequality is an inequality that contains one or more rational expressions. x x 3 by using a graph and a table. Use a graph. On a graphing calculator, Y1 = x and Y = 3. x The graph of Y1 is

More information

Practice Test - Chapter 2

Practice Test - Chapter 2 Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 1. f (x) = 0.25x 3 Evaluate the function for several

More information

Catholic Central High School

Catholic Central High School Catholic Central High School Algebra II Practice Examination I Instructions: 1. Show all work on the test copy itself for every problem where work is required. Points may be deducted if insufficient or

More information

CHAPTER EIGHT: SOLVING QUADRATIC EQUATIONS Review April 9 Test April 17 The most important equations at this level of mathematics are quadratic

CHAPTER EIGHT: SOLVING QUADRATIC EQUATIONS Review April 9 Test April 17 The most important equations at this level of mathematics are quadratic CHAPTER EIGHT: SOLVING QUADRATIC EQUATIONS Review April 9 Test April 17 The most important equations at this level of mathematics are quadratic equations. They can be solved using a graph, a perfect square,

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

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 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 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 Final Practice Exam Name: Student Number: For Marker

More information

PreCalculus: Semester 1 Final Exam Review

PreCalculus: Semester 1 Final Exam Review Name: Class: Date: ID: A PreCalculus: Semester 1 Final Exam Review Short Answer 1. Determine whether the relation represents a function. If it is a function, state the domain and range. 9. Find the domain

More information

Common Core Algebra 2. Chapter 5: Rational Exponents & Radical Functions

Common Core Algebra 2. Chapter 5: Rational Exponents & Radical Functions Common Core Algebra 2 Chapter 5: Rational Exponents & Radical Functions 1 Chapter Summary This first part of this chapter introduces radicals and nth roots and how these may be written as rational exponents.

More information

CHAPTER 4: Polynomial and Rational Functions

CHAPTER 4: Polynomial and Rational Functions MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial

More information

An equation is a statement that states that two expressions are equal. For example:

An equation is a statement that states that two expressions are equal. For example: Section 0.1: Linear Equations Solving linear equation in one variable: An equation is a statement that states that two expressions are equal. For example: (1) 513 (2) 16 (3) 4252 (4) 64153 To solve the

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

Math 1314 Lesson 1: Prerequisites

Math 1314 Lesson 1: Prerequisites Math 131 Lesson 1: Prerequisites Prerequisites are topics you should have mastered before you enter this class. Because of the emphasis on technology in this course, there are few skills which you will

More information

Final Exam C Name i D) 2. Solve the equation by factoring. 4) x2 = x + 72 A) {1, 72} B) {-8, 9} C) {-8, -9} D) {8, 9} 9 ± i

Final Exam C Name i D) 2. Solve the equation by factoring. 4) x2 = x + 72 A) {1, 72} B) {-8, 9} C) {-8, -9} D) {8, 9} 9 ± i Final Exam C Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 7 ) x + + 3 x - = 6 (x + )(x - ) ) A) No restrictions; {} B) x -, ; C) x -; {} D) x -, ; {2} Add

More information

Polynomial functions right- and left-hand behavior (end behavior):

Polynomial functions right- and left-hand behavior (end behavior): Lesson 2.2 Polynomial Functions For each function: a.) Graph the function on your calculator Find an appropriate window. Draw a sketch of the graph on your paper and indicate your window. b.) Identify

More information

Semester Review Packet

Semester Review Packet MATH 110: College Algebra Instructor: Reyes Semester Review Packet Remarks: This semester we have made a very detailed study of four classes of functions: Polynomial functions Linear Quadratic Higher degree

More information

Summer Packet for Students Taking Introduction to Calculus in the Fall

Summer Packet for Students Taking Introduction to Calculus in the Fall Summer Packet for Students Taking Introduction to Calculus in the Fall Algebra 2 Topics Needed for Introduction to Calculus Need to know: à Solve Equations Linear Quadratic Absolute Value Polynomial Rational

More information