MAC Rev.S Learning Objectives. Learning Objectives (Cont.)
|
|
- Eugenia Webster
- 5 years ago
- Views:
Transcription
1 MAC 1140 Module 6 Nonlinear Functions and Equations II Learning Objectives Upon completing this module, you should be able to 1. identify a rational function and state its domain.. find and interpret vertical asymptotes. 3. find and interpret horizontal asymptotes. 4. solve rational equations. 5. solve applications involving rational equations. 6. solve applications involving variations. 7. solve polynomial inequalities. 8. solve rational inequalities. Rev.S10 Learning Objectives (Cont.) 9. learn properties of rational exponents. 10. learn radical notation. 11. use power functions to model data. 1. solve equations involving rational exponents. 13. solve equations involving radical expressions. Rev.S
2 Nonlinear Functions and Equations II There are three sections in this module: 4.6 Rational Functions and Models 4.7 More Equations and Inequalities 4.8 Radical Equations and Power Functions Rev.S10 4 What is a Rational Function? A rational function is a nonlinear function. The domain of a rational function includes all real numbers except the zeros of the denominator q(x). Rev.S10 5 What is a Vertical Asymptote? In this graph, the line x is a vertical asymptote. Rev.S10 6
3 What is a Horizontal Asymptote? In this graph, the line y 5 is a horizontal asymptote. Rev.S10 7 Use the graph of Let s Look at an Example 1 to sketch the graph of x 1 g( x). ( x + 1)! Include all asymptotes in your graph. Write g(x) in terms of f(x). g(x) is a translation of f(x) left one unit and down units. The vertical asymptote is x 1 The horizontal asymptote is y g(x) f(x + 1) Rev.S10 8 How to Find Vertical and Horizontal Asymptotes? Rev.S
4 More Examples For each rational function, determine any horizontal or vertical asymptotes. a) x + 6 b) x! 1 c) x! 4 4x! 8 x! 9 x! To identify horizontal asymptote, look at the leading coefficient of the highest power term in both numerator and denominator. a) Horizontal Asymptote: If the Degree of numerator equals the degree of the denominator, y a/b is asymptote, so y /4 1/ To identify vertical asymptote, set the denominator to 0. Vertical Asymptote: 4x 8 0, x Rev.S10 10 Examples (Cont.) For each rational function, determine any horizontal or vertical asymptotes. x + 6 x! 1 x! 4 a) b) c) 4x! 8 x! 9 x! (Cont.) b) Horizontal Asymptote: Degree: numerator < denominator y 0 is the horizontal asymptote. Vertical Asymptote: x 9 0 x ± 3 are the vertical asymptotes. Rev.S10 11 (Cont.) Examples (Cont.) For each rational function, determine any horizontal or vertical asymptotes. a) x + 6 b) x! 1 c) x! 4 4x! 8 x! 9 x! c) Horizontal Asymptote: Degree: numerator > denominator no horizontal asymptotes x! 4 Vertical Asymptote: x! no vertical asymptotes ( x! )( x + ) (There will be a hole x! in the graph.) x + x " The graph is the line y x + with the point (,( 4) missing. Rev.S
5 What is a Slant/Oblique Asymptote? A third type of asymptote is neither horizontal or vertical. Occurs when the numerator of a rational function has a degree one more than the degree of the denominator. Rev.S10 13 Let x + 1. x! a) Use a calculator to graph f. Example b) Identify any asymptotes. c) Sketch a graph of f that includes the asymptotes. a) Reminder: Slant/Oblique Asymptote occurs when the numerator of a rational function has a degree one more than the degree of the denominator. Is that true in this rational function? Rev.S10 14 (Cont.) Example (Cont.) b) Asymptotes: The function is undefined when x 0 or when x. * Vertical asymptote at x * Oblique asymptote at y x + c) How about horizontal asymptote? Why don t we have it in this rational function? How can you tell from the function itself? Rev.S
6 Solve How to Solve Rational Equation? x 4. x! Symbolic Graphical Numerical x 4 x! x 4( x! ) x 4x! 8! x! 8 x 4 Rev.S10 16 One More Example Solve x! 1 x! 1 x + 1 Multiply by the LCD to clear the fractions x! 1 x! 1 x + 1 4( x! 1)( x + 1) ( x! 1)( x + 1) 3( x! 1)( x + 1) + x! 1 x! 1 x ( x + 1) + 3( x! 1) 4 x + + 3x! 3 4 5x! 1 1 x When 1 is substituted for x,, two expressions in the given equation are undefined. There are no solutions. Rev.S10 17 Direct Variation The nonzero number k is called the constant of variation or the constant of proportionality. In the area formula for a circle, A π r, the π is the constant of variation; so, k π in this case. Rev.S
7 Inverse Variation This inverse variation occurs when we have two quantities that vary inversely; the increase of one quantity will decrease the other quantity. Rev.S10 19 Example of Application At a distance of 3 meters, a 100-watt bulb produces an intensity of 0.88 watt per square meter. a) Find the constant of proportionality k. b) Determine the intensity at a distance of.5 meters. a) Substitute d 3 and I 0.88 into the equation and solve for k. 7.9 b) Let I and d.5. d 7.9 I d 7.9 I k I d k 0.88 or k The intensity at.5 meters is 1.7 watts per square meter. Rev.S10 0 How to Solve Polynomial Inequalities? An inequality says that one expression is greater than, greater than or equal to, less than, or less than or equal to, another expression. To solve Polynomial Inequalities, we need the following: Boundary numbers (x-values) are found where the inequality holds. A graph or a table of test values can be used to determine the intervals where the inequality holds. Rev.S
8 Solving Polynomial Inequalities in Four Steps Rev.S10 Let s Look at This Example 3 Solve x! " 7x " 10x symbolically and graphically. Symbolically 3 Step 1: Write the inequality as x + 7x + 10x! 0. Step : Replace the inequality symbol with an equal sign and solve. x x x ( x ) ( )( x ) x x x x x 0 or x! 5 or x! The boundary numbers are 5,, and 0. Rev.S10 3 Let s Look at This Example (Cont.) Step 3: The boundary numbers separate the number line into four disjoint intervals:!",! 5,! 5,!,!,0, and 0, " ( ) ( ) ( ) ( ) Rev.S
9 Let s Look at This Example (Cont.) Step 4: Complete a table of test values. Interval Test Value x x 3 + 7x + 10x Positive/Negative [ ) The solution set is [! 5,! ]! 0, ". Negative Positive Negative Positive Rev.S10 5 Let s Look at This Example (Cont.) Graphically: Rev.S10 6 How to Solve Rational Inequalities? Inequalities involving rational expressions are called rational inequalities. Rev.S
10 Example 5 Solve! 1. x + 4 Step 1: Rewrite the inequality in the form 5! 1 x " 1! 0 x "( x + 4)! 0 x + 4 1" x! 0 x + 4 p( x) 0. q( x )! Rev.S10 8 Example (Cont.) Step : Find the zeros of the numerator and the denominator. Numerator 1 x 0 x 1 Denominator x x 4 Step 3: The boundary numbers are 4 and 1, which separate the number line into three disjoint intervals: (!",! 4 ),(! 4,1 ) and ( 1, "). Rev.S10 9 Example (Cont.) Step 4: Use a table to solve the inequality. Interval Test Value x (1 x)/(x + 4) Positive/Negative 5 6 Negative 3/ Positive 1/6 Negative The interval notation is ( 4, 1]. Caution: When solving a rational inequality, it is essential not to multiply or divide each side of the inequality by the LCD if the LCD contains a variable. This techniques often leads to an incorrect solution set. Rev.S
11 Let s Review Some Properties of Rational Exponents Rev.S10 31 Let s Practice Some Simplification Simplify each expression by hand. a) 8 /3 b) ( 3) 4/5 s / ( )! 4/5! (! 3) (! 3) 4 (! ) 16 Rev.S10 3 Let s Practice Some Simplification (Cont.) Use positive rational exponents to write each expression. a) 5 4 b) 3 6 x xi x s 5 x 4 ( x 4 ) 1/5 x 4/5 1/ 1/ 3 6 1/3 1/ 6 1/3 + 1/ 6 xi x x ix x ( x ) 1/ 1/ 1/ 4 x ( ) ( ) ( ) ( ) Rev.S
12 What are Power Functions? Power functions typically have rational exponents. A special type of power function is a root function. Examples of power functions include: f 1 (x) x, f (x) x 3/4, f 3 (x) x 0.4, and f 4 (x) 3 x Rev.S10 34 What are Power Functions? (Cont.) Often, the domain of a power function f is restricted to nonnegative numbers. Suppose the rational number p/q is written in lowest terms. The the domain of f(x) x p/q is all real numbers whenever q is odd and all nonnegative numbers whenever q is even. The following graphs show 3 common power functions. Rev.S10 35 Example Modeling Wing Size of a Bird: Heavier birds have larger wings with more surface areas than do lighter birds. For some species the relationship can be modeled by S(w) 0.w /3, where w is the weight of the bird in kilograms and S is surface area of the wings in square meters. (Source: C. Pennycuick, Newton Rules Biology.) a) Approximate S(0.75) and interpret the result. b) What weight corresponds to a surface area of 0.45 square meter? Rev.S
13 Example (Cont.) a) S(0.75) 0.(0.75) /3! The wings of a bird that weighs about 0.75 kilogram have the surface area of about square meter. b) To answer this, we must solve the equation 0.w / Rev.S10 37 (cont.) Example (Cont.) /3 0.w 0.45 w /3 /3 3 ( w ) ! 0.45 " # $ % 0. &! 0.45 " w # $ % 0. &! 0.45 " w ± # $ % 0. & w ' ± 3.4 Since w must be positive, the wings of a 3.4 kilogram bird must have a surface area of about 0.45 square meter. Rev.S10 38 How to Solve Equations Involving Rational Exponents? Example Solve 4x 3/ 6 6. Approximate the answer to the nearest hundredth, and give graphical support. s Symbolic 4x 3/ 6 6 4x 3/ 1 (x 3/ ) 3 x 3 9 x 9 1/3 x.08 Graphical Rev.S
14 Check How to Solve Equations Involving Rational Exponents? (Cont.) 3 ( 1)! 1! 1 "! 1 ( ) 3 6! 6! 4 4 Substituting these values in the original equation shows that the value of 1 is an extraneous solution because it does not satisfy the given equation. Therefore, the only solution is 6. Rev.S10 40 How to Solve Equations Involving Radicals? Some equations may contain a cube root. Solve 4x! 4x + 1 x x! 4x + 1 x ( 4x! 4x + 1) ( x) 4x! 5x ( 4x! 1)( x! 1) 0 1 x or x 1 4 Both solutions check, so the solution set is! 1 " #, 1 $. % 4 & Rev.S10 41 How to Solve Equations Involving Negative Exponents?!! 1 Example Solve 6x + x.!! 1 6x + x 6u + u! 0 ( u )( u ) 3 +! u! or u 3 x! 3 or x 1 1 Since u, then x. x u Rev.S
15 We have learned to What have we learned? 1. identify a rational function and state its domain.. find and interpret vertical asymptotes. 3. find and interpret horizontal asymptotes. 4. solve rational equations. 5. solve applications involving rational equations. 6. solve applications involving variations. 7. solve polynomial inequalities. 8. solve rational inequalities. Rev.S10 43 What have we learned? (Cont.) 9. learn properties of rational exponents. 10. learn radical notation. 11. use power functions to model data. 1. solve equations involving rational exponents. 13. solve equations involving radical expressions Rev.S10 44 Credit Some of these slides have been adapted/modified in part/whole from the slides of the following textbook: Rockswold, Gary, Precalculus with Modeling and Visualization, 3th Edition Rev.S
MAC Learning Objectives. Module 7 Additional Equations and Inequalities. Let s Review Some Properties of Rational Exponents
MAC 1105 Module 7 Additional Equations and Inequalities Learning Objectives Upon completing this module, you should be able to: 1. Use properties of rational exponents (rational powers). 2. Understand
More informationMAC Module 7 Additional Equations and Inequalities. Rev.S08
MAC 1105 Module 7 Additional Equations and Inequalities Learning Objectives Upon completing this module, you should be able to: 1. Use properties of rational exponents (rational powers). 2. Understand
More informationMAC Module 10 Higher-Degree Polynomial Functions. Rev.S08
MAC 1105 Module 10 Higher-Degree Polynomial Functions Learning Objectives Upon completing this module, you should be able to 1. Identify intervals where a function is increasing or decreasing. 2. Find
More informationMAC Learning Objectives. Module 10. Higher-Degree Polynomial Functions. - Nonlinear Functions and Their Graphs - Polynomial Functions and Models
MAC 1105 Module 10 Higher-Degree Polynomial Functions Learning Objectives Upon completing this module, you should be able to 1. Identify intervals where a function is increasing or decreasing. 2. Find
More informationMAC Module 8 Exponential and Logarithmic Functions I. Rev.S08
MAC 1105 Module 8 Exponential and Logarithmic Functions I Learning Objectives Upon completing this module, you should be able to: 1. Distinguish between linear and exponential growth. 2. Model data with
More informationMAC Module 8. Exponential and Logarithmic Functions I. Learning Objectives. - Exponential Functions - Logarithmic Functions
MAC 1105 Module 8 Exponential and Logarithmic Functions I Learning Objectives Upon completing this module, you should be able to: 1. Distinguish between linear and exponential growth. 2. Model data with
More informationMAC Learning Objectives. Logarithmic Functions. Module 8 Logarithmic Functions
MAC 1140 Module 8 Logarithmic Functions Learning Objectives Upon completing this module, you should be able to 1. evaluate the common logarithmic function. 2. solve basic exponential and logarithmic equations.
More informationNAME 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 informationRational Functions. Elementary Functions. Algebra with mixed fractions. Algebra with mixed fractions
Rational Functions A rational function f (x) is a function which is the ratio of two polynomials, that is, Part 2, Polynomials Lecture 26a, Rational Functions f (x) = where and are polynomials Dr Ken W
More informationMAC Learning Objectives. Learning Objectives (Cont.) Module 10 System of Equations and Inequalities II
MAC 1140 Module 10 System of Equations and Inequalities II Learning Objectives Upon completing this module, you should be able to 1. represent systems of linear equations with matrices. 2. transform a
More informationCommon 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 informationMission 1 Simplify and Multiply Rational Expressions
Algebra Honors Unit 6 Rational Functions Name Quest Review Questions Mission 1 Simplify and Multiply Rational Expressions 1) Compare the two functions represented below. Determine which of the following
More information56 CHAPTER 3. POLYNOMIAL FUNCTIONS
56 CHAPTER 3. POLYNOMIAL FUNCTIONS Chapter 4 Rational functions and inequalities 4.1 Rational functions Textbook section 4.7 4.1.1 Basic rational functions and asymptotes As a first step towards understanding
More informationTo 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 informationMAC Learning Objectives. Learning Objectives. Module 1 Introduction to Functions and Graphs
MAC 1140 Module 1 Introduction to Functions and Graphs Learning Objectives Upon completing this module, you should be able to 1. recognize common sets of numbers. 2. understand scientific notation and
More informationMaking Connections with Rational Functions and Equations
Section 3.5 Making Connections with Rational Functions and Equations When solving a problem, it's important to read carefully to determine whether a function is being analyzed (Finding key features) or
More informationPolynomial Expressions and Functions
Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 1 - of 36 Topic 32: Polynomial Expressions and Functions Recall the definitions of polynomials and terms. Definition: A polynomial
More informationPreCalculus 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 informationPreCalculus: 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 informationReading Mathematical Expressions & Arithmetic Operations Expression Reads Note
Math 001 - Term 171 Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note x A x belongs to A,x is in A Between an element and a set. A B A is a subset of B Between two sets. φ
More information6.1 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 information8-5. 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 informationVocabulary: I. Inverse Variation: Two variables x and y show inverse variation if they are related as. follows: where a 0
8.1: Model Inverse and Joint Variation I. Inverse Variation: Two variables x and y show inverse variation if they are related as follows: where a 0 * In this equation y is said to vary inversely with x.
More informationCHAPTER 8A- RATIONAL FUNCTIONS AND RADICAL FUNCTIONS Section Multiplying and Dividing Rational Expressions
Name Objectives: Period CHAPTER 8A- RATIONAL FUNCTIONS AND RADICAL FUNCTIONS Section 8.3 - Multiplying and Dividing Rational Expressions Multiply and divide rational expressions. Simplify rational expressions,
More informationChapter 7 Rational Expressions, Equations, and Functions
Chapter 7 Rational Expressions, Equations, and Functions Section 7.1: Simplifying, Multiplying, and Dividing Rational Expressions and Functions Section 7.2: Adding and Subtracting Rational Expressions
More informationPolynomial and Rational Functions. Chapter 3
Polynomial and Rational Functions Chapter 3 Quadratic Functions and Models Section 3.1 Quadratic Functions Quadratic function: Function of the form f(x) = ax 2 + bx + c (a, b and c real numbers, a 0) -30
More informationMAC Module 1 Systems of Linear Equations and Matrices I
MAC 2103 Module 1 Systems of Linear Equations and Matrices I 1 Learning Objectives Upon completing this module, you should be able to: 1. Represent a system of linear equations as an augmented matrix.
More informationChapter 9 Notes SN AA U2C9
Chapter 9 Notes SN AA U2C9 Name Period Section 2-3: Direct Variation Section 9-1: Inverse Variation Two variables x and y show direct variation if y = kx for some nonzero constant k. Another kind of variation
More informationMAC Module 2 Modeling Linear Functions. Rev.S08
MAC 1105 Module 2 Modeling Linear Functions Learning Objectives Upon completing this module, you should be able to: 1. Recognize linear equations. 2. Solve linear equations symbolically and graphically.
More informationSection Properties of Rational Expressions
88 Section. - Properties of Rational Expressions Recall that a rational number is any number that can be written as the ratio of two integers where the integer in the denominator cannot be. Rational Numbers:
More information1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
Math120 - Precalculus. Final Review. Fall, 2011 Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
More informationMAT116 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 informationReteach Multiplying and Dividing Rational Expressions
8-2 Multiplying and Dividing Rational Expressions Examples of rational expressions: 3 x, x 1, and x 3 x 2 2 x 2 Undefined at x 0 Undefined at x 0 Undefined at x 2 When simplifying a rational expression:
More information11 /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 informationSolving Polynomial and Rational Inequalities Algebraically. Approximating Solutions to Inequalities Graphically
10 Inequalities Concepts: Equivalent Inequalities Solving Polynomial and Rational Inequalities Algebraically Approximating Solutions to Inequalities Graphically (Section 4.6) 10.1 Equivalent Inequalities
More informationMULTIPLE 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 informationYou analyzed parent functions and their families of graphs. (Lesson 1-5)
You analyzed parent functions and their families of graphs. (Lesson 1-5) Graph and analyze power functions. Graph and analyze radical functions, and solve radical equations. power function monomial function
More informationBell Ringer. 1. Make a table and sketch the graph of the piecewise function. f(x) =
Bell Ringer 1. Make a table and sketch the graph of the piecewise function f(x) = Power and Radical Functions Learning Target: 1. I can graph and analyze power functions. 2. I can graph and analyze radical
More informationPolynomial Functions
Polynomial Functions Polynomials A Polynomial in one variable, x, is an expression of the form a n x 0 a 1 x n 1... a n 2 x 2 a n 1 x a n The coefficients represent complex numbers (real or imaginary),
More informationPENNSYLVANIA. The denominator of a rational function is critical in the graph and solution of the function. Page 1 of 3.
Know: Understand: Do: 1 -- Essential Make sense of problems and persevere in solving them. The denominator of a rational function is critical in the graph and solution of the function. 1 -- Essential Make
More informationGUIDED 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 informationKing Fahd University of Petroleum and Minerals Prep-Year Math Program Math Term 161 Recitation (R1, R2)
Math 001 - Term 161 Recitation (R1, R) Question 1: How many rational and irrational numbers are possible between 0 and 1? (a) 1 (b) Finite (c) 0 (d) Infinite (e) Question : A will contain how many elements
More information2.1 Quadratic Functions
Date:.1 Quadratic Functions Precalculus Notes: Unit Polynomial Functions Objective: The student will sketch the graph of a quadratic equation. The student will write the equation of a quadratic function.
More informationChapter 1A -- Real Numbers. iff. Math Symbols: Sets of Numbers
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1A! Page 1 Chapter 1A -- Real Numbers Math Symbols: iff or Example: Let A = {2, 4, 6, 8, 10, 12, 14, 16,...} and let B = {3, 6, 9, 12, 15, 18,
More informationName: 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 informationStudy Guide for Math 095
Study Guide for Math 095 David G. Radcliffe November 7, 1994 1 The Real Number System Writing a fraction in lowest terms. 1. Find the largest number that will divide into both the numerator and the denominator.
More information#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 informationB.3 Solving Equations Algebraically and Graphically
B.3 Solving Equations Algebraically and Graphically 1 Equations and Solutions of Equations An equation in x is a statement that two algebraic expressions are equal. To solve an equation in x means to find
More informationMAC Module 9 Exponential and Logarithmic Functions II. Rev.S08
MAC 1105 Module 9 Exponential and Logarithmic Functions II Learning Objective Upon completing this module, you should be able to: 1. Learn and apply the basic properties of logarithms. 2. Use the change
More informationChapter P. Prerequisites. Slide P- 1. Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide P- 1 Chapter P Prerequisites 1 P.1 Real Numbers Quick Review 1. List the positive integers between -4 and 4.. List all negative integers greater than -4. 3. Use a calculator to evaluate the expression
More informationALGEBRA & TRIGONOMETRY FOR CALCULUS MATH 1340
ALGEBRA & TRIGONOMETRY FOR CALCULUS Course Description: MATH 1340 A combined algebra and trigonometry course for science and engineering students planning to enroll in Calculus I, MATH 1950. Topics include:
More informationEQ: What are limits, and how do we find them? Finite limits as x ± Horizontal Asymptote. Example Horizontal Asymptote
Finite limits as x ± The symbol for infinity ( ) does not represent a real number. We use to describe the behavior of a function when the values in its domain or range outgrow all finite bounds. For example,
More informationAlgebra 2 Segment 1 Lesson Summary Notes
Algebra 2 Segment 1 Lesson Summary Notes For each lesson: Read through the LESSON SUMMARY which is located. Read and work through every page in the LESSON. Try each PRACTICE problem and write down the
More informationModule 10 Polar Form of Complex Numbers
MAC 1114 Module 10 Polar Form of Complex Numbers Learning Objectives Upon completing this module, you should be able to: 1. Identify and simplify imaginary and complex numbers. 2. Add and subtract complex
More informationPolynomial 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 information2.6. Graphs of Rational Functions. Copyright 2011 Pearson, Inc.
2.6 Graphs of Rational Functions Copyright 2011 Pearson, Inc. Rational Functions What you ll learn about Transformations of the Reciprocal Function Limits and Asymptotes Analyzing Graphs of Rational Functions
More informationChapter Five Notes N P U2C5
Chapter Five Notes N P UC5 Name Period Section 5.: Linear and Quadratic Functions with Modeling In every math class you have had since algebra you have worked with equations. Most of those equations have
More informationCHAPTER 1: Review (See also the Precalculus notes at
CHAPTER 1: Review (See also the Precalculus notes at http://www.kkuniyuk.com) TOPIC 1: FUNCTIONS (Chapter 1: Review) 1.01 PART A: AN EXAMPLE OF A FUNCTION Consider a function f whose rule is given by f
More information1.2. Functions and Their Properties. Copyright 2011 Pearson, Inc.
1.2 Functions and Their Properties Copyright 2011 Pearson, Inc. What you ll learn about Function Definition and Notation Domain and Range Continuity Increasing and Decreasing Functions Boundedness Local
More informationComparison of Virginia s College and Career Ready Mathematics Performance Expectations with the Common Core State Standards for Mathematics
Comparison of Virginia s College and Career Ready Mathematics Performance Expectations with the Common Core State Standards for Mathematics February 17, 2010 1 Number and Quantity The Real Number System
More informationThe Graph of a Quadratic Function. Quadratic Functions & Models. The Graph of a Quadratic Function. The Graph of a Quadratic Function
8/1/015 The Graph of a Quadratic Function Quadratic Functions & Models Precalculus.1 The Graph of a Quadratic Function The Graph of a Quadratic Function All parabolas are symmetric with respect to a line
More informationSection 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 informationControlling 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 informationPre-Calculus: Functions and Their Properties (Solving equations algebraically and graphically, matching graphs, tables, and equations, and
Pre-Calculus: 1.1 1.2 Functions and Their Properties (Solving equations algebraically and graphically, matching graphs, tables, and equations, and finding the domain, range, VA, HA, etc.). Name: Date:
More informationMAT 129 Precalculus Chapter 5 Notes
MAT 129 Precalculus Chapter 5 Notes Polynomial and Rational Functions David J. Gisch and Models Example: Determine which of the following are polynomial functions. For those that are, state the degree.
More informationAlgebra 2 Honors Curriculum Pacing Guide
SOUTH CAROLINA ACADEMIC STANDARDS FOR MATHEMATICS The mathematical processes provide the framework for teaching, learning, and assessing in all high school mathematics core courses. Instructional programs
More informationSection 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 informationA Partial List of Topics: Math Spring 2009
A Partial List of Topics: Math 112 - Spring 2009 This is a partial compilation of a majority of the topics covered this semester and may not include everything which might appear on the exam. The purpose
More informationChapter 4: Radicals and Complex Numbers
Chapter : Radicals and Complex Numbers Section.1: A Review of the Properties of Exponents #1-: Simplify the expression. 1) x x ) z z ) a a ) b b ) 6) 7) x x x 8) y y y 9) x x y 10) y 8 b 11) b 7 y 1) y
More informationEquations and Inequalities
Algebra I SOL Expanded Test Blueprint Summary Table Blue Hyperlinks link to Understanding the Standards and Essential Knowledge, Skills, and Processes Reporting Category Algebra I Standards of Learning
More informationMath 115 Spring 11 Written Homework 10 Solutions
Math 5 Spring Written Homework 0 Solutions. For following its, state what indeterminate form the its are in and evaluate the its. (a) 3x 4x 4 x x 8 Solution: This is in indeterminate form 0. Algebraically,
More informationSection 2: Polynomial and Rational Functions
Section 2: Polynomial and Rational Functions The following maps the videos in this section to the Texas Essential Knowledge and Skills for Mathematics TAC 111.42(c). 2.01 Quadratic Functions Precalculus
More information2. 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 informationAlgebra 2 CP Curriculum Pacing Guide
SOUTH CAROLINA ACADEMIC STANDARDS FOR MATHEMATICS The mathematical processes provide the framework for teaching, learning, and assessing in all high school mathematics core courses. Instructional programs
More informationGrade 11 or 12 Pre-Calculus
Grade 11 or 12 Pre-Calculus Strands 1. Polynomial, Rational, and Radical Relationships 2. Trigonometric Functions 3. Modeling with Functions Strand 1: Polynomial, Rational, and Radical Relationships Standard
More informationRational 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 informationAlg Review/Eq & Ineq (50 topics, due on 01/19/2016)
Course Name: MAC 1140 Spring 16 Course Code: XQWHD-P4TU6 ALEKS Course: PreCalculus Instructor: Van De Car Course Dates: Begin: 01/11/2016 End: 05/01/2016 Course Content: 307 topics Textbook: Coburn: Precalculus,
More information10.7 Polynomial and Rational Inequalities
10.7 Polynomial and Rational Inequalities In this section we want to turn our attention to solving polynomial and rational inequalities. That is, we want to solve inequalities like 5 4 0. In order to do
More informationCURRICULUM GUIDE. Honors Algebra II / Trigonometry
CURRICULUM GUIDE Honors Algebra II / Trigonometry The Honors course is fast-paced, incorporating the topics of Algebra II/ Trigonometry plus some topics of the pre-calculus course. More emphasis is placed
More informationReference Material /Formulas for Pre-Calculus CP/ H Summer Packet
Reference Material /Formulas for Pre-Calculus CP/ H Summer Packet Week # 1 Order of Operations Step 1 Evaluate expressions inside grouping symbols. Order of Step 2 Evaluate all powers. Operations Step
More informationOBJECTIVES UNIT 1. Lesson 1.0
OBJECTIVES UNIT 1 Lesson 1.0 1. Define "set," "element," "finite set," and "infinite set," "empty set," and "null set" and give two examples of each term. 2. Define "subset," "universal set," and "disjoint
More informationPart 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 informationAlgebra 2 Khan Academy Video Correlations By SpringBoard Activity
SB Activity Activity 1 Creating Equations 1-1 Learning Targets: Create an equation in one variable from a real-world context. Solve an equation in one variable. 1-2 Learning Targets: Create equations in
More information4.3 Division of Polynomials
4.3 Division of Polynomials Learning Objectives Divide a polynomials by a monomial. Divide a polynomial by a binomial. Rewrite and graph rational functions. Introduction A rational epression is formed
More informationAlgebra 2 Khan Academy Video Correlations By SpringBoard Activity
SB Activity Activity 1 Creating Equations 1-1 Learning Targets: Create an equation in one variable from a real-world context. Solve an equation in one variable. 1-2 Learning Targets: Create equations in
More informationAdding and Subtracting Rational Expressions. Add and subtract rational expressions with the same denominator.
Chapter 7 Section 7. Objectives Adding and Subtracting Rational Expressions 1 3 Add and subtract rational expressions with the same denominator. Find a least common denominator. Add and subtract rational
More information1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
Math120 - Precalculus. Final Review Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents. (a) 5
More information2. 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 informationMathematics: Algebra II Honors Unit 6: Radical Functions
Understandings Questions Knowledge Vocabulary Skills Radical functions can be used to model real-life situations. What are the properties of Algebra and how are these used to solve radical equations? How
More informationPolynomial Functions and Models
1 CA-Fall 2011-Jordan College Algebra, 4 th edition, Beecher/Penna/Bittinger, Pearson/Addison Wesley, 2012 Chapter 4: Polynomial Functions and Rational Functions Section 4.1 Polynomial Functions and Models
More informationMath 0095: Developmental Emporium Mathematics
Math 0095: Developmental Emporium Mathematics Course Titles: Credit hours: Prerequisites: Math 0099: Early Foundations of College Mathematics Math 0100: Foundations of College Mathematics Math 0101: Foundations
More informationMATH 0960 ELEMENTARY ALGEBRA FOR COLLEGE STUDENTS (8 TH EDITION) BY ANGEL & RUNDE Course Outline
MATH 0960 ELEMENTARY ALGEBRA FOR COLLEGE STUDENTS (8 TH EDITION) BY ANGEL & RUNDE Course Outline 1. Real Numbers (33 topics) 1.3 Fractions (pg. 27: 1-75 odd) A. Simplify fractions. B. Change mixed numbers
More informationMATH: A2. ADE Summer Item Writing Institute. Performance-Based Assessment. x f(x) g(x)
Date: 6/11/2013 A-REI.11-2 Task Type: I II III Math Practice: 1 2 3 4 5 6 7 8 x f(x) g(x) If g(x) = 1 4 (x 2)2 + 1, find all values of x to the nearest tenth where f(x) = g(x). X= Click to enter Another
More informationChapter 2. Polynomial and Rational Functions. 2.6 Rational Functions and Their Graphs. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter Polynomial and Rational Functions.6 Rational Functions and Their Graphs Copyright 014, 010, 007 Pearson Education, Inc. 1 Objectives: Find the domains of rational functions. Use arrow notation.
More informationAlgebra 2 Honors: Final Exam Review
Name: Class: Date: Algebra 2 Honors: Final Exam Review Directions: You may write on this review packet. Remember that this packet is similar to the questions that you will have on your final exam. Attempt
More informationMTH30 Review Sheet. y = g(x) BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE
BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE MTH0 Review Sheet. Given the functions f and g described by the graphs below: y = f(x) y = g(x) (a)
More informationHorizontal 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 informationAn 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 informationAdvanced Algebra 2 - Assignment Sheet Chapter 1
Advanced Algebra - Assignment Sheet Chapter #: Real Numbers & Number Operations (.) p. 7 0: 5- odd, 9-55 odd, 69-8 odd. #: Algebraic Expressions & Models (.) p. 4 7: 5-6, 7-55 odd, 59, 6-67, 69-7 odd,
More informationMath ~ Exam #1 Review Guide* *This is only a guide, for your benefit, and it in no way replaces class notes, homework, or studying
Math 1050 2 ~ Exam #1 Review Guide* *This is only a guide, for your benefit, and it in no way replaces class notes, homework, or studying General Tips for Studying: 1. Review this guide, class notes, the
More information