9.1 Adding and Subtracting Rational Expressions
|
|
- Rafe Bruce
- 6 years ago
- Views:
Transcription
1 Name Class Date 9.1 Adding and Subtracting Rational Epressions Essential Question: How can you add and subtract rational epressions? Resource Locker Eplore Identifying Ecluded Values Given a rational epression, identify the ecluded values by finding the zeroes of the denominator. If possible, simplify the epression. A (1-2 ) _ - 1 The denominator of the epression is. B Since division by 0 is not defined, the ecluded values for this epression are all the values that would make the denominator equal to = 0 = C D E F Begin simplifying the epression by factoring the numerator. _ (1-2 ) ( )( ) - 1 = - 1 Divide out terms common to both the numerator and the denominator. _ (1-2 ) ( )( ) - 1 = -(1 - ) The simplified epression is _ (1-2 ) - 1 =, whenever = = What is the domain for this function? What is its range? Reflect 1. What factors can be divided out of the numerator and denominator? Module Lesson 1
2 Eplain 1 Writing Equivalent Rational Epressions Given a rational epression, there are different ways to write an equivalent rational epression. When common terms are divided out, the result is an equivalent but simplified epression. Eample 1 Rewrite the epression as indicated. A Write 3 as an equivalent rational epression that has a denominator of ( + 3) ( + 5). ( + 3) 3 The epression has a denominator of ( + 3). ( + 3) The factor missing from the denominator is ( + 5). Introduce a common factor, ( + 5). _ 3 ( + 3) = 3 ( + 5) ( + 3) ( + 5) 3 ( + 3) is equivalent to 3 ( + 5) ( + 3) ( + 5). B Simplify the epression ( ) ( ) ( + 3). Write the epression. ( ) ( ) ( + 3) Factor the numerator and denominator. Divide out like terms. Your Turn 2. Write 5 as an equivalent epression with a denominator of ( - 5) ( + 1) Simplify the epression ( + 3 )(1 2 ). ( 2 6 ) Module Lesson 1
3 Eplain 2 Identifying the LCD of Two Rational Epressions Given two or more rational epressions, the least common denominator (LCD) is found by factoring each denominator and finding the least common multiple (LCM) of the factors. This technique is useful for the addition and subtraction of epressions with unlike denominators. Least Common Denominator (LCD) of Rational Epressions To find the LCD of rational epressions: 1. Factor each denominator completely. Write any repeated factors as powers. 2. List the different factors. If the denominators have common factors, use the highest power of each common factor. Eample 2 Find the LCD for each set of rational epressions. _-2 A 3-15 and _ Factor each denominator completely = 3 ( - 5) = 4 ( + 7) List the different factors. 3, 4, - 5, + 7 The LCD is 3 4 ( - 5) ( + 7), or 12 ( - 5) ( + 7). -14 B and 9_ Factor each denominator completely = = List the different factors. and Taking the highest power of ( - 3), the LCD is. Reflect 4. Discussion When is the LCD of two rational epressions not equal to the product of their denominators? Your Turn Find the LCD for each set of rational epressions. 5. _ and _ _ = Module Lesson 1
4 Eplain 3 Adding and Subtracting Rational Epressions Adding and subtracting rational epressions is similar to adding and subtracting fractions. Eample 3 Add or subtract. Identify any ecluded values and simplify your answer. _ A _ Factor the denominators. _ _ ( + 1) Identify where the epression is not defined. The first epression is undefined when = 0. The second epression is undefined when = 0 and when = -1. Find a common denominator. The LCM for 2 and ( + 1) is 2 ( + 1). Write the epressions with a common denominator by multiplying both by the appropriate form of 1. Simplify each numerator. = _( + 1) ( + 1) _ _ 2 ( + 1) _ 3 2 ( + 1) 2 ( + 1) 2 Add = ( + 1) Since none of the factors of the denominator are factors of the numerator, the epression cannot be further simplified. _ 2 B _ Factor the denominators. _ 2 2 -_ Identify where the epression is not defined. The first epression is undefined when = 0 and when = 5. The second epression is undefined when = 0. Find a common denominator. The LCM for ( - 5) and 2 is. Write the epressions with a common denominator by multiplying both by the appropriate form of 1. 2 _ 2 ( - 5) - _ Simplify each numerator. 2 3 = _ 2 ( - 5) ( - 5) Subtract. = ( - 5) - 5 _ - 5 Since none of the factors of the denominator are factors of the numerator, the epression cannot be further simplified. Module Lesson 1
5 Your Turn Add each pair of epressions, simplifying the result and noting the combined ecluded values. Then subtract the second epression from the first, again simplifying the result and noting the combined ecluded values and (1-2 ) 8. 2 and 1 (4-2 ) (2 - ) Module Lesson 1
6 Eplain 4 Adding and Subtracting with Rational Models Rational epressions can model real-world phenomena, and can be used to calculate measurements of those phenomena. Eample 4 Find the sum or difference of the models to solve the problem. A B Two groups have agreed that each will contribute $2000 for an upcoming trip. Group A has 6 more people than group B. Let represent the number of people in group A. Write and simplify an epression in terms of that represents the difference between the number of dollars each person in group A must contribute and the number each person in group B must contribute. _ 2000 _ = 2000 ( 6) _ 2000 ( 6) ( 6) , = ( 6) 12, 000 = -_ ( - 6) A freight train averages 30 miles per hour traveling to its destination with full cars and 40 miles per hour on the return trip with empty cars. Find the total time in terms of d. Use the formula t = d r. Let d represent the one-way distance. Total time: _ d 30 + d_ 40 = _ d. + _ d Your Turn = d. + d. = _ d 9. A hiker averages 1.4 miles per hour when walking downhill on a mountain trail and 0.8 miles per hour on the return trip when walking uphill. Find the total time in terms of d. Use the formula t = d r. Image Credits: (t) Larry Lee Photography/Corbis; (b) Henrik Trygg/Corbis Module Lesson 1
7 10. Yvette ran at an average speed of 6.20 feet per second during the first two laps of a race and an average speed of 7.75 feet per second during the second two laps of a race. Find her total time in terms of d, the distance around the racecourse. Elaborate 11. Why do rational epressions have ecluded values? 12. How can you tell if your answer is written in simplest form? 13. Essential Question Check-In Why must the ecluded values of each epression in a sum or difference of rational epressions also be ecluded values for the simplified epression? Module Lesson 1
8 Evaluate: Homework and Practice Given a rational epression, identify the ecluded values by finding the zeroes of the denominator. 1. _ _ ( + 17) Online Homework Hints and Help Etra Practice Write the given epression as an equivalent rational epression that has the given denominator Epression: + 8 Denominator: Denominator: (2 - ) ( 2 + 9) 4. Epression: Simplify the given epression. 5. (-4-4) ( ) _ Find the LCD for each set of rational epressions. 9. _ and _ _ _ 5-30 and _ Module Lesson 1
9 _ and _ and and _ and Add or subtract the given epressions, simplifying each result and noting the combined ecluded values _ _ _ _ Module Lesson 1
10 17. 1_ _ _ _ _ _ _ _ _ _ _ _ Module Lesson 1
11 23. _ _ _ _ _ _ The owner of store A and store B wants to know the average cost of both stores q Store A has an average cost of q, and store B has an average cost q of, where both stores have the same output, q. Find an epression 2q to represent the cost of both stores. Image Credits: Chris Crisman/Corbis 26. An auto race consists of 8 laps. A driver completes the first 3 laps at an average speed of 185 miles per hour and the remaining laps at an average speed of 200 miles per hour. Let d represent the length of one lap. Find the time in terms of d that it takes the driver to complete the race. Module Lesson 1
12 27. The junior and senior classes of a high school are cleaning up a beach. Each class has pledged to clean 1600 meters of shoreline. The junior class has 12 more students than the senior class. Let s represent the number of students in the senior class. Write and simplify an epression in terms of s that represents the difference between the number of meters of shoreline each senior must clean and the number of meters each junior must clean. 28. Architecture The Renaissance architect Andrea Palladio believed that the height of a room with vaulted ceilings should be the harmonic mean of the length and width. The harmonic mean of two positive numbers a and b is equal to 2 1 a + 1 Simplify this. b epression. What are the ecluded values? What do they mean in this problem? 29. Match each epression with the correct ecluded value(s) a. _ + 2 no ecluded values 1 + b. _ 2-1 0, -2 c. _ , d. _ 2 ( + 2) -2 Image Credits: Klaus Vedfelt/Getty Images Module Lesson 1
13 H.O.T. Focus on Higher Order Thinking 30. Eplain the Error George was asked to write the epression 2-3 three times, once each with ecluded values at = 1, = 2, and = -3. He wrote the following epressions: a. _ b. _ c. _ What error did George make? Write the correct epressions, then write an epression that has all three ecluded values. 31. Communicate Mathematical Ideas Write a rational epression with ecluded values at = 0 and = Critical Thinking Sketch the graph of the rational equation y = Think about how to show graphically that a graph eists over a domain ecept at one point. 8 y Module Lesson 1
14 Lesson Performance Task A kayaker spends an afternoon paddling on a river. She travels 3 miles upstream and 3 miles downstream in a total of 4 hours. In still water, the kayaker can travel at an average speed of 2 miles per hour. Based on this information, can you estimate the average speed of the river s current? Is your answer reasonable? Net, assume the average speed of the kayaker is an unknown, k, and not necessarily 2 miles per hour. What is the range of possible average kayaker speeds under the rest of the constraints? Module Lesson 1
10.1 Adding and Subtracting Rational Expressions
Name Class Date 10.1 Adding and Subtracting Rational Epressions Essential Question: How can you add and subtract rational epressions? A2.7.F Determine the sum, difference of rational epressions with integral
More information9.1 Adding and Subtracting Rational Expressions
9.1 Adding and Subtracting Rational Epressions Essential Question: How can you add and subtract rational epressions? Resource Locker Eplore Identifying Ecluded Values Given a rational epression, identify
More informationAdding and Subtracting Rational Expressions
COMMON CORE Locker LESSON 9.1 Adding and Subtracting Rational Epressions Name Class Date 9.1 Adding and Subtracting Rational Epressions Essential Question: How can you add and subtract rational epressions?
More informationWhy? 2 3 times a week. daily equals + 8_. Thus, _ 38 or 38% eat takeout more than once a week. c + _ b c = _ a + b. Factor the numerator. 1B.
Then You added and subtracted polynomials. (Lesson 7-5) Now Add and subtract rational epressions with like denominators. 2Add and subtract rational epressions with unlike denominators. Adding and Subtracting
More informationRational and Radical Functions
Rational and Radical Functions 8A Rational Functions Lab Model Inverse Variation 8-1 Variation Functions 8- Multiplying and Dividing Rational Epressions 8- Adding and Subtracting Rational Epressions Lab
More information8.2 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 information9.3 Solving Rational Equations
Name Class Date 9.3 Solving Rational Equations Essential Question: What methods are there for solving rational equations? Explore Solving Rational Equations Graphically A rational equation is an equation
More informationRational and Radical Functions
Rational and Radical Functions 8A Rational Functions Lab Model Inverse Variation 8-1 Variation Functions 8- Multiplying and Dividing Rational Epressions 8- Adding and Subtracting Rational Epressions Lab
More informationAlgebra 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 informationName Class Date. Inverse of Function. Understanding Inverses of Functions
Name Class Date. Inverses of Functions Essential Question: What is an inverse function, and how do ou know it s an inverse function? A..B Graph and write the inverse of a function using notation such as
More information4.1 Identifying and Graphing Sequences
Name Class Date 4.1 Identifing and Graphing Sequences Essential Question: What is a sequence and how are sequences and functions related? Resource Locker Eplore Understanding Sequences A go-kart racing
More information20.2 Connecting Intercepts and Linear Factors
Name Class Date 20.2 Connecting Intercepts and Linear Factors Essential Question: How are -intercepts of a quadratic function and its linear factors related? Resource Locker Eplore Connecting Factors and
More informationInverse Variation Read 7.1 Examples 1-4
CC Algebra II HW #52 Name Period Row Date Inverse Variation Read 7.1 Eamples 1-4 Section 7.1 1. Vocabulary Eplain how direct variation equations and inverse variation equations are different. Tell whether
More informationUnit 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 information11.1 Solving Linear Systems by Graphing
Name Class Date 11.1 Solving Linear Sstems b Graphing Essential Question: How can ou find the solution of a sstem of linear equations b graphing? Resource Locker Eplore Tpes of Sstems of Linear Equations
More informationHonors 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 information5.3 Interpreting Rate of Change and Slope
Name Class Date 5.3 Interpreting Rate of Change and Slope Essential question: How can ou relate rate of change and slope in linear relationships? Resource Locker Eplore Determining Rates of Change For
More informationUnit 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 information15.4 Equation of a Circle
Name Class Date 1.4 Equation of a Circle Essential Question: How can ou write the equation of a circle if ou know its radius and the coordinates of its center? Eplore G.1.E Show the equation of a circle
More informationReady To Go On? Skills Intervention 12-1 Inverse Variation
12A Find this vocabular word in Lesson 12-1 and the Multilingual Glossar. Identifing Inverse Variation Tell whether the relationship is an inverse variation. Eplain. A. Read To Go On? Skills Intervention
More information7.1 Rational Expressions and Their Simplification
7.1 Rational Epressions and Their Simplification Learning Objectives: 1. Find numbers for which a rational epression is undefined.. Simplify rational epressions. Eamples of rational epressions: 3 and 1
More informationAlgebra 2 Chapter 9 Page 1
Section 9.1A 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
More information7.2 Connecting Intercepts and Linear Factors
Name Class Date 7.2 Connecting Intercepts and Linear Factors Essential Question: How are -intercepts of a quadratic function and its linear factors related? Resource Locker Eplore Connecting Factors and
More information4.7 Solutions of Rational Equations
www.ck1.org Chapter 4. Rational Equations and Functions 4.7 s of Rational Equations Learning Objectives Solve rational equations using cross products. Solve rational equations using lowest common denominators.
More information10.1 Inverses of Simple Quadratic and Cubic Functions
Name Class Date 10.1 Inverses of Simple Quadratic and Cubic Functions Essential Question: What functions are the inverses of quadratic functions and cubic functions, and how can ou find them? Resource
More informationFocusing on Linear Functions and Linear Equations
Focusing on Linear Functions and Linear Equations In grade, students learn how to analyze and represent linear functions and solve linear equations and systems of linear equations. They learn how to represent
More informationAbout the Portfolio Activities. About the Chapter Project
Galileo is credited as the first person to notice that the motion of a pendulum depends only upon its length. About the Chapter Project Finding an average is something that most people can do almost instinctively.
More information11.4 Solving Linear Systems by Multiplying First
Name Class Date 11. Solving Linear Systems by Multiplying First Essential Question: How can you solve a system of linear equations by using multiplication and elimination? Resource Locker Eplore 1 Understanding
More informationWhen a graph on a coordinate plane is a straight line that goes through the origin it is called a direct
DIRECT VARIATION TABLES AND SLOPE LESSON 3-B When a graph on a coordinate plane is a straight line that goes through the origin it is called a direct variation graph. In this lesson you will investigate
More informationLesson #9 Simplifying Rational Expressions
Lesson #9 Simplifying Rational Epressions A.A.6 Perform arithmetic operations with rational epressions and rename to lowest terms Factor the following epressions: A. 7 4 B. y C. y 49y Simplify: 5 5 = 4
More informationProject - Math 99 Final Practice Due Thursday 3 rd August
Project - Math 99 Final Practice Summer 017 Project - Math 99 Final Practice Due Thursday rd August Section 1: Multiple Choice Questions Students Name : 1. The epression 10 + when factored fully is:- A.
More informationProject 2 - Math 99 - Practice Final Due Tuesday 5 th December (100 points) Section 1: Multiple Choice Questions
Project - Math 99 Final Practice Fall 017 Project - Math 99 - Practice Final Due Tuesday 5 th December (100 points) Section 1: Multiple Choice Questions Students Name : 1. The epression 10 + when factored
More information5.1 Understanding Linear Functions
Name Class Date 5.1 Understanding Linear Functions Essential Question: What is a linear function? Resource Locker Eplore 1 Recognizing Linear Functions A race car can travel up to 210 mph. If the car could
More informationRational Expressions
CHAPTER 6 Rational Epressions 6. Rational Functions and Multiplying and Dividing Rational Epressions 6. Adding and Subtracting Rational Epressions 6.3 Simplifying Comple Fractions 6. Dividing Polynomials:
More information6.5 Comparing Properties of Linear Functions
Name Class Date 6.5 Comparing Properties of Linear Functions Essential Question: How can ou compare linear functions that are represented in different was? Resource Locker Eplore Comparing Properties of
More informationLESSON #34 - FINDING RESTRICTED VALUES AND SIMPLIFYING RATIONAL EXPRESSIONS COMMON CORE ALGEBRA II
LESSON #4 - FINDING RESTRICTED VALUES AND SIMPLIFYING RATIONAL EXPRESSIONS COMMON CORE ALGEBRA II A rational epression is a fraction that contains variables. A variable is very useful in mathematics. In
More informationMultiplying and Dividing Rational Expressions y y v 2 3 v 2-13v x z 25 x. n - 6 n 2-6n. 6x + 2 x 2. w y a 3 w.
8- Multiplying and Dividing Rational Epressions Simplify each epression.. 9 a b 7 a b c. ( m n ) -8 m 5 n. 0 y + 5y 5 y - 5y. k - k - 5 k - 9 5. 5 - v v - v - 0. + - - 7. - u y 5 z 5 5 u y 8. a + y y +
More informationEquations 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 informationLinear Relations and Functions
Linear Relations and Functions Why? You analyzed relations and functions. (Lesson 2-1) Now Identify linear relations and functions. Write linear equations in standard form. New Vocabulary linear relations
More information9.5 Solving Nonlinear Systems
Name Class Date 9.5 Solving Nonlinear Sstems Essential Question: How can ou solve a sstem of equations when one equation is linear and the other is quadratic? Eplore Determining the Possible Number of
More information13.2 Exponential Growth Functions
Name Class Date. Eponential Growth Functions Essential Question: How is the graph of g () = a b - h + k where b > related to the graph of f () = b? A.5.A Determine the effects on the ke attributes on the
More informationBasic Property: of Rational Expressions. Multiplication and Division of Rational Expressions. The Domain of a Rational Function: P Q WARNING:
Basic roperties of Rational Epressions A rational epression is any epression of the form Q where and Q are polynomials and Q 0. In the following properties, no denominator is allowed to be zero. The Domain
More informationSECTION P.5. Factoring Polynomials. Objectives. Critical Thinking Exercises. Technology Exercises
BLITMCPB.QXP.0599_48-74 2/0/02 0:4 AM Page 48 48 Chapter P Prerequisites: Fundamental Concepts of Algebra Technology Eercises 98. The common cold is caused by a rhinovirus. The polynomial -0.75 4 + + 5
More informationDefine a rational expression: a quotient of two polynomials. ..( 3 10) (3 2) Rational expressions have the same properties as rational numbers:
1 UNIT 7 RATIONAL EXPRESSIONS & EQUATIONS Simplifying Rational Epressions Define a rational epression: a quotient of two polynomials. A rational epression always indicates division EX: 10 means..( 10)
More informationFinding Complex Solutions of Quadratic Equations
COMMON CORE y - 0 y - - 0 - Locker LESSON 3.3 Finding Comple Solutions of Quadratic Equations Name Class Date 3.3 Finding Comple Solutions of Quadratic Equations Essential Question: How can you find the
More information4.3 Rational Inequalities and Applications
342 Rational Functions 4.3 Rational Inequalities and Applications In this section, we solve equations and inequalities involving rational functions and eplore associated application problems. Our first
More informationAlgebra II Notes Unit Nine: Rational Equations and Functions
Syllabus Objectives: 9. The student will solve a problem by applying inverse and joint variation. 9.6 The student will develop mathematical models involving rational epressions to solve realworld problems.
More information13.1 Exponential Growth Functions
Name Class Date 1.1 Eponential Growth Functions Essential Question: How is the graph of g () = a b - h + k where b > 1 related to the graph of f () = b? Resource Locker Eplore 1 Graphing and Analzing f
More information2.3 Solving Absolute Value Inequalities
Name Class Date.3 Solving Absolute Value Inequalities Essential Question: What are two was to solve an absolute value inequalit? Resource Locker Eplore Visualizing the Solution Set of an Absolute Value
More informationk y = where k is the constant of variation and
Syllabus Objectives: 9. The student will solve a problem by applying inverse and joint variation. 9.6 The student will develop mathematical models involving rational epressions to solve realworld problems.
More information7.3 Adding and Subtracting Rational Expressions
7.3 Adding and Subtracting Rational Epressions LEARNING OBJECTIVES. Add and subtract rational epressions with common denominators. 2. Add and subtract rational epressions with unlike denominators. 3. Add
More information10.1 Inverses of Simple Quadratic and Cubic Functions
COMMON CORE Locker LESSON 0. Inverses of Simple Quadratic and Cubic Functions Name Class Date 0. Inverses of Simple Quadratic and Cubic Functions Essential Question: What functions are the inverses of
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 information7.2 Multiplying Polynomials
Locker LESSON 7. Multiplying Polynomials Teas Math Standards The student is epected to: A.7.B Add, subtract, and multiply polynomials. Mathematical Processes A.1.E Create and use representations to organize,
More informationTable of Contents. Unit 3: Rational and Radical Relationships. Answer Key...AK-1. Introduction... v
These materials may not be reproduced for any purpose. The reproduction of any part for an entire school or school system is strictly prohibited. No part of this publication may be transmitted, stored,
More informationMath 102, Intermediate Algebra Author: Debra Griffin Circle one: Red Bluff Main Campus Burney Weaverville
Math 02, Intermediate Algebra Name Author: Debra Griffin Circle one: Red Bluff Main Campus Burney Weaverville Chapter 2 Rational Epressions 2. Rational Epressions and Allowed X-Values (Domain) 2 Determining
More information6.3 Standard Form. Comparing Forms of Linear Equations. Explore. The slope is. Circle true or false. You can read the slope from the equation.
Name Class Date 6.3 Standard Form Essential Question: How can you write a linear equation in standard form given properties of the line including its slope and points on the line? Resource Locker Explore
More informationEssential Question: How can you solve equations involving variable exponents? Explore 1 Solving Exponential Equations Graphically
6 7 6 y 7 8 0 y 7 8 0 Locker LESSON 1 1 Using Graphs and Properties to Solve Equations with Eponents Common Core Math Standards The student is epected to: A-CED1 Create equations and inequalities in one
More information11.1 Inverses of Simple Quadratic and Cubic Functions
Locker LESSON 11.1 Inverses of Simple Quadratic and Cubic Functions Teas Math Standards The student is epected to: A..B Graph and write the inverse of a function using notation such as f (). Also A..A,
More information6.2 Multiplying Polynomials
Locker LESSON 6. Multiplying Polynomials PAGE 7 BEGINS HERE Name Class Date 6. Multiplying Polynomials Essential Question: How do you multiply polynomials, and what type of epression is the result? Common
More information10.2 Graphing Exponential Functions
Name Class Date 10. Graphing Eponential Functions Essential Question: How do ou graph an eponential function of the form f () = ab? Resource Locker Eplore Eploring Graphs of Eponential Functions Eponential
More informationWords to Review. Give an example of the vocabulary word. Numerical expression. Variable. Evaluate a variable expression. Variable expression
1 Words to Review Give an example of the vocabulary word. Numerical expression 5 12 Variable x Variable expression 3x 1 Verbal model Distance Rate p Time Evaluate a variable expression Evaluate the expression
More information11.3 Solving Radical Equations
Name Class Date 11.3 Solving Radical Equations Essential Question: How can you solve equations involving square roots and cube roots? Explore Investigating Solutions of Square Root Equations Resource Locker
More informationWords to Review. Give an example of the vocabulary word. Numerical expression. Variable. Variable expression. Evaluate a variable expression
1 Words to Review Give an example of the vocabulary word. Numerical expression 5 1 Variable x Variable expression 3x 1 Verbal model Distance Rate p Time Evaluate a variable expression Evaluate the expression
More information2.2 Solving Absolute Value Equations
Name Class Date 2.2 Solving Absolute Value Equations Essential Question: How can you solve an absolute value equation? Resource Locker Explore Solving Absolute Value Equations Graphically Absolute value
More informationA2T. Rational Expressions/Equations. Name: Teacher: Pd:
AT Packet #1: Rational Epressions/Equations Name: Teacher: Pd: Table of Contents o Day 1: SWBAT: Review Operations with Polynomials Pgs: 1-3 HW: Pages -3 in Packet o Day : SWBAT: Factor using the Greatest
More informationSolving Multiplication Equations. Each friend pays $2. The solution of 3x = 6 is 2. Solve each equation using models or a drawing.
3-3 Solving Multiplication Equations MAIN IDEA Solve multiplication equations. New Vocabulary formula Math Online glencoe.com Concepts In Motion Etra Eamples Personal Tutor Self-Check Quiz MONEY Suppose
More informationUnit 9 Study Sheet Rational Expressions and Types of Equations
Algebraic Fractions: Unit 9 Study Sheet Rational Expressions and Types of Equations Simplifying Algebraic Fractions: To simplify an algebraic fraction means to reduce it to lowest terms. This is done by
More information10.2 Graphing Square Root Functions
Name Class Date. Graphing Square Root Functions Essential Question: How can ou use transformations of a parent square root function to graph functions of the form g () = a (-h) + k or g () = b (-h) + k?
More informationExploring Operations Involving Complex Numbers. (3 + 4x) (2 x) = 6 + ( 3x) + +
Name Class Date 11.2 Complex Numbers Essential Question: What is a complex number, and how can you add, subtract, and multiply complex numbers? Explore Exploring Operations Involving Complex Numbers In
More informationSolving Equations with Variables on Both Sides
1. Solving Equations with Variables on Both Sides Essential Question How can you solve an equation that has variables on both sides? Perimeter Work with a partner. The two polygons have the same perimeter.
More informationSimplifying Rational Expressions
.3 Simplifying Rational Epressions What are the ecluded values of a rational epression? How can you simplify a rational epression? ACTIVITY: Simplifying a Rational Epression Work with a partner. Sample:
More informationReview of Rational Expressions and Equations
Page 1 of 14 Review of Rational Epressions and Equations A rational epression is an epression containing fractions where the numerator and/or denominator may contain algebraic terms 1 Simplify 6 14 Identification/Analysis
More informationDefinition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.
8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections.6 and.) 8. Equivalent Inequalities Definition 8. Two inequalities are equivalent
More information5.2 Solving Linear-Quadratic Systems
Name Class Date 5. Solving Linear-Quadratic Sstems Essential Question: How can ou solve a sstem composed of a linear equation in two variables and a quadratic equation in two variables? Resource Locker
More informationReteach Variation Functions
8-1 Variation Functions The variable y varies directly as the variable if y k for some constant k. To solve direct variation problems: k is called the constant of variation. Use the known and y values
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 information4. (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 informationSYSTEMS 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 information20.3 Applying the Zero Product Property to Solve Equations
20.3 Applying the Zero Product Property to Solve Equations Essential Question: How can you use the Zero Product Property to solve quadratic equations in factored form? Resource Locker Explore Understanding
More informationNAME DATE PERIOD. Study Guide and Intervention. Solving Rational Equations and Inequalities
NAME DATE PERIOD Solve Rational Equations A rational equation contains one or more rational epressions. To solve a rational equation, first multipl each side b the least common denominator of all of the
More information3.1 Solving Quadratic Equations by Taking Square Roots
COMMON CORE -8-16 1 1 10 8 6 0 y Locker LESSON.1 Solving Quadratic Equations by Taking Square Roots Name Class Date.1 Solving Quadratic Equations by Taking Square Roots Essential Question: What is an imaginary
More informationRecall that when you multiply or divide both sides of an inequality by a negative number, you must
Unit 3, Lesson 5.3 Creating Rational Inequalities Recall that a rational equation is an equation that includes the ratio of two rational epressions, in which a variable appears in the denominator of at
More informationSlope as a Rate of Change
Find the slope of a line using two of its points. Interpret slope as a rate of change in real-life situations. FINDING THE SLOPE OF A LINE The slope m of the nonvertical line passing through the points
More information6.3 Interpreting Vertex Form and Standard Form
Name Class Date 6.3 Interpreting Verte Form and Standard Form Essential Question: How can ou change the verte form of a quadratic function to standard form? Resource Locker Eplore Identifing Quadratic
More informationFor 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 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 information( ) ( 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 informationSection 3.4 Rational Functions
3.4 Rational Functions 93 Section 3.4 Rational Functions In the last few sections, we have built polynomials based on the positive whole number power functions. In this section we eplore functions based
More information2.3 Solving Absolute Value Inequalities
.3 Solving Absolute Value Inequalities Essential Question: What are two was to solve an absolute value inequalit? Resource Locker Eplore Visualizing the Solution Set of an Absolute Value Inequalit You
More informationModel Inverse Variation
. Model Inverse Variation Rational Equations and Functions. Graph Rational Functions.3 Divide Polynomials.4 Simplify Rational Epressions. Multiply and Divide Rational Epressions.6 Add and Subtract Rational
More information4.5 Multiplication and Division of Rational Expressions
.5. Multiplication and Division of Rational Epressions www.ck2.org.5 Multiplication and Division of Rational Epressions Learning Objectives Multiply rational epressions involving monomials. Multiply rational
More informationChapter 9 Prerequisite Skills
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
More information11.3 Finding Complex Solutions of Quadratic Equations
Name Class Date 11.3 Finding Complex Solutions of Quadratic Equations Essential Question: How can you find the complex solutions of any quadratic equation? Resource Locker Explore Investigating Real Solutions
More information14.3 Constructing Exponential Functions
Name Class Date 1.3 Constructing Eponential Functions Essential Question: What are discrete eponential functions and how do ou represent them? Resource Locker Eplore Understanding Discrete Eponential Functions
More information3. A tennis field has length 78 feet and width of 12 yards. What is the area of the field (in square feet)?
Station 1: MSG9-12.A1.NQ.1: Use units of measure (linear, area, capacity, rates, and time) as a way to understand problems; identify, use and record appr opriate units of measure within context, within
More informationAlgebra. Robert Taggart
Algebra Robert Taggart Table of Contents To the Student.............................................. v Unit 1: Algebra Basics Lesson 1: Negative and Positive Numbers....................... Lesson 2: Operations
More information21.1 Solving Equations by Factoring
Name Class Date 1.1 Solving Equations by Factoring x + bx + c Essential Question: How can you use factoring to solve quadratic equations in standard form for which a = 1? Resource Locker Explore 1 Using
More informationAlgebra I Notes Unit Thirteen: Rational Expressions and Equations
Algebra I Notes Unit Thirteen: Rational Epressions and Equations Syllabus Objective: 10. The student will solve rational equations. (proportions and percents) Ratio: the relationship a b of two quantities,
More informationProblem 1 Oh Snap... Look at the Denominator on that Rational
Problem Oh Snap... Look at the Denominator on that Rational Previously, you learned that dividing polynomials was just like dividing integers. Well, performing operations on rational epressions involving
More information