Algebra I Notes Direct Variation Unit 04e

Size: px
Start display at page:

Download "Algebra I Notes Direct Variation Unit 04e"

Transcription

1 OBJECTIVES: F.IF.B.4 Interpret functions that arise in applications in terms of the contet. For a function that models a relationship between two quantities, interpret ke features of graphs and tables in terms of the quantities, and sketch graphs showing ke features given a verbal description of the relationship. F.IF.B. Understand the concept of a function and use function notation. b. Use function notation, evaluate functions for inputs of their domains, and interpret statements that use function notation in terms of a contet. BIG IDEA: When two variables are related in such a wa that the ratio of their values alwas remains the same, the two variables are said to be in direct variation. The algebraic interpretation for direct variation is that it should be seen as a proportional relationship in which multipling b some fied amount also multiplies b the same fied amount. PREREQUISITE SKILLS: students must have a clear understanding of a graphing functions students must be able to solve proportions VOCABULARY: constant of variation: the number that relates two variables that are directl proportional or inversel proportional to one another direct variation: two variables related in such a wa that their values alwas have a constant ratio slope: measure of steepness of a line origin: the point of intersection of the - and -aes. The (,) of the point of origin is alwas (0,0) SKILLS: students should understand the basic properties and operations of real numbers in order to simplif functions write linear equations that represent direct variation and to know when equations are not varied directl use a ratio to write an equation for direct variation relation: a set of ordered pairs Alg Unit 04e Notes Direct Variation Page of 8 6/5/3

2 REVIEW AND EXAMPLES: Recognizing and using Direct Variation Two variables and var directl if there is a nonzero number k such that the following is true. k where the k represents the constant of variation Two quantities that var directl are said to have direct variation. The properties of graphs of direct variation models indicate that the graph of k must run through the origin and that the slope of the graph E. k is k. If a gallon of milk costs $, and I bu gallon, the total cost is$. If I bu 5 gallons, the price is $0. The total cost of the milk and the number of gallons purchased are subject to direct variation the ratio of the cost to the number of gallons is alwas. If varies directl as, then the graph of all points that describe this relationship is a line going through the origin (0,0) whose slope is called the constant of variation. That s because each of the variables is a constant multiple of the other. If ou bought 0 gallons of milk what would be the total cost? $0 for 0 gallons of milk Epressing Direct Variation as an Equation The equation 4 states that varies directl as since the ratio of to (also written :) never changes. The number 4 in the equation 4 is called the constant of variation. The equation 4 can also be written in the equivalent form 4. That form shows ou that is alwas 4 times as much as. Alg Unit 04e Notes Direct Variation Page of 8 6/5/3

3 E. Find the constant of variation and the slope for the direct variation model. Solution: The equation is in the form k, where k. It is also in the form of = m + b, where m = and b = 0. So, the constant of variation is the same thing as the slope, and the both tell us that for an value, will alwas be as much. Writing a Direct Variation Equation E 3. The variables and var directl. When 5, 0. a. Write an equation that relates and. b. Find the value of when = 0. k 0 k(5) 4 k 4 Now substitute k = 4 when = 0 4(0) 40 when 0, 40 Alg Unit 04e Notes Direct Variation Page 3 of 8 6/5/3

4 Writing a Direct Variation Model E 4. The snack shack sold two bottles of a sports drink and made a profit of $3.50. The net da the sold four bottles of the same sports drink and made a profit of $7. If and var directl, how much profit will the make if the sold siteen bottles of that sports drink? Let d = the sports drink Let p = profit p = kd direct variation formula (d, p) will be the ordered pairs, therefore ou ma either choose (,3.50) or (4,7) substitute one of the ordered pairs into the direct variation formula 3.50 = k() solve for k; k =.75 Now use this constant of variation to solve for the 6 bottles sold. The profit made was $8. Using a Ratio to Model Direct Variation The model k for direct variation can be rewritten as follows. p.75( d) p 8 k This formula is the ratio form of a direct variation model. The ratio form tells ou that if and have direct variation, then the ratio of to is the same for all values of and. (Sometimes real-life data has to be approimated b a direct variation model, even though the data ma not fit the model eactl). Alg Unit 04e Notes Direct Variation Page 4 of 8 6/5/3

5 E 5. The table below gives sample cell phone bills, showing the toal monthl cost and the number of ninutes used that month. Tell whether the total cost and the number of minutes show direct variation. If so, write an equation that relates the quantities. Total cost, c(in dollars) Minutes used, m Solution: Find the ratio of total cost, c, to the minutes used, m, for each month Because the ratios are approimatel equal, the data show direct variation. The equation c relating the total cost to minutes used is 0.35 or c 0.35m m Alg Unit 04e Notes Direct Variation Page 5 of 8 6/5/3

6 ASSESSMENT ITEMS:. Find the constant of variation and the slope. 5 ANS: m k 5 5. If the equation. k represents direct variation between and, and is said to var directl with, the nonzero constant k is called what? ANS: Constant of variation. 3. Write and graph a direct variation equation that has the ordered pair ( 4, ) as a solution. ANS: 4. Hooke s law states that the force needed to stretch a spring varies directl with the amount the spring is stretched. If 64 lbs. of force F stretches a spring a distance d of 8 inches, what is the direct variation equation? A. F = 64 d B. F = 8 d C. F = 8d D. F = 64d ANS: C Alg Unit 04e Notes Direct Variation Page 6 of 8 6/5/3

7 5. If varies directl as, and the constant of variation is 5 k, what is when = 9? 3 A..67 B. 3 C. 5 D. 5 ANS: D 6. Which eample is not an eample of direct variation? A. = + B. = 3 C. = 9 D. cost/person ANS: A 7. Does the US shoe size var directl with the UK shoe size? US shoe size UK shoe size A. True B. False ANS: B 8. A deli sells 3 cookies for $0.60 or 5 cookies for $.00. What is the constant of variation? A. 0.0 B C..80 D ANS: A Alg Unit 04e Notes Direct Variation Page 7 of 8 6/5/3

8 9. Which situation represents direct variation? A. The area B of the base and the height h of a prism with a volume of 0 cubic meters. The related equation is Bh = 0. B. The mass m and the volume V of a substance are related b the equation V = m. C. Alicia cuts a pizza into 8 slices and eats d slices. The related equation is b 8 d. D. The temperature at the start of the da,t,increases 0 degrees b noon. The related equation is t + 0 = d 0. The variables and var directl. Given one pair of values for and, find the equation that relates the variables. 7 ANS: B A. 6 B. 6 C. 60 D. 864 ANS. B Alg Unit 04e Notes Direct Variation Page 8 of 8 6/5/3

Algebra I Notes Relations and Functions Unit 03a

Algebra I Notes Relations and Functions Unit 03a OBJECTIVES: F.IF.A.1 Understand the concept of a function and use function notation. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element

More information

Name Date. and y = 5.

Name Date. and y = 5. Name Date Chapter Fair Game Review Evaluate the epression when = and =.... 0 +. 8( ) Evaluate the epression when a = 9 and b =.. ab. a ( b + ) 7. b b 7 8. 7b + ( ab ) 9. You go to the movies with five

More information

Functions. Essential Question What is a function?

Functions. Essential Question What is a function? 3. Functions COMMON CORE Learning Standard HSF-IF.A. Essential Question What is a function? A relation pairs inputs with outputs. When a relation is given as ordered pairs, the -coordinates are inputs

More information

Functions. Essential Question What is a function? Work with a partner. Functions can be described in many ways.

Functions. Essential Question What is a function? Work with a partner. Functions can be described in many ways. . Functions Essential Question What is a function? A relation pairs inputs with outputs. When a relation is given as ordered pairs, the -coordinates are inputs and the -coordinates are outputs. A relation

More information

Unit 11 - Solving Quadratic Functions PART ONE

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

More information

Linear and Nonlinear Systems of Equations. The Method of Substitution. Equation 1 Equation 2. Check (2, 1) in Equation 1 and Equation 2: 2x y 5?

Linear and Nonlinear Systems of Equations. The Method of Substitution. Equation 1 Equation 2. Check (2, 1) in Equation 1 and Equation 2: 2x y 5? 3330_070.qd 96 /5/05 Chapter 7 7. 9:39 AM Page 96 Sstems of Equations and Inequalities Linear and Nonlinear Sstems of Equations What ou should learn Use the method of substitution to solve sstems of linear

More information

A calculator may be used on the exam.

A calculator may be used on the exam. The Algebra Semester A eamination has the following tpes of questions: Selected Response Student Produced Response (Grid-in) Brief Constructed Response (BCR) Etended Constructed Response (ECR) Short Answer

More information

Fair Game Review. Chapter of a mile the next day. How. far will you jog over the next two days? How many servings does the

Fair Game Review. Chapter of a mile the next day. How. far will you jog over the next two days? How many servings does the Name Date Chapter Evaluate the epression.. Fair Game Review 5 +. 3 3 7 3 8 4 3. 4 4. + 5 0 5 6 5. 3 6. 4 6 5 4 6 3 7. 5 8. 3 9 8 4 3 5 9. You plan to jog 3 4 of a mile tomorrow and 7 8 of a mile the net

More information

3.2 Understanding Relations and Functions-NOTES

3.2 Understanding Relations and Functions-NOTES Name Class Date. Understanding Relations and Functions-NOTES Essential Question: How do ou represent relations and functions? Eplore A1.1.A decide whether relations represented verball, tabularl, graphicall,

More information

3.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS

3.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS Section. Logarithmic Functions and Their Graphs 7. LOGARITHMIC FUNCTIONS AND THEIR GRAPHS Ariel Skelle/Corbis What ou should learn Recognize and evaluate logarithmic functions with base a. Graph logarithmic

More information

Introduction Direct Variation Rates of Change Scatter Plots. Introduction. EXAMPLE 1 A Mathematical Model

Introduction Direct Variation Rates of Change Scatter Plots. Introduction. EXAMPLE 1 A Mathematical Model APPENDIX B Mathematical Modeling B1 Appendi B Mathematical Modeling B.1 Modeling Data with Linear Functions Introduction Direct Variation Rates of Change Scatter Plots Introduction The primar objective

More information

Algebra 1 Midterm Name

Algebra 1 Midterm Name Algebra 1 Midterm 01-013 Name Stud Guide Date: Period: THIS REVIEW WILL BE COLLECTED JANUARY 4 th or Januar 7 th. One problem each page will be graded! You ma not turn this eam review in after the due

More information

Appendix D: Variation

Appendix D: Variation A96 Appendi D Variation Appendi D: Variation Direct Variation There are two basic types of linear models. The more general model has a y-intercept that is nonzero. y m b, b 0 The simpler model y k has

More information

Unit 11 - Solving Quadratic Functions PART TWO

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

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A) 5 B) 277 C) 126 D) 115

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A) 5 B) 277 C) 126 D) 115 MAC 1 Sullivan Practice for Chapter 2 Test (Kincade) Name Date Section MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the distance d(p1, P2)

More information

Analytic Geometry 300 UNIT 9 ANALYTIC GEOMETRY. An air traffi c controller uses algebra and geometry to help airplanes get from one point to another.

Analytic Geometry 300 UNIT 9 ANALYTIC GEOMETRY. An air traffi c controller uses algebra and geometry to help airplanes get from one point to another. UNIT 9 Analtic Geometr An air traffi c controller uses algebra and geometr to help airplanes get from one point to another. 00 UNIT 9 ANALYTIC GEOMETRY Copright 00, K Inc. All rights reserved. This material

More information

Chapter Start Thinking! For use before Activity 6.1. For use before Activity Start Thinking! For use before Lesson

Chapter Start Thinking! For use before Activity 6.1. For use before Activity Start Thinking! For use before Lesson . Enrichment and Etension. a =, b =. a =, b =. a =, b =. a =, b =. a =, b is an number ecept.. a =, b =. a =, b =. a =, b =. Check students work.. Puzzle PAY HIM Etension. Start Thinking! For use before

More information

Unit 10 - Graphing Quadratic Functions

Unit 10 - Graphing Quadratic Functions Unit - Graphing Quadratic Functions PREREQUISITE SKILLS: students should be able to add, subtract and multipl polnomials students should be able to factor polnomials students should be able to identif

More information

c. x x < 60 d. x x =9. What are the first four terms of the sequence? a. 12, 21, 30, 39 b.

c. x x < 60 d. x x =9. What are the first four terms of the sequence? a. 12, 21, 30, 39 b. Algebra I Unit Reasoning with Linear Equations and Inequalities Post Test... A famil s cell phone plan costs $ per month for, minutes and cents per minute over the limit. This month, the famil paid $..

More information

Chapter 5: Systems of Equations

Chapter 5: Systems of Equations Chapter : Sstems of Equations Section.: Sstems in Two Variables... 0 Section. Eercises... 9 Section.: Sstems in Three Variables... Section. Eercises... Section.: Linear Inequalities... Section.: Eercises.

More information

Systems of Linear Equations: Solving by Graphing

Systems of Linear Equations: Solving by Graphing 8.1 Sstems of Linear Equations: Solving b Graphing 8.1 OBJECTIVE 1. Find the solution(s) for a set of linear equations b graphing NOTE There is no other ordered pair that satisfies both equations. From

More information

Say It With Symbols Answers

Say It With Symbols Answers Investigation Additional Practice. p w and p ( w). a. p w () () b. p (w) w and p w. (.) m. A w Q Properties used for items will var, but all include the Distributive Propert.. Possible answer: 7 and ().

More information

DMA 50 Worksheet #1 Introduction to Graphs: Analyzing, Interpreting, and Creating Graphs

DMA 50 Worksheet #1 Introduction to Graphs: Analyzing, Interpreting, and Creating Graphs DMA 0 Worksheet #1 Introduction to Graphs: Analzing, Interpreting, and Creating Graphs A graph will be given followed b a set of questions to answer. Show our work. The bar graph below shows the number

More information

7.1 Guided Practice (p. 401) 1. to find an ordered pair that satisfies each of the equations in the system. solution of the system.

7.1 Guided Practice (p. 401) 1. to find an ordered pair that satisfies each of the equations in the system. solution of the system. CHAPTER 7 Think and Discuss (p. 9). 6,00,000 units. 0,00,000 6,00,000 4,400,000 renters Skill Review (p. 96) 9r 4r 6r. 8.. 0.d.d d 4. w 4 w 4 w 4 w 4 w. 6. 7 g g 9 g 7 g 6 g 0 7 8 40 40 40 7. 6 8. 8 9....

More information

LESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II

LESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II 1 LESSON #4 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The

More information

THIS IS A CLASS SET - DO NOT WRITE ON THIS PAPER

THIS IS A CLASS SET - DO NOT WRITE ON THIS PAPER THIS IS A CLASS SET - DO NOT WRITE ON THIS PAPER ALGEBRA EOC PRACTICE Which situation can be represented b =? A The number of eggs,, in dozen eggs for sale after dozen eggs are sold B The cost,, of buing

More information

Summary and Vocabulary

Summary and Vocabulary Chapter 2 Chapter 2 Summar and Vocabular The functions studied in this chapter are all based on direct and inverse variation. When k and n >, formulas of the form = k n define direct-variation functions,

More information

A11.1 Areas under curves

A11.1 Areas under curves Applications 11.1 Areas under curves A11.1 Areas under curves Before ou start You should be able to: calculate the value of given the value of in algebraic equations of curves calculate the area of a trapezium.

More information

8.4. If we let x denote the number of gallons pumped, then the price y in dollars can $ $1.70 $ $1.70 $ $1.70 $ $1.

8.4. If we let x denote the number of gallons pumped, then the price y in dollars can $ $1.70 $ $1.70 $ $1.70 $ $1. 8.4 An Introduction to Functions: Linear Functions, Applications, and Models We often describe one quantit in terms of another; for eample, the growth of a plant is related to the amount of light it receives,

More information

SYSTEMS OF THREE EQUATIONS

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

More information

Algebra I. Slide 1 / 176 Slide 2 / 176. Slide 3 / 176. Slide 4 / 176. Slide 6 / 176. Slide 5 / 176. System of Linear Equations.

Algebra I. Slide 1 / 176 Slide 2 / 176. Slide 3 / 176. Slide 4 / 176. Slide 6 / 176. Slide 5 / 176. System of Linear Equations. Slide 1 / 176 Slide 2 / 176 Algebra I Sstem of Linear Equations 21-11-2 www.njctl.org Slide 3 / 176 Slide 4 / 176 Table of Contents Solving Sstems b Graphing Solving Sstems b Substitution Solving Sstems

More information

b(n) = 4n, where n represents the number of students in the class. What is the independent

b(n) = 4n, where n represents the number of students in the class. What is the independent Which situation can be represented b =? A The number of eggs,, in dozen eggs for sale after dozen eggs are sold B The cost,, of buing movie tickets that sell for $ each C The cost,, after a $ discount,

More information

What You ll Learn Identify direct variation. Use direct variation to solve problems.

What You ll Learn Identify direct variation. Use direct variation to solve problems. AM_S_C_L_3.indd Page // 3: PM s-user /Volumes//GO/CORE_READING/TENNESSEE/ANCILLARY... Proportionalit and Linear Relationships Teach the Concept Lesson - Direct Variation Interactive Stud Guide See pages

More information

14.1 Systems of Linear Equations in Two Variables

14.1 Systems of Linear Equations in Two Variables 86 Chapter 1 Sstems of Equations and Matrices 1.1 Sstems of Linear Equations in Two Variables Use the method of substitution to solve sstems of equations in two variables. Use the method of elimination

More information

Chapter 9 BUILD YOUR VOCABULARY

Chapter 9 BUILD YOUR VOCABULARY C H A P T E R 9 BUILD YUR VCABULARY Chapter 9 This is an alphabetical list of new vocabular terms ou will learn in Chapter 9. As ou complete the stud notes for the chapter, ou will see Build Your Vocabular

More information

CHAPTER 3 Graphs and Functions

CHAPTER 3 Graphs and Functions CHAPTER Graphs and Functions Section. The Rectangular Coordinate Sstem............ Section. Graphs of Equations..................... 7 Section. Slope and Graphs of Linear Equations........... 7 Section.

More information

Ordered pair: Domain: Range:

Ordered pair: Domain: Range: Sec 2.1 Relations Learning Objectives: 1. Understand relations. 2. Find the domain and the range of a relation. 3. Graph a relation defined b an equation. 1. Understand relations Relation eists when the

More information

LESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II

LESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II 1 LESSON #8 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The

More information

Answers. Chapter 4 A15

Answers. Chapter 4 A15 . =. Sample answer:. a. B is congruent to itself. A and D have the same line of sight, and so the are congruent. Because two angles are congruent, the third angles are congruent. Because the triangles

More information

Graph Quadratic Functions in Standard Form

Graph Quadratic Functions in Standard Form TEKS 4. 2A.4.A, 2A.4.B, 2A.6.B, 2A.8.A Graph Quadratic Functions in Standard Form Before You graphed linear functions. Now You will graph quadratic functions. Wh? So ou can model sports revenue, as in

More information

Section 3.1 Solving Linear Systems by Graphing

Section 3.1 Solving Linear Systems by Graphing Section 3.1 Solving Linear Sstems b Graphing Name: Period: Objective(s): Solve a sstem of linear equations in two variables using graphing. Essential Question: Eplain how to tell from a graph of a sstem

More information

2.2 Equations of Lines

2.2 Equations of Lines 660_ch0pp07668.qd 10/16/08 4:1 PM Page 96 96 CHAPTER Linear Functions and Equations. Equations of Lines Write the point-slope and slope-intercept forms Find the intercepts of a line Write equations for

More information

1.4 Assess Your Understanding

1.4 Assess Your Understanding 40 CHAPTER Graphs.4 Assess Your Understanding Concepts and Vocabular. The slope of a vertical line is ; the slope of a horizontal line is.. Two nonvertical lines have slopes m and m, respectivel. The lines

More information

A calculator may be used on the exam.

A calculator may be used on the exam. The Algebra Semester A eamination will have the following tpes of questions: Selected Response Student Produced Response (Grid-in) Brief Constructed Response (BCR) Etended Constructed Response (ECR) Short

More information

Instructor: KATHRYN SCHRADER Course: A Kathryn Elizabeth Schrader - Alg 1 Hon (2018 / DARNELL COOKMAN-INTEGRATED)

Instructor: KATHRYN SCHRADER Course: A Kathryn Elizabeth Schrader - Alg 1 Hon (2018 / DARNELL COOKMAN-INTEGRATED) Student: Date: Instructor: KATHRYN SCHRADER Course: 3 3 - A - 37 - Kathrn Elizabeth Schrader - Alg Hon (28 / DARNELL COOKMAN-INTEGRATED) Assignment: Chapter Review. What are the variables of the graph

More information

MATH 1710 College Algebra Final Exam Review

MATH 1710 College Algebra Final Exam Review MATH 7 College Algebra Final Eam Review MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. ) There were 80 people at a pla. The admission price was $

More information

Maintaining Mathematical Proficiency

Maintaining Mathematical Proficiency Name Date Chapter 5 Maintaining Mathematical Proficienc Graph the equation. 1. + =. = 3 3. 5 + = 10. 3 = 5. 3 = 6. 3 + = 1 Solve the inequalit. Graph the solution. 7. a 3 > 8. c 9. d 5 < 3 10. 8 3r 5 r

More information

Chapter 11. Systems of Equations Solving Systems of Linear Equations by Graphing

Chapter 11. Systems of Equations Solving Systems of Linear Equations by Graphing Chapter 11 Sstems of Equations 11.1 Solving Sstems of Linear Equations b Graphing Learning Objectives: A. Decide whether an ordered pair is a solution of a sstem of linear equations. B. Solve a sstem of

More information

Answers Investigation 4

Answers Investigation 4 Answers Investigation Applications. a. 7 gallons are being pumped out each hour; students may make a table and notice the constant rate of change, which is - 7, or they may recognize that - 7 is the coefficient

More information

Lesson 5.1 Exponential Functions

Lesson 5.1 Exponential Functions Lesson.1 Eponential Functions 1. Evaluate each function at the given value. Round to four decimal places if necessar. a. r (t) 2(1 0.0) t, t 8 b. j() 9.(1 0.09), 10 2. Record the net three terms for each

More information

M122 College Algebra Review for Final Exam

M122 College Algebra Review for Final Exam M1 College Algebra Review for Final Eam Revised Fall 017 for College Algebra - Beecher All answers should include our work (this could be a written eplanation of the result, a graph with the relevant feature

More information

5.6 RATIOnAl FUnCTIOnS. Using Arrow notation. learning ObjeCTIveS

5.6 RATIOnAl FUnCTIOnS. Using Arrow notation. learning ObjeCTIveS CHAPTER PolNomiAl ANd rational functions learning ObjeCTIveS In this section, ou will: Use arrow notation. Solve applied problems involving rational functions. Find the domains of rational functions. Identif

More information

Applications of Quadratic Equations

Applications of Quadratic Equations 33 Chapter 6 Quadratic Equations and Inequalities Section 6. Applications of Quadratic Equations. Verbal model: Selling price per doz eggs.6 Number eggs sold Number eggs purchased 6.6 6.6.3 6.6 9.6.6.3.8

More information

c) domain {x R, x 3}, range {y R}

c) domain {x R, x 3}, range {y R} Answers Chapter 1 Functions 1.1 Functions, Domain, and Range 1. a) Yes, no vertical line will pass through more than one point. b) No, an vertical line between = 6 and = 6 will pass through two points..

More information

Lecture Guide. Math 90 - Intermediate Algebra. Stephen Toner. Intermediate Algebra, 2nd edition. Miller, O'Neill, & Hyde. Victor Valley College

Lecture Guide. Math 90 - Intermediate Algebra. Stephen Toner. Intermediate Algebra, 2nd edition. Miller, O'Neill, & Hyde. Victor Valley College Lecture Guide Math 90 - Intermediate Algebra to accompan Intermediate Algebra, 2nd edition Miller, O'Neill, & Hde Prepared b Stephen Toner Victor Valle College Last updated: 11/24/10 0 1.1 Sets of Numbers

More information

Ready To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions

Ready To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions Read To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions Find these vocabular words in Lesson 5-1 and the Multilingual Glossar. Vocabular quadratic function parabola verte

More information

MATH 021 UNIT 1 HOMEWORK ASSIGNMENTS

MATH 021 UNIT 1 HOMEWORK ASSIGNMENTS MATH 01 UNIT 1 HOMEWORK ASSIGNMENTS General Instructions You will notice that most of the homework assignments for a section have more than one part. Usuall, the part (A) questions ask for eplanations,

More information

Answers to All Exercises

Answers to All Exercises Answers to All Eercises CHAPTER 5 CHAPTER 5 CHAPTER 5 CHAPTER REFRESHING YOUR SKILLS FOR CHAPTER 5 1a. between 3 and 4 (about 3.3) 1b. between 6 and 7 (about 6.9) 1c. between 7 and 8 (about 7.4) 1d. between

More information

CHAPTER 5 LESSON 5.1 SUPPORT EXAMPLE EXERCISES REFRESHING YOUR SKILLS WORKING WITH POSITIVE SQUARE ROOTS

CHAPTER 5 LESSON 5.1 SUPPORT EXAMPLE EXERCISES REFRESHING YOUR SKILLS WORKING WITH POSITIVE SQUARE ROOTS 5. Sample answer: Using the graphs of f () and g () 1, ou can find f (g ()) 1. Start at on the -ais, move to the corresponding point on the graph of g (), then move to, then to the graph of f (), and lastl

More information

13.2 Exponential Growth Functions

13.2 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 information

2.1 The Rectangular Coordinate System

2.1 The Rectangular Coordinate System . The Rectangular Coordinate Sstem In this section ou will learn to: plot points in a rectangular coordinate sstem understand basic functions of the graphing calculator graph equations b generating a table

More information

Algebra I. Administered May 2014 RELEASED

Algebra I. Administered May 2014 RELEASED STAAR State of Teas Assessments of Academic Readiness Algebra I Administered Ma 0 RELEASED Copright 0, Teas Education Agenc. All rights reserved. Reproduction of all or portions of this work is prohibited

More information

Algebra II Notes Rational Functions Unit Rational Functions. Math Background

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

More information

Chapter 3. 2, which matches a typical. Worked-Out Solutions. Chapter 3 Maintaining Mathematical Proficiency (p.101)

Chapter 3. 2, which matches a typical. Worked-Out Solutions. Chapter 3 Maintaining Mathematical Proficiency (p.101) Chapter Chapter Maintaining Mathematical Proficienc (p.). C A B E F D. This point is in Quadrant I.. This point is in Quadrant II.. This point is on the positive -ais.. This point is in Quadrant III. 5.

More information

Algebra 1. Standard Linear Functions. Categories Graphs Tables Equations Context. Summative Assessment Date: Friday, September 14 th.

Algebra 1. Standard Linear Functions. Categories Graphs Tables Equations Context. Summative Assessment Date: Friday, September 14 th. Algebra 1 Standard Linear Functions Categories Graphs Tables Equations Contet Summative Assessment Date: Friday, September 14 th Page 1 Page 2 Page 3 Linear Functions DAY 1 Notesheet Topic Increasing and

More information

f(x)= x about the y axis.

f(x)= x about the y axis. Practice Eam 2 CH 1 Functions, transformations and graphs Math 3ML FALL 2016 TRUE/FALSE. Write 'T' if the statement is true and 'F' if the statement is false. Provide reasoning. NO EXPLANATION NO CREDIT.

More information

Can you. 1. 3xy y. Inverse inverse neither direct direct. 6. x y x y

Can you. 1. 3xy y. Inverse inverse neither direct direct. 6. x y x y Salisbur CP Unit 5 You Can (No Calculator) You should be able to demonstrate the following skills b completing the associated problems. It is highl suggested that ou read over our notes before attempting

More information

7-1. Exploring Exponential Models. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary. 1. Cross out the expressions that are NOT powers.

7-1. Exploring Exponential Models. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary. 1. Cross out the expressions that are NOT powers. 7-1 Eploring Eponential Models Vocabular Review 1. Cross out the epressions that are NOT powers. 16 6a 1 7. Circle the eponents in the epressions below. 5 6 5a z Vocabular Builder eponential deca (noun)

More information

Ch 3 Alg 2 Note Sheet.doc 3.1 Graphing Systems of Equations

Ch 3 Alg 2 Note Sheet.doc 3.1 Graphing Systems of Equations Ch 3 Alg Note Sheet.doc 3.1 Graphing Sstems of Equations Sstems of Linear Equations A sstem of equations is a set of two or more equations that use the same variables. If the graph of each equation =.4

More information

proportion, p. 163 cross product, p. 168 scale drawing, p. 170

proportion, p. 163 cross product, p. 168 scale drawing, p. 170 REVIEW KEY VOCABULARY classzone.com Multi-Language Glossary Vocabulary practice inverse operations, p. 14 equivalent equations, p. 14 identity, p. 156 ratio, p. 162 proportion, p. 16 cross product, p.

More information

c. Find the slope and y-intercept of the graph of the linear equation. Then sketch its graph.

c. Find the slope and y-intercept of the graph of the linear equation. Then sketch its graph. Name Solve. End-of-Course. 7 =. 5 c =. One cell phone plan charges $0 per month plus $0.5 per minute used. A second cell phone plan charges $5 per month plus $0.0 per minute used. Write and solve an equation

More information

The Remainder and Factor Theorems

The Remainder and Factor Theorems Page 1 of 7 6.5 The Remainder and Factor Theorems What you should learn GOAL 1 Divide polynomials and relate the result to the remainder theorem and the factor theorem. GOAL 2 Use polynomial division in

More information

LESSON #11 - FORMS OF A LINE COMMON CORE ALGEBRA II

LESSON #11 - FORMS OF A LINE COMMON CORE ALGEBRA II LESSON # - FORMS OF A LINE COMMON CORE ALGEBRA II Linear functions come in a variet of forms. The two shown below have been introduced in Common Core Algebra I and Common Core Geometr. TWO COMMON FORMS

More information

118 Intermediate Algebra Fall Intermediate Algebra FNMT Time: 6:30-9:35 pm, Thursday. Room 228

118 Intermediate Algebra Fall Intermediate Algebra FNMT Time: 6:30-9:35 pm, Thursday. Room 228 118 Intermediate Algebra Fall 2018 Intermediate Algebra - 40428 - FNMT 118-122 Time: 6:30-9:3 pm, Thursda Room 228 SYLLABUS Catalog description. Real numbers, polnomials, rational epressions, algebraic

More information

1Write and graph. 2Solve problems. Now. Then. Why? New Vocabulary

1Write and graph. 2Solve problems. Now. Then. Why? New Vocabulary Direct Variation Then You found rates of change of linear functions. (Lesson -) Now Write and graph direct variation equations. Solve problems involving direct variation. Wh? Bianca is saving her mone

More information

Ready To Go On? Skills Intervention 2-1 Solving Linear Equations and Inequalities

Ready To Go On? Skills Intervention 2-1 Solving Linear Equations and Inequalities A Read To Go n? Skills Intervention -1 Solving Linear Equations and Inequalities Find these vocabular words in Lesson -1 and the Multilingual Glossar. Vocabular equation solution of an equation linear

More information

Unit 9: Rational Functions

Unit 9: Rational Functions Date Period Unit 9: Rational Functions DAY TOPIC Direct, Inverse and Combined Variation Graphs of Inverse Variation Page 484 In Class 3 Rational Epressions Multipling and Dividing 4 Adding and Subtracting

More information

Essential Question How can you solve a system of linear equations? $15 per night. Cost, C (in dollars) $75 per Number of. Revenue, R (in dollars)

Essential Question How can you solve a system of linear equations? $15 per night. Cost, C (in dollars) $75 per Number of. Revenue, R (in dollars) 5.1 Solving Sstems of Linear Equations b Graphing Essential Question How can ou solve a sstem of linear equations? Writing a Sstem of Linear Equations Work with a partner. Your famil opens a bed-and-breakfast.

More information

2.1 Intercepts; Symmetry; Graphing Key Equations

2.1 Intercepts; Symmetry; Graphing Key Equations Ch. Graphs.1 Intercepts; Smmetr; Graphing Ke Equations 1 Find Intercepts from an Equation MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. List the

More information

Modeling Linear Relationships In the Patterns of Change unit, you studied a variety of

Modeling Linear Relationships In the Patterns of Change unit, you studied a variety of LESSON 1 Modeling Linear Relationships In the Patterns of Change unit, you studied a variety of relationships between quantitative variables. Among the most common were linear functions those with straight-line

More information

Essential Question How can you use a quadratic function to model a real-life situation?

Essential Question How can you use a quadratic function to model a real-life situation? 3. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..A A..E A..A A..B A..C Modeling with Quadratic Functions Essential Question How can ou use a quadratic function to model a real-life situation? Work with a partner.

More information

C) C) 5 D) 13

C) C) 5 D) 13 MATH 03 FINAL EXAM REVIEW Simplif. 1) 8-4 2 72 + 62 16 8 4 13-8 13-8 8 Solve. 2) + 2 = 9 1 2 4 13 3) 4 = a -9-6 -1-36 - 4) -4.0 = b 13 8.0-8.0 7.0-3.00 ) 1 f - = 1-12 12-30 30 6) 2-1 3 = -7-10 7 10 7)

More information

6. 4 : 5 is in simplest form because 4 and 5 have no. common factors. 4 : : 5 3. and 7 : : 14 2

6. 4 : 5 is in simplest form because 4 and 5 have no. common factors. 4 : : 5 3. and 7 : : 14 2 9. Let the number of balloons be. Total cost of balloons: $(.0) Cost of banner: $ Total ependiture of committee: $(.0 ) The most the committee can spend is $:.0.0.0 90.0.0 90.0.7 The can bu at most balloons.

More information

Interpret Linear Graphs

Interpret Linear Graphs Interpret Linear Graphs Objectives: -Interpret the meaning of the and intercepts, slope, and points on and off the line of a graph, in the contet of a real world situation. Common Core Standards: N.Q.1

More information

For use after the chapter Graphing Linear Equations and Functions 3 D. 7. 4y 2 3x 5 4; (0, 1) x-intercept: 6 y-intercept: 3.

For use after the chapter Graphing Linear Equations and Functions 3 D. 7. 4y 2 3x 5 4; (0, 1) x-intercept: 6 y-intercept: 3. Chapter Test A Write the coordinates of the point.. A. B. D. C. A. D C B.... Tell whether the ordered pair is a solution of the equation.. ; (, ) 7.. ; (, ). 7. ; (, ). Draw the line that has the given

More information

XIV. Mathematics, Grade 8

XIV. Mathematics, Grade 8 XIV. Mathematics, Grade 8 Grade 8 Mathematics Test The spring 2015 grade 8 Mathematics test was based on standards in the five domains for grade 8 in the Massachusetts Curriculum Framework for Mathematics

More information

Practice « «3. ` «- -2« 5. ` « « « $ «$ -7« $ 16.!2 $

Practice « «3. ` «- -2« 5. ` « « « $ «$ -7« $ 16.!2 $ Practice - Properties of Real Numbers Simplif.. - 4.«. - 6«. ` 7 4. «- -«6 `. ` 6. 0. -6«7. 4-8«8. -0.0«` Replace each $ with the smbol R, S,or to make the sentence true. 9.!6 $!0 0. $.. 0.06 $ 0.6. 4

More information

Name Class Date. Solving by Graphing and Algebraically

Name Class Date. Solving by Graphing and Algebraically Name Class Date 16-4 Nonlinear Sstems Going Deeper Essential question: How can ou solve a sstem of equations when one equation is linear and the other is quadratic? To estimate the solution to a sstem

More information

Fair Game Review. Chapter 2. and y = 5. Evaluate the expression when x = xy 2. 4x. Evaluate the expression when a = 9 and b = 4.

Fair Game Review. Chapter 2. and y = 5. Evaluate the expression when x = xy 2. 4x. Evaluate the expression when a = 9 and b = 4. Name Date Chapter Fair Game Review Evaluate the epression when = and =.... 0 +. 8( ) Evaluate the epression when a = 9 and b =.. ab. a ( b + ) 7. b b 7 8. 7b + ( ab ) 9. You go to the movies with five

More information

11.1 Solving Linear Systems by Graphing

11.1 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 information

Chapter 2: Differentiation 1. Find the slope of the tangent line to the graph of the function below at the given point.

Chapter 2: Differentiation 1. Find the slope of the tangent line to the graph of the function below at the given point. Chapter : Differentiation 1. Find the slope of the tangent line to the graph of the function below at the given point. f( ) 10, (, ) 10 1 E) none of the above. Find the slope of the tangent line to the

More information

Modeling with Exponential and Logarithmic Functions 6.7. Essential Question How can you recognize polynomial, exponential, and logarithmic models?

Modeling with Exponential and Logarithmic Functions 6.7. Essential Question How can you recognize polynomial, exponential, and logarithmic models? .7 Modeling with Eponential and Logarithmic Functions Essential Question How can ou recognize polnomial, eponential, and logarithmic models? Recognizing Different Tpes of Models Work with a partner. Match

More information

4.2 Start Thinking. 4.2 Warm Up. 4.2 Cumulative Review Warm Up

4.2 Start Thinking. 4.2 Warm Up. 4.2 Cumulative Review Warm Up . Start Thinking How can ou find a linear equation from a graph for which ou do not know the -intercept? Describe a situation in which ou might know the slope but not the -intercept. Provide a graph of

More information

Algebra I Practice Exam

Algebra I Practice Exam Algebra I This practice assessment represents selected TEKS student expectations for each reporting category. These questions do not represent all the student expectations eligible for assessment. Copyright

More information

Review of Exponent Rules

Review of Exponent Rules Page Review of Eponent Rules Math : Unit Radical and Rational Functions Rule : Multipling Powers With the Same Base Multipl Coefficients, Add Eponents. h h h. ( )( ). (6 )(6 ). (m n )(m n ). ( 8ab)( a

More information

[B] 2. Which of these numbers is a solution for 12 x 7? [A] 5 [B] 1 [C] 2 [D] 3

[B] 2. Which of these numbers is a solution for 12 x 7? [A] 5 [B] 1 [C] 2 [D] 3 . Which graph shows the solution of + 7? 9 6 0 6 9 9 6 0 6 9 9 6 0 6 9 9 6 0 6 9. Which of these numbers is a solution for 7? 5. Which graph shows the solution to +

More information

Linear Relations and Functions

Linear 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 information

Bridge-Thickness Experiment. Student 2

Bridge-Thickness Experiment. Student 2 Applications 1. Below are some results from the bridge-thickness eperiment. Bridge-Thickness Eperiment Thickness (laers) Breaking Weight (pennies) 15 5 5 a. Plot the (thickness, breaking weight) data.

More information

NAME DATE PERIOD. Study Guide and Intervention. Ax + By = C, where A 0, A and B are not both zero, and A, B, and C are integers with GCF of 1.

NAME DATE PERIOD. Study Guide and Intervention. Ax + By = C, where A 0, A and B are not both zero, and A, B, and C are integers with GCF of 1. NAME DATE PERID 3-1 Stud Guide and Intervention Identif Linear Equations and Intercepts A linear equation is an equation that can be written in the form A + B = C. This is called the standard form of a

More information

NAME DATE PERIOD. Study Guide and Intervention. Ax + By = C, where A 0, A and B are not both zero, and A, B, and C are integers with GCF of 1.

NAME DATE PERIOD. Study Guide and Intervention. Ax + By = C, where A 0, A and B are not both zero, and A, B, and C are integers with GCF of 1. NAME DATE PERID 3-1 Stud Guide and Intervention Graphing Linear Equations Identif Linear Equations and Intercepts A linear equation is an equation that can be written in the form A + B = C. This is called

More information