RELATIONS AND FUNCTIONS through

Size: px
Start display at page:

Download "RELATIONS AND FUNCTIONS through"

Transcription

1 RELATIONS AND FUNCTIONS through Relations and Functions establish a correspondence between the input values (usuall ) and the output values (usuall ) according to the particular relation or function. Often this correspondence can be described b an equation and a graph. Relations are functions if, for each input value, there is onl one output value. Relations and functions work like a machine, as shown in the diagram in Eample 1. A relation or function is given a name that is usuall a letter, such as f or g. The notation f () represents the output when is processed b the machine. (Note: f () is read, f of. ) When is put into the machine, f (), the value of a relation or function for a specific -value, comes out. See the Math Notes bo on page 73. Eample 1 Numbers are put into the relation or function machine (in this case, the function f () = 2 + 1) one at a time, and then the relation or function performs the operation(s) on each input to determine each output. For eample, when = 3 is put into the function f () = 2 + 1, the machine multiplies 3 b 2 and adds 1 to get the output, which is 7. The notation f (3) = 7 shows that the function named f connects the input (3) with the output (7). This also means the point (3, 7) lies on the graph of the function. inputs = 3 f () = f (3) = 7 outputs Eample 2 a. If f () =! 2 then f (11) =? f (11) = 11! 2 = 9 = 3 b. If g() = 3! 2 then g(5) =? g(5) = 3! (5) 2 = 3! 25 =!22 c. If f () = + 3 2! 5 then f (2) =? f (2) = ! 2 " 5 = 5 "1 = "5 Chapter 11: Functions and Relations 93

2 Eample 3 A relation in which each input has onl one output is called a function. Eamining the graphs below, g() is a function while f () is not. f () g() Each input () has onl one output (). g(!2) = 1,!g(0) = 3,!g() =!1 and so on. Each input greater than!3 has two -values associated with it. f (1) = 2 and f (1) =!2 Eample The set of all possible inputs of a relation is called the domain, while the set of all possible outputs of a relation is called the range. For g() in Eample 3 the domain is all numbers from!2 through and the range is all numbers from!1 through 3. For f () in Eample 3, since the graph starts at!3 and continues forever to the right, the domain is all numbers greater than or equal to!3. The graph of f () etends in both the positive and negative directions forever, so the range is all numbers. Eample 5 For the graph at right, since the -values etend forever in both directions the domain is all numbers. The -values start at 1 and go higher so the range is all numbers greater or equal to 1. 9 Algebra Connections Parent Guide

3 Problems Determine the outputs for the following relations and the given inputs. 1. = 2 2. =!6 3. = 9 f () =!2 +. f () = (5! ) 2 f (8) =? 7. h() = 5! h(9) =? f () =? f () =! 2 5. g() = 2! 5 g(!3) =? 8. h() = 5! h(9) =? f () =? f () = f () = ! 9 f (3) =? 9. f () =! 2 f () =? f () =? Determine if each relation is a function. Then state its domain and range Chapter 11: Functions and Relations 95

4 Answers not possible not possible es, each input has one output; domain is all numbers, range is!1 " " 13. no; -1 has two outputs; domain is -,-3, -1, 0, 1, 2, 3,, range is -, -3, -2, -1, 0, 1, no, for eample =0 has two outputs; domain is! "3, range is all numbers 1. es; domain is all numbers, range is! "2 12. es; domain all numbers, range is!2 " " no, man inputs have two outputs; domain is!2 " " range is!2 " " 96 Algebra Connections Parent Guide

5 TRANSFORMATIONS OF A FUNCTION The simplest equation of one shape (e.g., line, parabola, absolute value) is called a parent equation. Changing the parent equation b addition or multiplication moves and changes the size and orientation of the parent graph but does not change the basic shape. These changes are called transformations. The first set of eamples shows how the parent graph of a parabola can be moved on an -coordinate sstem. In later courses, ou will learn how to make it wider or narrower. Transformations of other functions are done in a similar manner. Eamples = 2 the parent graph = (! 3) 2 right 3 units = ( + 2) 2 left 2 units = 2! 2 down 2 units = ( + 3) 2! 2 left 3 and down 2 =!(! 2) right 2, up 5, and flipped Problems Predict how each parabola is different from the parent graph. 1. = ( + 5) 2 2. = =! 2. = 2! 5. = =!( + 2) 2 7. = (! 5) 2! 3 8. =!( + 2) = (! 3) 2! 5 Chapter 11: Functions and Relations 97

6 The parent graphs for absolute value and square root are shown at right. The are transformed eactl the same wa as parabolas. Predict how the graph of each equation below is different from the parent graph. = = 10. =! 11. = = = =! =! 16. =! =!! =!! = =! =! Answers 1. left 5 2. up 5 3. flipped. down 5. up 5 6. left 2, flipped 7. right 5, down 3 8. left 2, up 1, flipped 9. right 3, down right 11. up up left 2 1. up 5, flipped 15. flipped 16. left 3, flipped 17. right 5, flipped 18. right 2, up 5, flipped 19. up right left 3, up 7, flipped 98 Algebra Connections Parent Guide

7 INTERCEPTS AND INTERSECTIONS and Intercepts and Intersections both represent a point at which two paths cross, but an intercept specifies where a curve or line crosses an ais, whereas an intersection refers to an point where the graphs of the two equations cross. When the graph is present, the points ma be estimated from the graph. For more accurac, intercepts and intersections ma be found using algebra in most cases. Eample 1 Use the graphs of the parabola = 2! 3!10 and the line =!2 + 2 at right. The graph shows that the parabola has two -intercepts (-2, 0) and (5, 0) and one -intercept (0, -10). The line has -intercept (1, 0) and -intercept (0, 2). The parabola and the line cross each other twice ielding two points of intersection: (-3, 8) and (, -6). Recall that -intercepts are alwas of the form (, 0) so the ma be found b making = 0 and solving for. Y-intercepts are alwas of the form (0, ) so the ma be found b making = 0 and solving for. To find the points of intersection, solve the sstem of equations. See the algebraic solutions below. Intercepts To find the -intercepts of the parabola, make = 0 and solve for. 0 = 2! 3! 10 Factor and use the zero-product propert. 0 = (! 5)( + 2) = 5 or =!2 so (5, 0) and (-2, 0). To find the -intercepts of the parabola, make = 0 and solve for. = 0 2! 3" 0! 10 =!10 so (0, -10). To find the - and -intercepts of the line follow the same procedure. If = 0, then =!2 " = 2 so (0, 2). If = 0, then 0 =!2 + 2, = 1 so (1, 0). Intersection To find the point(s) of intersection of the sstem of equations use the Equal Values Method or Substitution. 2! 3! 10 =!2 + 2 Make one side equal to zero, factor, and use the zero product propert to solve for. (The quadratic formula is also possible.) 2!! 12 = 0 (! )( + 3) = 0 = or =!3 Using = in either equation ields =!6 so (, -6). Using =!3 in either equation ields = 8 so (-3, 8). Chapter 11: Functions and Relations 99

8 Eample 2 Given the lines = and = 1! 3 determine the intercepts and intersection without 2 graphing. Using the same methods as above: Intercepts Intersection For the line = : If = 0, then = 1! = 8 so (0, 8). 3 If = 0, 0 = 1 + 8, =!2 so (-2, 0). 3 For the line = 1 2! 3 : If = 0, then = 1! 0 " 3 = "3, so (0, -3). 2 If = 0, then 0 = 1! 3, = 6 so (6, 0) = 1 2! 3 6! 1 ( = 1 2 " 3 ) = 3! = Substituting = 66 into either original equation ields = 30. The point of intersection is (66, 30). Problems For each equation, determine the - and -intercepts. 1. = 3! 2 2. = 1! = 12.! + 3 = = 15! 3 6. = = 2! 3! = 2! 11! 3 9. = 2 + 3! 2 Determine the point(s) of intersection 10. = =!3! = + 7 =! = = =! = 31 1.! 3 =!10 = 1! ! = 6! = = 2! 3! 8 = = 2! 7 = = =!! Algebra Connections Parent Guide

9 Answers 1. (0.6,!0),!(0,!!2) 2. (8, 0), (0, -2) 3. (, 0), (0, 6). (-15, 0), (0, 5) 5. (5, 0), (0, 7.5) 6. (-2, 0), (-3, 0), (0, 6) 7. (5, 0), (-2, 0) (0, -10) 8. (!0.25,!0),!(3,!0) (0, -3) 9. (0.56,!0),!(!3.56,!0) (0, -2) 10. (-2, -9) 11. (, 11) 12. (13, 2) 13. (3.5, 1) 1. (-0.25, 3) 15. (3, 0) 16. (5, 2), (-2, 2) 17. (5, 18), (-3, 2) 18. (-3, 11) Chapter 11: Functions and Relations 101

( 7, 3) means x = 7 and y = 3. ( 7, 3) works in both equations so. Section 5 1: Solving a System of Linear Equations by Graphing

( 7, 3) means x = 7 and y = 3. ( 7, 3) works in both equations so. Section 5 1: Solving a System of Linear Equations by Graphing Section 5 : Solving a Sstem of Linear Equations b Graphing What is a sstem of Linear Equations? A sstem of linear equations is a list of two or more linear equations that each represents the graph of a

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

MAT 1275: Introduction to Mathematical Analysis. Graphs and Simplest Equations for Basic Trigonometric Functions. y=sin( x) Function

MAT 1275: Introduction to Mathematical Analysis. Graphs and Simplest Equations for Basic Trigonometric Functions. y=sin( x) Function MAT 275: Introduction to Mathematical Analsis Dr. A. Rozenblum Graphs and Simplest Equations for Basic Trigonometric Functions We consider here three basic functions: sine, cosine and tangent. For them,

More information

Vertex. March 23, Ch 9 Guided Notes.notebook

Vertex. March 23, Ch 9 Guided Notes.notebook March, 07 9 Quadratic Graphs and Their Properties A quadratic function is a function that can be written in the form: Verte Its graph looks like... which we call a parabola. The simplest quadratic function

More information

Unit 12 Study Notes 1 Systems of Equations

Unit 12 Study Notes 1 Systems of Equations You should learn to: Unit Stud Notes Sstems of Equations. Solve sstems of equations b substitution.. Solve sstems of equations b graphing (calculator). 3. Solve sstems of equations b elimination. 4. Solve

More information

SOLVING SYSTEMS OF EQUATIONS

SOLVING SYSTEMS OF EQUATIONS SOLVING SYSTEMS OF EQUATIONS 3.. 3..4 In this course, one focus is on what a solution means, both algebraicall and graphicall. B understanding the nature of solutions, students are able to solve equations

More information

12x y (4) 2x y (4) 5x y is the same as

12x y (4) 2x y (4) 5x y is the same as Name: Unit #6 Review Quadratic Algebra Date: 1. When 6 is multiplied b the result is 0 1 () 9 1 () 9 1 () 1 0. When is multiplied b the result is 10 6 1 () 7 1 () 7 () 10 6. Written without negative eponents

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

SOLVING SYSTEMS OF EQUATIONS

SOLVING SYSTEMS OF EQUATIONS SOLVING SYSTEMS OF EQUATIONS 4.. 4..4 Students have been solving equations even before Algebra. Now the focus on what a solution means, both algebraicall and graphicall. B understanding the nature of solutions,

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

3 Polynomial and Rational Functions

3 Polynomial and Rational Functions 3 Polnomial and Rational Functions 3.1 Quadratic Functions and Models 3.2 Polnomial Functions and Their Graphs 3.3 Dividing Polnomials 3.4 Real Zeros of Polnomials 3.5 Comple Zeros and the Fundamental

More information

Systems of Linear and Quadratic Equations. Check Skills You ll Need. y x. Solve by Graphing. Solve the following system by graphing.

Systems of Linear and Quadratic Equations. Check Skills You ll Need. y x. Solve by Graphing. Solve the following system by graphing. NY- Learning Standards for Mathematics A.A. Solve a sstem of one linear and one quadratic equation in two variables, where onl factoring is required. A.G.9 Solve sstems of linear and quadratic equations

More information

= x. Algebra II Notes Quadratic Functions Unit Graphing Quadratic Functions. Math Background

= x. Algebra II Notes Quadratic Functions Unit Graphing Quadratic Functions. Math Background Algebra II Notes Quadratic Functions Unit 3.1 3. Graphing Quadratic Functions Math Background Previousl, ou Identified and graphed linear functions Applied transformations to parent functions Graphed quadratic

More information

Learning Targets: Standard Form: Quadratic Function. Parabola. Vertex Max/Min. x-coordinate of vertex Axis of symmetry. y-intercept.

Learning Targets: Standard Form: Quadratic Function. Parabola. Vertex Max/Min. x-coordinate of vertex Axis of symmetry. y-intercept. Name: Hour: Algebra A Lesson:.1 Graphing Quadratic Functions Learning Targets: Term Picture/Formula In your own words: Quadratic Function Standard Form: Parabola Verte Ma/Min -coordinate of verte Ais of

More information

Mth Quadratic functions and quadratic equations

Mth Quadratic functions and quadratic equations Mth 0 - Quadratic functions and quadratic equations Name Find the product. 1) 8a3(2a3 + 2 + 12a) 2) ( + 4)( + 6) 3) (3p - 1)(9p2 + 3p + 1) 4) (32 + 4-4)(2-3 + 3) ) (4a - 7)2 Factor completel. 6) 92-4 7)

More information

Unit 2 Notes Packet on Quadratic Functions and Factoring

Unit 2 Notes Packet on Quadratic Functions and Factoring Name: Period: Unit Notes Packet on Quadratic Functions and Factoring Notes #: Graphing quadratic equations in standard form, verte form, and intercept form. A. Intro to Graphs of Quadratic Equations: a

More information

7.5 Solve Special Types of

7.5 Solve Special Types of 75 Solve Special Tpes of Linear Sstems Goal p Identif the number of of a linear sstem Your Notes VOCABULARY Inconsistent sstem Consistent dependent sstem Eample A linear sstem with no Show that the linear

More information

Mathematics. Polynomials and Quadratics. hsn.uk.net. Higher. Contents. Polynomials and Quadratics 1. CfE Edition

Mathematics. Polynomials and Quadratics. hsn.uk.net. Higher. Contents. Polynomials and Quadratics 1. CfE Edition Higher Mathematics Contents 1 1 Quadratics EF 1 The Discriminant EF 3 3 Completing the Square EF 4 4 Sketching Parabolas EF 7 5 Determining the Equation of a Parabola RC 9 6 Solving Quadratic Inequalities

More information

Mathematics. Polynomials and Quadratics. hsn.uk.net. Higher. Contents. Polynomials and Quadratics 52 HSN22100

Mathematics. Polynomials and Quadratics. hsn.uk.net. Higher. Contents. Polynomials and Quadratics 52 HSN22100 Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 5 1 Quadratics 5 The Discriminant 54 Completing the Square 55 4 Sketching Parabolas 57 5 Determining the Equation

More information

TRANSFORMATIONS OF f(x) = x Example 1

TRANSFORMATIONS OF f(x) = x Example 1 TRANSFORMATIONS OF f() = 2 2.1.1 2.1.2 Students investigate the general equation for a famil of quadratic functions, discovering was to shift and change the graphs. Additionall, the learn how to graph

More information

Higher. Polynomials and Quadratics. Polynomials and Quadratics 1

Higher. Polynomials and Quadratics. Polynomials and Quadratics 1 Higher Mathematics Contents 1 1 Quadratics EF 1 The Discriminant EF 3 3 Completing the Square EF 4 4 Sketching Parabolas EF 7 5 Determining the Equation of a Parabola RC 9 6 Solving Quadratic Inequalities

More information

Algebra 1 Skills Needed for Success in Math

Algebra 1 Skills Needed for Success in Math Algebra 1 Skills Needed for Success in Math A. Simplifing Polnomial Epressions Objectives: The student will be able to: Appl the appropriate arithmetic operations and algebraic properties needed to simplif

More information

LESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II

LESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II LESSON #4 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART COMMON CORE ALGEBRA II You will recall from unit 1 that in order to find the inverse of a function, ou must switch and and solve for. Also,

More information

Glossary. Also available at BigIdeasMath.com: multi-language glossary vocabulary flash cards. An equation that contains an absolute value expression

Glossary. Also available at BigIdeasMath.com: multi-language glossary vocabulary flash cards. An equation that contains an absolute value expression Glossar This student friendl glossar is designed to be a reference for ke vocabular, properties, and mathematical terms. Several of the entries include a short eample to aid our understanding of important

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

MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED

MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED FOM 11 T GRAPHING LINEAR INEQUALITIES & SET NOTATION - 1 1 MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED 1) INEQUALITY = a mathematical statement that contains one of these four inequalit signs: ,.

More information

Section 5.1: Functions

Section 5.1: Functions Objective: Identif functions and use correct notation to evaluate functions at numerical and variable values. A relationship is a matching of elements between two sets with the first set called the domain

More information

Using Intercept Form

Using Intercept Form 8.5 Using Intercept Form Essential Question What are some of the characteristics of the graph of f () = a( p)( q)? Using Zeros to Write Functions Work with a partner. Each graph represents a function of

More information

TABLES, GRAPHS, AND RULES

TABLES, GRAPHS, AND RULES TABLES, GRAPHS, AND RULES 3.1.1 3.1.7 Three was to write relationships for data are tables, words (descriptions), and rules. The pattern in tables between input () and output () values usuall establishes

More information

MA123, Chapter 1: Equations, functions and graphs (pp. 1-15)

MA123, Chapter 1: Equations, functions and graphs (pp. 1-15) MA123, Chapter 1: Equations, functions and graphs (pp. 1-15) Date: Chapter Goals: Identif solutions to an equation. Solve an equation for one variable in terms of another. What is a function? Understand

More information

Vertex form of a quadratic equation

Vertex form of a quadratic equation Verte form of a quadratic equation Nikos Apostolakis Spring 017 Recall 1. Last time we looked at the graphs of quadratic equations in two variables. The upshot was that the graph of the equation: k = a(

More information

8.1 Exponents and Roots

8.1 Exponents and Roots Section 8. Eponents and Roots 75 8. Eponents and Roots Before defining the net famil of functions, the eponential functions, we will need to discuss eponent notation in detail. As we shall see, eponents

More information

Algebra Concepts Equation Solving Flow Chart Page 1 of 6. How Do I Solve This Equation?

Algebra Concepts Equation Solving Flow Chart Page 1 of 6. How Do I Solve This Equation? Algebra Concepts Equation Solving Flow Chart Page of 6 How Do I Solve This Equation? First, simplify both sides of the equation as much as possible by: combining like terms, removing parentheses using

More information

APPENDIX D Rotation and the General Second-Degree Equation

APPENDIX D Rotation and the General Second-Degree Equation APPENDIX D Rotation and the General Second-Degree Equation Rotation of Aes Invariants Under Rotation After rotation of the - and -aes counterclockwise through an angle, the rotated aes are denoted as the

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

Chapter 18 Quadratic Function 2

Chapter 18 Quadratic Function 2 Chapter 18 Quadratic Function Completed Square Form 1 Consider this special set of numbers - the square numbers or the set of perfect squares. 4 = = 9 = 3 = 16 = 4 = 5 = 5 = Numbers like 5, 11, 15 are

More information

a. plotting points in Cartesian coordinates (Grade 9 and 10), b. using a graphing calculator such as the TI-83 Graphing Calculator,

a. plotting points in Cartesian coordinates (Grade 9 and 10), b. using a graphing calculator such as the TI-83 Graphing Calculator, GRADE PRE-CALCULUS UNIT C: QUADRATIC FUNCTIONS CLASS NOTES FRAME. After linear functions, = m + b, and their graph the Quadratic Functions are the net most important equation or function. The Quadratic

More information

Chapter(5( (Quadratic(Equations( 5.1 Factoring when the Leading Coefficient Equals 1

Chapter(5( (Quadratic(Equations( 5.1 Factoring when the Leading Coefficient Equals 1 .1 Factoring when the Leading Coefficient Equals 1 1... x 6x 8 x 10x + 9 x + 10x + 1 4. (x )( x + 1). (x + 6)(x 4) 6. x(x 6) 7. (x + )(x + ) 8. not factorable 9. (x 6)(x ) 10. (x + 1)(x ) 11. (x + 7)(x

More information

Mt. Douglas Secondary

Mt. Douglas Secondary Foundations of Math 11 Section 7.1 Quadratic Functions 31 7.1 Quadratic Functions Mt. Douglas Secondar Quadratic functions are found in everda situations, not just in our math classroom. Tossing a ball

More information

QUADRATIC FUNCTION REVIEW

QUADRATIC FUNCTION REVIEW Name: Date: QUADRATIC FUNCTION REVIEW Linear and eponential functions are used throughout mathematics and science due to their simplicit and applicabilit. Quadratic functions comprise another ver important

More information

Lesson Goals. Unit 4 Polynomial/Rational Functions Quadratic Functions (Chap 0.3) Family of Quadratic Functions. Parabolas

Lesson Goals. Unit 4 Polynomial/Rational Functions Quadratic Functions (Chap 0.3) Family of Quadratic Functions. Parabolas Unit 4 Polnomial/Rational Functions Quadratic Functions (Chap 0.3) William (Bill) Finch Lesson Goals When ou have completed this lesson ou will: Graph and analze the graphs of quadratic functions. Solve

More information

1.3. Absolute Value and Piecewise-Defined Functions Absolutely Piece-ful. My Notes ACTIVITY

1.3. Absolute Value and Piecewise-Defined Functions Absolutely Piece-ful. My Notes ACTIVITY Absolute Value and Piecewise-Defined Functions Absolutel Piece-ful SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Create Representations, Quickwrite. Graph both = - for < 3 and = - + 7 for

More information

4Cubic. polynomials UNCORRECTED PAGE PROOFS

4Cubic. polynomials UNCORRECTED PAGE PROOFS 4Cubic polnomials 4.1 Kick off with CAS 4. Polnomials 4.3 The remainder and factor theorems 4.4 Graphs of cubic polnomials 4.5 Equations of cubic polnomials 4.6 Cubic models and applications 4.7 Review

More information

12.1 Systems of Linear equations: Substitution and Elimination

12.1 Systems of Linear equations: Substitution and Elimination . Sstems of Linear equations: Substitution and Elimination Sstems of two linear equations in two variables A sstem of equations is a collection of two or more equations. A solution of a sstem in two variables

More information

Derivatives 2: The Derivative at a Point

Derivatives 2: The Derivative at a Point Derivatives 2: The Derivative at a Point 69 Derivatives 2: The Derivative at a Point Model 1: Review of Velocit In the previous activit we eplored position functions (distance versus time) and learned

More information

4. exponential decay; 20% 9.1 Practice A found square root instead of cube root 16 =

4. exponential decay; 20% 9.1 Practice A found square root instead of cube root 16 = 9.. eponential deca; 0% 9. Practice A.. 7. 7.. 6. 9. 0 7.. 9. 0. found square root instead of cube root 6 = = = 9. = 7, 9. =,.. 7n 7n. 96. =, 97. =, 9. linear function: = + 0 99. quadratic function: =

More information

Math 3201 UNIT 5: Polynomial Functions NOTES. Characteristics of Graphs and Equations of Polynomials Functions

Math 3201 UNIT 5: Polynomial Functions NOTES. Characteristics of Graphs and Equations of Polynomials Functions 1 Math 301 UNIT 5: Polnomial Functions NOTES Section 5.1 and 5.: Characteristics of Graphs and Equations of Polnomials Functions What is a polnomial function? Polnomial Function: - A function that contains

More information

Essential Question How can you solve a nonlinear system of equations?

Essential Question How can you solve a nonlinear system of equations? .5 Solving Nonlinear Sstems Essential Question Essential Question How can ou solve a nonlinear sstem of equations? Solving Nonlinear Sstems of Equations Work with a partner. Match each sstem with its graph.

More information

Answers. Chapter Warm Up. Sample answer: The graph of f is a translation 3 units right of the parent linear function.

Answers. Chapter Warm Up. Sample answer: The graph of f is a translation 3 units right of the parent linear function. Chapter. Start Thinking As the string V gets wider, the points on the string move closer to the -ais. This activit mimics a vertical shrink of a parabola... Warm Up.. Sample answer: The graph of f is a

More information

A. Simplifying Polynomial Expressions

A. Simplifying Polynomial Expressions A. Simplifing Polnomial Epressions I. Combining Like Terms - You can add or subtract terms that are considered "like", or terms that have the same variable(s) with the same eponent(s). E. 1: 5-7 + 10 +

More information

EOC Review. Algebra I

EOC Review. Algebra I EOC Review Algebra I Order of Operations PEMDAS Parentheses, Eponents, Multiplication/Division, Add/Subtract from left to right. A. Simplif each epression using appropriate Order of Operations.. 5 6 +.

More information

Module 3, Section 4 Analytic Geometry II

Module 3, Section 4 Analytic Geometry II Principles of Mathematics 11 Section, Introduction 01 Introduction, Section Analtic Geometr II As the lesson titles show, this section etends what ou have learned about Analtic Geometr to several related

More information

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

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

More information

Summer Math Packet (revised 2017)

Summer Math Packet (revised 2017) Summer Math Packet (revised 07) In preparation for Honors Math III, we have prepared a packet of concepts that students should know how to do as these concepts have been taught in previous math classes.

More information

UNIT 2 QUADRATIC FUNCTIONS AND MODELING Lesson 2: Interpreting Quadratic Functions Instruction

UNIT 2 QUADRATIC FUNCTIONS AND MODELING Lesson 2: Interpreting Quadratic Functions Instruction Prerequisite Skills This lesson requires the use of the following skills: knowing the standard form of quadratic functions using graphing technolog to model quadratic functions Introduction The tourism

More information

Essential Question What is the equation of a circle with center (h, k) and radius r in the coordinate plane?

Essential Question What is the equation of a circle with center (h, k) and radius r in the coordinate plane? 10.7 Circles in the Coordinate Plane Essential Question What is the equation of a circle with center (h, k) and radius r in the coordinate plane? The Equation of a Circle with Center at the Origin Work

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

INEQUALITIES

INEQUALITIES Chapter 4 INEQUALITIES 4.2.1 4.2.4 Once the students understand the notion of a solution, the can etend their understanding to inequalities and sstems of inequalities. Inequalities tpicall have infinitel

More information

PRINCIPLES OF MATHEMATICS 11 Chapter 2 Quadratic Functions Lesson 1 Graphs of Quadratic Functions (2.1) where a, b, and c are constants and a 0

PRINCIPLES OF MATHEMATICS 11 Chapter 2 Quadratic Functions Lesson 1 Graphs of Quadratic Functions (2.1) where a, b, and c are constants and a 0 PRINCIPLES OF MATHEMATICS 11 Chapter Quadratic Functions Lesson 1 Graphs of Quadratic Functions (.1) Date A. QUADRATIC FUNCTIONS A quadratic function is an equation that can be written in the following

More information

Algebra 2 Honors Summer Packet 2018

Algebra 2 Honors Summer Packet 2018 Algebra Honors Summer Packet 018 Solving Linear Equations with Fractional Coefficients For these problems, ou should be able to: A) determine the LCD when given two or more fractions B) solve a linear

More information

20.2 Connecting Intercepts and Linear Factors

20.2 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 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

1. Solutions to Systems of Linear Equations. Determine whether the ordered pairs are solutions to the system. x y 6. 3x y 2

1. Solutions to Systems of Linear Equations. Determine whether the ordered pairs are solutions to the system. x y 6. 3x y 2 78 Chapter Sstems of Linear Equations Section. Concepts. Solutions to Sstems of Linear Equations. Dependent and Inconsistent Sstems of Linear Equations. Solving Sstems of Linear Equations b Graphing Solving

More information

Pre-AP Algebra 2 Lesson 1-1 Basics of Functions

Pre-AP Algebra 2 Lesson 1-1 Basics of Functions Lesson 1-1 Basics of Functions Objectives: The students will be able to represent functions verball, numericall, smbolicall, and graphicall. The students will be able to determine if a relation is a function

More information

QUADRATIC FUNCTIONS AND MODELS

QUADRATIC FUNCTIONS AND MODELS QUADRATIC FUNCTIONS AND MODELS What You Should Learn Analyze graphs of quadratic functions. Write quadratic functions in standard form and use the results to sketch graphs of functions. Find minimum and

More information

Summer Review For Students Entering Algebra 2

Summer Review For Students Entering Algebra 2 Summer Review For Students Entering Algebra Teachers and administrators at Tuscarora High School activel encourage parents and communit members to engage in children s learning. This Summer Review For

More information

Algebra 2 CPA Summer Assignment 2018

Algebra 2 CPA Summer Assignment 2018 Algebra CPA Summer Assignment 018 This assignment is designed for ou to practice topics learned in Algebra 1 that will be relevant in the Algebra CPA curriculum. This review is especiall important as ou

More information

INEQUALITIES

INEQUALITIES INEQUALITIES 3.2.1 3.2.4 Once the meaning of a solution is understood, it can be applied to understanding solutions of inequalities and sstems of inequalities. Inequalities tpicall have infinitel man solutions,

More information

3.7 InveRSe FUnCTIOnS

3.7 InveRSe FUnCTIOnS CHAPTER functions learning ObjeCTIveS In this section, ou will: Verif inverse functions. Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.

More information

7-6. nth Roots. Vocabulary. Geometric Sequences in Music. Lesson. Mental Math

7-6. nth Roots. Vocabulary. Geometric Sequences in Music. Lesson. Mental Math Lesson 7-6 nth Roots Vocabular cube root n th root BIG IDEA If is the nth power of, then is an nth root of. Real numbers ma have 0, 1, or 2 real nth roots. Geometric Sequences in Music A piano tuner adjusts

More information

8 f(8) = 0 (8,0) 4 f(4) = 4 (4, 4) 2 f(2) = 3 (2, 3) 6 f(6) = 3 (6, 3) Outputs. Inputs

8 f(8) = 0 (8,0) 4 f(4) = 4 (4, 4) 2 f(2) = 3 (2, 3) 6 f(6) = 3 (6, 3) Outputs. Inputs In the previous set of notes we covered how to transform a graph by stretching or compressing it vertically. In this lesson we will focus on stretching or compressing a graph horizontally, which like the

More information

Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs

Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs Ch 5 Alg Note Sheet Ke Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs Definition: Standard Form of a Quadratic Function The

More information

Cubic and quartic functions

Cubic and quartic functions 3 Cubic and quartic functions 3A Epanding 3B Long division of polnomials 3C Polnomial values 3D The remainder and factor theorems 3E Factorising polnomials 3F Sum and difference of two cubes 3G Solving

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Name Date Chapter Polnomial and Rational Functions Section.1 Quadratic Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Important Vocabular Define

More information

Mathematics 10 Page 1 of 7 The Quadratic Function (Vertex Form): Translations. and axis of symmetry is at x a.

Mathematics 10 Page 1 of 7 The Quadratic Function (Vertex Form): Translations. and axis of symmetry is at x a. Mathematics 10 Page 1 of 7 Verte form of Quadratic Relations The epression a p q defines a quadratic relation called the verte form with a horizontal translation of p units and vertical translation of

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

ASSOCIATION IN A TWO-WAY TABLE

ASSOCIATION IN A TWO-WAY TABLE ASSOCIATION IN A TWO-WAY TABLE 10.1.1 Data based on measurements such as height, speed, and temperature is numerical. In Chapter 6 you described associations between two numerical variables. Data can also

More information

Northwest High School s Algebra 2/Honors Algebra 2

Northwest High School s Algebra 2/Honors Algebra 2 Northwest High School s Algebra /Honors Algebra Summer Review Packet 0 DUE Frida, September, 0 Student Name This packet has been designed to help ou review various mathematical topics that will be necessar

More information

Graphing Calculator Computations 2

Graphing Calculator Computations 2 Graphing Calculator Computations A) Write the graphing calculator notation and B) Evaluate each epression. 4 1) 15 43 8 e) 15 - -4 * 3^ + 8 ^ 4/ - 1) ) 5 ) 8 3 3) 3 4 1 8 3) 7 9 4) 1 3 5 4) 5) 5 5) 6)

More information

UNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives

UNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives Chapter 3 3Quadratics Objectives To recognise and sketch the graphs of quadratic polnomials. To find the ke features of the graph of a quadratic polnomial: ais intercepts, turning point and ais of smmetr.

More information

MA 15800, Summer 2016 Lesson 25 Notes Solving a System of Equations by substitution (or elimination) Matrices. 2 A System of Equations

MA 15800, Summer 2016 Lesson 25 Notes Solving a System of Equations by substitution (or elimination) Matrices. 2 A System of Equations MA 800, Summer 06 Lesson Notes Solving a Sstem of Equations b substitution (or elimination) Matrices Consider the graphs of the two equations below. A Sstem of Equations From our mathematics eperience,

More information

STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE. Functions & Graphs

STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE. Functions & Graphs STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE Functions & Graphs Contents Functions and Relations... 1 Interval Notation... 3 Graphs: Linear Functions... 5 Lines and Gradients... 7 Graphs: Quadratic

More information

Solving Systems of Linear and Quadratic Equations

Solving Systems of Linear and Quadratic Equations 9.5 Solving Systems of Linear and Quadratic Equations How can you solve a system of two equations when one is linear and the other is quadratic? ACTIVITY: Solving a System of Equations Work with a partner.

More information

Power Functions. A polynomial expression is an expression of the form a n. x n 2... a 3. ,..., a n. , a 1. A polynomial function has the form f(x) a n

Power Functions. A polynomial expression is an expression of the form a n. x n 2... a 3. ,..., a n. , a 1. A polynomial function has the form f(x) a n 1.1 Power Functions A rock that is tossed into the water of a calm lake creates ripples that move outward in a circular pattern. The area, A, spanned b the ripples can be modelled b the function A(r) πr,

More information

+ = + + = x = + = + = 36x

+ = + + = x = + = + = 36x Ch 5 Alg L Homework Worksheets Computation Worksheet #1: You should be able to do these without a calculator! A) Addition (Subtraction = add the opposite of) B) Multiplication (Division = multipl b the

More information

Chapter 6: Systems of Equations and Inequalities

Chapter 6: Systems of Equations and Inequalities Chapter 6: Sstems of Equations and Inequalities 6-1: Solving Sstems b Graphing Objectives: Identif solutions of sstems of linear equation in two variables. Solve sstems of linear equation in two variables

More information

Algebra 1 Skills Needed to be Successful in Algebra 2

Algebra 1 Skills Needed to be Successful in Algebra 2 Algebra 1 Skills Needed to be Successful in Algebra A. Simplifing Polnomial Epressions Objectives: The student will be able to: Appl the appropriate arithmetic operations and algebraic properties needed

More information

INVESTIGATE the Math

INVESTIGATE the Math . Graphs of Reciprocal Functions YOU WILL NEED graph paper coloured pencils or pens graphing calculator or graphing software f() = GOAL Sketch the graphs of reciprocals of linear and quadratic functions.

More information

7.2 Connecting Intercepts and Linear Factors

7.2 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 information

Review Topics for MATH 1400 Elements of Calculus Table of Contents

Review Topics for MATH 1400 Elements of Calculus Table of Contents Math 1400 - Mano Table of Contents - Review - page 1 of 2 Review Topics for MATH 1400 Elements of Calculus Table of Contents MATH 1400 Elements of Calculus is one of the Marquette Core Courses for Mathematical

More information

Solving Linear-Quadratic Systems

Solving Linear-Quadratic Systems 36 LESSON Solving Linear-Quadratic Sstems UNDERSTAND A sstem of two or more equations can include linear and nonlinear equations. In a linear-quadratic sstem, there is one linear equation and one quadratic

More information

Quadratic Functions. The graph of the function shifts right 3. The graph of the function shifts left 3.

Quadratic Functions. The graph of the function shifts right 3. The graph of the function shifts left 3. Quadratic Functions The translation o a unction is simpl the shiting o a unction. In this section, or the most part, we will be graphing various unctions b means o shiting the parent unction. We will go

More information

SOLVING INEQUALITIES and 9.1.2

SOLVING INEQUALITIES and 9.1.2 SOLVING INEQUALITIES 9.1.1 and 9.1.2 To solve an inequality in one variable, first change it to an equation and solve. Place the solution, called a boundary point, on a number line. This point separates

More information

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

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

More information

CK- 12 Algebra II with Trigonometry Concepts 1

CK- 12 Algebra II with Trigonometry Concepts 1 10.1 Parabolas with Verte at the Origin Answers 1. up. left 3. down 4.focus: (0, -0.5), directri: = 0.5 5.focus: (0.065, 0), directri: = -0.065 6.focus: (-1.5, 0), directri: = 1.5 7.focus: (0, ), directri:

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

Maintaining Mathematical Proficiency

Maintaining Mathematical Proficiency Chapter Maintaining Mathematical Proficienc Find the -intercept of the graph of the linear equation. 1. = + 3. = 3 + 5 3. = 10 75. = ( 9) 5. 7( 10) = +. 5 + 15 = 0 Find the distance between the two points.

More information

9-1. The Function with Equation y = ax 2. Vocabulary. Graphing y = x 2. Lesson

9-1. The Function with Equation y = ax 2. Vocabulary. Graphing y = x 2. Lesson Chapter 9 Lesson 9-1 The Function with Equation = a BIG IDEA The graph of an quadratic function with equation = a, with a 0, is a parabola with verte at the origin. Vocabular parabola refl ection-smmetric

More information

MATH 115: Review for Chapter 6

MATH 115: Review for Chapter 6 MATH 115: Review for Chapter 6 In order to prepare for our test on Chapter 6, ou need to understand and be able to work problems involving the following topics: I SYSTEMS OF LINEAR EQUATIONS CONTAINING

More information

Solve each system by substitution or elimination. Check your solutions. b.

Solve each system by substitution or elimination. Check your solutions. b. Algebra: 10.3.1: Intersect or Intercept? Name Solutions Block Date Bell Work: a. = 4 2 3 = 3 2 3(4 ) = 3 2 12 + 3 = 3 5 12 = 3 5 = 15 Solve each sstem b substitution or elimination. Check our solutions.

More information