SOLVING SYSTEMS OF EQUATIONS

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "SOLVING SYSTEMS OF EQUATIONS"

Transcription

1 SOLVING SYSTEMS OF EQUATIONS 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, students are able to solve equations in new and different was. Their understanding also provides opportunities to solve some challenging applications. In this section the will etend their knowledge about solving one and two variable equations to solve sstems with three variables. Eample The graph of = (! 5) 2! 4 is shown below right. Solve each of the following equations. Eplain how the graph can be used to solve the equations. a. (! 5) 2! 4 = 2 b. (! 5) 2! 4 =!3 c. (! 5) 2 = 4 Students can determine the correct answers through a variet of different was. For part (a), most students would do the following: (! 5) 2! 4 = 2 (! 5) 2 = 6! 5 = ±4 = 5 ± 4 = 9, This is correct and a standard procedure. However, with the graph of the parabola provided, the student can find the solution b inspecting the graph. Since we alread have the graph of = (! 5) 2! 4, we can add the graph of = 2 which is a horizontal line. These two graphs cross at two points, and the -coordinates of these points are the solutions. The intersection points are (, 2) and (9, 2). Therefore the solutions to the equation are = and = 9. We can use this method for part (b) as well. Draw the graph of = 3 to find that the graphs intersect at (4, 3) and (6, 3). Therefore the solutions to part (b) are = 4 and = 6. The equation in part (c) might look as if we cannot solve it with the graph, but we can. Granted, this is eas to solve algebraicall: (! 5) 2 = 4! 5 = ±2 = 7, 3 But, we want students to be looking for alternative approaches to solving problems. B looking for and eploring alternative solutions, students are epanding their repertoire for solving problems. This ear the will encounter equations that can onl be solved through alternative methods. B recognizing the equation in part (c) as equivalent to (! 5) 2! 4 = 0 (subtract four from both sides), we can use the graph to find where the parabola crosses the line = 0 (the -ais). The graph tells us the solutions are = 7 and = Core Connections Algebra 2

2 Chapter 4 Eample 2 Solve the equation + 2 = 2 + using at least two different approaches. Eplain our methods and the implications of the solution(s). Most students will begin b using algebra to solve this equation. This includes squaring both sides and solving a quadratic equation as shown at right. A problem arises if the students do not check their work. If the students substitute each -value back into the original equation, onl one -value checks: =. This is the onl solution. We 4 can see wh the other solution does not work if we use a graph to solve the equation. The graphs of = + 2 and = 2 + are shown at right. Notice that the graphs onl intersect at one point, namel =. This is the onl point where the are 4 equal. The solution to the equation is this one value; the other value is called an etraneous solution. Remember that a solution is a value that makes the equation true. In our original equation, this would mean that both sides of the equation would be equal for certain values of. Using the graphs, the solution is the -value that has the same -value for both graphs, or the point(s) at which the graphs intersect. + 2 = 2 + ( + 2 ) 2 = (2 + ) = ! = 0 (4! )( + ) = 0 = 4,! Eample 3 Solve each sstem of equations below without graphing. For each one, eplain what the solution (or lack thereof) tells ou about the graph of the sstem. a. =! b. =!2(! 2) = 3 5! 2 =!2 + 5 c. = 6 2! = 25 The two equations in part (a) are written in = form, which makes the Substitution method the most efficient method for solving. Since both epressions in are equal to, we set the epressions equal to each other, and solve for. =! = 3 5! 2 =! = 3 5! 2 ( ) = 5 ( 5 3! 2 )!2 +5 = 3!0 5! !5 =!25 = 5 Parent Guide and Etra Practice 4

3 We substitute this value for back into either one of the original equations to determine the value of. Finall, we must check that the solution satisfies both equations. Check Therefore the solution is the point (5, ), which means that the graphs of these two lines intersect in one point, the point (5, ). The equations in part (b) are written in the same form so we will solve this sstem the same wa we did in part (a). We substitute each -value into either equation to find the corresponding -value. Here we will use the simpler equation. = 6, =!2 + 5 =!2(6) + 5 =!2 + 5 = 3 Solution to check: (5, ) Solution: (6, 3) =!, =!2 + 5 =!2(!) + 5 = = 7 Solution: (!, 7) Lastl, we need to check each point in both equations to make sure we do not have an etraneous solutions. =!2(! 2) =!2 +5!2(! 2) 2 =!2! 20!2( 2! 4 + 4) =!2! 20! ! 8 =!2! 20! = 0 2! 5! 6 = 0 (! 6)( +) = 0 = 6, =! (6, 3) : =!2(! 2) (6, 3) : =! =!2(6! 2) =!2(6) +5 Check 3 =!2(6) + 35 Check (!, 7) : =!2(! 2) =!2(!! 2) =!2(9) + 35 Check (!, 7) : =! =!2(!) +5 Check In solving these two equations with two unknowns, we found two solutions, both of which check in the original equations. This means that the graphs of the equations, a parabola and a line, intersect in eactl two distinct points. 42 Core Connections Algebra 2

4 Chapter 4 Part (c) requires substitution to solve. We can replace in the second equation with what the first equation tells us it equals, but that will require us to solve an equation of degree four (an eponent of 4). Instead, we will first rewrite the first equation without fractions in an effort to simplif. We do this b multipling both sides of the equation b si. = 6 2! = 2! = 2 Now we can replace the 2 in the second equation with Net we substitute this value back into either equation to find the corresponding -value = 25 (6 + 34) + 2 = = = 0 ( + 3)( + 3) = 0 =!3 =!3 : = 2 6(!3) + 34 = 2! = 2 6 = 2 = ±4 This gives us two possible solutions: (4,!3 ) and (!4,!3 ). Be sure to check these points for etraneous solutions! (4,!3) : = 6 2! 34 6,! 3 = 6 (4)2! 34 6 = 6 6! 34 6 =!8 6, check. (4,!3) : = 25, (4) 2 + (!3) 2 = = 25, check. (!4,!3) : = 6 2! 34 6,! 3 = 6 (!4)2! 34 6 = 6 6! 34 6 =!8 6, check. (!4,!3) : = 25, (!4) 2 + (!3) 2 = = 25, check. Since there are two points that make this sstem true, the graphs of this parabola and this circle intersect in onl two points, (4,!3) and (!4,!3). Parent Guide and Etra Practice 43

5 Eample 4 Jo has small containers of lemonade and lime soda. She once mied one lemonade container with three containers of lime soda to make 7 ounces of a tast drink. Another time, she combined five containers of lemonade with si containers of lime soda to produce 58 ounces of another splendid beverage. Given this information, how man ounces are in each small container of lemonade and lime soda? We can solve this problem b using a sstem of equations. To start, we let equal the number of ounces of lemonade in each small container, and equal the number of ounces of lime soda in each of its small containers. We can write an equation that describes each miture Jo created. The first miture used one ounce of liquid and three ounces of liquid to produce 7 ounces. This can be represented as + 3 = 7. The second miture used five ounces of liquid and si ounces of liquid to produce 58 ounces. This can be represented b the equation = 58. We can solve this sstem to find the values for and. + 3 = 7! " = = 58! = (3) = = 7 = 8 9 = 27 = 3 (Note: ou should check these values!) Therefore each container of Jo s lemonade has 8 ounces, and each container of her lime soda has onl 3 ounces. 44 Core Connections Algebra 2

6 Chapter 4 Problems Solve each of the following sstems for and. Then eplain what the answer tells ou about the graphs of the equations. Be sure to check our work.. + = 3! = ! 3 =!9!5 + 2 = = = = 8!6 + = ! 6 = 24 = 3 4! ! 7 =!5 2 3! 4 = 24 The graph of = 2 (! 4)2 + 3 is shown at right. Use the graph to solve each of the following equations. Eplain how ou get our answer (! 4)2 + 3 = (! 4)2 + 3 = (! 4)2 + 3 = 0. 2 (! 4)2 = 8 Solve each equation below. Think about rewriting, looking inside, or undoing to simplif the process.. 3(! 4) = = 9! ( 0!3 2 ) = 5 4.!3 2! + 4 = 0 Solve each of the following sstems of equations algebraicall. What does the solution tell ou about the graph of the sstem? 5. =! = = ( + ) = =!3(! 4) 2! 2 =! = 0 = (! 4) 2! 6 9. Adult tickets for the Mr. Moose s Fantas Show on Ice are $6.50 while a child s ticket is onl $2.50. At Tuesda night s performance, 435 people were in attendance. The bo office brought in $ for that evening. How man of each tpe of ticket were sold? Parent Guide and Etra Practice 45

7 20. The net math test will contain 50 questions. Some will be worth three points while the rest will be worth si points. If the test is worth 95 points, how man three-point questions are there, and how man si-point questions are there? 2. Reread Eample 3 from Chapter 4 about Dudle s water balloon fight. If ou did this problem, ou found that Dudle s water balloons followed the path described b the equation =! 8 25 (! 0) Suppose Dudle s nemesis, in a mad dash to save his 5 base from total water balloon bombardment, ran to the wall and set up his launcher at its base. Dudle s nemesis launches his balloons to follow the path =! (! ) in an effort to knock Dudle s water bombs out of the air. Is Dudle s nemesis successful? Eplain. Answers. (4, 7) 2. (!2, 5) 3. no solution 4. (! 2, ) 5. All real numbers 6. (2, 3) 7. = 4. The horizontal line = 3 crosses the parabola in onl one point, at the verte. 9. No solution. The horizontal line = does not cross the parabola. 8. = 2, = 6 0. = 0, = 8. Add three to both sides to rewrite the equation as 2 (! 4)2 + 3 =. The horizontal line = crosses at these two points.. = 7, = 2. no solution 3. = 2 4. No solution. (A square root must have a positive result.) 5. All real numbers. When graphed, these equations give the same line. 7. No solution. This parabola and this line do not intersect adult tickets were sold, while 290 child tickets were sold. 6. (0, 4). The parabola and the line intersect onl once. 8. (2,!2 ) and (5,!5 ). The line and the parabola intersect twice. 20. There are 35 three-point questions and 5 si-point questions on the test. 2. B graphing we see that the nemesis balloon when launched at the base of the wall (the -ais), hits the path of the Dudle s water balloon. Therefore, if timed correctl, the nemesis is successful. 46 Core Connections Algebra 2

Math 1400/1650 (Cherry): Quadratic Functions

Math 1400/1650 (Cherry): Quadratic Functions Math 100/1650 (Cherr): Quadratic Functions A quadratic function is a function Q() of the form Q() = a + b + c with a 0. For eample, Q() = 3 7 + 5 is a quadratic function, and here a = 3, b = 7 and c =

More information

Graphing Nonlinear Systems

Graphing Nonlinear Systems 10.4 Graphing Nonlinear Sstems 10.4 OBJECTIVES 1. Graph a sstem of nonlinear equations 2. Find ordered pairs associated with the solution set of a nonlinear sstem 3. Graph a sstem of nonlinear inequalities

More information

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1 Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.

More information

Florida Algebra I EOC Online Practice Test

Florida Algebra I EOC Online Practice Test Florida Algebra I EOC Online Practice Test 1 Directions: This practice test contains 65 multiple-choice questions. Choose the best answer for each question. Detailed answer eplanations appear at the end

More information

8.7 Systems of Non-Linear Equations and Inequalities

8.7 Systems of Non-Linear Equations and Inequalities 8.7 Sstems of Non-Linear Equations and Inequalities 67 8.7 Sstems of Non-Linear Equations and Inequalities In this section, we stud sstems of non-linear equations and inequalities. Unlike the sstems of

More information

Solving Systems of Equations

Solving Systems of Equations Solving Sstems of Equations When we have or more equations and or more unknowns, we use a sstem of equations to find the solution. Definition: A solution of a sstem of equations is an ordered pair that

More information

Higher. Polynomials and Quadratics 64

Higher. Polynomials and Quadratics 64 hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining

More information

Section C Non Linear Graphs

Section C Non Linear Graphs 1 of 8 Section C Non Linear Graphs Graphic Calculators will be useful for this topic of 8 Cop into our notes Some words to learn Plot a graph: Draw graph b plotting points Sketch/Draw a graph: Do not plot,

More information

Graphing Linear Equations

Graphing Linear Equations 6.3 Graphing Linear Equations 6.3 OBJECTIVES 1. Graph a linear equation b plotting points 2. Graph a linear equation b the intercept method 3. Graph a linear equation b solving the equation for We are

More information

MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60

MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets

More information

Simplification of Rational Expressions and Functions

Simplification of Rational Expressions and Functions 7.1 Simplification of Rational Epressions and Functions 7.1 OBJECTIVES 1. Simplif a rational epression 2. Identif a rational function 3. Simplif a rational function 4. Graph a rational function Our work

More information

Mathematical goals. Starting points. Materials required. Time needed

Mathematical goals. Starting points. Materials required. Time needed Level A7 of challenge: C A7 Interpreting functions, graphs and tables tables Mathematical goals Starting points Materials required Time needed To enable learners to understand: the relationship between

More information

Why? Describe how the graph of each function is related to the graph of f(x) = x 2. a. h(x) = x 2 + 3

Why? Describe how the graph of each function is related to the graph of f(x) = x 2. a. h(x) = x 2 + 3 Transformations of Quadratic Functions Then You graphed quadratic functions b using the verte and ais of smmetr. (Lesson 9-1) Now 1Appl translations to quadratic functions. 2Appl dilations and reflections

More information

Identify a pattern and find the next three numbers in the pattern. 5. 5(2s 2 1) 2 3(s 1 2); s 5 4

Identify a pattern and find the next three numbers in the pattern. 5. 5(2s 2 1) 2 3(s 1 2); s 5 4 Chapter 1 Test Do ou know HOW? Identif a pattern and find the net three numbers in the pattern. 1. 5, 1, 3, 7, c. 6, 3, 16, 8, c Each term is more than the previous Each term is half of the previous term;

More information

1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model

1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model . Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses

More information

SECTION 2.2. Distance and Midpoint Formulas; Circles

SECTION 2.2. Distance and Midpoint Formulas; Circles SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation

More information

10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED

10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations

More information

Systems of linear equations (simultaneous equations)

Systems of linear equations (simultaneous equations) Before starting this topic ou should review how to graph equations of lines. The link below will take ou to the appropriate location on the Academic Skills site. http://www.scu.edu.au/academicskills/numerac/inde.php/1

More information

6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions:

6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions: Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(-, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions - 4, use the graph at the right.. Eplain wh the graph

More information

Polynomial and Rational Functions

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

More information

Exponential Functions

Exponential Functions CHAPTER Eponential Functions 010 Carnegie Learning, Inc. Georgia has two nuclear power plants: the Hatch plant in Appling Count, and the Vogtle plant in Burke Count. Together, these plants suppl about

More information

5.2 Inverse Functions

5.2 Inverse Functions 78 Further Topics in Functions. Inverse Functions Thinking of a function as a process like we did in Section., in this section we seek another function which might reverse that process. As in real life,

More information

2.2 Absolute Value Functions

2.2 Absolute Value Functions . Absolute Value Functions 7. Absolute Value Functions There are a few was to describe what is meant b the absolute value of a real number. You ma have been taught that is the distance from the real number

More information

Families of Quadratics

Families of Quadratics Families of Quadratics Objectives To understand the effects of a, b, and c on the graphs of parabolas of the form a 2 b c To use quadratic equations and graphs to analze the motion of projectiles To distinguish

More information

Alex and Morgan were asked to graph the equation y = 2x + 1

Alex and Morgan were asked to graph the equation y = 2x + 1 Which is better? Ale and Morgan were asked to graph the equation = 2 + 1 Ale s make a table of values wa Morgan s use the slope and -intercept wa First, I made a table. I chose some -values, then plugged

More information

The Graph of a Linear Equation

The Graph of a Linear Equation 4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that

More information

1.2 GRAPHS OF EQUATIONS

1.2 GRAPHS OF EQUATIONS 000_00.qd /5/05 : AM Page SECTION. Graphs of Equations. GRAPHS OF EQUATIONS Sketch graphs of equations b hand. Find the - and -intercepts of graphs of equations. Write the standard forms of equations of

More information

Example 1: Model A Model B Total Available. Gizmos. Dodads. System:

Example 1: Model A Model B Total Available. Gizmos. Dodads. System: Lesson : Sstems of Equations and Matrices Outline Objectives: I can solve sstems of three linear equations in three variables. I can solve sstems of linear inequalities I can model and solve real-world

More information

Rational Functions. 7.1 A Rational Existence. 7.2 A Rational Shift in Behavior. 7.3 A Rational Approach. 7.4 There s a Hole In My Function, Dear Liza

Rational Functions. 7.1 A Rational Existence. 7.2 A Rational Shift in Behavior. 7.3 A Rational Approach. 7.4 There s a Hole In My Function, Dear Liza Rational Functions 7 The ozone laer protects Earth from harmful ultraviolet radiation. Each ear, this laer thins dramaticall over the poles, creating ozone holes which have stretched as far as Australia

More information

Graphing Quadratic Equations

Graphing Quadratic Equations .4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph first-degree equations. Similar methods will allow ou to graph quadratic equations

More information

3.1 Quadratic Functions

3.1 Quadratic Functions Section 3.1 Quadratic Functions 1 3.1 Quadratic Functions Functions Let s quickl review again the definition of a function. Definition 1 A relation is a function if and onl if each object in its domain

More information

Downloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x

Downloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x Functions and equations Assessment statements. Concept of function f : f (); domain, range, image (value). Composite functions (f g); identit function. Inverse function f.. The graph of a function; its

More information

Essential Question: What are two ways to solve an absolute value inequality? A2.6.F Solve absolute value linear inequalities.

Essential Question: What are two ways to solve an absolute value inequality? A2.6.F Solve absolute value linear inequalities. Locker LESSON.3 Solving Absolute Value Inequalities Name Class Date.3 Solving Absolute Value Inequalities Teas Math Standards The student is epected to: A.6.F Essential Question: What are two was to solve

More information

Translating Points. Subtract 2 from the y-coordinates

Translating Points. Subtract 2 from the y-coordinates CONDENSED L E S S O N 9. Translating Points In this lesson ou will translate figures on the coordinate plane define a translation b describing how it affects a general point (, ) A mathematical rule that

More information

3.1 Graphically Solving Systems of Two Equations

3.1 Graphically Solving Systems of Two Equations 3.1 Graphicall Solving Sstems of Two Equations (Page 1 of 24) 3.1 Graphicall Solving Sstems of Two Equations Definitions The plot of all points that satisf an equation forms the graph of the equation.

More information

LESSON EIII.E EXPONENTS AND LOGARITHMS

LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential

More information

Quadratic Functions and Models. The Graph of a Quadratic Function. These functions are examples of polynomial functions. Why you should learn it

Quadratic Functions and Models. The Graph of a Quadratic Function. These functions are examples of polynomial functions. Why you should learn it 0_00.qd 8 /7/05 Chapter. 9:0 AM Page 8 Polnomial and Rational Functions Quadratic Functions and Models What ou should learn Analze graphs of quadratic functions. Write quadratic functions in standard form

More information

More Equations and Inequalities

More Equations and Inequalities Section. Sets of Numbers and Interval Notation 9 More Equations and Inequalities 9 9. Compound Inequalities 9. Polnomial and Rational Inequalities 9. Absolute Value Equations 9. Absolute Value Inequalities

More information

Introduction. Introduction

Introduction. Introduction Introduction Solving Sstems of Equations Let s start with an eample. Recall the application of sales forecasting from the Working with Linear Equations module. We used historical data to derive the equation

More information

Rational Functions, Equations, and Inequalities

Rational Functions, Equations, and Inequalities Chapter 5 Rational Functions, Equations, and Inequalities GOALS You will be able to Graph the reciprocal functions of linear and quadratic functions Identif the ke characteristics of rational functions

More information

4 Non-Linear relationships

4 Non-Linear relationships NUMBER AND ALGEBRA Non-Linear relationships A Solving quadratic equations B Plotting quadratic relationships C Parabolas and transformations D Sketching parabolas using transformations E Sketching parabolas

More information

Contents. How You May Use This Resource Guide

Contents. How You May Use This Resource Guide Contents How You Ma Use This Resource Guide ii 9 Fractional and Quadratic Equations 1 Worksheet 9.1: Similar Figures.......................... 5 Worksheet 9.: Stretch of a Spring........................

More information

Applications of Quadratic Functions Word Problems ISU. Part A: Revenue and Numeric Problems

Applications of Quadratic Functions Word Problems ISU. Part A: Revenue and Numeric Problems Mr. Gardner s MPMD Applications of Quadratic Functions Word Problems ISU Part A: Revenue and Numeric Problems When ou solve problems using equations, our solution must have four components: 1. A let statement,

More information

4-1. Quadratic Functions and Transformations. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary

4-1. Quadratic Functions and Transformations. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary 4-1 Quadratic Functions and Transformations Vocabular Review 1. Circle the verte of each absolute value graph. Vocabular Builder parabola (noun) puh RAB uh luh Related Words: verte, ais of smmetr, quadratic

More information

2.2 Absolute Value Functions

2.2 Absolute Value Functions . Absolute Value Functions 7. Absolute Value Functions There are a few was to describe what is meant b the absolute value of a real number. You ma have been taught that is the distance from the real number

More information

17.1 Connecting Intercepts and Zeros

17.1 Connecting Intercepts and Zeros Locker LESSON 7. Connecting Intercepts and Zeros Teas Math Standards The student is epected to: A.7.A Graph quadratic functions on the coordinate plane and use the graph to identif ke attributes, if possible,

More information

Lesson 6: Linear Functions and their Slope

Lesson 6: Linear Functions and their Slope Lesson 6: Linear Functions and their Slope A linear function is represented b a line when graph, and represented in an where the variables have no whole number eponent higher than. Forms of a Linear Equation

More information

Review of Essential Skills and Knowledge

Review of Essential Skills and Knowledge Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope

More information

3.1 Quadratic Functions

3.1 Quadratic Functions 33337_030.qp 252 2/27/06 Chapter 3 :20 PM Page 252 Polnomial and Rational Functions 3. Quadratic Functions The Graph of a Quadratic Function In this and the net section, ou will stud the graphs of polnomial

More information

MULTIPLE REPRESENTATIONS through 4.1.7

MULTIPLE REPRESENTATIONS through 4.1.7 MULTIPLE REPRESENTATIONS 4.1.1 through 4.1.7 The first part of Chapter 4 ties together several was to represent the same relationship. The basis for an relationship is a consistent pattern that connects

More information

To Be or Not To Be a Linear Equation: That Is the Question

To Be or Not To Be a Linear Equation: That Is the Question To Be or Not To Be a Linear Equation: That Is the Question Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form A + B C where A and B are not

More information

Quadratic Functions and Parabolas

Quadratic Functions and Parabolas MATH 11 Quadratic Functions and Parabolas A quadratic function has the form Dr. Neal, Fall 2008 f () = a 2 + b + c where a 0. The graph of the function is a parabola that opens upward if a > 0, and opens

More information

Solving Systems of Linear Equations

Solving Systems of Linear Equations 5 Solving Sstems of Linear Equations 5. Solving Sstems of Linear Equations b Graphing 5. Solving Sstems of Linear Equations b Substitution 5.3 Solving Sstems of Linear Equations b Elimination 5. Solving

More information

6.3 Polar Coordinates

6.3 Polar Coordinates 6 Polar Coordinates Section 6 Notes Page 1 In this section we will learn a new coordinate sstem In this sstem we plot a point in the form r, As shown in the picture below ou first draw angle in standard

More information

Quadratic Functions. Academic Vocabulary justify derive verify

Quadratic Functions. Academic Vocabulary justify derive verify Quadratic Functions 015 College Board. All rights reserved. Unit Overview This unit focuses on quadratic functions and equations. You will write the equations of quadratic functions to model situations.

More information

SYSTEMS OF LINEAR EQUATIONS

SYSTEMS OF LINEAR EQUATIONS SYSTEMS OF LINEAR EQUATIONS Sstems of linear equations refer to a set of two or more linear equations used to find the value of the unknown variables. If the set of linear equations consist of two equations

More information

Answers (Lesson 3-1) Study Guide and Intervention. Study Guide and Intervention (continued) Solving Systems of Equations by Graphing

Answers (Lesson 3-1) Study Guide and Intervention. Study Guide and Intervention (continued) Solving Systems of Equations by Graphing Glencoe/McGraw-Hill A Glencoe Algebra - NAME DATE PERID Stud Guide and Intervention Solving Sstems of Equations b Graphing Graph Sstems of Equations A sstem of equations is a set of two or more equations

More information

Quadratic Functions. Unit

Quadratic Functions. Unit Quadratic Functions Unit 5 Unit Overview In this unit ou will stud a variet of was to solve quadratic functions and appl our learning to analzing real world problems. Academic Vocabular Add these words

More information

Chapter 6 Quadratic Functions

Chapter 6 Quadratic Functions Chapter 6 Quadratic Functions Determine the characteristics of quadratic functions Sketch Quadratics Solve problems modelled b Quadratics 6.1Quadratic Functions A quadratic function is of the form where

More information

Essential Question How can you describe the graph of the equation Ax + By = C? Number of adult tickets. adult

Essential Question How can you describe the graph of the equation Ax + By = C? Number of adult tickets. adult 3. Graphing Linear Equations in Standard Form Essential Question How can ou describe the graph of the equation A + B = C? Using a Table to Plot Points Work with a partner. You sold a total of $16 worth

More information

Deriving the Standard-Form Equation of a Parabola

Deriving the Standard-Form Equation of a Parabola Name Class Date. Parabolas Essential Question: How is the distance formula connected with deriving equations for both vertical and horizontal parabolas? Eplore Deriving the Standard-Form Equation of a

More information

Rising Algebra 2 Honors

Rising Algebra 2 Honors Rising Algebra Honors Complete the following packet and return it on the first da of school. You will be tested on this material the first week of class. 50% of the test grade will be completion of this

More information

Name Class Date. Deriving the Equation of a Parabola

Name Class Date. Deriving the Equation of a Parabola Name Class Date Parabolas Going Deeper Essential question: What are the defining features of a parabola? Like the circle, the ellipse, and the hperbola, the parabola can be defined in terms of distance.

More information

Inverse Relations and Functions

Inverse Relations and Functions 11.1 Inverse Relations and Functions 11.1 OBJECTIVES 1. Find the inverse of a relation 2. Graph a relation and its inverse 3. Find the inverse of a function 4. Graph a function and its inverse 5. Identif

More information

Course 2 Answer Key. 1.1 Rational & Irrational Numbers. Defining Real Numbers Student Logbook. The Square Root Function Student Logbook

Course 2 Answer Key. 1.1 Rational & Irrational Numbers. Defining Real Numbers Student Logbook. The Square Root Function Student Logbook Course Answer Ke. Rational & Irrational Numbers Defining Real Numbers. integers; 0. terminates; repeats 3. two; number 4. ratio; integers 5. terminating; repeating 6. rational; irrational 7. real 8. root

More information

Are You Ready? Circumference and Area of Circles

Are You Ready? Circumference and Area of Circles SKILL 39 Are You Read? Circumference and Area of Circles Teaching Skill 39 Objective Find the circumference and area of circles. Remind students that perimeter is the distance around a figure and that

More information

2 Analysis of Graphs of

2 Analysis of Graphs of ch.pgs1-16 1/3/1 1:4 AM Page 1 Analsis of Graphs of Functions A FIGURE HAS rotational smmetr around an ais I if it coincides with itself b all rotations about I. Because of their complete rotational smmetr,

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Chapter 3 Eponential and Logarithmic Functions Section 3.1 Eponential Functions and Their Graphs Objective: In this lesson ou learned how to recognize, evaluate, and graph eponential functions. Course

More information

Quadratic Equations and Functions

Quadratic Equations and Functions Quadratic Equations and Functions. Square Root Propert and Completing the Square. Quadratic Formula. Equations in Quadratic Form. Graphs of Quadratic Functions. Verte of a Parabola and Applications In

More information

Objectives. By the time the student is finished with this section of the workbook, he/she should be able

Objectives. By the time the student is finished with this section of the workbook, he/she should be able QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula....99 The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of a

More information

Solving Absolute Value Equations and Inequalities Graphically

Solving Absolute Value Equations and Inequalities Graphically 4.5 Solving Absolute Value Equations and Inequalities Graphicall 4.5 OBJECTIVES 1. Draw the graph of an absolute value function 2. Solve an absolute value equation graphicall 3. Solve an absolute value

More information

CHAPTER 9. Polynomials

CHAPTER 9. Polynomials CHAPTER 9 In this chapter ou epand our knowledge of families of functions to include polnomial functions. As ou investigate the equation! graph connection for polnomials, ou will learn how to search for

More information

Solving inequalities. Jackie Nicholas Jacquie Hargreaves Janet Hunter

Solving inequalities. Jackie Nicholas Jacquie Hargreaves Janet Hunter Mathematics Learning Centre Solving inequalities Jackie Nicholas Jacquie Hargreaves Janet Hunter c 6 Universit of Sdne Mathematics Learning Centre, Universit of Sdne Solving inequalities In these nots

More information

CHAPTER 5 EXPONENTS AND POLYNOMIALS

CHAPTER 5 EXPONENTS AND POLYNOMIALS Chapter Five Additional Eercises 1 CHAPTER EXPONENTS AND POLYNOMIALS Section.1 The Product Rule and Power Rules for Eponents Objective 1 Use eponents. Write the epression in eponential form and evaluate.

More information

SECTION 5-1 Exponential Functions

SECTION 5-1 Exponential Functions 354 5 Eponential and Logarithmic Functions Most of the functions we have considered so far have been polnomial and rational functions, with a few others involving roots or powers of polnomial or rational

More information

8. Bilateral symmetry

8. Bilateral symmetry . Bilateral smmetr Our purpose here is to investigate the notion of bilateral smmetr both geometricall and algebraicall. Actuall there's another absolutel huge idea that makes an appearance here and that's

More information

2.4 Inequalities with Absolute Value and Quadratic Functions

2.4 Inequalities with Absolute Value and Quadratic Functions 08 Linear and Quadratic Functions. Inequalities with Absolute Value and Quadratic Functions In this section, not onl do we develop techniques for solving various classes of inequalities analticall, we

More information

Imagine a cube with any side length. Imagine increasing the height by 2 cm, the. Imagine a cube. x x

Imagine a cube with any side length. Imagine increasing the height by 2 cm, the. Imagine a cube. x x OBJECTIVES Eplore functions defined b rddegree polnomials (cubic functions) Use graphs of polnomial equations to find the roots and write the equations in factored form Relate the graphs of polnomial equations

More information

Linear Equations and Arithmetic Sequences

Linear Equations and Arithmetic Sequences CONDENSED LESSON 3.1 Linear Equations and Arithmetic Sequences In this lesson ou will write eplicit formulas for arithmetic sequences write linear equations in intercept form You learned about recursive

More information

Problem Set Chapter 3 & 4 Solutions

Problem Set Chapter 3 & 4 Solutions Problem Set Chapter 3 & 4 Solutions. Graph a tpical indifference curve for the following utilit functions and determine whether the obe the assumption of diminishing MRS: a. (, ) 3 + 3 Slope -3 3 Since

More information

THE POWER RULES. Raising an Exponential Expression to a Power

THE POWER RULES. Raising an Exponential Expression to a Power 8 (5-) Chapter 5 Eponents and Polnomials 5. THE POWER RULES In this section Raising an Eponential Epression to a Power Raising a Product to a Power Raising a Quotient to a Power Variable Eponents Summar

More information

North Carolina Community College System Diagnostic and Placement Test Sample Questions

North Carolina Community College System Diagnostic and Placement Test Sample Questions North Carolina Communit College Sstem Diagnostic and Placement Test Sample Questions 0 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College

More information

FACTORING QUADRATICS 8.1.1 through 8.1.4

FACTORING QUADRATICS 8.1.1 through 8.1.4 Chapter 8 FACTORING QUADRATICS 8.. through 8..4 Chapter 8 introduces students to rewriting quadratic epressions and solving quadratic equations. Quadratic functions are any function which can be rewritten

More information

CHAPTER 10 SYSTEMS, MATRICES, AND DETERMINANTS

CHAPTER 10 SYSTEMS, MATRICES, AND DETERMINANTS CHAPTER 0 SYSTEMS, MATRICES, AND DETERMINANTS PRE-CALCULUS: A TEACHING TEXTBOOK Lesson 64 Solving Sstems In this chapter, we re going to focus on sstems of equations. As ou ma remember from algebra, sstems

More information

Graphing Quadratic Functions

Graphing Quadratic Functions A. THE STANDARD PARABOLA Graphing Quadratic Functions The graph of a quadratic function is called a parabola. The most basic graph is of the function =, as shown in Figure, and it is to this graph which

More information

3.4 The Point-Slope Form of a Line

3.4 The Point-Slope Form of a Line Section 3.4 The Point-Slope Form of a Line 293 3.4 The Point-Slope Form of a Line In the last section, we developed the slope-intercept form of a line ( = m + b). The slope-intercept form of a line is

More information

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review D0 APPENDIX D Precalculus Review APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b

More information

1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered

1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,

More information

Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science

Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science Colorado College has two all college requirements (QR and SI) which can be satisfied in full, or part, b taking

More information

7.3 Parabolas. 7.3 Parabolas 505

7.3 Parabolas. 7.3 Parabolas 505 7. Parabolas 0 7. Parabolas We have alread learned that the graph of a quadratic function f() = a + b + c (a 0) is called a parabola. To our surprise and delight, we ma also define parabolas in terms of

More information

{ } Sec 3.1 Systems of Linear Equations in Two Variables

{ } Sec 3.1 Systems of Linear Equations in Two Variables Sec.1 Sstems of Linear Equations in Two Variables Learning Objectives: 1. Deciding whether an ordered pair is a solution.. Solve a sstem of linear equations using the graphing, substitution, and elimination

More information

Q (x 1, y 1 ) m = y 1 y 0

Q (x 1, y 1 ) m = y 1 y 0 . Linear Functions We now begin the stud of families of functions. Our first famil, linear functions, are old friends as we shall soon see. Recall from Geometr that two distinct points in the plane determine

More information

Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form

Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving

More information

The Quadratic Function

The Quadratic Function 0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral

More information

Filling in Coordinate Grid Planes

Filling in Coordinate Grid Planes Filling in Coordinate Grid Planes A coordinate grid is a sstem that can be used to write an address for an point within the grid. The grid is formed b two number lines called and that intersect at the

More information

Review of Intermediate Algebra Content

Review of Intermediate Algebra Content Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6

More information

Solving Systems of Linear Equations

Solving Systems of Linear Equations 5 Solving Sstems of Linear Equations 5. Solving Sstems of Linear Equations b Graphing 5. Solving Sstems of Linear Equations b Substitution 5.3 Solving Sstems of Linear Equations b Elimination 5. Solving

More information

2.3 Quadratic Functions

2.3 Quadratic Functions . Quadratic Functions 9. Quadratic Functions You ma recall studing quadratic equations in Intermediate Algebra. In this section, we review those equations in the contet of our net famil of functions: the

More information