Graphing Nonlinear Systems

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Graphing Nonlinear Systems"

Transcription

1 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 4. Use substitution to find the solution set for a nonlinear sstem. 5. Identif the solution set of a sstem of nonlinear inequalities In Section 5.1, we solved a sstem of linear equations b graphing the lines corresponding to those equations, and then recording the point of intersection. That point represented the solution to the sstem of equations. We will use a similar method to find the solution set for a nonlinear sstem. A sstem with two or more conic curves can have zero, one, two, three, or four solutions. The following graphs represent each of those possibilities. Zero Solutions One Solution Two Solutions Three Solutions Four Solutions For the remainder of this section, we will restrict our discussion to a sstem that has as its graph a line and a parabola. Such a sstem has either zero, one, or two solutions. Eample 1 Solving a Sstem of Nonlinear Equations Solve the following sstem of equations

2 796 CHAPTER 10 GRAPHS OF CONIC SECTIONS First, we will graph the sstem. From this graph we will be able to see the number of solutions. The graph will also give us a wa to check the reasonableness of our algebraic results. NOTE Use our calculator to approimate the solutions for the sstem. Let s use the method of substitution to solve the sstem. Substituting 6, from the second equation, for in the first equation, we get ( 4)( 1) The values for the solutions are 1 and 4. We can substitute these values for in either equation to solve for, but we know from the second equation that 6. The solution set is ( 1, 6), (4, 6). Looking at the graph, we see that this is a reasonable solution set for the sstem. CHECK YOURSELF 1 Solve the following sstem of equations Of course, not ever quadratic epression is factorable. In Eample 2, we must use the quadratic formula. Eample 2 Solving a Nonlinear Sstem Solve the following sstem of equations

3 GRAPHING NONLINEAR SYSTEMS SECTION Let s look at the graph of the sstem. We see two points of intersection, but neither seems to be an integer value for. Let s solve the sstem algebraicall. Using the method of substitution, we find The result is not factorable, so we use the quadratic formula to find the solutions The two points of intersection are, 7 It is difficult to 2 and 2. check these points against the graph, so we will approimate them. The approimate solutions (to the nearest tenth) are ( 2.6, 7) and (1.6, 7). The graph indicates that these are reasonable answers. CHECK YOURSELF 2 Solve the following sstem of equations As was stated earlier, not ever sstem has two solutions. In Eample 3, we will see a sstem with no real solution. Eample 3 Solving a Sstem of Nonlinear Equations Solve the following sstem of equations

4 798 CHAPTER 10 GRAPHS OF CONIC SECTIONS As we did with the previous sstems, we will first look at the graph of the sstem. Using the method of substitution, we get Using the quadratic formula, we can confirm that there are no real solutions to this sstem. ( 2) 2( 2) 2 4(1)(3) 2(1) CHECK YOURSELF Solve the following sstem of equations Consider the sstem consisting of the following two equations: The graph of the sstem indicates there are four solutions.

5 GRAPHING NONLINEAR SYSTEMS SECTION We could approimate the solutions, then check those approimations b substitution. But how could we find the solutions algebraicall? Eample 4 illustrates the elimination method. Eample 4 Solving a Nonlinear Sstem b Elimination Solve the following sstem algebraicall As was the case with linear sstems, we can eliminate one of the variables. In this case, adding the equations eliminates the variable Dividing b 4, we have 2 9, so 3 Substituting the value 3 into the first equation (3) Two of the ordered pairs in the solution set are (3, 4) and (3, 4). Substituting the value 3 into the first equation ( 3) The other two pairs in the solution set are ( 3, 4) and ( 3, 4).

6 800 CHAPTER 10 GRAPHS OF CONIC SECTIONS The solutions set is ( 3, 4),( 3, 4), (3, 4), (3, 4) ( 3, 4) (3, 4) ( 3, 4) (3, 4) CHECK YOURSELF 4 Solve b the elimination method Recall that a sstem of inequalities has as its solutions the set of all ordered pairs that make ever inequalit in the sstem a true statement. We almost alwas epress the solutions to a sstem of inequalities graphicall. We will do the same thing with nonlinear sstems. Eample 5 Solving a Sstem of Nonlinear Inequalities Solve the following sstem From Eample 1, we have the graph of the related sstem of equations.

7 GRAPHING NONLINEAR SYSTEMS SECTION The first inequalit has as its solution set ever ordered pair with a value that is greater than (above) the graph of the parabola. The second statement has as its solution set ever ordered pair with a value that is less than (below) the graph of the line. The solution set to the sstem is the set of ordered pairs that meet both of those criteria. Here is the graph of the solution set. NOTE The solution set is the shaded area above the parabola and below the line. CHECK YOURSELF 5 Solve the following sstem Eample 6 demonstrates that, even if the related sstem of equations has no solution, the sstem of inequalities could have a solution. Eample 6 Solving a Sstem of Nonlinear Inequalities Solve the following sstem As we did with the previous sstems, we will first look at the graph of the related sstem of equations (from Eample 3.)

8 802 CHAPTER 10 GRAPHS OF CONIC SECTIONS The solution set is now the set of all ordered pairs below the parabola ( 2 2 1) and above the line ( 2). Here is the graph of the solution set. NOTE The solution continues beond the borders of the grid. 2 CHECK YOURSELF 6 Solve the following sstem CHECK YOURSELF ANSWERS ( 1, 10), (6, 10) 2. (1.3, 8), ( 2.3, 8) 2 3. No real solution 4. ( 1, 3), ( 1, 3), (1, 3), (1, 3) 5. 6.,

9 Name 10.4 Eercises Section Date In eercises 1 to 8, the graph of a sstem of equations is given. Determine how man real solutions each sstem has ANSWERS

10 ANSWERS In eercises 9 to 12, draw the graph of a sstem that has the indicated number of solutions. Use the conic sections indicated solutions: (a) use a circle and an ellipse, and (b) use a parabola and a line. (a) (b) solution: (a) use a parabola and a circle, and (b) use a line and an ellipse. (a) (b) solutions: (a) use a parabola and a circle, and (b) use an ellipse and a parabola. (a) (b) 804

11 ANSWERS solutions: (a) use a circle and an ellipse, and (b) use a parabola and a circle. (a) (b) In eercises 13 to 24, graph each sstem and estimate the solutions

12 ANSWERS

13 ANSWERS In eercises 25 to 32, solve using algebraic methods. (Note: These eercises have been solved graphicall in eercises 13 to 24.) (See eercise 13.) (See eercise 14.) (See eercise 15.) (See eercise 16.) (See eercise 19.) (See eercise 20.) (See eercise 23.) (See eercise 24.) In eercises 33 to 40, solve the sstems of inequalities graphicall. (Note: These have alread been graphed as sstems of equations in eercises 13 to 24.) (See eercise 13.) (See eercise 14.) 807

14 ANSWERS (See eercise 15.) (See eercise 16.) (See eercise 17.) (See eercise 18.) (See eercise 23.) (See eercise 24.) 808

15 ANSWERS In eercises 41 to 44, (a) graph each sstem and estimate the solution, and (b) use algebraic methods to solve each sstem Solve the following applications. 45. The manager of a large apartment comple has found that the profit, in dollars, is given b the equation P in which is the number of apartments rented. How man apartments must be rented to produce a profit of $3600? 46. The manager of a biccle shop has found that the revenue (in dollars) from the sale of biccles is given b the following equation. R How man biccles must be sold to produce a revenue of $12,500? 809

16 ANSWERS Find the equation of the line passing through the points of intersection of the graphs 2 and Write a sstem of inequalities to describe the following set of points: The points are in the interior of a circle whose center is the origin with a radius of 4, and above the line We are asked to solve the following sstem of equations Eplain how we can determine, before doing an work, that this sstem cannot have more than two solutions. 50. Without graphing, how can ou tell that the following sstem of inequalities has no solution? Solve the following sstems algebraicall Answers (a) (b) 810

17 11. (a) (b) 13. (3, 4) and ( 2, 4) 15. (4, 3) and (1, 3) 17. ( 3, 4) and ( 1, 4) 19. (0.6, 6) and ( 1.6, 6) 21. (6.2, 6) and (0.8, 6) 23. No solution 811

18 25. (3, 4) and ( 2, 4) 27. (4, 3) and (1, 3) , 6 or (0.618, 6) and ( 1.62, 6) 2 and, No solution (1, 1) and ( 2, 4) 43. ( 2, 1) and 38 25, ( 4, 1), ( 4, 1), (4, 1), (4, 1) 53. ( 2, 2), ( 2, 2), (2, 2), (2, 2) 812

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

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

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

4.9 Graph and Solve Quadratic

4.9 Graph and Solve Quadratic 4.9 Graph and Solve Quadratic Inequalities Goal p Graph and solve quadratic inequalities. Your Notes VOCABULARY Quadratic inequalit in two variables Quadratic inequalit in one variable GRAPHING A QUADRATIC

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

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

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

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

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

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

Graphing Linear Inequalities in Two Variables

Graphing Linear Inequalities in Two Variables 5.4 Graphing Linear Inequalities in Two Variables 5.4 OBJECTIVES 1. Graph linear inequalities in two variables 2. Graph a region defined b linear inequalities What does the solution set look like when

More information

Systems of Linear Equations: Solving by Substitution

Systems of Linear Equations: Solving by Substitution 8.3 Sstems of Linear Equations: Solving b Substitution 8.3 OBJECTIVES 1. Solve sstems using the substitution method 2. Solve applications of sstems of equations In Sections 8.1 and 8.2, we looked at graphing

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

ax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 )

ax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 ) SECTION 1. The Circle 1. OBJECTIVES The second conic section we look at is the circle. The circle can be described b using the standard form for a conic section, 1. Identif the graph of an equation as

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

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

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

Systems of Equations. from Campus to Careers Fashion Designer

Systems of Equations. from Campus to Careers Fashion Designer Sstems of Equations from Campus to Careers Fashion Designer Radius Images/Alam. Solving Sstems of Equations b Graphing. Solving Sstems of Equations Algebraicall. Problem Solving Using Sstems of Two Equations.

More information

7.3 Solving Systems by Elimination

7.3 Solving Systems by Elimination 7. Solving Sstems b Elimination In the last section we saw the Substitution Method. It turns out there is another method for solving a sstem of linear equations that is also ver good. First, we will need

More information

Ax 2 Cy 2 Dx Ey F 0. Here we show that the general second-degree equation. Ax 2 Bxy Cy 2 Dx Ey F 0. y X sin Y cos P(X, Y) X

Ax 2 Cy 2 Dx Ey F 0. Here we show that the general second-degree equation. Ax 2 Bxy Cy 2 Dx Ey F 0. y X sin Y cos P(X, Y) X Rotation of Aes ROTATION OF AES Rotation of Aes For a discussion of conic sections, see Calculus, Fourth Edition, Section 11.6 Calculus, Earl Transcendentals, Fourth Edition, Section 1.6 In precalculus

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

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

The Distance Formula and the Circle

The Distance Formula and the Circle 10.2 The Distance Formula and the Circle 10.2 OBJECTIVES 1. Given a center and radius, find the equation of a circle 2. Given an equation for a circle, find the center and radius 3. Given an equation,

More information

Reasoning with Equations and Inequalities

Reasoning with Equations and Inequalities Instruction Goal: To provide opportunities for students to develop concepts and skills related to solving linear sstems of equations b graphing Common Core Standards Algebra: Solve sstems of equations.

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

Anytime plan TalkMore plan

Anytime plan TalkMore plan CONDENSED L E S S O N 6.1 Solving Sstems of Equations In this lesson ou will represent situations with sstems of equations use tables and graphs to solve sstems of linear equations A sstem of equations

More information

Systems of Equations Involving Circles and Lines

Systems of Equations Involving Circles and Lines Name: Systems of Equations Involving Circles and Lines Date: In this lesson, we will be solving two new types of Systems of Equations. Systems of Equations Involving a Circle and a Line Solving a system

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

Zeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system.

Zeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system. _.qd /7/ 9:6 AM Page 69 Section. Zeros of Polnomial Functions 69. Zeros of Polnomial Functions What ou should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polnomial

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

7.5 SYSTEMS OF INEQUALITIES. Copyright Cengage Learning. All rights reserved.

7.5 SYSTEMS OF INEQUALITIES. Copyright Cengage Learning. All rights reserved. 7.5 SYSTEMS OF INEQUALITIES Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch the graphs of inequalities in two variables. Solve systems of inequalities. Use systems of inequalities

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

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

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

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

Inequalities and Absolute Values. Assignment Guide: EOO = every other odd, 1, 5, 9, 13, EOP = every other pair, 1, 2, 5, 6, 9, 10,

Inequalities and Absolute Values. Assignment Guide: EOO = every other odd, 1, 5, 9, 13, EOP = every other pair, 1, 2, 5, 6, 9, 10, Chapter 4 Inequalities and Absolute Values Assignment Guide: E = ever other odd,, 5, 9, 3, EP = ever other pair,, 2, 5, 6, 9, 0, Lesson 4. Page 75-77 Es. 4-20. 23-28, 29-39 odd, 40-43, 49-52, 59-73 odd

More information

POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

More information

THE PARABOLA 13.2. section

THE PARABOLA 13.2. section 698 (3 0) Chapter 3 Nonlinear Sstems and the Conic Sections 49. Fencing a rectangle. If 34 ft of fencing are used to enclose a rectangular area of 72 ft 2, then what are the dimensions of the area? 50.

More information

Solving Systems of Linear Equations by Graphing

Solving Systems of Linear Equations by Graphing . Solving Sstems of Linear Equations b Graphing How can ou solve a sstem of linear equations? ACTIVITY: Writing a Sstem of Linear Equations Work with a partner. Your famil starts a bed-and-breakfast. It

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

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

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

Section 7.2 Linear Programming: The Graphical Method

Section 7.2 Linear Programming: The Graphical Method Section 7.2 Linear Programming: The Graphical Method Man problems in business, science, and economics involve finding the optimal value of a function (for instance, the maimum value of the profit function

More information

y intercept Gradient Facts Lines that have the same gradient are PARALLEL

y intercept Gradient Facts Lines that have the same gradient are PARALLEL CORE Summar Notes Linear Graphs and Equations = m + c gradient = increase in increase in intercept Gradient Facts Lines that have the same gradient are PARALLEL If lines are PERPENDICULAR then m m = or

More information

Solving Special Systems of Linear Equations

Solving Special Systems of Linear Equations 5. Solving Special Sstems of Linear Equations Essential Question Can a sstem of linear equations have no solution or infinitel man solutions? Using a Table to Solve a Sstem Work with a partner. You invest

More information

STRAND: ALGEBRA Unit 3 Solving Equations

STRAND: ALGEBRA Unit 3 Solving Equations CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic

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

When I was 3.1 POLYNOMIAL FUNCTIONS

When I was 3.1 POLYNOMIAL FUNCTIONS 146 Chapter 3 Polnomial and Rational Functions Section 3.1 begins with basic definitions and graphical concepts and gives an overview of ke properties of polnomial functions. In Sections 3.2 and 3.3 we

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

P1. Plot the following points on the real. P2. Determine which of the following are solutions

P1. Plot the following points on the real. P2. Determine which of the following are solutions Section 1.5 Rectangular Coordinates and Graphs of Equations 9 PART II: LINEAR EQUATIONS AND INEQUALITIES IN TWO VARIABLES 1.5 Rectangular Coordinates and Graphs of Equations OBJECTIVES 1 Plot Points in

More information

Lesson 9.1 Solving Quadratic Equations

Lesson 9.1 Solving Quadratic Equations Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One -intercept and all nonnegative y-values. b. The verte in the third quadrant and no -intercepts. c. The verte

More information

Methods to Solve Quadratic Equations

Methods to Solve Quadratic Equations Methods to Solve Quadratic Equations We have been learning how to factor epressions. Now we will apply factoring to another skill you must learn solving quadratic equations. a b c 0 is a second-degree

More information

7.7 Solving Rational Equations

7.7 Solving Rational Equations Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate

More information

Introduction - Algebra I

Introduction - Algebra I LIFORNI STNRS TEST lgebra I Introduction - lgebra I The following released test questions are taken from the lgebra I Standards Test. This test is one of the alifornia Standards Tests administered as part

More information

6.3 PARTIAL FRACTIONS AND LOGISTIC GROWTH

6.3 PARTIAL FRACTIONS AND LOGISTIC GROWTH 6 CHAPTER 6 Techniques of Integration 6. PARTIAL FRACTIONS AND LOGISTIC GROWTH Use partial fractions to find indefinite integrals. Use logistic growth functions to model real-life situations. Partial Fractions

More information

Zero and Negative Exponents and Scientific Notation. a a n a m n. Now, suppose that we allow m to equal n. We then have. a am m a 0 (1) a m

Zero and Negative Exponents and Scientific Notation. a a n a m n. Now, suppose that we allow m to equal n. We then have. a am m a 0 (1) a m 0. E a m p l e 666SECTION 0. OBJECTIVES. Define the zero eponent. Simplif epressions with negative eponents. Write a number in scientific notation. Solve an application of scientific notation We must have

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

Classifying Solutions to Systems of Equations

Classifying Solutions to Systems of Equations CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Classifing Solutions to Sstems of Equations Mathematics Assessment Resource Service Universit of Nottingham

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

Section 0.2 Set notation and solving inequalities

Section 0.2 Set notation and solving inequalities Section 0.2 Set notation and solving inequalities (5/31/07) Overview: Inequalities are almost as important as equations in calculus. Man functions domains are intervals, which are defined b inequalities.

More information

Section 10-5 Parametric Equations

Section 10-5 Parametric Equations 88 0 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY. A hperbola with the following graph: (2, ) (0, 2) 6. A hperbola with the following graph: (, ) (2, 2) C In Problems 7 2, find the coordinates of an foci relative

More information

The majority of college students hold credit cards. According to the Nellie May

The majority of college students hold credit cards. According to the Nellie May CHAPTER 6 Factoring Polynomials 6.1 The Greatest Common Factor and Factoring by Grouping 6. Factoring Trinomials of the Form b c 6.3 Factoring Trinomials of the Form a b c and Perfect Square Trinomials

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

FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether or not the relationship shown in the table is a function. 1) -

More information

I think that starting

I think that starting . Graphs of Functions 69. GRAPHS OF FUNCTIONS One can envisage that mathematical theor will go on being elaborated and etended indefinitel. How strange that the results of just the first few centuries

More information

Solution of the System of Linear Equations: any ordered pair in a system that makes all equations true.

Solution of the System of Linear Equations: any ordered pair in a system that makes all equations true. Definitions: Sstem of Linear Equations: or more linear equations Sstem of Linear Inequalities: or more linear inequalities Solution of the Sstem of Linear Equations: an ordered pair in a sstem that makes

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

Partial Fractions. and Logistic Growth. Section 6.2. Partial Fractions

Partial Fractions. and Logistic Growth. Section 6.2. Partial Fractions SECTION 6. Partial Fractions and Logistic Growth 9 Section 6. Partial Fractions and Logistic Growth Use partial fractions to find indefinite integrals. Use logistic growth functions to model real-life

More information

SAMPLE. Polynomial functions

SAMPLE. Polynomial functions Objectives C H A P T E R 4 Polnomial functions To be able to use the technique of equating coefficients. To introduce the functions of the form f () = a( + h) n + k and to sketch graphs of this form through

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

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

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

Polynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter

Polynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter Mathematics Learning Centre Polnomials Jackie Nicholas Jacquie Hargreaves Janet Hunter c 26 Universit of Sdne Mathematics Learning Centre, Universit of Sdne 1 1 Polnomials Man of the functions we will

More information

Business and Economic Applications

Business and Economic Applications Appendi F Business and Economic Applications F1 F Business and Economic Applications Understand basic business terms and formulas, determine marginal revenues, costs and profits, find demand functions,

More information

Chapter 3. Curve Sketching. By the end of this chapter, you will

Chapter 3. Curve Sketching. By the end of this chapter, you will Chapter 3 Curve Sketching How much metal would be required to make a -ml soup can? What is the least amount of cardboard needed to build a bo that holds 3 cm 3 of cereal? The answers to questions like

More information

Graphing Linear Equations in Slope-Intercept Form

Graphing Linear Equations in Slope-Intercept Form 4.4. Graphing Linear Equations in Slope-Intercept Form equation = m + b? How can ou describe the graph of the ACTIVITY: Analzing Graphs of Lines Work with a partner. Graph each equation. Find the slope

More information

Math 152, Intermediate Algebra Practice Problems #1

Math 152, Intermediate Algebra Practice Problems #1 Math 152, Intermediate Algebra Practice Problems 1 Instructions: These problems are intended to give ou practice with the tpes Joseph Krause and level of problems that I epect ou to be able to do. Work

More information

Functions and Graphs CHAPTER INTRODUCTION. The function concept is one of the most important ideas in mathematics. The study

Functions and Graphs CHAPTER INTRODUCTION. The function concept is one of the most important ideas in mathematics. The study Functions and Graphs CHAPTER 2 INTRODUCTION The function concept is one of the most important ideas in mathematics. The stud 2-1 Functions 2-2 Elementar Functions: Graphs and Transformations 2-3 Quadratic

More information

3 Rectangular Coordinate System and Graphs

3 Rectangular Coordinate System and Graphs 060_CH03_13-154.QXP 10/9/10 10:56 AM Page 13 3 Rectangular Coordinate Sstem and Graphs In This Chapter 3.1 The Rectangular Coordinate Sstem 3. Circles and Graphs 3.3 Equations of Lines 3.4 Variation Chapter

More information

Algebra 1 Course Title

Algebra 1 Course Title Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

More information

3. Solve the equation containing only one variable for that variable.

3. Solve the equation containing only one variable for that variable. Question : How do you solve a system of linear equations? There are two basic strategies for solving a system of two linear equations and two variables. In each strategy, one of the variables is eliminated

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

Learning Objectives for Section 1.2 Graphs and Lines. Linear Equations in Two Variables. Linear Equations

Learning Objectives for Section 1.2 Graphs and Lines. Linear Equations in Two Variables. Linear Equations Learning Objectives for Section 1.2 Graphs and Lines After this lecture and the assigned homework, ou should be able to calculate the slope of a line. identif and work with the Cartesian coordinate sstem.

More information

The Rectangular Coordinate System

The Rectangular Coordinate System 3.2 The Rectangular Coordinate Sstem 3.2 OBJECTIVES 1. Graph a set of ordered pairs 2. Identif plotted points 3. Scale the aes NOTE In the eighteenth centur, René Descartes, a French philosopher and mathematician,

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

Algebra II. Administered May 2013 RELEASED

Algebra II. Administered May 2013 RELEASED STAAR State of Teas Assessments of Academic Readiness Algebra II Administered Ma 0 RELEASED Copright 0, Teas Education Agenc. All rights reserved. Reproduction of all or portions of this work is prohibited

More information

8.9 Intersection of Lines and Conics

8.9 Intersection of Lines and Conics 8.9 Intersection of Lines and Conics The centre circle of a hockey rink has a radius of 4.5 m. A diameter of the centre circle lies on the centre red line. centre (red) line centre circle INVESTIGATE &

More information

135 Final Review. Determine whether the graph is symmetric with respect to the x-axis, the y-axis, and/or the origin.

135 Final Review. Determine whether the graph is symmetric with respect to the x-axis, the y-axis, and/or the origin. 13 Final Review Find the distance d(p1, P2) between the points P1 and P2. 1) P1 = (, -6); P2 = (7, -2) 2 12 2 12 3 Determine whether the graph is smmetric with respect to the -ais, the -ais, and/or the

More information

Section 3-7. Marginal Analysis in Business and Economics. Marginal Cost, Revenue, and Profit. 202 Chapter 3 The Derivative

Section 3-7. Marginal Analysis in Business and Economics. Marginal Cost, Revenue, and Profit. 202 Chapter 3 The Derivative 202 Chapter 3 The Derivative Section 3-7 Marginal Analysis in Business and Economics Marginal Cost, Revenue, and Profit Application Marginal Average Cost, Revenue, and Profit Marginal Cost, Revenue, and

More information

2.3 Domain and Range of a Function

2.3 Domain and Range of a Function Section Domain and Range o a Function 1 2.3 Domain and Range o a Function Functions Recall the deinition o a unction. Deinition 1 A relation is a unction i and onl i each object in its domain is paired

More information

Reteaching Masters. To jump to a location in this book. 1. Click a bookmark on the left. To print a part of the book. 1. Click the Print button.

Reteaching Masters. To jump to a location in this book. 1. Click a bookmark on the left. To print a part of the book. 1. Click the Print button. Reteaching Masters To jump to a location in this book. Click a bookmark on the left. To print a part of the book. Click the Print button.. When the Print window opens, tpe in a range of pages to print.

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 SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its

More information

PROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS

PROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS PROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS A ver important set of curves which has received considerabl attention in recent ears in connection with the factoring of large numbers

More information

In this this review we turn our attention to the square root function, the function defined by the equation. f(x) = x. (5.1)

In this this review we turn our attention to the square root function, the function defined by the equation. f(x) = x. (5.1) Section 5.2 The Square Root 1 5.2 The Square Root In this this review we turn our attention to the square root function, the function defined b the equation f() =. (5.1) We can determine the domain and

More information

3 e) x f) 2. Precalculus Worksheet P.1. 1. Complete the following questions from your textbook: p11: #5 10. 2. Why would you never write 5 < x > 7?

3 e) x f) 2. Precalculus Worksheet P.1. 1. Complete the following questions from your textbook: p11: #5 10. 2. Why would you never write 5 < x > 7? Precalculus Worksheet P.1 1. Complete the following questions from your tetbook: p11: #5 10. Why would you never write 5 < > 7? 3. Why would you never write 3 > > 8? 4. Describe the graphs below using

More information

9.1 9.1. Graphical Solutions to Equations. A Graphical Solution to a Linear Equation OBJECTIVES

9.1 9.1. Graphical Solutions to Equations. A Graphical Solution to a Linear Equation OBJECTIVES SECTION 9.1 Graphical Solutions to Equations in One Variable 9.1 OBJECTIVES 1. Rewrite a linear equation in one variable as () () 2. Find the point o intersection o () and () 3. Interpret the point o intersection

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

DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS

DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the

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