Use Square Roots to Solve Quadratic Equations


 June Thomas
 4 years ago
 Views:
Transcription
1 10.4 Use Square Roots to Solve Quadratic Equations Before You solved a quadratic equation by graphing. Now You will solve a quadratic equation by finding square roots. Why? So you can solve a problem about a falling object, as in Example 5. FPO Key Vocabulary square root, p. 110 perfect square, p. 111 To use square roots to solve a quadratic equation of the form ax 2 1 c 5 0, first isolate x 2 on one side to obtain x 2 5 d. Then use the following information about the solutions of x 2 5 d to solve the equation. KEY CONCEPT For Your Notebook READING Recall that in this course, solutions refers to realnumber solutions. Solving x 2 5 d by Taking Square Roots If d. 0, then x 2 5 d has two solutions: x 56Ï } d. If d 5 0, then x 2 5 d has one solution: x 5 0. If d, 0, then x 2 5 d has no solution. d > 0 d 5 0 d <0 y x E XAMPLE 1 Solve quadratic equations Solve the equation. a. 2x b. m c. b ANOTHER WAY You can also use factoring to solve 2x : 2x (x 2 2 4) 5 0 2(x 2 2)(x 1 2) 5 0 x 5 2 or x 522 Solution a. 2x Write original equation. x Divide each side by 2. x 56Ï } Take square roots of each side. Simplify. c The solutions are 22 and 2. b. m Write original equation. m Add 18 to each side. m 5 0 The square root of 0 is 0. c The solution is 0. c. b Write original equation. b Subtract 12 from each side. c Negative real numbers do not have real square roots. So, there is no solution. 652 Chapter 10 Quadratic Equations and Functions
2 SIMPLIFYING SQUARE ROOTS In cases where you need to take the square root of a fraction whose numerator and denominator are perfect squares, the radical can be written as a fraction. For example, Î } 16 } can be written 25 as } 4 because 5 1 } } E XAMPLE 2 Take square roots of a fraction Solve 4z Solution 4z Write original equation. z } 4 Divide each side by 4. z 56Î } 9 }4 z 56 3 } 2 Take square roots of each side. Simplify. c The solutions are 2 3 } 2 and 3 } 2. APPROXIMATING SQUARE ROOTS In cases where d in the equation x 2 5 d is not a perfect square or a fraction whose numerator and denominator are not perfect squares, you need to approximate the square root. A calculator can be used to find an approximation. E XAMPLE 3 Approximate solutions of a quadratic equation Solve 3x Round the solutions to the nearest hundredth. Solution 3x x Write original equation. Add 11 to each side. x Divide each side by 3. x 56Ï } 6 Take square roots of each side. x ø Use a calculator. Round to the nearest hundredth. c The solutions are about and about GUIDED PRACTICE for Examples 1, 2, and 3 Solve the equation. 1. c w x x m b Solve the equation. Round the solutions to the nearest hundredth. 7. x k p Use Square Roots to Solve Quadratic Equations 653
3 E XAMPLE 4 Solve a quadratic equation Solve 6(x 2 4) Round the solutions to the nearest hundredth. 6(x 2 4) Write original equation. (x 2 4) Divide each side by 6. x Ï } 7 x Ï } 7 Take square roots of each side. Add 4 to each side. c The solutions are 4 1 Ï } 7 ø 6.65 and 4 2 Ï } 7 ø CHECK To check the solutions, first write the equation so that 0 is on one side as follows: 6(x 2 4) Then graph the related function y 5 6(x 2 4) The xintercepts appear to be about 6.6 and about 1.3. So, each solution checks E XAMPLE 5 Solve a multistep problem ANOTHER WAY For alternative methods for solving the problem in Example 5, turn to page 659 for the Problem Solving Workshop. SPORTS EVENT During an ice hockey game, a remotecontrolled blimp flies above the crowd and drops a numbered tabletennis ball. The number on the ball corresponds to a prize. Use the information in the diagram to find the amount of time that the ball is in the air. Solution DETERMINE VELOCITY When an object is dropped, it has an initial vertical velocity of 0 feet per second. STEP 1 Use the vertical motion model to write an equation for the height h (in feet) of the ball as a function of time t (in seconds). h 5216t 2 1 vt 1 s Vertical motion model h 5216t 2 1 0t 1 45 Substitute for v and s. STEP 2 Find the amount of time the ball is in the air by substituting 17 for h and solving for t. h 5216t Write model. 45 ft 17 ft Not drawn to scale t Substitute 17 for h t 2 Subtract 45 from each side. 28 } 16 5 t 2 Divide each side by 216. INTERPRET SOLUTION Because the time cannot be a negative number, ignore the negative square root. Î } 28 } 16 5 t Take positive square root ø t Use a calculator. c The ball is in the air for about 1.32 seconds. 654 Chapter 10 Quadratic Equations and Functions
4 GUIDED PRACTICE for Examples 4 and 5 Solve the equation. Round the solutions to the nearest hundredth, if necessary (x 2 2) (q 2 3) (t 1 5) WHAT IF? In Example 5, suppose the tabletennis ball is released 58 feet above the ground and is caught 12 feet above the ground. Find the amount of time that the ball is in the air. Round your answer to the nearest hundredth of a second EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKEDOUT SOLUTIONS on p. WS1 for Exs. 25 and 59 5 STANDARDIZED TEST PRACTICE Exs. 2, 15, 16, 29, 51, 52, 57, and 60 5 MULTIPLE REPRESENTATIONS Ex VOCABULARY Copy and complete: If b 2 5 a, then b is a(n)? of a. 2. WRITING Describe two methods for solving a quadratic equation of the form ax 2 1 c 5 0. EXAMPLES 1 and 2 on pp for Exs SOLVING EQUATIONS Solve the equation. 3. 3x x x m d a g w q b z n MULTIPLE CHOICE Which of the following is a solution of the equation n ? A 5 B 10 C 25 D MULTIPLE CHOICE Which of the following is a solution of the equation x ? A 2 6 } 5 B 1 } 6 C 5 } 6 D 5 EXAMPLE 3 on p. 653 for Exs APPROXIMATING SQUARE ROOTS Solve the equation. Round the solutions to the nearest hundredth. 17. x x x a k p m z c d b n MULTIPLE CHOICE The equation 17 2 } 1 4 x has a solution between which two integers? A 1 and 2 B 2 and 3 C 3 and 4 D 4 and Use Square Roots to Solve Quadratic Equations 655
5 ERROR ANALYSIS Describe and correct the error in solving the equation x d x x x x 5 Ï } 36 x 5 6 The solution is 6. 7d d d 2 52} 11 7 d ø The solutions are about and about EXAMPLE 4 on p. 654 for Exs SOLVING EQUATIONS Solve the equation. Round the solutions to the nearest hundredth. 32. (x 2 7) (x 2 3) (x 1 4) (m 1 5) (a 2 2) (z 1 14) } 2 ( c 2 8) } 2 (n 1 1) } 3 (k 2 6) SOLVING EQUATIONS Solve the equation. Round the solutions to the nearest hundredth, if necessary x x (x 2 1 5) x (4x 2 2 3) t 2 5 } w 2 7 } (4m 2 2 6) GEOMETRY Use the given area A of the circle to find the radius r or the diameter d to the nearest hundredth. 47. A 5 144π in A 5 21π m A 5 34π ft 2 r r d 50. REASONING An equation of the graph shown is y 5 } 1 (x 2 2) Two points on the parabola have 2 ycoordinates of 9. Find the xcoordinates of these points. 1 y 1 x 51. SHORT RESPONSE Solve x without using a calculator. Explain your reasoning. 52. OPEN ENDED Give values for a and c so that ax 2 1 c 5 0 has (a) two solutions, (b) one solution, and (c) no solution. CHALLENGE Solve the equation without graphing. 53. x x x x x x WORKEDOUT SOLUTIONS on p. WS1 5 STANDARDIZED TEST PRACTICE
6 PROBLEM SOLVING EXAMPLE 5 on p. 654 for Exs FALLING OBJECT Fenway Park is a Major League Baseball park in Boston, Massachusetts. The park offers seats on top of the left field wall. A person sitting in one of these seats accidentally drops his sunglasses on the field. The height h (in feet) of the sunglasses can be modeled by the function h 5216t where t is the time (in seconds) since the sunglasses were dropped. Find the time it takes for the sunglasses to reach the field. Round your answer to the nearest hundredth of a second. 57. MULTIPLE CHOICE Which equation can be used to find the time it takes for an object to hit the ground after it was dropped from a height of 68 feet? A 216t B 216t C 216t D 216t INTERNET USAGE For the period , the number y (in thousands) of Internet users worldwide can be modeled by the function y 5 12,697x ,722 where x is the number of years since Between which two years did the number of Internet users worldwide reach 100,000,000? 59. GEMOLOGY To find the weight w (in carats) of round faceted gems, gemologists use the formula w D 2 ds where D is the diameter (in millimeters) of the gem, d is the depth (in millimeters) of the gem, and s is the specific gravity of the gem. Find the diameter to the nearest tenth of a millimeter of each round faceted gem in the table. Gem Weight (carats) Depth (mm) Specific gravity Diameter (mm) a. b. c. Amethyst ? Diamond ? Ruby ? 60. SHORT RESPONSE In deep water, the speed s (in meters per second) of a series of waves and the wavelength L (in meters) of the waves are related by the equation 2πs L. L Crest Crest The wavelength L is the distance between one crest and the next. a. Find the speed to the nearest hundredth of a meter per second of a series of waves with the following wavelengths: 6 meters, 10 meters, and 25 meters. (Use 3.14 for π.) b. Does the speed of a series of waves increase or decrease as the wavelength of the waves increases? Explain Use Square Roots to Solve Quadratic Equations 657
7 61. MULTISTEP PROBLEM The Doyle log rule is a formula used to estimate the amount of lumber that can be sawn from logs of various sizes. The amount of lumber L(D 2 4)2 V (in board feet) is given by V 5} where L is 16 the length (in feet) of a log and D is the smallend diameter (in inches) of the log. a. Solve the formula for D. b. Use the rewritten formula to find the diameters, to the nearest tenth of a foot, of logs that will yield 50 board feet and have the following lengths: 16 feet, 18 feet, 20 feet, and 22 feet. Diameter Boards 62. MULTIPLE REPRESENTATIONS A ride at an amusement park lifts seated riders 250 feet above the ground. Then the riders are dropped. They experience free fall until the brakes are activated at 105 feet above the ground. a. Writing an Equation Use the vertical motion model to write an equation for the height h (in feet) of the riders as a function of the time t (in seconds) into the free fall. b. Making a Table Make a table that shows the height of the riders after 0, 1, 2, 3, and 4 seconds. Use the table to estimate the amount of time the riders experience free fall. c. Solving an Equation Use the equation to find the amount of time, to the nearest tenth of a second, that the riders experience free fall. 63. CHALLENGE The height h (in feet) of a dropped object on any planet can be modeled by h 5 2} g t 2 1 s where g is the acceleration (in feet per 2 second per second) due to the planet s gravity, t is the time (in seconds) after the object is dropped, and s is the initial height (in feet) of the object. Suppose the same object is dropped from the same height on Earth and Mars. Given that g is 32 feet per second per second on Earth and 12 feet per second per second on Mars, on which planet will the object hit the ground first? Explain. MIXED REVIEW PREVIEW Prepare for Lesson 10.5 in Exs Evaluate the power. (p. 2) } } } } Write an equation of the line with the given slope and yintercept. (p. 283) 68. slope: slope: slope: 3 yintercept: 11 yintercept: 27 yintercept: 22 Write an equation of the line that passes through the given point and is perpendicular to the given line. (p. 319) 71. (1, 21), y 5 2x 72. (0, 8), y 5 4x (29, 24), y 523x EXTRA PRACTICE for Lesson 10.4, p. 947 ONLINE QUIZ at classzone.com
8 LESSON 10.4 Using ALTERNATIVE METHODS Another Way to Solve Example 5, page 654 MULTIPLE REPRESENTATIONS In Example 5 on page 654, you saw how to solve a problem about a dropped tabletennis ball by using a square root. You can also solve the problem by using factoring or by using a table. P ROBLEM SPORTS EVENT During an ice hockey game, a remotecontrolled blimp flies above the crowd and drops a numbered tabletennis ball. The number on the ball corresponds to a prize. Use the information in the diagram to find the amount of time that the ball is in the air. 45 ft 17 ft Not drawn to scale M ETHOD 1 Using Factoring One alternative approach is to use factoring. STEP 1 Write an equation for the height h (in feet) of the ball as a function of time t (in seconds) after it is dropped using the vertical motion model. h 5216t 2 1 vt 1 s Vertical motion model h 5216t 2 1 0t 1 45 Substitute 0 for v and 45 for s. STEP 2 Substitute 17 for h to find the time it takes the ball to reach a height of 17 feet. Then write the equation so that 0 is on one side t Substitute 17 for h t Subtract 17 from each side. USE AN APPROXIMATION By replacing 28 with 25, you will obtain an answer that is an approximation of the amount of time that the ball is in the air. STEP 3 Solve the equation by factoring. Replace 28 with the closest perfect square, 25, so that the right side of the equation is factorable as a difference of two squares t Use 25 as an approximation for (16t ) Factor out (4t 2 5)(4t 1 5) Difference of two squares pattern 4t or 4t Zeroproduct property t 5 5 } 4 or t 52 5 } 4 Solve for t. c The ball is in the air about 5 } 4, or 1.25, seconds. Using Alternative Methods 659
9 M ETHOD 2 Using a Table Another approach is to make and use a table. STEP 1 Make a table that shows the height h (in feet) of the ball by substituting values for time t (in seconds) in the function h 5216t Use increments of 1 second. STEP 2 Identify the time interval in which the height of the ball is 17 feet. This happens between 1 and 2 seconds. STEP 3 Make a second table using increments of 0.1 second to get a closer approximation. c The ball is in the air about 1.3 seconds. Time t (seconds) Height h (feet) Time t (seconds) Height h (feet) P RACTICE 1. WHAT IF? In the problem on page 659, suppose the ball is caught at a height of 10 feet. For how many seconds is the ball in the air? Solve this problem using two different methods. 2. OPENENDED Describe a problem about a dropped object. Then solve the problem and explain what your solution means in this situation. 3. GEOMETRY The box below is a rectangular prism with the dimensions shown. 5x in. 5 in. x in. a. Write an equation that gives the volume V (in cubic inches) of the box as a function of x. b. The volume of the box is 83 cubic inches. Find the dimensions of the box. Use factoring to solve the problem. c. Make a table to check your answer from part (b). 4. TRAPEZE You are learning how to perform on a trapeze. While hanging from a still trapeze bar, your shoe comes loose and falls to a safety net that is 6 feet off the ground. If your shoe falls from a height of 54 feet, how long does it take your shoe to hit the net? Choose any method for solving the problem. Show your steps. 5. ERROR ANALYSIS A student solved the problem in Exercise 4 as shown below. Describe and correct the error. Let t be the time (in seconds) that the shoe is in the air t t Replace 60 with the closest perfect square, t (t 2 2)(t 1 2) t 5 2 or t 522 It takes about 2 seconds. 660 Chapter 10 Quadratic Equations and Functions
10 MIXED REVIEW of Problem Solving STATE TEST PRACTICE classzone.com Lessons MULTISTEP PROBLEM A company s yearly profits from 1996 to 2006 can be modeled by the function y 5 x 2 2 8x 1 80 where y is the profit (in thousands of dollars) and x is the number of years since a. In what year did the company experience its lowest yearly profit? b. What was the lowest yearly profit? 2. MULTISTEP PROBLEM Use the rectangle below. 2x ft (14 2 x) ft a. Find the value of x that gives the greatest possible area of the rectangle. b. What is the greatest possible area of the rectangle? 3. EXTENDED RESPONSE You throw a lacrosse ball twice using a lacrosse stick. 4. OPENENDED Describe a realworld situation of an object being dropped. Then write an equation that models the height of the object as a function of time. Use the equation to determine the time it takes the object to hit the ground. 5. SHORT RESPONSE A football player is attempting a field goal. The path of the kicked football can be modeled by the graph of y x x where x is the horizontal distance (in yards) traveled by the football and y is the corresponding height (in feet) of the football. Will the football pass over the goal post that is 10 feet above the ground and 45 yards away? Explain. 6. GRIDDED ANSWER The force F (in newtons) a rider feels while a train goes around a curve is given by F 5 } mv2 r where m is the mass (in kilograms) of the rider, v is the velocity (in meters per second) of the train, and r is the radius (in meters) of the curve. A rider with a mass of 75 kilograms experiences a force of 18,150 newtons, while going around a curve that has a radius of 8 meters. Find the velocity (in meters per second) the train travels around the curve. 7. SHORT RESPONSE The opening of the tunnel shown can be modeled by the graph of the equation y x x 2 12 where x and y are measured in feet. a. For your first throw, the ball is released 8 feet above the ground with an initial vertical velocity of 35 feet per second. Use the vertical motion model to write an equation for the height h (in feet) of the ball as a function of time t (in seconds). b. For your second throw, the ball is released 7 feet above the ground with an initial vertical velocity of 45 feet per second. Use the vertical motion model to write an equation for the height h (in feet) of the ball as a function of time t (in seconds). c. If no one catches either throw, for which throw is the ball in the air longer? Explain. a. Find the maximum height of the tunnel. b. A semi trailer is 7.5 feet wide, and the top of the trailer is 10.5 feet above the ground. Given that traffic travels one way on one lane through the center of the tunnel, will the semi trailer fit through the opening of the tunnel? Explain. Mixed Review of Problem Solving 661
11 Investigating Algebra ACTIVITY Use before Lesson 10.5 Algebra classzone.com 10.5 Completing the Square Using Algebra Tiles MATERIALS algebra tiles QUESTION How can you use algebra tiles to complete the square? For an expression of the form x 2 1 bx, you can add a constant c to the expression so that the expression x 2 1 bx 1 c is a perfect square trinomial. This process is called completing the square. E XPLORE Complete the square Find the value of c that makes x 2 1 4x 1 c a perfect square trinomial. STEP 1 Model expression Use algebra tiles to model the expression x 2 1 4x. You will need one x 2 tile and four xtiles for this expression. STEP 2 Rearrange tiles Arrange the tiles to form a square. The arrangement will be incomplete in one of the corners. STEP 3 Complete the square Determine the number of 1tiles needed to complete the square. The number of 1tiles is the value of c. So, the perfect square trinomial is x 2 1 4x 1 4 or (x 1 2) 2. DRAW CONCLUSIONS Use your observations to complete these exercises 1. Copy and complete the table using algebra tiles. Expression Number of 1tiles needed to complete the square Expression written as a square x 2 1 4x 4 x 2 1 4x (x 1 2) 2 x 2 1 6x?? x 2 1 8x?? x x?? 2. In the statement x 2 1 bx 1 c 5 (x 1 d) 2, how are b and d related? How are c and d related? 3. Use your answer to Exercise 2 to predict the number of 1tiles you would need to add to complete the square for the expression x x. 662 Chapter 10 Quadratic Equations and Functions
Solve Quadratic Equations by the Quadratic Formula. The solutions of the quadratic equation ax 2 1 bx 1 c 5 0 are. Standardized Test Practice
10.6 Solve Quadratic Equations by the Quadratic Formula Before You solved quadratic equations by completing the square. Now You will solve quadratic equations using the quadratic formula. Why? So you can
More informationFactor Polynomials Completely
9.8 Factor Polynomials Completely Before You factored polynomials. Now You will factor polynomials completely. Why? So you can model the height of a projectile, as in Ex. 71. Key Vocabulary factor by grouping
More informationFactor and Solve Polynomial Equations. In Chapter 4, you learned how to factor the following types of quadratic expressions.
5.4 Factor and Solve Polynomial Equations Before You factored and solved quadratic equations. Now You will factor and solve other polynomial equations. Why? So you can find dimensions of archaeological
More informationALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form
ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola
More information1.6. Solve Linear Inequalities E XAMPLE 1 E XAMPLE 2. Graph simple inequalities. Graph compound inequalities
.6 Solve Linear Inequalities Before You solved linear equations. Now You will solve linear inequalities. Why? So you can describe temperature ranges, as in Ex. 54. Key Vocabulary linear inequality compound
More information10.1. 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 informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationHow do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.
The verbal answers to all of the following questions should be memorized before completion of prealgebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More information10 7, 8. 2. 6x + 30x + 36 SOLUTION: 89 Perfect Squares. The first term is not a perfect square. So, 6x + 30x + 36 is not a perfect square trinomial.
Squares Determine whether each trinomial is a perfect square trinomial. Write yes or no. If so, factor it. 1.5x + 60x + 36 SOLUTION: The first term is a perfect square. 5x = (5x) The last term is a perfect
More informationScope and Sequence KA KB 1A 1B 2A 2B 3A 3B 4A 4B 5A 5B 6A 6B
Scope and Sequence Earlybird Kindergarten, Standards Edition Primary Mathematics, Standards Edition Copyright 2008 [SingaporeMath.com Inc.] The check mark indicates where the topic is first introduced
More informationReview 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 informationSect 6.7  Solving Equations Using the Zero Product Rule
Sect 6.7  Solving Equations Using the Zero Product Rule 116 Concept #1: Definition of a Quadratic Equation A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0 (referred
More informationAlgebra I. In this technological age, mathematics is more important than ever. When students
In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,
More informationMA107 Precalculus Algebra Exam 2 Review Solutions
MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write
More informationSolving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
More informationThis unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide.
COLLEGE ALGEBRA UNIT 2 WRITING ASSIGNMENT This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide. 1) What is the
More informationIn this section, you will develop a method to change a quadratic equation written as a sum into its product form (also called its factored form).
CHAPTER 8 In Chapter 4, you used a web to organize the connections you found between each of the different representations of lines. These connections enabled you to use any representation (such as a graph,
More informationWarmUp Oct. 22. Daily Agenda:
Evaluate y = 2x 3x + 5 when x = 1, 0, and 2. Daily Agenda: Grade Assignment Go over Ch 3 Test; Retakes must be done by next Tuesday 5.1 notes / assignment Graphing Quadratic Functions 5.2 notes / assignment
More informationReadings this week. 1 Parametric Equations Supplement. 2 Section 10.1. 3 Sections 2.12.2. Professor Christopher Hoffman Math 124
Readings this week 1 Parametric Equations Supplement 2 Section 10.1 3 Sections 2.12.2 Precalculus Review Quiz session Thursday equations of lines and circles worksheet available at http://www.math.washington.edu/
More informationVocabulary Cards and Word Walls Revised: June 29, 2011
Vocabulary Cards and Word Walls Revised: June 29, 2011 Important Notes for Teachers: The vocabulary cards in this file match the Common Core, the math curriculum adopted by the Utah State Board of Education,
More informationSection 1.1 Linear Equations: Slope and Equations of Lines
Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of
More information1.3. Maximum or Minimum of a Quadratic Function. Investigate A
< P16 photo of a large arched bridge, similar to the one on page 292 or p 360361of the fish book> Maximum or Minimum of a Quadratic Function 1.3 Some bridge arches are defined by quadratic functions.
More informationMATH 60 NOTEBOOK CERTIFICATIONS
MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5
More informationAlgebra 1 EndofCourse Exam Practice Test with Solutions
Algebra 1 EndofCourse Exam Practice Test with Solutions For Multiple Choice Items, circle the correct response. For Fillin Response Items, write your answer in the box provided, placing one digit in
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationSummer Math Exercises. For students who are entering. PreCalculus
Summer Math Eercises For students who are entering PreCalculus It has been discovered that idle students lose learning over the summer months. To help you succeed net fall and perhaps to help you learn
More informationMATH 10034 Fundamental Mathematics IV
MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.
More informationParallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.
CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes
More informationLesson 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 yvalues. b. The verte in the third quadrant and no intercepts. c. The verte
More informationof surface, 569571, 576577, 578581 of triangle, 548 Associative Property of addition, 12, 331 of multiplication, 18, 433
Absolute Value and arithmetic, 730733 defined, 730 Acute angle, 477 Acute triangle, 497 Addend, 12 Addition associative property of, (see Commutative Property) carrying in, 11, 92 commutative property
More informationFlorida Math for College Readiness
Core Florida Math for College Readiness Florida Math for College Readiness provides a fourthyear math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness
More information7.1 Graphs of Quadratic Functions in Vertex Form
7.1 Graphs of Quadratic Functions in Vertex Form Quadratic Function in Vertex Form A quadratic function in vertex form is a function that can be written in the form f (x) = a(x! h) 2 + k where a is called
More informationCharlesworth School Year Group Maths Targets
Charlesworth School Year Group Maths Targets Year One Maths Target Sheet Key Statement KS1 Maths Targets (Expected) These skills must be secure to move beyond expected. I can compare, describe and solve
More informationBrunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 20142015 school year.
Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 20142015 school year. Goal The goal of the summer math program is to help students
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More informationACTIVITY: Finding a Formula Experimentally. Work with a partner. Use a paper cup that is shaped like a cone.
8. Volumes of Cones How can you find the volume of a cone? You already know how the volume of a pyramid relates to the volume of a prism. In this activity, you will discover how the volume of a cone relates
More informationGraphing Quadratic Functions
Problem 1 The Parabola Examine the data in L 1 and L to the right. Let L 1 be the x value and L be the yvalues for a graph. 1. How are the x and yvalues related? What pattern do you see? To enter the
More information7.2 Quadratic Equations
476 CHAPTER 7 Graphs, Equations, and Inequalities 7. Quadratic Equations Now Work the Are You Prepared? problems on page 48. OBJECTIVES 1 Solve Quadratic Equations by Factoring (p. 476) Solve Quadratic
More informationCharacteristics of the Four Main Geometrical Figures
Math 40 9.7 & 9.8: The Big Four Square, Rectangle, Triangle, Circle Pre Algebra We will be focusing our attention on the formulas for the area and perimeter of a square, rectangle, triangle, and a circle.
More informationMath and FUNDRAISING. Ex. 73, p. 111 1.3 0. 7
Standards Preparation Connect 2.7 KEY VOCABULARY leading digit compatible numbers For an interactive example of multiplying decimals go to classzone.com. Multiplying and Dividing Decimals Gr. 5 NS 2.1
More informationPerimeter, Area, and Volume
Perimeter, Area, and Volume Perimeter of Common Geometric Figures The perimeter of a geometric figure is defined as the distance around the outside of the figure. Perimeter is calculated by adding all
More informationVeterans Upward Bound Algebra I Concepts  Honors
Veterans Upward Bound Algebra I Concepts  Honors Brenda Meery Kaitlyn Spong Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) www.ck12.org Chapter 6. Factoring CHAPTER
More information1.1 Practice Worksheet
Math 1 MPS Instructor: Cheryl Jaeger Balm 1 1.1 Practice Worksheet 1. Write each English phrase as a mathematical expression. (a) Three less than twice a number (b) Four more than half of a number (c)
More information3 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 information1. Which of the 12 parent functions we know from chapter 1 are power functions? List their equations and names.
Pre Calculus Worksheet. 1. Which of the 1 parent functions we know from chapter 1 are power functions? List their equations and names.. Analyze each power function using the terminology from lesson 1.
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationFree PreAlgebra Lesson 55! page 1
Free PreAlgebra Lesson 55! page 1 Lesson 55 Perimeter Problems with Related Variables Take your skill at word problems to a new level in this section. All the problems are the same type, so that you can
More informationUnit 7 Quadratic Relations of the Form y = ax 2 + bx + c
Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c Lesson Outline BIG PICTURE Students will: manipulate algebraic expressions, as needed to understand quadratic relations; identify characteristics
More informationInterpreting Graphs. Interpreting a Bar Graph
1.1 Interpreting Graphs Before You compared quantities. Now You ll use graphs to analyze data. Why? So you can make conclusions about data, as in Example 1. KEY VOCABULARY bar graph, p. 3 data, p. 3 frequency
More informationAlgebra 2 Chapter 5 Practice Test (Review)
Name: Class: Date: Algebra 2 Chapter 5 Practice Test (Review) Multiple Choice Identify the choice that best completes the statement or answers the question. Determine whether the function is linear or
More informationMTH 100 College Algebra Essex County College Division of Mathematics Sample Review Questions 1 Created June 6, 2011
MTH 00 College Algebra Essex County College Division of Mathematics Sample Review Questions Created June 6, 0 Math 00, Introductory College Mathematics, covers the mathematical content listed below. In
More informationSOLVING EQUATIONS WITH RADICALS AND EXPONENTS 9.5. section ( 3 5 3 2 )( 3 25 3 10 3 4 ). The OddRoot Property
498 (9 3) Chapter 9 Radicals and Rational Exponents Replace the question mark by an expression that makes the equation correct. Equations involving variables are to be identities. 75. 6 76. 3?? 1 77. 1
More informationIntroduction to Quadratic Functions
Introduction to Quadratic Functions The St. Louis Gateway Arch was constructed from 1963 to 1965. It cost 13 million dollars to build..1 Up and Down or Down and Up Exploring Quadratic Functions...617.2
More informationFlorida Algebra 1 EndofCourse Assessment Item Bank, Polk County School District
Benchmark: MA.912.A.2.3; Describe the concept of a function, use function notation, determine whether a given relation is a function, and link equations to functions. Also assesses MA.912.A.2.13; Solve
More informationUnit #3: Investigating Quadratics (9 days + 1 jazz day + 1 summative evaluation day) BIG Ideas:
Unit #3: Investigating Quadratics (9 days + 1 jazz day + 1 summative evaluation day) BIG Ideas: Developing strategies for determining the zeroes of quadratic functions Making connections between the meaning
More informationFirst published in 2013 by the University of Utah in association with the Utah State Office of Education.
First published in 201 by the University of Utah in association with the Utah State Office of Education. Copyright 201, Utah State Office of Education. Some rights reserved. This work is published under
More information2After completing this chapter you should be able to
After completing this chapter you should be able to solve problems involving motion in a straight line with constant acceleration model an object moving vertically under gravity understand distance time
More informationMath 120 Final Exam Practice Problems, Form: A
Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,
More informationFREE FALL. Introduction. Reference Young and Freedman, University Physics, 12 th Edition: Chapter 2, section 2.5
Physics 161 FREE FALL Introduction This experiment is designed to study the motion of an object that is accelerated by the force of gravity. It also serves as an introduction to the data analysis capabilities
More informationArea is a measure of how much space is occupied by a figure. 1cm 1cm
Area Area is a measure of how much space is occupied by a figure. Area is measured in square units. For example, one square centimeter (cm ) is 1cm wide and 1cm tall. 1cm 1cm A figure s area is the number
More informationMathematics PreTest Sample Questions A. { 11, 7} B. { 7,0,7} C. { 7, 7} D. { 11, 11}
Mathematics PreTest Sample Questions 1. Which of the following sets is closed under division? I. {½, 1,, 4} II. {1, 1} III. {1, 0, 1} A. I only B. II only C. III only D. I and II. Which of the following
More informationALGEBRA I (Common Core) Thursday, January 28, 2016 1:15 to 4:15 p.m., only
ALGEBRA I (COMMON CORE) The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA I (Common Core) Thursday, January 28, 2016 1:15 to 4:15 p.m., only Student Name: School Name: The
More informationNorth Carolina Community College System Diagnostic and Placement Test Sample Questions
North Carolina Community College System Diagnostic and Placement Test Sample Questions 01 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College
More informationFACTORING QUADRATICS 8.1.1 and 8.1.2
FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.
More informationArea of Parallelograms (pages 546 549)
A Area of Parallelograms (pages 546 549) A parallelogram is a quadrilateral with two pairs of parallel sides. The base is any one of the sides and the height is the shortest distance (the length of a perpendicular
More informationStudent Outcomes. Lesson Notes. Classwork. Exercises 1 3 (4 minutes)
Student Outcomes Students give an informal derivation of the relationship between the circumference and area of a circle. Students know the formula for the area of a circle and use it to solve problems.
More information83 Dot Products and Vector Projections
83 Dot Products and Vector Projections Find the dot product of u and v Then determine if u and v are orthogonal 1u =, u and v are not orthogonal 2u = 3u =, u and v are not orthogonal 6u = 11i + 7j; v
More informationPOLYNOMIAL 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 information4Unit 2 Quadratic, Polynomial, and Radical Functions
CHAPTER 4Unit 2 Quadratic, Polnomial, and Radical Functions Comple Numbers, p. 28 f(z) 5 z 2 c Quadratic Functions and Factoring Prerequisite Skills... 234 4. Graph Quadratic Functions in Standard Form...
More information1.6 A LIBRARY OF PARENT FUNCTIONS. Copyright Cengage Learning. All rights reserved.
1.6 A LIBRARY OF PARENT FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Identify and graph linear and squaring functions. Identify and graph cubic, square root, and reciprocal
More informationMATH 21. College Algebra 1 Lecture Notes
MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a
More informationCOMPETENCY TEST SAMPLE TEST. A scientific, nongraphing calculator is required for this test. C = pd or. A = pr 2. A = 1 2 bh
BASIC MATHEMATICS COMPETENCY TEST SAMPLE TEST 2004 A scientific, nongraphing calculator is required for this test. The following formulas may be used on this test: Circumference of a circle: C = pd or
More informationFactoring Polynomials
UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can
More information1.2 GRAPHS OF EQUATIONS. Copyright Cengage Learning. All rights reserved.
1.2 GRAPHS OF EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch graphs of equations. Find x and yintercepts of graphs of equations. Use symmetry to sketch graphs
More information1) (3) + (6) = 2) (2) + (5) = 3) (7) + (1) = 4) (3)  (6) = 5) (+2)  (+5) = 6) (7)  (4) = 7) (5)(4) = 8) (3)(6) = 9) (1)(2) =
Extra Practice for Lesson Add or subtract. ) (3) + (6) = 2) (2) + (5) = 3) (7) + () = 4) (3)  (6) = 5) (+2)  (+5) = 6) (7)  (4) = Multiply. 7) (5)(4) = 8) (3)(6) = 9) ()(2) = Division is
More informationInv 1 5. Draw 2 different shapes, each with an area of 15 square units and perimeter of 16 units.
Covering and Surrounding: Homework Examples from ACE Investigation 1: Questions 5, 8, 21 Investigation 2: Questions 6, 7, 11, 27 Investigation 3: Questions 6, 8, 11 Investigation 5: Questions 15, 26 ACE
More informationFlorida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper
Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic
More informationShow that when a circle is inscribed inside a square the diameter of the circle is the same length as the side of the square.
Week & Day Week 6 Day 1 Concept/Skill Perimeter of a square when given the radius of an inscribed circle Standard 7.MG:2.1 Use formulas routinely for finding the perimeter and area of basic twodimensional
More information12) 13) 14) (5x)2/3. 16) x5/8 x3/8. 19) (r1/7 s1/7) 2
DMA 080 WORKSHEET # (8.8.2) Name Find the square root. Assume that all variables represent positive real numbers. ) 6 2) 8 / 2) 9x8 ) 00 ) 8 27 2/ Use a calculator to approximate the square root to decimal
More informationFactoring Polynomials
Factoring Polynomials 8A Factoring Methods 81 Factors and Greatest Common Factors Lab Model Factoring 82 Factoring by GCF Lab Model Factorization of Trinomials 83 Factoring x 2 + bx + c 84 Factoring
More informationFlorida Math 0018. Correlation of the ALEKS course Florida Math 0018 to the Florida Mathematics Competencies  Lower
Florida Math 0018 Correlation of the ALEKS course Florida Math 0018 to the Florida Mathematics Competencies  Lower Whole Numbers MDECL1: Perform operations on whole numbers (with applications, including
More informationFP1. HiSET TM Mathematics Practice Test
FP1 HiSET TM Mathematics Practice Test Copyright 013 Educational Testing Service. All rights reserved. E T S and the E T S logo are registered trademarks of Educational Testing Service (E T S) in the United
More informationBig Bend Community College. Beginning Algebra MPC 095. Lab Notebook
Big Bend Community College Beginning Algebra MPC 095 Lab Notebook Beginning Algebra Lab Notebook by Tyler Wallace is licensed under a Creative Commons Attribution 3.0 Unported License. Permissions beyond
More informationSolutions to old Exam 1 problems
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
More informationSQUARESQUARE ROOT AND CUBECUBE ROOT
UNIT 3 SQUAREQUARE AND CUBEUBE (A) Main Concepts and Results A natural number is called a perfect square if it is the square of some natural number. i.e., if m = n 2, then m is a perfect square where m
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationQuick Reference ebook
This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed
More informationAlgebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test
Dear Parents, Based on the results of the High School Placement Test (HSPT), your child should forecast to take Algebra 1 this fall. If you are okay with that placement then you have no further action
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS B. Thursday, January 29, 2004 9:15 a.m. to 12:15 p.m.
The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS B Thursday, January 9, 004 9:15 a.m. to 1:15 p.m., only Print Your Name: Print Your School s Name: Print your name and
More informationMAIN IDEA The rectangle at the right has an area of 20 square units. The distance around the rectangle is 5 + 4 + 5 + 4, or 18 units.
19 Algebra: Area Formulas MAIN IDEA The rectangle at the right has an area of 20 square units. The distance around the rectangle is 5 + 4 + 5 + 4, or 1. Find the areas of rectangles and squares. New Vocabulary
More informationMATH 108 REVIEW TOPIC 10 Quadratic Equations. B. Solving Quadratics by Completing the Square
Math 108 T10Review Topic 10 Page 1 MATH 108 REVIEW TOPIC 10 Quadratic Equations I. Finding Roots of a Quadratic Equation A. Factoring B. Quadratic Formula C. Taking Roots II. III. Guidelines for Finding
More information6.4 Factoring Polynomials
Name Class Date 6.4 Factoring Polynomials Essential Question: What are some ways to factor a polynomial, and how is factoring useful? Resource Locker Explore Analyzing a Visual Model for Polynomial Factorization
More informationExam 1 Review Questions PHY 2425  Exam 1
Exam 1 Review Questions PHY 2425  Exam 1 Exam 1H Rev Ques.doc  1  Section: 1 7 Topic: General Properties of Vectors Type: Conceptual 1 Given vector A, the vector 3 A A) has a magnitude 3 times that
More informationMathematics as Problem Solving The students will demonstrate the ability to gather information from a graphical representation of an equation.
Title: Another Way of Factoring Brief Overview: Students will find factors for quadratic equations with a leading coefficient of one. The students will then graph these equations using a graphing calculator
More informationFor additional information, see the Math Notes boxes in Lesson B.1.3 and B.2.3.
EXPONENTIAL FUNCTIONS B.1.1 B.1.6 In these sections, students generalize what they have learned about geometric sequences to investigate exponential functions. Students study exponential functions of the
More informationOpenEnded ProblemSolving Projections
MATHEMATICS OpenEnded ProblemSolving Projections Organized by TEKS Categories TEKSING TOWARD STAAR 2014 GRADE 7 PROJECTION MASTERS for PROBLEMSOLVING OVERVIEW The Projection Masters for ProblemSolving
More informationSAT Math Facts & Formulas Review Quiz
Test your knowledge of SAT math facts, formulas, and vocabulary with the following quiz. Some questions are more challenging, just like a few of the questions that you ll encounter on the SAT; these questions
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More information