Quadratic and Exponential Functions

Transcription

1 9 Quadratic and Eponential Functions Graph quadratic functions. Solve quadratic equations. Graph eponential functions. Solve problems involving growth and deca. Ke Vocabular completing the square (p. 487) eponential function (p. 50) parabola (p. 471) Quadratic Formula (p. 493) Real-World Link Dinosaurs Eponential deca is one tpe of eponential function. Carbon dating uses eponential deca to determine the age of fossils and dinosaurs. Quadratic and Eponential Functions Make this Foldable to help ou organize our notes. Begin with three sheets of grid paper. 1 Fold each sheet in half along the width. Unfold each sheet and tape to form one long piece. 3 Label each page with the lesson number as shown. Refold to form a booklet Vocabular 468 Chapter 9 Quadratic and Eponential Functions

2 GET READY for Chapter 9 Diagnose Readiness You have two options for checking Prerequisite Skills. ption ption 1 Take the nline Readiness Quiz at algebra1.com. Take the Quick Check below. Refer to the Quick Review for help. Use a table of values to graph each equation. (Lesson 3-3) 1. = + 5. = = = = = SAVINGS Suppose ou have alread saved $00 toward the cost of a car. You plan to save $35 each month for the net several months. Graph the equation for the total amount T ou will have in m months. Determine whether each trinomial is a perfect square trinomial. If so factor it. (Lesson 8-6) 8. t + 1t a - 14a m - 18m b - 6b p + 1p s - 4s + 9 Find the net three terms of each arithmetic sequence. (Lesson 3-4) 16. 5, 9, 13, 17, , 5, -, -9, , -1,, 5, , 3, 40, 48, GEMETRY Write a formula that can be used to find the number of sides of a pattern containing n triangles. EXAMPLE 1 Use a table of values to graph = -. = - -1 (-1) (0) (1) - 0 () - EXAMPLE Determine whether is a perfect square trinomial. If so, factor it. 1. Is the first term a perfect square? es. Is the last term a perfect square? es 3. Is the middle term equal to (1)(11)? es is a perfect square trinomial = ( - 11) EXAMPLE 3 Find the net three terms of the arithmetic sequence -104, -4, 96, 196,.... Find the common difference b subtracting successive terms (-104) = 100 The common difference is 100. Add to find the net three terms = 96, = 396, = 496 The net three terms are 96, 396, 496. Chapter 9 Get Read For Chapter 9 469

3 EXPLRE 9-1 Graphing Calculator Lab Eploring Graphs of Quadratic Functions Not all functions are linear. The graphs of nonlinear functions have different shapes. ne tpe of nonlinear function is a quadratic function. The graph of a quadratic function is a parabola. You use a data collection device to conduct an eperiment and investigate quadratic functions. Set up the data collection device to collect data ever 0. second for 4 seconds. Connect the motion sensor to our data collection device. Position the motion detector on the floor pointed upward. ACTIVITY Step 1 Have one group member hold a ball about 3 feet above the motion detector. Another group member will operate the data collection device. Step When the person operating the data collection device sas go, he or she should press the start button to begin data collection. At the same time, the ball should be tossed straight upward. Step 3 Tr to catch the ball at about the same height at which it was tossed. Stop collecting data when the ball is caught. 470 Chapter 9 Quadratic and Eponential Functions ANALYZE THE RESULTS 1. The domain contains values represented b the independent variable, time. The range contains values represented b the dependent variable, distance. Use the graphing calculator to graph the data.. Write a sentence that describes the shape of the graph. Is the graph linear? Eplain. 3. Describe the position of the point on the graph that represents the starting position of the ball. 4. Use the TRACE feature of the calculator to find the maimum height of the ball. At what time was the maimum height achieved? 5. Repeat the eperiment and toss the ball higher. Compare and contrast the new graph and the first graph. 6. Conduct an eperiment in which the motion detector is held at a height of 4 feet and pointed downward at a dropped ball. How does the graph for this eperiment compare to the other graphs?

4 range 9-1 Graphing Quadratic Functions Main Ideas Graph quadratic functions. Find the equation of the ais of smmetr and the coordinates of the verte of a parabola. New Vocabular quadratic function parabola minimum maimum verte smmetr ais of smmetr Boston s Fourth of Jul celebration includes a fireworks displa set to music. If a rocket (firework) is launched with an initial velocit of 39. meters per second at a height of 1.6 meters above the ground, the equation h = -4.9 t + 39.t represents the rocket s height h in meters after t seconds. The rocket will eplode at approimatel the highest point. Graph Quadratic Functions The function describing the height of the rocket is an eample of a quadratic function. A quadratic function can be written in the form = a + b + c, where a 0. This form of equation is called standard form. The graph of a quadratic function is called a parabola. Quadratic Function Words Models A quadratic function can be described b an equation of the form = a + b + c, where a 0. Parent Graph The parent graph of the famil of quadratic functions is =. EXAMPLE Graph pens Upward Use a table of values to graph = What are the domain and range of this function? domain Graph these ordered pairs and connect them with a smooth curve Because the parabola 0-5 etends infinitel, the 1-7 domain is all real numbers The range is all real numbers greater than 4 11 or equal to Lesson 9-1 Graphing Quadratic Functions 471

5 1. Use a table of values to graph = + 3. What are the domain and range of this function? Consider the standard form = a + b + c. Notice that the value of a in Eample 1 is positive and the curve opens upward. The graph of an quadratic function in which a is positive opens upward. The lowest point, or minimum, of this graph is located at (1, -7). Graph pens Downward FLYING DISKS The equation = represents the height of a fling disk seconds after it is tossed. a. Use a table of values to graph = Graph these ordered pairs and 0 3 connect them with a smooth curve. 1 6 b. What are the domain and range 7 of this function? 3 6 D: { is a real number.} 4 3 R: { 7} c. Describe reasonable domain and range values for this situation. The fling disk is in the air for about 4.6 seconds, so a reasonable domain is { 0 4.6}. The height of the fling disk ranges from 0 to 7 feet, so a reasonable range is { 0 7}.. Use a table of values to graph = What are the domain and range of this function? Reading Math Verte The plural of verte is vertices. In math, verte has several meanings. For eample, there are the verte of an angle, the vertices of a polgon, and the verte of a parabola. Notice that the value of a in Eample is negative and the curve opens downward. The graph of an quadratic function in which a is negative opens downward. The highest point, or maimum, of the graph is located at (, 3). The maimum or minimum point of a parabola is called the verte. Smmetr and Vertices Parabolas possess a geometric propert called smmetr. Smmetrical figures are those in which each half of the figure matches the other eactl. The line that divides a parabola into two halves is called the ais of smmetr. Each point on the parabola that is on one side of the ais of smmetr has a corresponding point on the parabola on the other side of the ais. The verte is the onl point on the parabola that is on the ais of smmetr. Notice the relationship between the values a and b and the equation of the ais of smmetr. 47 Chapter 9 Quadratic and Eponential Functions

6 Ais of Smmetr of a Parabola Words The equation of the ais of smmetr for the graph of = a + b + c, where a 0, is = - b_ a. Model b a EXAMPLE Verte and Ais of Smmetr Coordinates of Verte Notice that ou can find the -coordinate b knowing the ais of smmetr. However, to find the -coordinate, ou must substitute the value of into the quadratic equation. Consider the graph of = a. Write the equation of the ais of smmetr. In = , a = -3 and b = -6. = - b_ Equation for the ais of smmetr of a parabola a -6 = -_ or -1 a = -3 and b = -6 (-3) The equation of the ais of smmetr is = -1. b. Find the coordinates of the verte. Since the equation of the ais of smmetr is = -1 and the verte lies on the ais, the -coordinate for the verte is -1. = riginal equation = -3 (-1) - 6 (-1) + 4 = -1 = = 7 Simplif. Add. The verte is at (-1, 7). c. Identif the verte as a maimum or minimum. Since the coefficient of the term is negative, the parabola opens downward and the verte is a maimum point. d. Graph the function. You can use the smmetr of the parabola ( 1, 7) to help ou draw its graph. n a coordinate plane, graph the verte and the ais of smmetr. Choose a value for other than -1. For eample, choose 1 and find the -coordinate that satisfies the equation. = riginal equation = -3 (1) - 6 (1) + 4 Let = 1. = -5 Simplif. ( 3, 5) 1 (1, 5) Graph (1, -5). Since the graph is smmetrical about its ais of smmetr = -1, ou can find another point on the other side of the ais of smmetr. The point at (1, -5) is units to the right of the ais. Go units to the left of the ais and plot the point (-3, -5). Repeat this for several other points. Then sketch the parabola. Etra Eamples at algebra1.com Lesson 9-1 Graphing Quadratic Functions 473

7 Consider the graph of = A. Write the equation of the ais of smmetr. 3B. Find the coordinates of the verte. 3C. Identif the verte as a maimum or minimum. 3D. Graph the function. Match Equations and Graphs Which is the graph of + 1 = ( + 1)? A B C D Substituting Values The ordered pair (0, 0) satisfies the equation + 1 = ( + 1). Since the point at (0, 0) is on the graph, choices A and D can be eliminated. Read the Test Item You are given a quadratic function, and ou are asked to choose its graph. Solve the Test Item Step 1 First write the equation in standard form. + 1 = ( + 1) riginal equation + 1 = ( + 1) = = Subtract 1 from each side. = + Simplif. Step Then find the ais of smmetr of the graph of = +. = - b_ a Equation for the ais of smmetr = - _ or -1 a = 1 and b = ( 1) The ais of smmetr is = -1. Look at the graphs. Since onl choices C and D have = -1 as their ais of smmetr, ou can eliminate choices A and B. Since the coefficient of the term is positive, the graph opens upward. Eliminate choice D. The answer is C. 4. Which is the equation of the graph? F - 1 = ( + ) G - 1 = ( - ) H + = ( - 1) J - = ( + 1) Personal Tutor at algebra1.com 474 Chapter 9 Quadratic and Eponential Functions

8 Eamples 1, (pp ) Eample 3 (pp ) Eample 4 (p. 474) Use a table of values to graph each function. 1. = - 5. = + 3. = = Write the equation of the ais of smmetr, and find the coordinates of the verte of the graph of each function. Identif the verte as a maimum or minimum. Then graph the function. 5. = = = -( - ) = ( + 3) STANDARDIZED TEST PRACTICE Which is the graph of = - 1_ + 1? A C B D HELP HMEWRK For See Eercises Eamples , Use a table of values to graph each function. 10. = = = = = = Write the equation of the ais of smmetr, and find the coordinates of the verte of the graph of each function. Identif the verte as a maimum or minimum. Then graph the function. 16. = = = = = = = = What is the equation of the ais of smmetr of the graph of = ? 5. Find the equation of the ais of smmetr of the graph of = Lesson 9-1 Graphing Quadratic Functions 475

9 ENTERTAINMENT For Eercises 6 and 7, use the following information. A carnival game involves striking a lever that forces a weight up a tube. If the weight reaches 0 feet to ring the bell, the contestant wins a prize. The equation h = -16 t + 3t + 3 gives the height of the weight if the initial velocit is 3 feet per second. 6. Find the maimum height of the weight. 7. Will a prize be won? Eplain. winner PETS For Eercises 8 and 9, use the following information. Miriam has 40 meters of fencing to build a pen for her dog. 8. Use the diagram to write an equation for the area A of the pen. Describe a reasonable domain and range for this situation. 9. What value of will result in the greatest area? What is the greatest possible area of the pen? 0 0 Real-World Link The Gatewa Arch is part of a tribute to Thomas Jefferson, the Louisiana Purchase, and the pioneers who settled the West. Each ear about.5 million people visit the arch. Source: World Book Encclopedia Write the equation of the ais of smmetr, and find the coordinates of the verte of the graph of each function. Identif the verte as a maimum or minimum. Then graph the function. 30. = -( - 4) = = _ 1 3 ( + ) = _ 1 ( + 1) The verte of a parabola is at (-4, -3). If one -intercept is -11, what is the other -intercept? 35. What is the equation of the ais of smmetr of a parabola if its -intercepts are -6 and 4? ARCHITECTURE For Eercises 36 38, use the following information. The shape of the Gatewa Arch in St. Louis, Missouri, is a catenar curve. It resembles a parabola with the equation h = , where h is the height in feet and is the distance from one base in feet. 36. What is the equation of the ais of smmetr? 37. What is the distance from one end of the arch to the other? 38. What is the maimum height of the arch? PRACTICE EXTRA See pages 735, 75. Self-Check Quiz at algebra1.com H..T. Problems FTBALL For Eercises 39 41, use the following information. A football is kicked from ground level at an initial velocit of 90 feet per second. The equation h = -16 t + 90t gives the height h of the football after t seconds. 39. What is the height of the ball after one second? 40. When is the ball 16 feet high? 41. When is the height of the ball zero feet? Describe the events these represent. 4. PEN ENDED Sketch a parabola that models a real-life situation and describe what the verte represents. Determine reasonable domain and range values for this tpe of situation. 43. REASNING Sketch the parent graph of the function = Chapter 9 Quadratic and Eponential Functions

10 REASNING Let f() = What is the domain of f()? 45. What is the range of f()? 46. For what values of is f() negative? 47. When is a real number, what are the domain and range of f() = - 9? 48. REASNING Determine the range of f() = ( - 5) CHALLENGE Write and graph a quadratic equation whose graph has the ais of smmetr = - 3_. Summarize the steps that ou took to determine 8 the equation. 50. Writing in Math Use the information about a rocket s path on page 471 to eplain how a fireworks displa can be coordinated with recorded music. Include an eplanation of how to determine when the rocket will eplode and how to find the height of the rocket when it eplodes. 51. In the graph of the function = - 3, which describes the shift in the verte of the parabola if, in the function, -3 is changed to 1? A units up B 4 units up C units down D 4 units down 5. REVIEW The costs of two packs of Brand A gum and two packs of Brand B gum are shown in the table. What percent of the cost of Brand B gum does James save b buing two packs of Brand A gum? Gum Cost of Two Packs Brand A $1.98 Brand B $.50 F 11.6% H 6.3% G 0.8% J 79.% Factor each polnomial, if possible. (Lessons 8-5 and 8-6) a + a m - 4m q a g Find each sum or difference. (Lesson 7-5) 59. (13 + 9) (8 - c ) + (1 + c ) 61. (7 p - p - 7) - ( p + 11) 6. RECREATIN At a recreation facilit, 3 members and 3 nonmembers pa a total of $180 to take an aerobics class. A group of 5 members and 3 nonmembers pa $10 to take the same class. How much does it cost each to take an aerobics class? (Lesson 5-3) PREREQUISITE SKILL Find the -intercept of the graph of each equation. (Lesson 3-3) = = = 7 Lesson 9-1 Graphing Quadratic Functions 477

11 EXTEND 9-1 Graphing Calculator Lab The Famil of Quadratic Functions The parent function of the famil of quadratic functions is =. ACTIVITY 1 Graph each group of equations on the same screen. Use the standard viewing window. Compare and contrast the graphs. KEYSTRKES: Review graphing equations on pages 16 and 163. Animation algebra1.com a. =, =, = 4 4 b. =, = 0.5, = Each graph opens upward and has its verte at the origin. The graphs of = and = 4 are narrower than the graph of =. Each graph opens upward and has its verte at the origin. The graphs of = 0.5 and = 0. are wider than the graph of =. 1A. H ow does the value of a in = a affect the shape of the graph? c. =, = + 3, = -, = - 4 d. =, = ( - 3), = ( + ), = ( + 4) 3 4 ( 4) ( ) ( 3) Each graph opens upward and has the same shape as =. However, each parabola has a different verte, located along the -ais. Each graph opens upward and has the same shape as =. However, each parabola has a different verte located along the -ais. 1B. How does the value of the constant affect the position of the graph? 1C. How is the location of the verte related to the equation of the graph? 478 Chapter 9 Quadratic and Eponential Functions ther Calculator Kestrokes at algebra1.com

12 Suppose ou graph the same equation using different windows. How will the appearance of the graph change? ACTIVITY Graph = - 7 in each viewing window. What conclusions can ou draw about the appearance of a graph in the window used? a. standard viewing window b. [-10, 10] scl: 1 b [-00, 00] scl: 50 c. [-50, 50] scl: 5 b [-10, 10] scl: 1 d. [-0.5, 0.5] scl: 0.1 b [-10, 10] scl: 1 Without knowing the window, graph b might be of the famil = a, where 0 < a < 1. Graph c looks like a member of = a - 7, where a > 1. Graph d looks more like a line. However, all are graphs of the same equation. EXERCISES Graph each famil of equations on the same screen. Compare and contrast the graphs. 1. = -. = - 3. = - 4. = - = -3 = -0.6 = - ( + 5) = = -6 = -0.4 = - ( - 4) = Use the graphs on page 478 and Eercises 1 4 above to predict the appearance of the graph of each equation. Then draw the graph. 5. = = ( + 1) 7. = 4 8. = - 6 Describe how each change in = would affect the graph of =. Be sure to consider all values of a, h, and k. 9. = a 10. = ( + h) 11. = + k 1. = ( + h) + k Etend 9 1 Graphing Calculator Lab: The Famil of Quadratic Functions 479

13 9- Solving Quadratic Equations b Graphing Main Idea Solve quadratic equations b graphing. Estimate solutions of quadratic equations b graphing. New Vocabular quadratic equation roots zeros double root A golf ball follows a path much like a parabola. Because of this propert, quadratic functions can be used to simulate parts of a computer golf game. ne of the -intercepts of the quadratic function represents the location where the ball will hit the ground. Solve b Graphing A quadratic equation is an equation that can be written in the form a + b + c = 0, where a 0. The value of the related quadratic function is 0. Quadratic Equation = 0 Related Quadratic Function f() = The solutions of a quadratic equation are called the roots of the equation. The roots of a quadratic equation can be found b finding the -intercepts or zeros of the related quadratic function. EXAMPLETwo Roots Animation algebra1.com -intercepts The -intercepts of a graph are also called the horizontal intercepts. Solve = 0 b graphing. Graph the related function f() = The equation of the ais of smmetr is = -_ 6 or = -3. When equals -3, (1) f() equals (-3) + 6(-3) - 7 or -16. So, the coordinates of the verte are (-3, -16). Make a table of values to find other points to sketch the graph. f() f() To solve = 0, ou need to know where the value of f() is 0. This occurs at the -intercepts. The -intercepts of the parabola appear to be -7 and Chapter 9 Quadratic and Eponential Functions

14 CHECK Solve b factoring. Common Misconception Although solutions found b graphing ma appear to be eact, ou cannot be sure that the are eact. Solutions need to be verified b substituting into the equation and checking, or b using the algebraic methods that ou will learn in this chapter = 0 ( + 7)( - 1) = 0 Factor. riginal equation + 7 = 0 or - 1 = 0 Zero Product Propert = -7 = 1 The solutions are -7 and Solve - c + 5c - 4 = 0 b graphing. Quadratic equations alwas have two roots. However, these roots are not alwas two distinct numbers. Sometimes the two roots are the same number, called a double root. In other cases the roots are not real numbers. EXAMPLE A Double Root Solve b + 4b = -4 b graphing. First rewrite the equation so one side is equal to zero. b + 4b = -4 riginal equation b + 4b + 4 = 0 Add 4 to each side. Graph the related function f(b) = b + 4b + 4. Notice that the verte of the parabola is the b-intercept. Thus, one solution is -. What is the other solution? Tr solving the equation b factoring. b + 4b + 4 = 0 riginal equation (b + )(b + ) = 0 Factor. b + = 0 or b + = 0 Zero Product Propert b = - b = - The solution is -. f (b) b 4b 4 f (b) b. Solve 0 = b graphing. EXAMPLE No Real Roots Empt Set The smbol, indicating an empt set, is often used to represent no real solutions. Solve = 0 b graphing. Graph the related function f() = The graph has no -intercept. Thus, there are no real number solutions for this equation. f() f () f () 4 3. Solve - t - 3t = 5 b graphing. Etra Eamples at algebra1.com Lesson 9- Solving Quadratic Equations b Graphing 481

15 Factoring can be used to determine whether the graph of a quadratic function intersects the -ais in zero, one, or two points. EXAMPLEFactoring Use factoring to determine how man times the graph of f() = intersects the -ais. Identif each root. The graph intersects the -ais when f() = = 0 riginal equation ( - 3)( + 4) = 0 Factor. Since the trinomial factors into two distinct factors, the graph of the function intersects the -ais times. The roots are = 3 and = Use factoring to determine how man times the graph of f() = intersects the -ais. Identif each root. Estimate Solutions In Eamples 1 and, the roots of the equation were integers. Usuall the roots of a quadratic equation are not integers. In these cases, use estimation to approimate the roots of the equation. EXAMPLERational Roots Solve n + 6n + 7 = 0 b graphing. If integral roots cannot be found, estimate the roots b stating the consecutive integers between which the roots lie. Graph the related function f(n) = n + 6n + 7. Location of Roots Since quadratic functions are continuous, there must be a zero between -values when their function values have opposite signs. n f(n) Notice that the value of the function changes from negative to positive between the n values of 5 and 4 and between and 1. f(n) f(n) n 6n 7 n The n-intercepts are between -5 and -4 and between - and -1. So, one root is between -5 and -4, and the other root is between - and Solve a + 6a - 3 = 0 b graphing. If integral roots cannot be found, estimate the roots b stating the consecutive integers between which the roots lie. Personal Tutor at algebra1.com 48 Chapter 9 Quadratic and Eponential Functions

16 Real-World Link The game of soccer, called football in countries other than North America, began in 1857 in Britain. It is plaed on ever continent of the world. Source: worldsoccer.about.com SCCER If a goalie kicks a soccer ball with an upward velocit of 65 feet per second and his foot meets the ball 3 feet off the ground, the function = -16 t + 65t + 3 represents the height of the ball in feet after t seconds. Approimatel how long is the ball in the air? You need to find the solution of the equation 0 = -16t + 65t + 3. Use a graphing calculator to graph the related function = -16t + 65t + 3. The -intercept is about 4. Therefore, the ball is in the air about 4 seconds. [, 7] scl: 1 b [ 0, 80] scl: NUMBER THERY Use a quadratic equation to find two numbers whose sum is 5 and whose product is -4. Eamples 1 3 (pp ) Eample 4 (p. 48) Eample 5 (p. 48) Eample 6 (p. 483) Solve each equation b graphing = 0. - a - 10a = 5 3. c + 3 = 0 Use factoring to determine how man times the graph of each function intersects the -ais. Identif each root. 4. f() = f() = Solve each equation b graphing. If integral roots cannot be found, estimate the roots b stating the consecutive integers between which the roots lie t - 5t + 1 = = w - 3w = 5 9. NUMBER THERY Two numbers have a sum of 4 and a product of -1. Use a quadratic equation to determine the two numbers. HELP HMEWRK For See Eercises Eamples , 3 6 Solve each equation b graphing. 10. c - 5c - 4 = n + n + 6 = = b + 4b = = r - 6r = The roots of a quadratic equation are - and -6. The minimum point of the graph of its related function is at (-4, -). Sketch the graph of the function and compare the graph to the graph of the parent function =. 17. The roots of a quadratic equation are -6 and 0. The maimum point of the graph of its related function is at (-3, 4). Sketch the graph of the function and compare the graph to the graph of the parent function =. Use factoring to determine how man times the graph of each function intersects the -ais. Identif each root. 18. g() = h() = f() = g() = Lesson 9- Solving Quadratic Equations b Graphing 483

17 Solve each equation b graphing. If integral roots cannot be found, estimate the roots b stating the consecutive integers between which the roots lie.. a - 1 = n + 7 = 0 4. c + 0c + 3 = s + 9s - 1 = = = a + 8a = = m - 10m = NUMBER THERY Use a quadratic equation to find two numbers whose sum is 9 and whose product is 0. The graph of the surface areas of the planets can be modeled b a quadratic equation. Visit algebra1.com to continue work on our project. 3. CMPUTER GAMES In a computer football game, the function d + 0.d = h simulates the path of a football at the kickoff. In this equation, h is the height of the ball and d is the horizontal distance in ards. What is the horizontal distance the ball will travel before it hits the ground? 33. HIKING While hiking in the mountains, Mona and Kishi stop for lunch on a ledge 1000 feet above a valle. Kishi decides to climb to another ledge 0 feet above Mona. Mona throws an apple up to Kishi, but Kishi misses it. The equation h = - 16t + 30t represents the height in feet of the apple t seconds after it was thrown. How long did it take for the apple to reach the ground? THEATER For Eercises 34 37, use the following information. The drama club is building a backdrop using arches whose shape can be represented b the function f() = , where is the length in feet. The area under each arch is to be covered with fabric. 34. Graph the quadratic function and determine its -intercepts. 35. What is the length of the segment along the floor of each arch? 36. What is the height of the arch? 37. The formula A = _ bh can be used to estimate the area A under a parabola. In 3 this formula, b represents the length of the base, and h represents the height. If there are five arches, calculate the total amount of fabric that is needed. PRACTICE EXTRA See pages 735, 75. Self-Check Quiz at algebra1.com H..T. Problems WRK For Eercises 38 40, use the following information. Kirk and Montega mow the soccer plaing fields. 1 area The must mow an area 500 feet long and 400 feet wide. The agree that each will mow half the area. Kirk will mow around the edge in a path of equal width until half the area is left. 500 ft 38. What is the area each person will mow? 39. Write a quadratic equation that could be used to find the width that Kirk should mow. What width should Kirk mow? 40. The mower can mow a path 5 feet wide. To the nearest whole number, how man times should Kirk go around the field? 41. PEN ENDED Draw a graph to show a countereample to the following statement. Eplain. All quadratic equations have two different solutions. 4. CHALLENGE Describe the zeros of f() = Eplain our reasoning ft 484 Chapter 9 Quadratic and Eponential Functions

18 Look Back To review linear inequalities, see Lesson CHALLENGE The graph shown is a quadratic inequalit. Similar to a linear inequalit, the quadratic equation is a boundar between two half-planes. Analze the graph and determine whether the inequalit is alwas, sometimes, or never greater than. Eplain. 44. Writing in Math Use the information about computer games on page 480 to eplain how quadratic equations can be used in computer simulations. Describe what the roots of a simulation equation for a computer golf game represent. 45. The graph of the equation = is shown. For what value or values of is = 0? A = -4 C = 7 and = 3 B = -5 D = -7 and = REVIEW Q-Mart has 100 blue towels in stock. If the sell half of their towels ever three months and do not receive an more shipments of towels, how man towels will the have left after a ear? F 60 H 150 G 75 J 300 Write the equation of the ais of smmetr, and find the coordinates of the verte of the graph of each equation. Identif the verte as a maimum or minimum. Then graph the function. (Lesson 9-1) 47. = = = Solve each equation. Check our solutions. (Lesson 8-6) 50. m - 4m = r = 70r d + 9 = -1d Simplif. Assume that no denominator is equal to zero. (Lesson 7-) 53. _ 10m _ a b 5 c 7 30m -11abc 56. SHIPPING An empt book crate weighs 30 pounds. The weight of a book is 1.5 pounds. For shipping, the crate must weigh at least 55 pounds and no more than 60 pounds. What is the acceptable number of books that can be packed in the crate? (Lesson 6-4) 55. _-9m 3 n 5 7m - n 5-4 PREREQUISITE SKILL Determine whether each trinomial is a perfect square trinomial. If so, factor it. (Lesson 8-6) 57. a m - 10m t + 16t Lesson 9- Solving Quadratic Equations b Graphing 485

19 9-3 Solving Quadratic Equations b Completing the Square Main Ideas Solve quadratic equations b finding the square root. Solve quadratic equations b completing the square. New Vocabular completing the square Al-Khwarizmi, born in Baghdad in 780, is considered to be one of the foremost mathematicians of all time. He wrote algebra in sentences instead of using equations, and he eplained the work with geometric sketches. Al-Khwarizmi would have described + 8 = 35 as A square and 8 roots are equal to 35 units. He would solve the problem using the following sketch The area of the shaded portion is 8 or 35. Four squares each with an area of 4 are used to complete the square. To solve problems this wa toda, ou might use algebra tiles or a method called completing the square. Find the Square Root Some equations can be solved b taking the square root of each side. EXAMPLE Irrational Roots Solve = 7 b taking the square root of each side. Round to the nearest tenth if necessar = 7 riginal equation ( - 5) = is a perfect square trinomial. ( - 5) = 7 Take the square root of each side. - 5 = 7 Simplif. - 5 = ± 7 Definition of absolute value = ± Add 5 to each side. = 5 ± 7 Simplif. Use a calculator to evaluate each value of. = or = 5-7 Write each solution Simplif. The solution set is {.4, 7.6}. 1. Solve m + 18m + 81 = 90 b taking the square root of each side. Round to the nearest tenth if necessar. 486 Chapter 9 Quadratic and Eponential Functions

20 Review Vocabular Perfect Square Trinomial a trinomial that is the square of a binomial; Eample: is a perfect square trinomial because it is the square of ( + 6). (Lesson 8-6) Complete the Square In Eample 1, the quadratic epression on one side of the equation was a perfect square. However, few quadratic epressions are perfect squares. To make an quadratic epression a perfect square, a method called completing the square ma be used. Consider the pattern for squaring a binomial such as + 6. ( + 6) = + (6)() + 6 = _ ( 1 ) 6 Notice that one half of 1 is 6 and 6 is 36. Completing the Square To complete the square for a quadratic epression of the form + b, ou can follow the steps below. Step 1 Find 1_ of b, the coefficient of. Step Square the result of Step 1. Step 3 Add the result of Step to + b, the original epression. EXAMPLE Complete the Square Find the value of c that makes c a perfect square. Method 1 Use algebra tiles. Animation algebra1.com is a perfect square. Method Complete the square. Step 1 Find _ 1 of 6. 6_ = 3 Step Square the result of Step 1. 3 = 9 Step 3 Add the result of Step to Thus, c = 9. Notice that = ( + 3).. Find the value of c that makes r + 8r + c a perfect square. You can use the technique of completing the square to solve quadratic equations. Etra Eamples at algebra1.com Lesson 9-3 Solving Quadratic Equations b Completing the Square 487

21 EXAMPLE Solve an Equation b Completing the Square Solve a - 14a + 3 = -10 b completing the square. Isolate the a and a terms. Then complete the square and solve. a - 14a + 3 = -10 a - 14a = riginal equation Subtract 3 from each side. a - 14a = -13 Simplif. a - 14a + 49 = Since ( _-14 ) = 49, add 49 to each side. (a - 7) = 36 Factor a - 14a a - 7 = ±6 a = 7 ± 6 a = or a = 7-6 = 13 = 1 Simplif. The solution set is {1, 13}. Take the square root of each side. Add 7 to each side. Separate the solutions. 3. Solve - 8 = 4 b completing the square. Round to the nearest tenth if necessar. To solve a quadratic equation in which the leading coefficient is not 1, first divide each term b the coefficient. Then complete the square. Solve a Quadratic Equation in Which a 1 ENTERTAINMENT The path of debris from fireworks when the wind is about 15 miles per hour can be modeled b the quadratic function h = , where h is the height and is the horizontal distance in feet. How far awa from the launch site will the debris land? Eplore You know the function that relates the horizontal and vertical distances. You want to know how far awa the debris will land. Real-World Link ne of the eploded fireworks for the Lake Toa Festival in Japan on Jul 15, 1988, broke a world record. The diameter of the burst was 3937 feet. Source: The Guinness Book of Records Plan The debris will hit the ground when h = 0. Complete the square to solve = 0. Solve = 0 Equation for where debris will land = = 0 0_ = = 00 Divide each side b Simplif. Add 00 to each side. Simplif = Since ( 50_ ) = 65, add 65 to each side = 85 Simplif. ( - 5) = 85 Factor = ± 85 Take the square root of each side. = 5 ± 85 Add 5 to each side. 488 Chapter 9 Quadratic and Eponential Functions

22 Use a calculator to evaluate each value of. = or = 5-85 Separate the solutions Evaluate. Check Since ou are looking for a distance, the negative number is not reasonable. The debris will land about 53.7 feet from the launch site. 4. Solve 3 n - 18n = 30 b completing the square. Round to the nearest tenth if necessar. Personal Tutor at algebra1.com Eample 1 (p. 486) Eample (p. 487) Eample 3 (p. 488) Eample 4 (pp ) Solve each equation b taking the square root of each side. Round to the nearest tenth if necessar. 1. b - 6b + 9 = 5. m + 14m + 49 = 0 Find the value of c that makes each trinomial a perfect square. 3. a - 1a + c 4. t + 5t + c Solve each equation b completing the square. Round to the nearest tenth if necessar. 5. c - 6c = = v + 14v - 9 = 6 8. r - 4r = 9. 4 a + 9a - 1 = = p - 5p GEMETRY The area of a square can be doubled b increasing the length b 6 inches and the width b 4 inches. What is the length of the side of the square? HELP HMEWRK For See Eercises Eamples Solve each equation b taking the square root of each side. Round to the nearest tenth if necessar. 1. b - 4b + 4 = t + t + 1 = g - 8g + 16 = 15. w + 16w + 64 = 18 Find the value of c that makes each trinomial a perfect square. 16. s - 16s + c c 18. p - 7p + c 19. c + 11k + k Solve each equation b completing the square. Round to the nearest tenth if necessar. 0. s - 4s - 1 = 0 1. d + 3d - 10 = = d + 0d + 11 = a - 5a = p - 4p = = 0 7. d - 8d + 7 = s - 10s = r + 49 = 4r h + 5 = 0h w - 1w - 1 = 0 Lesson 9-3 Solving Quadratic Equations b Completing the Square 489

23 3. PARK PLANNING A rectangular garden of wild flowers is 9 meters long b 6 meters wide. A pathwa of constant width goes around the garden. If the area of the path equals the area of the garden, what is the width of the path? 6m 9m m 33. NUTRITIN The consumption of bread and cereal in the United States is increasing and can be modeled b the function = , where represents the consumption of bread and cereal in pounds and represents the number of ears since If this trend continues, in what future ear will the average American consume 300 pounds of bread and cereal? Solve each equation b completing the square. Round to the nearest tenth if necessar t + 0.1t = v +.5 = v 1 _ d - 5d - 3 = _ For more information, go to algebra1.com. EXTRA PRACTICE See pages 735, Find all values of c that make + c + 81 a perfect square. 39. Find all values of c that make + c a perfect square. Solve each equation for in terms of c b completing the square c = c = 0 4. PHTGRAPHY Emilio is placing a photograph behind a 1-inch-b-1-inch piece of matting. The photograph is to be positioned so that the matting is twice as wide at the top and bottom as it is at the sides. If the area of the photograph is to be 54 square inches, what are the dimensions? 1 in. in. in. Photographer Photographers must consider lighting, lens setting, and composition to create the best photograph. 3 in. Real-World Career 1 _ _ f - 7f + _ =0 4 1 in. in. Self-Check Quiz at algebra1.com H..T. Problems 43. PEN ENDED Make a square using one or more of each of the following tpes of tiles. -tile -tile 1-tile Describe the area of our square using an algebraic epression. 44. REASNING Compare and contrast the following strategies for solving = 0: completing the square, graphing the related function, and factoring. 45. CHALLENGE Without graphing, describe the solution of = 0. Eplain our reasoning. Then describe the graph of the related function. 46. Which ne Doesn t Belong? Identif the epression that does not belong with the other three. Eplain our reasoning. 1 n - n + _ 4 1 n + n + _ Chapter 9 Quadratic and Eponential Functions 1 n - _ n+_ n + _ n+_ 9 3

24 47. Writing in Math Use the information about Al-Khwarizmi on page 486 to eplain how ancient mathematicians used squares to solve algebraic equations. Include an eplanation of Al-Khwarizmi s drawings for + 8 = 35 and a step-b-step algebraic solution with justification for each step of the equation. 48. What are the solutions to the quadratic equation p - 14p = 3? A 16 C -, 16 B -3, 14 D -4, REVIEW If a = -5 and b = 6, then 3a - ab = F -75 H 30 G -55 J 45 Solve each equation b graphing. (Lesson 9-) = = = 0 PARKS For Eercises 53 and 54, use the following information. (Lesson 9-1) A cit is building a dog park that is rectangular in shape and measures 80 feet around three of the four sides as shown in the diagram. 53. If the width of the park in feet is, write an equation that models the area A of the park. 54. Analze the graph of the related function b finding the coordinates of the verte and describing what this point represents. Find the GCF for each set of monomials. (Lesson 8-1) a b 3, 0a 3 b 3 c, 35ab c m n 3, 8m n, 56 m 3 n Write an inequalit for each graph. (Lesson 6-4) Use substitution to solve each sstem of equations. If the sstem does not have eactl one solution, state whether it has no solution or infinitel man solutions. (Lesson 5-) 59. = 60. = = 3 + = 9-3 = = 3 PREREQUISITE SKILL Evaluate b - 4ac for each set of values. Round to the nearest tenth if necessar. (Lesson 1 ) 6. a = 1, b = -, c = a =, b = 7, c = a = 1, b = 5, c = a = -, b = 7, c = 5 Lesson 9-3 Solving Quadratic Equations b Completing the Square 491

25 CHAPTER 9 Mid-Chapter Quiz Lessons 9-1 through 9-3 Write the equation of the ais of smmetr, and find the coordinates of the verte of the graph of each function. Identif the verte as a maimum or minimum. Then graph the function. (Lesson 9-1) 1. = = = MULTIPLE CHICE Which graph shows a function = + b when b > 1? (Lesson 9-1) A B Solve each equation b graphing. If integral roots cannot be found, estimate the roots b stating the consecutive integers between which the roots lie. (Lesson 9-) = = = 0 8. SFTBALL In a softball game, Lola hit the ball straight up with an initial upward velocit of 47 feet per second. The height h of the softball in feet above ground after t seconds can be modeled b the equation h = -16 t + 47t + 3. How long was the softball in the air before it hit the ground? (Lesson 9-) Solve each equation b completing the square. Round to the nearest tenth if necessar. (Lesson 9-3) 9. s + 8s = a - 10a = = 5 1. b - b - 7 = 14 C 13. RCKETS A model rocket is launched from the ground with an initial upward velocit of 475 feet per second. About how man seconds will it take to reach the ground? Use the formula h = -16 t + 175t, where h is the height of the rocket and t is the time in seconds. Round to the nearest tenth if necessar. (Lesson 9-3) D 14. GEMETRY The length and width of the rectangle are increased b the same amount so that the new area is 154 square centimeters. Find the dimensions of the new rectangle. (Lesson 9-3) cm 5 cm 8 cm 49 Chapter 9 Quadratic and Eponential Functions

26 9-4 Solving Quadratic Equations b Using the Quadratic Formula Main Ideas Solve quadratic equations b using the Quadratic Formula. Use the discriminant to determine the number of solutions for a quadratic equation. New Vocabular Quadratic Formula discriminant In the past few decades, there has been a dramatic increase in the percent of people living in the United States who were born in other countries. This trend can be modeled b the quadratic function P = t t , where P is the percent born outside the United States and t is the number of ears since Percent Percent Born utside the U.S P P 0.006t 0.080t Years Since 1960 To predict when 15% of the population will be people who were born outside of the U.S., ou can solve the equation 15 = t t This equation would be impossible or difficult to solve using factoring, graphing, or completing the square. Quadratic Formula You can solve the standard form of the quadratic equation a + b + c = 0 for. The result is the Quadratic Formula. The Quadratic Formula The solutions of a quadratic equation in the form a + b + c = 0, where a 0, are given b the Quadratic Formula. = -b ± b - 4ac a You can solve quadratic equations b factoring, graphing, completing the square, or using the Quadratic Formula. EXAMPLE Solve Quadratic Equations Quadratic Formula The Quadratic Formula is proved in Lesson Solve each equation. Round to the nearest tenth if necessar. a = 0 Method 1 Factoring = 0 riginal equation ( + 4)( - 6) = 0 Factor = 0 or - 6 = 0 Zero Product Propert = -4 = 6 Solve for. (continued on the net page) Lesson 9-4 Solving Quadratic Equations b Using the Quadratic Formula 493

27 Method Quadratic Formula For this equation, a = 1, b = -, and c = -4. = -b ± b - 4ac a Quadratic Formula = - (-) ± (-) - 4 (1)(-4) a = 1, b = -, and c = -4 (1) = ± Multipl. = _ ± or _ ± 10 Add and simplif. = _ - 10 or = _ + 10 Separate the solutions. = -4 = 6 Simplif. The solution set is {-4, 6}. The Quadratic Formula You ma want to simplif this equation b dividing each side b before appling the Quadratic Formula. However, the Quadratic Formula can help ou find the solution of an quadratic equation. b = 6 Step 1 Rewrite the equation in standard form = 6 riginal equation = 0 Subtract 6 from each side. Step Appl the Quadratic Formula. = -b ± b - 4ac a = - (-14) ± (-14) - 4 (4)(-6) (4) 14 ± = ± = _ = _ or 14 + = _ Quadratic Formula a = 4, b = -14, and c = -6 Multipl. Add Simplif. Separate the solutions. Check the solutions b using the CALC menu on a graphing calculator to determine the zeros of the related quadratic function. To the nearest tenth, the solution set is {-0.3, 0.9}. Personal Tutor at algebra1.com 494 Chapter 9 Quadratic and Eponential Functions 1A = 0 1B. 4 + = 17

28 The table summarizes the five methods for solving quadratic equations. Method Can Be Used Comments Lesson(s) factoring sometimes Use if constant term is 0 or factors are easil determined. 8- to 8-6 using a table sometimes Not alwas eact; use onl when an approimate solution is sufficient. graphing alwas Not alwas eact; use onl when an approimate solution is sufficient. completing the square Quadratic Formula alwas Useful for equations of the form + b + c = 0, where b is an even number. alwas Solving Quadratic Equations ther methods ma be easier to use in some cases, but this method alwas gives accurate solutions Use the Quadratic Formula to Solve a Problem Real-World Link Astronauts have found walking on the Moon to be ver different from walking on Earth because the gravitational pull of the Moon is onl 1.6 meters per second squared. The gravitational pull on Earth is 9.8 meters per second squared. Source: World Book Encclopedia SPACE TRAVEL The height H of an object t seconds after it is propelled upward with an initial velocit v is represented b H = - 1_ gt + vt + h, where g is the gravitational pull and h is the initial height. Suppose an astronaut on the Moon throws a baseball upward with an initial velocit of 10 meters per second, letting go of the ball meters above the ground. Use the information at the left to find how much longer the ball will sta in the air than a similarl thrown baseball on Earth. In order to find when the ball hits the ground, ou must find when H = 0. Write two equations to represent the situation on the Moon and on Earth. Baseball Thrown on the Moon H = - 1_ gt + vt + h 0 = - 1_ ( 1.6) t + 10t + 0 = -0.8 t + 10t + Baseball Thrown on Earth H = - 1_ gt + vt + h 0 = - 1_ ( 9.8) t + 10t + 0 = - 4.9t + 10t + To find accurate solutions, use the Quadratic Formula. t = -b ± b - 4ac t = -b ± b - 4ac a a = -10 ± 10-4 (-0.8) () (-0.8) -10 ± = t 1.7 or t -0. = -10 ± 10-4 (-4.9) () (-4.9) -10 ± = t. or t -0. Since a negative time is not reasonable, use the positive solutions. The ball will sta in the air about or 10.5 seconds longer on the Moon.. GEMETRY The perimeter of a rectangle is 60 inches. Find the dimensions of the rectangle if its area is 1 square inches. Lesson 9-4 Solving Quadratic Equations b Using the Quadratic Formula 495

29 The Discriminant In the Quadratic Formula, the epression under the radical sign, b - 4ac, is called the discriminant. The value of the discriminant can be used to determine the number of real roots for a quadratic equation. Using the Discriminant Discriminant negative zero positive Eample = = = 0-1 ± 1-4()(3) = -6 ± = 6-4(1)(9) -(-5) ± = (-5) - 4(1)() () (1) (1) -1 ± = -3-6 ± = _ 0 = _ 5 ± 4 There are no real roots = _ -6 or -3 There are two roots, since no real number can be the square root There is a double root, -3. _ and _ 5 -. of a negative number. Graph of Related Function f () f() 6 9 f () f() 5 f () f() 3 Number of Real Roots The graph does not cross the -ais. The graph touches the -ais in one place. The graph crosses the -ais twice. 0 1 EXAMPLE Use the Discriminant 496 Chapter 9 Quadratic and Eponential Functions State the value of the discriminant for each equation. Then determine the number of real roots of the equation. a = 0 b - 4ac = 10-4 () (11) a =, b = 10, and c = 11 = 1 Simplif. Since the discriminant is positive, the equation has two real roots. b. 3m + 4m = - Step 1 Rewrite the equation in standard form. 3m + 4m = - riginal equation 3m + 4m + = - + 3m + 4m + = 0 Step Find the discriminant. Add to each side. Simplif. b - 4ac = 4-4 (3) () a = 3, b = 4, and c = = -8 Simplif. Since the discriminant is negative, the equation has no real roots.

30 3A. 4 n - 0n + 5 = 0 3B = 0 3C = 0 Eample 1 (pp ) Eample (p. 495) Eample 3 (p. 496) Solve each equation b using the Quadratic Formula. Round to the nearest tenth if necessar = 0. t + 11t = 1 3. r + 10r + 1 = v + 5v + 11 = 0 5. MANUFACTURING A pan is to be formed b cutting -centimeter-b--centimeter squares from each corner of a square piece of sheet metal and then folding the sides. If the volume of the pan is to be square centimeters, what should the dimensions of the original piece of sheet metal be? State the value of the discriminant for each equation. Then determine the number of real roots of the equation. 6. m + 5m - 6 = 0 7. s + 8s + 16 = 0 8. z + z = -50 HELP HMEWRK For See Eercises Eamples 9 0 1, 1, Solve each equation b using the Quadratic Formula. Round to the nearest tenth if necessar. 9. v + 1v + 0 = t - 7t - 0 = = = r + 5 = = s = 40s 16. r + r - 14 = n - 7n - 3 = v - 7v = z - z = 3 0. w = -(7w + 3) 1. GEMETRY What are the dimensions of Rectangle ABCD rectangle ABCD? perimeter 4 cm area 80 cm. PHYSICAL SCIENCE A projectile is shot verticall up in the air from ground level. Its distance s, in feet, after t seconds is given b s = 96t - 16 t. Find the values of t when s is 96 feet. State the value of the discriminant for each equation. Then determine the number of real roots of the equation = = p + 10p = m + m = r = _ 1 r - _ _ 3 n + 4n = -3 Solve each equation b using the Quadratic Formula. Round to the nearest tenth if necessar d - 1.1d = = _ 5 4 = _ 1 3. w + _ 5 = 3 _ 5 w Lesson 9-4 Solving Quadratic Equations b Using the Quadratic Formula 497