Quadratic Functions 5A Quadratic Functions and Complex Numbers Lab Explore Parameter Changes 5-1 Using Transformations to Graph Quadratic Functions 5-2 Properties of Quadratic Functions in Standard Form Lab Explore Graphs and Factors 5-3 Solving Quadratic Equations by Graphing and Factoring 5-4 Completing the Square 5-5 Complex Numbers and Roots 5-6 The Quadratic Formula 5B Applying Quadratic Functions 5-7 Solving Quadratic Inequalities 5-8 Curve Fitting with Quadratic Models 5-9 Operations with Complex Numbers • Make connections among representations of quadratic functions. • Use various methods to solve quadratic equations and apply them to realworld problems. Planetary Pass How far could you throw a football if you were on Mars or Saturn? You can find the answer by using quadratic functions. KEYWORD: MB7 ChProj 310 Chapter 5 Vocabulary Match each term on the left with a definition on the right. A. a change in a function rule and its graph 1. linear equation 2. solution set 3. transformation 4. x-intercept B. the x-coordinate of the point where a graph crosses the x-axis C. the group of values that make an equation or inequality true D. a letter or symbol that represents a number E. an equation whose graph is a line Squares and Square Roots Simplify each expression. 5. 3.2 2 6. (_25 ) 2 7. √ 121 8. 1 _ √ 16 Simplify Radical Expressions Simplify each expression. 9. √ 72 10. 2( √ 144 - 4) 11. √ 33 · √ 75 √ 54 12. _ √ 3 Multiply Binomials Multiply. 13. (x - 2)(x - 6) 14. (x + 9)(x - 9) 15. (x + 2)(x + 7) 16. (2x - 3)(5x + 1) Solve Multi-Step Equations Solve each equation. 17. 2x + 10 = - 32 2 (x - 1) = 11 19. _ 3 18. 2x - (1 - x) = 2 20. 2(x + 5) - 5x = 1 Graph Linear Functions Graph each function. 21. y = -x 23. y = -3x + 6 22. y = 2x - 1 1x + 2 24. y = _ 3 Quadratic Functions 311 Previously, you • graphed and transformed • • • linear functions. solved linear equations and inequalities. fit data using linear models. used and performed operations with real numbers. You will study • graphing and transforming • • • quadratic functions. solving quadratic equations and inequalities. fitting data to quadratic models. using and performing operations with imaginary and other complex numbers. You can use the skills in this chapter • in advanced math classes, • • 312 Chapter 5 including Precalculus. in other classes, such as Chemistry, Physics, and Economics. outside of school to identify patterns and model data. Key Vocabulary/Vocabulario absolute value of a complex number valor absoluto de un número complejo complex conjugate conjugado complejo complex number número complejo imaginary number número imaginario maximum value valor máximo minimum value valor mínimo parabola parábola quadratic function función cuadrática vertex form forma en vértice zero of a function cero de una función Thinking About Vocabulary To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. Quadratic is from the Latin quadrum, which means “square.” A quadratic function always contains a square of the variable, such as x 2. What is a quadrilateral, and how does it relate to a square? What are some other words that use the root quad-, and what do they mean? 2. The word conjugate can mean “joined together, especially in pairs.” Name some mathematical relationships that involve pairs. 3. What might the terms maximum value or minimum value of a function refer to? 4. The word vertex can mean “highest point.” What might the vertex form of a quadratic function indicate about the function’s graph? � ���� ������������������������ ����������������������� ���������� ����������������� �������������������������� ����������������� ������������������������� �������������� ������������������������������������ ��������������������������������������� ��������������������������������� ���������������� Study Strategy: Use Multiple Representations ���������� ������������ The explanation ����������� � and example problems used to introduce new math concepts ���������������������������������������������� � � ������������������������������������������������ often include various representations of information. Different representations of ������������������������������������������������ the same idea help you���������������������������������������������� fully understand the material. As you study, take �note of � the tables, lists, graphs,������������������������������������������������������ diagrams, symbols, and/or words used to clarify a concept. �� � � � ��������������������������������������������������� From Lesson 3-2 ���������������������������������������� �� �� ������� � � �������������������������������������� ��������������������������������������������������� ����������� ������� � � �� �� � � ��������� ������������������������������������������� ������������������������������������������������������ Symbols ��������������������������������������������������������� � ������������������� ������������������� ������������������� ��������������������� ������������������ �������������������� ���������� ��������� ����������������� ������������������������������������������������ �� ����������� ������������������� �� ������ �� ����� ��������������������������������������������������������������������� ������������ ���������� �������������� ����������������� ������ ����������������������������������������� �������������������������������������������� Table Graph ��� ���������� �������������� Try This Describe two representations you could use to solve each problem. 1. A triangle with coordinates A(3, 5), B(2, 2), and C(3, -2) is translated 3 units left and 2 units up. Give the coordinates of the image. 2. A bottle of juice from a vending machine costs $1.50. Hiroshi buys a bottle by inserting 8 coins in quarters and dimes. If Hiroshi receives 5 cents in change, how many quarters did he use? how many dimes? 3. What is the slope of the line that passes through the point (6, 9) and has a y-intercept of 3? Quadratic Functions 313 5-1 Explore Parameter Changes You can use a graphing calculator to explore how changes in the parameters of a quadratic function affect its graph. Recall from Lesson 1-9 that the quadratic parent function is f (x) = x 2 and that its graph is a parabola. Use with Lesson 5-1 KEYWORD: MB7 Lab5 Activity Describe what happens when you change the value of k in the quadratic function g(x) = x 2 + k. 1 Choose three values for k. Use 0, -5 (a negative value), and 4 (a positive value). Press , and enter X2 for Y1, X2 - 5 for Y2, and X2 + 4 for Y3. 2 Change the style of the graphs of Y1 and Y2 so that you can tell which graph represents which function. To do this, move the cursor to the graph style indicator next to Y1. Press to cycle through the options. For Y1, which represents the parent function, choose the thick line. ������������ ��������� 3 Next, change the line style for Y2 to the dotted line. 4 Graph the functions in the square window by pressing and choosing 5 : ZSquare. Notice that the graphs are identical except that the graph of Y2 is shifted 5 units down and the graph of Y3 has been shifted 4 units up from the graph of Y1. You can conclude that the parameter k in the function g(x) = x 2 + k has the effect of translating the parent function f (x) = x 2 k units up if k is positive and ⎜k⎟ units down if k is negative. Try This Use your graphing calculator to compare the graph of each function to the graph of f (x) = x 2. Describe how the graphs differ. 1. g(x) = (x - 4 )2 2. g(x) = (x + 3)2 3. g(x) = -x 2 4. Make a Conjecture Use your graphing calculator to determine what happens 2 when you change the value of h in the quadratic function g(x) = (x - h ) . Check both positive and negative values of h. 5. Make a Conjecture Use your graphing calculator to determine what happens when you change the value of a in the quadratic function g(x) = ax 2. Check values of a that are greater than 1 and values of a that are between 0 and 1. 314 Chapter 5 Quadratic Functions 5-1 */ Using Transformations to Graph Quadratic Functions A2.3.3 andgraphs sketch of quadratic functions including the lines IA-2.8 Carry outdescribe, a the procedure to graph transformations of parent functions 2.1a Analyze, Recognize parent ofgraphs polynomial, exponential, radical, quadratic, and logarithmic 2 of symmetry. (including y = predict x, y = xthe , and y = ⎪xof⎥).transformations on the parent graphs, using various . . . functions and effects Why learn this? You can use transformations of quadratic functions to analyze changes in braking distance. (See Example 5.) Objectives Transform quadratic functions. Describe the effects of changes in the coefficients of 2 y = a(x - h) + k. Vocabulary quadratic function parabola vertex of a parabola vertex form In Chapters 2 and 3, you studied linear functions of the form f (x) = mx + b. A quadratic function is a function that can be written in the form 2 f (x) = a(x - h) + k (a ≠ 0). In a quadratic function, the variable is always squared. The table shows the linear and quadratic parent functions. Linear and Quadratic Parent Functions ALGEBRA NUMBERS GRAPH Linear Parent Function f(x) = x . . . methods and tools which may include graphing calculators. x -2 -1 0 1 2 f(x) = x -2 -1 0 1 2 x -2 -1 0 1 2 4 1 0 1 4 Quadratic Parent Function f(x) = x 2 f(x) = x 2 Notice that the graph of the parent function f (x) = x 2 is a U-shaped curve called a parabola . As with other functions, you can graph a quadratic function by plotting points with coordinates that make the equation true. EXAMPLE 1 Graphing Quadratic Functions Using a Table Graph f (x) = x 2 - 6x + 8 by using a table. Make a table. Plot enough ordered pairs to see both sides of the curve. x f(x) = x 2 - 6x + 8 (x, f(x)) 1 f(1) = 1 2 - 6(1) + 8 = 3 2 f(2) = 2 2 - 6(2) + 8 = 0 3 f(3) = 3 2 - 6(3) + 8 = -1 4 f(4) = 4 2 - 6(4) + 8 = 0 5 f(5) = 5 2 - 6(5) + 8 = 3 (1, 3) (2, 0) (3, -1) (4, 0) (5, 3) Þ È { £]ÊÎ® x]ÊÎ® Ó Ó]Êä® {]Êä® ä Î]Ê£® Ý È 5-1 Using Transformations to Graph Quadratic Functions 315 1. Graph g(x) = -x 2 + 6x - 8 by using a table. You can also graph quadratic functions by applying transformations to the parent function f (x) = x 2. Transforming quadratic functions is similar to transforming linear functions (Lesson 2-6). Translations of Quadratic Functions Horizontal Translations Vertical Translations Horizontal Shift of ⎪h⎥ Units Vertical Shift of ⎪k⎥ Units f(x) = x 2 f (x - h ) = (x - h ) EXAMPLE 2 f(x) = x 2 2 f(x) + k = x 2 + k Moves left for h<0 Moves down for k<0 Moves right for h>0 Moves up for k > 0 Translating Quadratic Functions Using the graph of f (x) = x 2 as a guide, describe the transformations, and then graph each function. A g(x) = (x + 3) 2 + 1 B g(x) = (x - 2) 2 - 1 Identify h and k. Identify h and k. g(x) = (x -(-3)) + 1 2 h k Because h = -3, the graph is translated 3 units left. Because k = 1, the graph is translated 1 unit up. Therefore, g is f translated 3 units left and 1 unit up. } h k Because h = 2, the graph is translated 2 units right. Because k = -1, the graph is translated 1 unit down. Therefore, g is f translated 2 units right and 1 unit down. Þ Þ v { { Ó Ó Î]Ê£® { g(x) = (x - 2) 2 + (-1) v } Ý Ý Ó ä Ó Ó ä Ó]Ê£® { Using the graph of f (x) = x 2 as a guide, describe the transformations, and then graph each function. 2a. g(x) = x 2 - 5 2b. g(x) = (x + 3) 2 - 2 Recall that functions can also be reflected, stretched, or compressed. 316 Chapter 5 Quadratic Functions Reflections, Stretches, and Compressions of Quadratic Functions Reflections Reflection Across x-axis Reflection Across y-axis Input values change. Output values change. f(x) = x 2 f(x) = x 2 -f(x) = -(x 2) = -x 2 f(-x) = (-x) 2 = x 2 The function f(x) = x 2 is its own reflection across the y-axis. The function is flipped across the x-axis. Stretches and Compressions Horizontal Stretch/Compression by a Factor of b Vertical Stretch/Compression by a Factor of a Output values change. Input values change. f(x) = x 2 f 1x = 1x b b f(x) = x 2 (_ ) (_ ) EXAMPLE 2 a · f(x) = ax 2 b > 1 stretches away from the y-axis. a > 1 stretches away from the x-axis. 0 < b < 1 compresses toward the y-axis. 0 < a < 1 compresses toward the x-axis. 3 Reflecting, Stretching, and Compressing Quadratic Functions Using the graph of f (x) = x 2 as a guide, describe the transformations, and then graph each function. 2 A g (x) = -4x 2 B g(x) = 1 x 2 Because a is negative, g is a Because b = 2, g is a reflection of f across the x-axis. horizontal stretch of Because a = 4, g is a vertical f by a factor of 2. stretch of f by a factor of 4. (_ ) { Þ v { Þ v } Ó Ó Ý Ý { Ó Ó { { Ó ä Ó { Ó { } { Using the graph of f (x) = x 2 as a guide, describe the transformations, and then graph each function. 1 x2 3b. g(x) = -_ 3a. g (x) = (2x) 2 2 5-1 Using Transformations to Graph Quadratic Functions 317 If a parabola opens upward, it has a lowest point. If a parabola opens downward, it has a highest point. This lowest or highest point is the vertex of a parabola . The parent function f (x) = x 2 has its vertex at the origin. You can identify the vertex of other quadratic functions by analyzing the function in vertex form. 2 The vertex form of a quadratic function is f (x) = a(x - h) + k, where a, h, and k are constants. Vertex Form of a Quadratic Function a indicates a reflection across the x-axis and/or a vertical stretch or compression. h indicates a horizontal translation. k indicates a vertical translation. Because the vertex is translated h horizontal units and k vertical units from the origin, the vertex of the parabola is at (h, k ). EXAMPLE 4 Writing Transformed Quadratic Functions Use the description to write the quadratic function in vertex form. The parent function f (x) = x 2 is reflected across the x-axis, vertically stretched by a factor of 6, and translated 3 units left to create g. Step 1 Identify how each transformation affects the constants in vertex form. reflection across x-axis: ProjectbyTitle vertical stretch 6: When the quadratic parent function f(x) = x 2 is written in vertex form, y = a(x - h) 2 + k, a = 1, h = 0, and k = 0. a is negative ⎫ ⎬ a = -6 ⎭ ⎜a⎟ = 6 Algebra II 2007 Student translation left 3 units: h = -3Edition Spec Number Step 2 Write the transformed function. A207Se2 c05l01002 g(x) = a(x - h) + k Vertex form of a quadratic function Created By2 = -6 ) + 0 Substitute -6 for a, -3 for h, and 0 for k. (x -(-3)Corporation Krosscore Creation (x + 3) 2 Date = -6 Simplify. 01/26/2005 Check Graph both functions on a graphing calculator. Enter f as Y1 and g as Y2. The graph indicates the identified transformations. � � Use the description to write the quadratic function in vertex form. 4a. The parent function f (x) = x 2 is vertically compressed by a factor of __13 and translated 2 units right and 4 units down to create g. 4b. The parent function f (x) = x 2 is reflected across the x-axis and translated 5 units left and 1 unit up to create g. 318 Chapter 5 Quadratic Functions EXAMPLE 5 Automotive Application The minimum braking distance d in feet for a vehicle on dry concrete is approximated by the function d(v) = 0.045v 2, where v is the vehicle’s speed in miles per hour. If the vehicle’s tires are in poor condition, the braking-distance function is d p(v) = 0.068v 2. What kind of transformation describes this change, and what does the transformation mean? Examine both functions in vertex form. d(v) = 0.045(v - 0) 2 + 0 d p(v) = 0.068(v - 0) 2 + 0 The value of a has increased from 0.045 to 0.068. The increase indicates a vertical stretch. Find the stretch factor by comparing the new a-value to the old a-value: a from d p(v) _ __ = 0.068 ≈ 1.5 0.045 a from d(v) The function d p represents a vertical stretch of d by a factor of approximately 1.5. Because the value of each function approximates braking distance, a vehicle with tires in poor condition takes about 1.5 times as many feet to stop as a vehicle with good tires does. Check Graph both functions on a graphing calculator. The graph of d p appears to be vertically stretched compared with the graph of d. £ä `« ` ä £x ä Use the information above to answer the following. 5. The minimum braking distance d n in feet for a vehicle with 2 new tires at optimal inflation is d n(v) = 0.039v , where v is the vehicle’s speed in miles per hour. What kind of transformation describes this change from d(v) = 0.045v 2, and what does this transformation mean? THINK AND DISCUSS 1. Explain how the values of a, h, and k in the vertex form of a quadratic function affect the function’s graph. 2. Explain how to determine which of two quadratic functions expressed in vertex form has a narrower graph. 3. GET ORGANIZED Copy and complete the graphic organizer. In each row, write an equation that represents the indicated transformation of the quadratic parent function, and show its graph. /À>ÃvÀ>Ì µÕ>Ì À>« 6iÀÌV>ÊÌÀ>Ã>Ì ÀâÌ>ÊÌÀ>Ã>Ì ,iviVÌ 6iÀÌV>ÊÃÌÀiÌV 6iÀÌV>ÊV«ÀiÃÃ 5-1 Using Transformations to Graph Quadratic Functions 319 5-1 Exercises KEYWORD: MB7 5-1 KEYWORD: MB7 Parent GUIDED PRACTICE 1. Vocabulary The highest or lowest point on the graph of a quadratic function is the ? . (vertex or parabola) −−− SEE EXAMPLE 1 2. f (x) = -2x 2 - 4 p. 315 SEE EXAMPLE Graph each function by using a table. 2 p. 316 3 p. 317 8. g(x) = 3x 2 11. h(x) = -(5x) 2 SEE EXAMPLE 4 4. h(x) = x 2 + 2x Using the graph of f(x) = x 2 as a guide, describe the transformations, and then graph each function. 5. d(x) = (x - 4) 2 SEE EXAMPLE 3. g(x) = -x 2 + 3x - 2 6. g(x) = (x - 3) 2 + 2 ( ) 1x 2 9. h(x) = _ 8 12. g(x) = 4.2x 2 7. h(x) = (x + 1) 2 - 3 10. p(x) = 0.25x 2 2 x2 13. d(x) = -_ 3 Use the description to write each quadratic function in vertex form. 14. The parent function f (x) = x 2 is vertically stretched by a factor of 2 and translated 3 units left to create g. p. 318 15. The parent function f (x) = x 2 is reflected across the x-axis and translated 6 units down to create h. SEE EXAMPLE 5 p. 319 16. Physics The safe working load L in pounds for a natural rope can be estimated by L(r) = 5920r 2, where r is the radius of the rope in inches. For an old rope, the function L o(r) = 4150r 2 is used to estimate its safe working load. What kind of transformation describes this change, and what does this transformation mean? PRACTICE AND PROBLEM SOLVING Independent Practice For See Exercises Example 17–19 20–25 26–28 29–30 31 1 2 3 4 5 Extra Practice Skills Practice p. S12 Application Practice p. S36 Graph each function by using a table. 17. f (x) = -x 2 + 4 18. g(x) = x 2 - 2x + 1 19. h(x) = 2x 2 + 4x - 1 Using the graph of f(x) = x 2 as a guide, describe the transformations, and then graph each function. 20. g(x) = x 2 - 2 21. h(x) = (x + 5) 2 22. j(x) = (x - 1) 2 23. g(x) = (x + 4) 2 - 3 4 x2 26. g(x) = _ 7 24. h(x) = (x + 2) 2 + 2 25. j(x) = (x - 4) 2 - 9 1x 2 28. j(x) = _ 3 27. h(x) = -20x 2 ( ) Use the description to write each quadratic function in vertex form. 29. The parent function f (x) = x 2 is reflected across the x-axis, vertically compressed by a factor of __12 , and translated 1 unit right to create g. 30. The parent function f (x) = x 2 is vertically stretched by a factor of 2.5 and translated 2 units left and 1 unit up to create h. 31. Consumer Economics The average gas mileage m in miles per gallon for a compact car is modeled by m(s) = -0.015(s - 47) 2 + 33, where s is the car’s speed in miles per hour. The average gas mileage for an SUV is modeled by m u(s) = -0.015(s - 47) 2 + 15. What kind of transformation describes this change, and what does this transformation mean? 320 Chapter 5 Quadratic Functions Using f (x) = x 2 as a guide, describe the transformations for each function. 33. p(x) = -(x - 4) 2 34. g(x) = 8(x + 2) 2 35. h(x) = 4x 2 - 2 1 x2 + 2 36. p(x) = _ 4 1x 2 38. h(x) = - _ 3 37. g(x) = (3x) 2 + 1 ( ) Ài>ÊvÌÓ® 32. Pets Keille is building a rectangular pen for a pet rabbit. She can buy wire fencing in a roll of 40 ft or a roll of 80 ft. The graph shows the area of pens she can build with each type of roll. a. Describe the function for an 80 ft roll of fencing as a transformation of the function for a 40 ft roll *ÃÃLiÊÀi>ÊvÊ*i of fencing. b. Is the largest pen Keille can build with an 80 ft roll {ää of fencing twice as large as the largest pen she can Îxä build with a 40 ft roll of fencing? Explain. Îää Óxä Óää £xä £ää xä ä £ä Óä Îä {ä 7`Ì ÊvÌ® {äÊvÌÊÀ näÊvÌÊÀ Match each graph with one of the following functions. A. a(x) = 4(x + 8) 2 - 3 39. Î]Ên® n 40. Þ { n { { { ä £]Êä® Ý ä B. b(x) = -2(x - 8) 2 + 3 Þ n]ÊÎ® ]Ê£® { n 1 (x + 3) 2 + 8 C. c(x) = -_ 2 41. Þ n Ý { Ç]Ê£® £Ó { { n 42. Geometry The area A of the circle in the figure can be represented by A(r) = πr 2, where r is the radius. a. Write a function B in terms of r that represents the area of the shaded portion of the figure. b. Describe B as a transformation of A. c. What are the reasonable domain and range for each function? Explain. £Ó { n]ÊÎ® À Ý ä { x x 43. Critical Thinking What type of graph would a function of the form 2 f (x) = a(x - h) + k have if a = 0? What type of function would it be? 44. Write About It Describe the graph of f (x) = 999,999(x + 5) 2 + 5 without graphing it. 45. This problem will prepare you for the Multi-Step Test Prep on page 364. The height h in feet of a baseball on Earth after t seconds can be modeled by the function h(t) = -16(t - 1.5) 2 + 36, where -16 is a constant in ft/s 2 due to Earth’s gravity. a. What if...? The gravity on Mars is only 0.38 times that on Earth. If the same baseball were thrown on Mars, it would reach a maximum height 59 feet higher and 2.5 seconds later than on Earth. Describe the transformations that must be applied to make the function model the height of the baseball on Mars. b. Write a height function for the baseball thrown on Mars. 5-1 Using Transformations to Graph Quadratic Functions 321 Use the graph for Exercises 46 and 47. 46. Which best describes how the graph of the function y = -x 2 was transformed to produce the graph shown? Þ Ó Translation 2 units right and 2 units up Translation 2 units right and 2 units down Translation 2 units left and 2 units up Translation 2 units left and 2 units down ä Ý Ó { Ó]ÊÓ® Ó { È ä]ÊÈ® n 47. Which gives the function rule for the parabola shown? f(x) = (x + 2) 2 - 2 f(x) = (x - 2) 2 - 2 f(x) = -(x - 2) 2 - 2 f(x) = -(x + 2) 2 - 2 48. Which shows the functions below in order from widest to narrowest of their corresponding graphs? 1 x2 1 x2 m(x) = _ n(x) = 4x 2 p(x) = 6x 2 q(x) = -_ 6 2 m, n, p, q m, q, n, p q, p, n, m q, m, n, p 49. Which of the following functions has its vertex below the x-axis? f(x) = (x - 7) 2 f(x) = -2x 2 f(x) = x 2 - 8 f(x) = -(x + 3) 2 50. Gridded Response What is the y-coordinate of the vertex of the graph of f(x) = -3(x - 1) 2 + 5? CHALLENGE AND EXTEND 51. Identify the transformations of the graph of f (x) = -3(x + 3) 2 - 3 that would cause the graph’s image to have a vertex at (3, 3). Then write the transformed function. 52. Consider the functions f (x) = (2x) 2 - 2 and g(x) = 4x 2 - 2. a. Describe each function as a transformation of the quadratic parent function. b. Graph both functions on the coordinate plane. c. Make a conjecture about the relationship between the two functions. d. Write the rule for a horizontal compression of the parent function that would give the same graph as f (x) = 9x 2. SPIRAL REVIEW 53. Packaging Peanuts are packaged in cylindrical containers. A small container is 7 in. tall and has a radius of 2 in. A large container is 5.5 in. tall and has a radius twice that of the small container. The price of the large container is three times the price of the small container. Is this price justified? Explain. (Previous course) Identify the parent function for g from its function rule. (Lesson 1-9) 54. g(x) = 4x + √3 55. g(x) = 3 √ x+4 Write each function in slope-intercept form. Then graph the function. (Lesson 2-3) 1 y + 4 = -1 56. 2y + 5x = 14 57. x - _ 2 322 Chapter 5 Quadratic Functions 5-2 */ Properties of Quadratic Functions in Standard Form A2.3.3 describe, function and sketch ofthe quadratic functions including 2.3b Analyze, Graph a quadratic andgraphs identify x- and y-intercepts and the lines of symmetry. maximum or minimum value, using various methods and tools . . . Why learn this? Quadratic functions can be used to find the maximum power generated by the engine of a speedboat. (See Example 4.) Objectives Define, identify, and graph quadratic functions. Identify and use maximums and minimums of quadratic functions to solve problems. Vocabulary axis of symmetry standard form minimum value maximum value When you transformed quadratic functions in the previous lesson, you saw that reflecting the parent function across the y-axis results in the same function. f (x) = x 2 g(x) = (-x) 2 = x 2 . . . which may include a graphing calculator. This shows that parabolas are symmetric curves. The axis of symmetry is the line through the vertex of a parabola that divides the parabola into two congruent halves. Axis of Symmetry Quadratic Functions WORDS ALGEBRA GRAPH The axis of symmetry is a The quadratic function 2 vertical line through the f(x) = a(x - h) + k has the vertex of the function’s axis of symmetry x = h. graph. ]Ê® EXAMPLE 1 Identifying the Axis of Symmetry Identify the axis of symmetry for the graph of f (x) = 2(x + 2) 2 - 3. Rewrite the function to find the value of h. f (x) = 2⎡⎣x - (-2)⎦⎤2 - 3 Because h = -2, the axis of symmetry is the vertical line x = -2. Check Analyze the graph on a graphing calculator. The parabola is symmetric about the vertical line x = -2. 1. Identify the axis of symmetry for the graph of f (x) = (x - 3) 2 + 1. 5- 2 Properties of Quadratic Functions in Standard Form 323 Another useful form of writing quadratic functions is the standard form. The standard form of a quadratic function is f (x) = ax 2 + bx + c, where a ≠ 0. The coefficients a, b, and c can show properties of the graph of the function. You can determine these properties by expanding the vertex form. f (x) = a(x - h) + k 2 f (x) = a(x 2 - 2xh + h 2) + k Multiply to expand (x - h) . f (x) = a(x 2) - a(2hx) + a(h 2) + k Distribute a. f (x) = ax 2 + (-2ah)x + (ah 2 + k) Simplify and group like terms. a=a 2 -2ah = b ah 2 + k = c a in standard form is the same as in vertex form. It indicates a = a whether a reflection and/or vertical stretch or compression has been applied. b = -2ah b b Solving for h gives h = ____ = - ___ . Therefore, the axis of -2a 2a symmetry, x = h, for a quadratic function in standard form is x = -__ b . 2a Notice that the value of c is the same value given by the vertex y-intercept. c = ah 2 + k form of f when x = 0: f (0) = a(0 - h) 2 + k = ah 2 + k. So c is the These properties can be generalized to help you graph quadratic functions. Properties of a Parabola For f(x) = ax 2 + bx + c, where a, b, and c are real numbers and a ≠ 0, the parabola has these properties: The parabola opens upward if a > 0 and downward if a < 0. ÝÃÊvÊÃÞiÌÀÞ Þ _ b The axis of symmetry is the vertical line x = - . 2a b b . The vertex is the point - , f 2a 2a The y-intercept is c. ( EXAMPLE 2 _ ( _)) Graphing Quadratic Functions in Standard Form A Consider the function f (x) = x 2 - 4x + 6. a. Determine whether the graph opens upward or downward. Because a is positive, the parabola opens upward. b. Find the axis of symmetry. b. The axis of symmetry is given by x = - _ 2a (-4) x = - _ = 2 Substitute -4 for b and 1 for a. 2(1) The axis of symmetry is the line x = 2. 324 Chapter 5 Quadratic Functions Ý c. Find the vertex. The vertex lies on the axis of symmetry, so the x-coordinate is 2. The y-coordinate is the value of the function at this x-value, or f (2). f (2) = (2)2 - 4(2) + 6 = 2 The vertex is (2, 2). d. Find the y-intercept. Because c = 6, the y-intercept is 6. e. Graph the function. Graph by sketching the axis of symmetry and then plotting the vertex and the intercept point, (0, 6). Use the axis of symmetry to find another point on the parabola. Notice that (0, 6) is 2 units left of the axis of symmetry. The point on the parabola symmetrical to (0, 6) is 2 units right of the axis at (4, 6). ÝÊÊÓ Þ ä]ÊÈ® {]ÊÈ® { Ó Ó]ÊÓ® Ý ä { È B Consider the function f (x) = -4x 2 - 12x - 3. a. Determine whether the graph opens upward or downward. Because a is negative, the parabola opens downward. When a is positive, the parabola is happy ( ). When a is negative, the parabola is sad ( ). b. Find the axis of symmetry. b. The axis of symmetry is given by x = -_ 2a (-12) 3 Substitute -12 for b and -4 for a. x = -_ = -_ 2 2(-4) 3 , or x = -1.5. The axis of symmetry is the line x = -_ 2 c. Find the vertex. The vertex lies on the axis of symmetry, so the x-coordinate is -1.5. The y-coordinate is the value of the function at this x-value, or f (-1.5). f (-1.5) = -4(-1.5)2 - 12(-1.5) - 3 = 6 The vertex is (-1.5, 6). d. Find the y-intercept. Because c = -3, the y-intercept is -3. e. Graph the function. Graph by sketching the axis of symmetry and then plotting the vertex and the intercept point, (0, -3). Use the axis of symmetry to find another point on the parabola. Notice that (0, -3) is 1.5 units right of the axis of symmetry. The point on the parabola symmetrical to (0, -3) is 1.5 units left of the axis at (-3, -3). ÝÊÊ£°x £°x]ÊÈ® È Þ Ý { Î]ÊÎ® Ó Ó ä]ÊÎ® For each function, (a) determine whether the graph opens upward or downward, (b) find the axis of symmetry, (c) find the vertex, (d) find the y-intercept, and (e) graph the function. 2a. f (x) = -2x 2 - 4x 2b. g(x) = x 2 + 3x - 1 5- 2 Properties of Quadratic Functions in Standard Form 325 Substituting any real value of x into a quadratic equation results in a real number. Therefore, the domain of any quadratic function is all real numbers, . The range of a quadratic function depends on its vertex and the direction that the parabola opens. Minimum and Maximum Values OPENS UPWARD OPENS DOWNWARD When a parabola opens downward, the y-value of the vertex is the maximum value. When a parabola opens upward, the y-value of the vertex is the minimum value. R: y|y ≥ k R: y|y ≤ k D: x|x D: x|x ]Ê® ]Ê® The domain is all real numbers, . The The domain is all real numbers, . The range is all values greater than or equal range is all values less than or equal to the maximum. to the minimum. EXAMPLE 3 Finding Minimum or Maximum Values Find the minimum or maximum value of f (x) = 2x 2 - 2x + 5. Then state the domain and range of the function. Step 1 Determine whether the function has a minimum or maximum value. Because a is positive, the graph opens upward and has a minimum value. Step 2 Find the x-value of the vertex. (-2) _ b = -_ 1 x = -_ = 2 =_ 4 2 2a 2(2) The minimum (or maximum) value is the y-value of the vertex. It is not the ordered pair representing the vertex. Substitute -2 for b and 2 for a. ( ) b . Step 3 Then find the y-value of the vertex, f -_ 2a 2 1 f 1 = 2 1 - 2 1 + 5 = 4_ 2 2 2 2 (_) (_) (_) The minimum value is 4__12 , or 4.5. The domain is all real numbers, . The range is all real numbers greater than or equal to 4.5, or y | y ≥ 4.5. Check Graph f (x) = 2x 2 - 2x + 5 on a graphing calculator. The graph and table support the answer. n Î Î Î Find the minimum or maximum value of each function. Then state the domain and range of the function. 3a. f (x) = x 2 - 6x + 3 3b. g(x) = -2x 2 - 4 326 Chapter 5 Quadratic Functions EXAMPLE 4 Transportation Application Steering wheel The power p in horsepower (hp) generated by a high-performance speedboat engine operating at r revolutions per minute (rpm) can be modeled by the function p(r) = -0.0000147r 2 + 0.18r - 251. What is the maximum power of this engine to the nearest horsepower? At how many revolutions per minute must the engine be operating to achieve this power? Hull Engine Propeller The maximum value will be at the vertex (r, p(r)). Step 1 Find the r-value of the vertex using a = -0.0000147 and b = 0.18. b = -__ 0.18 r = -_ ≈ 6122 2a 2(-0.0000147) Step 2 Substitute this r-value into p to find the corresponding maximum, p(r). p(r) = -0.0000147r 2 + 0.18r - 251 p(6122) = -0.0000147(6122) 2 + 0.18(6122) - 251 Substitute 6122 for r. p(6122) ≈ 300 Use a calculator. The maximum power is about 300 hp at 6122 rpm. Check Graph the function on a graphing calculator. Use the maximum feature under the CALCULATE menu to approximate the maximum. The graph supports your answer. Îxä {äää Óää näää 4. The highway mileage m in miles per gallon for a compact car is approximated by m(s) = -0.025s 2 + 2.45s - 30, where s is the speed in miles per hour. What is the maximum mileage for this compact car to the nearest tenth of a mile per gallon? What speed results in this mileage? THINK AND DISCUSS 1. Explain whether a quadratic function can have both a maximum value and a minimum value. 2. Explain why the value of f (x) = x 2 + 2x - 1 increases as the value of x decreases from -1 to -10. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write the criteria or equation to find each property of the parabola for f (x) = ax 2 + bx + c. "«iÃÊÕ«Ü>À`ÊÀ `ÜÜ>À` ÝÃÊvÊÃÞiÌÀÞ *À«iÀÌiÃ vÊ*>À>L>Ã ÞÌiÀVi«Ì 6iÀÌiÝ 5- 2 Properties of Quadratic Functions in Standard Form 327 5-2 Exercises KEYWORD: MB7 5-2 KEYWORD: MB7 Parent GUIDED PRACTICE 1. Vocabulary If the graph of a quadratic function opens upward, the y-value of the vertex is a ? value. (maximum or minimum) −−− SEE EXAMPLE 1 2. f (x) = -2(x - 2) 2 - 4 p. 323 SEE EXAMPLE Identify the axis of symmetry for the graph of each function. 2 p. 324 For each function, (a) determine whether the graph opens upward or downward, (b) find the axis of symmetry, (c) find the vertex, (d) find the y-intercept, and (e) graph the function. 5. f (x) = -x 2 - 2x - 8 SEE EXAMPLE 3 p. 326 p. 327 6. g(x) = x 2 - 3x + 2 7. h(x) = 4x - x 2 - 1 Find the minimum or maximum value of each function. Then state the domain and range of the function. 8. f (x) = x 2 - 1 SEE EXAMPLE 4 4. h(x) = (x + 5) 2 3. g(x) = 3x 2 + 4 9. g(x) = -x2 + 3x - 2 10. h(x) = -16x 2 + 32x + 4 11. Sports The path of a soccer ball is modeled by the function h(x) = -0.005x 2 + 0.25x, where h is the height in meters and x is the horizontal distance that the ball travels in meters. What is the maximum height that the ball reaches? PRACTICE AND PROBLEM SOLVING Independent Practice For See Exercises Example 12–14 15–23 24–29 30 1 2 3 4 Extra Practice Skills Practice p. S12 Application Practice p. S36 Identify the axis of symmetry for the graph of each function. 12. f (x) = -x 2 + 4 14. h(x) = 2(x + 1)2 - 3 13. g(x) = (x - 1) 2 For each function, (a) determine whether the graph opens upward or downward, (b) find the axis of symmetry, (c) find the vertex, (d) find the y-intercept, and (e) graph the function. 15. f (x) = x 2 + x - 2 16. g(x) = -3x 2 + 6x 17. h(x) = 0.5x 2 - 2x - 4 18. f (x) = -2x 2 + 8x + 5 19. g(x) = 3x 2 + 2x - 8 21. f (x) = -(2 + x 2) 22. g(x) = 0.5x 2 + 3x - 5 20. h(x) = 2x - 1 + x 2 1 x2 + x + 2 23. h(x) = _ 4 Find the minimum or maximum value of each function. Then state the domain and range of the function. 25. g(x) = 6x - x 2 26. h(x) = x 2 - 4x + 3 24. f (x) = -2x 2 + 7x - 3 1 x2 - 4 27. f (x) = -_ 28. g(x) = -x 2 - 6x + 1 29. h(x) = x 2 + 8x + 16 2 30. Weather The daily high temperature in Death Valley, California, in 2003 can be modeled by T (d) = -0.0018d 2 + 0.657d + 50.95, where T is temperature in degrees Fahrenheit and d is the day of the year. What was the maximum temperature in 2003 to the nearest degree? 31. Sports The height of a golf ball over time can be represented by a quadratic function. Graph the data in the table. What is the maximum height that the ball will reach? Explain your answer in terms of the axis of symmetry and vertex of the graph. 328 Chapter 5 Quadratic Functions Golf Ball Height Time (s) 0 0.5 1 2 3 Height (ft) 0 28 48 64 48 Biology Spittlebugs are insects that feed on the sap of plants. Young spittlebug larvae use sap to produce a layer of bubbles around themselves. The bubbles help to keep the larvae from drying out and may protect them from predators. 32. Manufacturing A roll of aluminum with a width of 32 cm is to be bent into rain gutters by folding up two sides at 90° angles. A rain gutter’s greatest capacity, or volume, is determined by the gutter’s greatest cross-sectional area, as shown. a. Write a function C to describe (32 – 2x ) cm x cm the cross-sectional area in terms of the width of the bend x. b. Make a table, and graph the x cm function. Cross-sectional c. Identify the meaningful domain area and range of the function. 32 cm d. Find the value of x that maximizes the cross-sectional area. 33. Biology The spittlebug is the world’s highest jumping animal relative to its body length of about 6 mm. The height h of a spittlebug’s jump in millimeters can be modeled by the function h(t) = -4000t 2 + 3000t, where t is the time in seconds. a. What is the maximum height that the spittlebug will reach? b. What is the ratio of a spittlebug’s maximum jumping height to its body length? In the best human jumpers, this ratio is about 1.38. Compare the ratio for spittlebugs with the ratio for the best human jumpers. c. What if...? Suppose humans had the same ratio of maximum jumping height to body length as spittlebugs. How high would a person with a height of 1.8 m be able to jump? 34. Gardening The function A(x) = x(10 - x) describes the area A of a rectangular flower garden, where x is its width in yards. What is the maximum area of the garden? Graphing Calculator Once you have graphed a function, the graphing calculator can automatically find the minimum or maximum value. From the CALC menu, choose the minimum or maximum feature. Use a graphing calculator to find the approximate minimum or maximum value of each function. 35. f (x) = 5.23x 2 - 4.84x - 1.91 36. g(x) = -12.8x 2 + 8.73x + 11.69 5 x2 + _ 9 x+_ 1 x2 - _ 4x + _ 2 21 37. h(x) = _ 38. j(x) = -_ 5 4 3 12 3 10 39. Critical Thinking Suppose you are given a parabola with two points that have the same y-value, such as (-7, 11) and (3, 11). Explain how to find the equation for the axis of symmetry of this parabola, and then determine this equation. 40. Write About It Can a maximum value for a quadratic function be negative? Can a minimum value for a quadratic function be positive? Explain by using examples. 41. This problem will prepare you for the Multi-Step Test Prep on page 364. A baseball is thrown with a vertical velocity of 50 ft/s from an initial height of 6 ft. The height h in feet of the baseball can be modeled by h(t) = -16t 2 + 50t + 6, where t is the time in seconds since the ball was thrown. a. Approximately how many seconds does it take the ball to reach its maximum height? b. What is the maximum height that the ball reaches? 5- 2 Properties of Quadratic Functions in Standard Form 329 Use the graph for exercises 42 and 43. 42. What is the range of the function graphed? All real numbers y ≥ -2 { y≤2 -2 ≤ y ≤ 2 Þ Ó Ý 43. The graph shown represents which quadratic function? f(x) = x 2 + 2x - 2 { ä Ó Ó Ó f(x) = -x 2 + 4x - 2 f(x) = x 2 - 4x - 2 f(x) = -x 2 - 2x + 2 44. Which of the following is NOT true of the graph of the function f(x) = -x 2 - 6x + 5? Its vertex is at (-3, 14). Its maximum value is 14. Its axis of symmetry is x = 14. Its y-intercept is 5. 45. Which equation represents the axis of symmetry for f(x) = 2x 2 - 4x + 5? x = -4 x=5 x=1 x=2 46. Short Response Explain how to find the maximum value or minimum value of a quadratic function such as f(x) = -x 2 - 8x + 4. CHALLENGE AND EXTEND 47. Write the equations in standard form for two quadratic functions that have the same vertex but open in different directions. 48. The graph of a quadratic function passes through the point (-5, 8), and its axis of symmetry is x = 3. a. What are the coordinates of another point on the graph of the function? Explain how you determined your answer. b. Can you determine whether the graph of the function opens upward or downward? Explain. 49. Critical Thinking What conclusions can you make about the axis of symmetry and the vertex of a quadratic function of the form f (x) = ax 2 + c? 50. Critical Thinking Given the quadratic function f and the fact that f (-1) = f (2), how can you find the axis of symmetry of this function? SPIRAL REVIEW Simplify each expression. (Lesson 1-3) · √180 51. √40 52. 2 √ 8 · 4 √ 3 (_) 53. √ 54 ÷ √ 30 54. √ 304 For each function, evaluate f (0), f 1 , and f (-2). (Lesson 1-7) 2 1 2 55. f (x) = (x - 3) 2 + 1 56. g(x) = 2 x - _ 2 3 57. f (x) = -4(x + 5) 58. g(x) = x - 4x + 8 ( ) Write the equation of each line with the given properties. (Lesson 2-4) 59. a slope of 3 passing through (1, -4) 61. a slope of -2 passing through (3, 5) 330 Chapter 5 Quadratic Functions 60. passing through (-3, 5) and (-1, -7) 62. passing through (4, 6) and (-2, 1) Factoring Quadratic Expressions Previous Courses Review the methods of factoring quadratic expressions in the examples below. Recall that the standard form of a quadratic expression is ax 2 + bx + c. Examples Factor each expression. 1 x 2 - 3x - 10 Because a = 1, use a table to find the factors of -10 that have a sum of -3. These factors are 2 and -5. Factors of -10 Sum Rewrite the expression as a product of binomial factors with 2 and -5 as constants. -2 and 5 -3 ✘ -1 and 10 -9 ✘ x 2 - 3x - 10 = (x + 2)(x - 5) Check your answer by multiplying. 1 and -10 -9 ✘ 2 and -5 -3 ✔ (x + 2)(x - 5) = x 2 - 5x + 2x - 10 = x 2 - 3x - 10 ✔ 3 -x 2 + 3x + 4 2 6x 2 - 15x Find the greatest common factor (GCF) of the terms. 6x = 2 · 3 · x · x 2 15x = 3 · 5 · x The GCF is 3x. Because a is negative, factor out -1. -x 2 + 3x + 4 = -1(x 2 - 3x - 4) Use the method from Example 1 to factor the expression in parentheses. Factor 3x from both terms. -(x 2 - 3x - 4) = -(x + 1)(x - 4) 6x 2 - 15x = 3x (2x - 5) Check your answer by multiplying. Check your answer by multiplying. 3x(2x - 5) = 3x (2x) - 3x (5) -(x + 1)(x - 4) = -(x 2 - 3x - 4) = -x 2 + 3x + 4 ✔ = 6x 2 - 15x ✔ Try This Factor each expression. 1. 4x 2 + 10x 2. 16x - 2x 2 3. x 2 - 6x + 8 4. x 2 + 4x + 3 5. x 2 - 8x + 15 6. x 2 + 10x - 24 7. x 2 - x - 56 8. x 2 - 6x + 9 9. x 2 + 48x - 100 10. -x 2 + 12x - 32 11. -x 2 + x + 20 12. -x 2 - 14x - 13 13. 4x 2 + 6x 14. x 2 + 14x + 24 15. x 2 - 16 16. 2x 2 - x - 3 17. 3x 2 + 16x + 5 18. 2x 2 - 9x + 7 Connecting Algebra to Previous Courses 331 5-3 Explore Graphs and Factors You can use graphs and linear factors to find the x-intercepts of a parabola. Use with Lesson 5-3 KEYWORD: MB7 Lab5 Activity Graph the lines y = x + 4 and y = x - 2. 1. Press , and enter X + 4 for Y1 and X - 2 for Y2. Graph the and choosing functions in the square window by pressing 5 : ZSquare. {]Êä® 2. Identify the x-intercept of each line. The x-intercepts are -4 and 2. Ó]Êä® 3. Find the x-value halfway between the two x-intercepts. This -4 + 2 x-value is the average of the x-intercepts: _____ = -1. 2 Graph the quadratic function y = (x + 4)(x - 2), which is the product of the two linear factors graphed above. 4. Press and enter (X + 4)(X - 2) for Y3. Press . 5. Identify the x-intercepts of the parabola. The x-intercepts are -4 and 2. Notice that they are the same as those of the two linear factors. 6. Examine the parabola at x = -1 (the x-value that is halfway between the x-intercepts). The axis of symmetry and the vertex of the parabola occur at this x-value. Try This Graph each quadratic function and each of its linear factors. Then identify the x-intercepts and the axis of symmetry of each parabola. 1. y = (x - 2)(x - 6) 2. y = (x + 3)(x - 1) 3. y = (x - 5)(x + 2) 4. y = (x + 4)(x - 4) 5. y = (x - 5)(x - 5) 6. y = (2x - 1)(2x + 3) 7. Critical Thinking Use a graph to determine whether the quadratic function y = 2x 2 + 5x - 12 is the product of the linear factors 2x - 3 and x + 4. 8. Make a Conjecture Make a conjecture about the linear factors, x-intercepts, and axis of symmetry of a quadratic function. 332 Chapter 5 Quadratic Functions 5-3 */ Solving Quadratic Equations by Graphing and Factoring IA-3.3 Carry out a procedure to solve quadratic equationscompleting algebraically A2.1.1 zeros, domain and range of a function. 2.3a Find Solvethe quadratic equations by graphing, factoring, the square (including factoring, completing the square, and applying the quadratic formula). and quadratic formula. Why learn this? You can use quadratic functions to model the height of a football, baseball, or soccer ball. (See Example 3.) Objectives Solve quadratic equations by graphing or factoring. Determine a quadratic function from its roots. Vocabulary zero of a function root of an equation binomial trinomial When a soccer ball is kicked into the air, how long will the ball take to hit the ground? The height h in feet of the ball after t seconds can be modeled by the quadratic function h(t) = -16t 2 + 32t. In this situation, the value of the function represents the height of the soccer ball. When the ball hits the ground, the value of the function is zero. A zero of a function is a value of the input x that makes the output f (x) equal zero. The zeros of a function are the x-intercepts. vÝ®ÊÊ>ÝÓÊ ÊLÝÊ ÊV Unlike linear functions, which have no more than one zero, quadratic functions can have two zeros, as shown at right. These zeros are always symmetric about the axis of symmetry. EXAMPLE 1 / iÊÝVÀ`>ÌiÃ >ÀiÊÌ iÊâiÀÃ° Finding Zeros by Using a Graph or Table Find the zeros of f (x) = x 2 + 2x - 3 by using a graph and table. Method 1 Graph the function f (x) = x 2 + 2x - 3. The graph opens upward because a > 0. The y-intercept is -3 because c = -3. b = -_ 2 = -1 The x-coordinate of the Find the vertex: x = -_ 2a b 2 (1) vertex is -__ . 2a Find f (-1): f (x) = x 2 + 2x - 3 Recall that for the graph of a quadratic function, any pair of points with the same y-value are symmetric about the axis of symmetry. f (-1) = (-1) 2 + 2(-1) - 3 Substitute -1 for x. f (-1) = -4 The vertex is (-1, -4). Plot the vertex and the y-intercept. Use symmetry and a table of values to find additional points. x f(x) -3 -2 0 -3 -1 0 -4 -3 { Þ 1 0 Ý Î]Êä® ä £]Êä® { The table and the graph indicate that the zeros are -3 and 1. 5- 3 Solving Quadratic Equations by Graphing and Factoring 333 Find the zeros of f (x) = x 2 + 2x - 3 by using a graph and table. Method 2 Use a calculator. Enter y = x 2 + 2x - 3 into a graphing calculator. Both the table and the graph show that y = 0 at x = -3 and x = 1. These are the zeros of the function. 1. Find the zeros of g(x) = -x 2 - 2x + 3 by using a graph and a table. • Functions have zeros or x-intercepts. • Equations have solutions or roots. You can also find zeros by using algebra. For example, to find the zeros of f (x) = x 2 + 2x - 3, you can set the function equal to zero. The solutions to the related equation x 2 + 2x - 3 = 0 represent the zeros of the function. The solutions to a quadratic equation of the form ax 2 + bx + c = 0 are roots. The roots of an equation are the values of the variable that make the equation true. You can find the roots of some quadratic equations by factoring and applying the Zero Product Property. Zero Product Property For all real numbers a and b, WORDS If the product of two quantities equals zero, at least one of the quantities equals zero. EXAMPLE 2 NUMBERS ALGEBRA 3(0 ) = 0 If ab = 0, then a = 0 or b = 0. 0 (4) = 0 Finding Zeros by Factoring Find the zeros of each function by factoring. A f (x) = x 2 - 8x + 12 x 2 - 8x + 12 = 0 (x - 2 )(x - 6 ) = 0 x - 2 = 0 or x - 6 = 0 x = 2 or x = 6 Set the function equal to 0. Factor: Find factors of 12 that add to -8. Apply the Zero Product Property. Solve each equation. Check x 2 - 8x + 12 = 0 −−−−−−−−−−−−− (2) 2 - 8(2) + 12 0 4 - 16 + 12 0 0 0✔ 334 Chapter 5 Quadratic Functions 2 x - 8x + 12 = 0 −−−−−−−−−−− −−−−−−−−−−− 2 (6) - 8(6) + 12 0 36 - 48 + 12 0 0 0✔ Substitute each value into the original equation. Find the zeros of each function by factoring. B g(x) = 3x 2 + 12x 3x 2 + 12x = 0 Set the function equal to 0. 3x(x + 4) = 0 Factor: The GCF is 3x. 3x = 0 or x + 4 = 0 x = 0 or x = -4 Apply the Zero Product Property. Solve each equation. n Check Check algebraically and by graphing. 3x 2 + 12x = 0 3x 2 + 12x = 0 −−−−−−−−−−−− −−−−−−−−−−−−−− 2 2 3(0) + 12(0) 0 3(-4) + 12(-4) 0 0+0 0✔ 48 - 48 0 ✔ £x°Ó £x°Ó £Ó Find the zeros of each function by factoring. 2a. f (x) = x 2 - 5x - 6 2b. g(x) = x 2 - 8x Any object that is thrown or launched into the air, such as a baseball, basketball, or soccer ball, is a projectile. The general function that approximates the height h in feet of a projectile on Earth after t seconds is given below. Constant due to Earth's gravity in ft/s2 Initial height in ft (at t 0) Initial vertical velocity in ft/s (at t 0) Note that this model has limitations because it does not account for air resistance, wind, and other real-world factors. EXAMPLE 3 Sports Application A soccer ball is kicked from ground level with an initial vertical velocity of 32 ft/s. After how many seconds will the ball hit the ground? h(t) = -16t 2 + v 0t + h 0 Write the general projectile function. h(t) = -16t + 32t + 0 Substitute 32 for v 0 and 0 for h 0. 2 The ball will hit the ground when its height is zero. -16t 2 + 32t = 0 Set h(t) equal to 0. -16t(t - 2) = 0 Factor: The GCF is -16t. -16t = 0 or (t - 2) = 0 Apply the Zero Product Property. t = 0 or t = 2 Solve each equation. The ball will hit the ground in 2 seconds. Notice that the height is also zero when t = 0, the instant that the ball is kicked. Check The graph of the function h(t) = -16t 2 + 32t shows its zeros at 0 and 2. Óä £ { x 5- 3 Solving Quadratic Equations by Graphing and Factoring 335 3. A football is kicked from ground level with an initial vertical velocity of 48 ft/s. How long is the ball in the air? Quadratic expressions can have one, two, or three terms, such as -16t 2, -16t 2 + 25t, or -16t 2 + 25t + 6. Quadratic expressions with two terms are binomials . Quadratic expressions with three terms are trinomials . Some quadratic expressions with perfect squares have special factoring rules. Special Products and Factors Perfect-Square Trinomial Difference of Two Squares 2 a 2 - 2ab + b 2 = (a - b) a 2 - b 2 = (a + b)(a - b) EXAMPLE 4 a 2 + 2ab + b 2 = (a + b) 2 Finding Roots by Using Special Factors Find the roots of each equation by factoring. A 9x 2 = 1 9x 2 - 1 = 0 Rewrite in standard form. (3x ) - ( 1 ) = 0 2 2 Write the left side as a 2 - b 2. (3x + 1)(3x - 1) = 0 Factor the difference of squares. 3x + 1 = 0 or 3x - 1 = 0 Apply the Zero Product Property. 1 or x = _ 1 x = -_ 3 3 Solve each equation. Check Graph each side of the equation on a graphing calculator. Let Y1 equal 9x 2, and let Y2 equal 1. The graphs appear to intersect at x = -__13 and at x = __13 . 3 -4.5 B 40x = 8x 2 + 50 -3 8x 2 - 40x + 50 = 0 2(4x 2 - 20x + 25) = 0 A quadratic equation can have two roots that are equal, such as x = __52 and x = __52 . Two equal roots are sometimes called a double root. 4x - 20x + 25 = 0 2 (2x) - 2(2x)(5) + (5) = 0 2 2 (2x - 5) 2 = 0 Rewrite in standard form. Factor. The GCF is 2. Divide both sides by 2. Write the left side as a 2 - 2ab + b 2. Factor the perfect-square trinomial: (a - b) . 2x - 5 = 0 or 2x - 5 = 0 Apply the Zero Product Property. 5 5 _ _ x = or x = Solve each equation. 2 2 5 into the original equation. Check Substitute the root _ 2 40x = 8x 2 + 50 −−−−−−−−−−− () () 5 8 _ 5 2 + 50 40 _ 2 2 100 100 ✔ Find the roots of each equation by factoring. 4a. x 2 - 4x = -4 4b. 25x 2 = 9 336 4.5 Chapter 5 Quadratic Functions 2 If you know the zeros of a function, you can work backward to write a rule for the function. EXAMPLE 5 Using Zeros to Write Function Rules Write a quadratic function in standard form with zeros 2 and -1. x = 2 or x = -1 Write the zeros as solutions for two equations. x - 2 = 0 or x + 1 = 0 Rewrite each equation so that it equals 0. (x - 2)(x + 1) = 0 Apply the converse of the Zero Product Property to write a product that equals 0. x2 - x - 2 = 0 Multiply the binomials. f (x) = x - x - 2 2 Replace 0 with f(x). x Check Graph the function f (x) = x 2 - x - 2 on a calculator. The graph shows the original zeros of 2 and -1. Ç°È Ç°È x 5. Write a quadratic function in standard form with zeros 5 and -5. Note that there are many quadratic functions with the same zeros. For example, the functions f (x) = x 2 - x - 2, g(x) = -x 2 + x + 2, and h(x) = 2x 2 - 2x - 4 all have zeros at 2 and -1. x Ç°È Ç°È x THINK AND DISCUSS 1. Describe the zeros of a function whose terms form a perfect square trinomial. 2. Compare the x- and y-intercepts of a quadratic function with those of a linear function. 3. A quadratic equation has no real solutions. Describe the graph of the related quadratic function. 4. GET ORGANIZED Copy and complete the graphic organizer. In each box, give information about special products and factors. >i ,Õi Ý>«i À>« vviÀiViÊvÊ /ÜÊ-µÕ>ÀiÃ *iÀviVÌ-µÕ>ÀiÊ /À> 5- 3 Solving Quadratic Equations by Graphing and Factoring 337 5-3 Exercises KEYWORD: MB7 5-3 KEYWORD: MB7 Parent GUIDED PRACTICE 1. Vocabulary The solutions of the equation 3x 2 + 2x + 5 = 0 are its ? . −−− (roots or zeros) SEE EXAMPLE 1 2. f (x) = x 2 + 4x - 5 p. 333 SEE EXAMPLE 2 p. 334 SEE EXAMPLE 3 p. 335 SEE EXAMPLE 4 5 p. 337 3. g(x) = -x 2 + 6x - 8 4. f (x) = x 2 - 1 Find the zeros of each function by factoring. 5. f (x) = x 2 - 7x + 6 6. g(x) = 2x 2 - 5x + 2 8. f (x) = x 2 + 9x + 20 9. g(x) = x 2 - 6x - 16 7. h(x) = x 2 + 4x 10. h(x) = 3x 2 + 13x + 4 11. Archery The height h of an arrow in feet is modeled by h(t) = -16t 2 + 63t + 4, where t is the time in seconds since the arrow was shot. How long is the arrow in the air? Find the roots of each equation by factoring. 12. x 2 - 6x = -9 p. 336 SEE EXAMPLE Find the zeros of each function by using a graph and table. 13. 5x 2 + 20 = 20x 14. x 2 = 49 Write a quadratic function in standard form for each given set of zeros. 15. 3 and 4 16. -4 and -4 17. 3 and 0 PRACTICE AND PROBLEM SOLVING Independent Practice For See Exercises Example 18–20 21–26 27 28–33 34–36 1 2 3 4 5 Extra Practice Skills Practice p. S12 Application Practice p. S36 Find the zeros of each function by using a graph and table. 18. f (x) = -x 2 + 4x - 3 19. g(x) = x 2 + x - 6 20. f (x) = x 2 - 9 Find the zeros of each function by factoring. 21. f (x) = x 2 + 11x + 24 22. g(x) = 2x 2 + x - 10 23. h(x) = -x 2 + 9x 24. f (x) = x 2 - 15x + 54 25. g(x) = x 2 + 7x - 8 26. h(x) = 2x 2 - 12x + 18 27. Biology A bald eagle snatches a fish from a lake and flies to an altitude of 256 ft. The fish manages to squirm free and falls back down into the lake. Its height h in feet can be modeled by h(t) = 256 - 16t 2, where t is the time in seconds. How many seconds will the fish fall before hitting the water? Find the roots of each equation by factoring. 28. x 2 + 8x = -16 29. 4x 2 = 81 30. 9x 2 + 12x + 4 = 0 31. 36x 2 - 9 = 0 32. x 2 - 10x + 25 = 0 33. 49x 2 = 28x - 4 Write a quadratic function in standard form for each given set of zeros. 34. 5 and -1 35. 6 and 2 36. 3 and 3 Find the zeros of each function. 338 37. f (x) = 6x - x 2 38. g(x) = x 2 - 25 39. h(x) = x 2 - 12x + 36 40. f (x) = 3x 2 - 12 41. g(x) = x 2 - 22x + 121 42. h(x) = 30 + x - x 2 43. f (x) = x 2 - 11x + 30 44. g(x) = x 2 - 8x - 20 45. h(x) = 2x 2 + 18x + 28 Chapter 5 Quadratic Functions Entertainment The Guinness world record for the greatest number of people juggling at one time was set in 1998 by 1508 people, each of whom juggled at least 3 objects for 10 seconds. 46. Movies A stuntwoman jumps from a building 73 ft high and lands on an air bag that is 9 ft tall. Her height above ground h in feet can be modeled by h (t) = 73 - 16t 2, where t is the time in seconds. a. Multi-Step How many seconds will the stuntwoman fall before touching the air bag? (Hint: Find the time t when the stuntwoman’s height above ground is 9 ft.) b. What if...? Suppose the stuntwoman jumps from a building that is half as tall. Will she be in the air for half as long? Explain. 47. Entertainment A juggler throws a ball into the air from a height of 5 ft with an initial vertical velocity of 16 ft/s. a. Write a function that can be used to model the height h of the ball in feet t seconds after the ball is thrown. b. How long does the juggler have to catch the ball before it hits the ground? Find the roots of each equation. 48. x 2 - 2x + 1 = 0 49. x 2 + 6x = -5 50. 25x 2 + 40x = -16 51. 9x 2 + 6x = -1 52. 5x 2 = 45 53. x 2 - 6 = x For each function, (a) find its vertex, (b) find its y-intercept, (c) find its zeros, and (d) graph it. 54. f (x) = x 2 + 2x - 8 55. g(x) = x 2 - 16 56. h(x) = x 2 - x - 12 57. f (x) = -2x 2 + 4x 58. g(x) = x 2 - 5x - 6 59. h(x) = 3x 2 + x - 4 60. Geometry The hypotenuse of a right triangle is 2 cm longer than one leg and 4 cm longer than the other leg. a. Let x represent the length of the hypotenuse. Use the Pythagorean Theorem to write an equation that can be solved for x. b. Find the solutions of the equation from part a. c. Are both solutions reasonable in the context of the problem situation? Explain. Geometry Find the dimensions of each rectangle. 61. ÊÊnäÊvÌÓ ÝÊ Ê£È Ý 62. 63. ÊÊÓ£äÊVÓ Ý ÝÊ Ê£ ÊÊxäÊÓ ÝÊÊÎ ÝÊ ÊÓ 64. Critical Thinking Will a function whose rule can be factored as a binomial squared ever have two different zeros? Explain. 65. Write About It Explain how the Zero Product Property can be used to help determine the zeros of quadratic functions. 66. This problem will prepare you for the Multi-Step Test Prep on page 364. A baseball player hits a ball toward the outfield. The height h of the ball in feet is modeled by h(t) = -16t 2 + 22t + 3, where t is the time in seconds. In addition, the function d(t) = 85t models the horizontal distance d traveled by the ball. a. If no one catches the ball, how long will it stay in the air? b. What is the horizontal distance that the ball travels before it hits the ground? 5- 3 Solving Quadratic Equations by Graphing and Factoring 339 68. Which function has -7 as its only zero? f(x) = x(x - 7) h(x) = (x - 7) 2 >Êi} Ì i} ÌÊvÌ® 67. Use the graph provided to choose the best description of what the graph represents. A ball is dropped from a height of 42 feet and lands on the ground after 3 seconds. A ball is dropped from a height of 42 feet and lands on the ground after 1.5 seconds. A ball is shot up in the air and reaches a height of 42 feet after 1 second. A ball is shot up in the air, reaches a height of 42 feet, and lands on the ground after 1.5 seconds. {ä Îä Óä £ä ä°x £ £°x /iÊÃ® g(x) = (x + 1)(x + 7) j(x) = (x + 7) 2 69. Which expression is a perfect square trinomial? 25y 2 - 40y + 16 25y 2 - 16 25y 2 - 20y + 16 25y 2 - 10y + 16 70. Gridded Response Find the positive root of x 2 + 4x - 21 = 0. CHALLENGE AND EXTEND Find the roots of each equation by factoring. 71. 3(x 2 - x) = x 2 1x 72. x 2 = _ 3 3x + _ 1 =0 73. x 2 - _ 4 8 74. x 2 + x + 0.21 = 0 75. Another special factoring case involves perfect cubes. The sum of two cubes can be factored by using the formula a 3 + b 3 = (a + b)(a 2 - ab + b 2). a. Verify the formula by multiplying the right side of the equation. b. Factor the expression 8x 3 + 27. c. Use multiplication and guess and check to find the factors of a 3 - b 3. d. Factor the expression x 3 - 1. SPIRAL REVIEW Evaluate each expression. Write the answer in scientific notation. (Lesson 1-5) 76. 78. (1.4 × 10 8)(6.1 × 10 -3) (3.5 × 10 6) __ (1.4 × 10 -4) 77. 79. Solve each proportion. (Lesson 2-2) n w 1.2 = _ 12 = _ 80. _ 81. _ 7.5 5 4.8 8.8 (2.7 × 10 10)(3.2 × 10 2) (3.12 × 10 -6) __ (4.8 × 10 3) 6.8 = _ r 82. _ 4.5 90 Using the graph of f (x) = x 2 as a guide, describe the transformations, and then graph each function. (Lesson 5-1) 83. h(x) = 0.5x 2 340 Chapter 5 Quadratic Functions 84. d(x) = x 2 + 2 85. g(x) = (x + 1) 2 Ó 5-4 */ Completing the Square IA-3.3 Carry out a procedure to be solve quadratic equations algebraically A2.3.5 problems that can modeled using quadratic equations and functions, 2.3a Solve quadratic equations by graphing, factoring, completing the square (including factoring, completing the square, andthe applying theare quadratic formula). interpret the solutions, and determine whether solutions reasonable. and quadratic formula. Objectives Solve quadratic equations by completing the square. Why learn this? You can solve quadratic equations to find how long water takes to fall from the top to the bottom of a waterfall. (See Exercise 39.) Write quadratic equations in vertex form. Vocabulary completing the square Many quadratic equations contain expressions that cannot be easily factored. For equations containing these types of expressions, you can use square roots to find roots. Square-Root Property WORDS NUMBERS x = 15 2 To solve a quadratic equation, you can take the square root of both sides. Be sure to consider the positive and negative square roots. EXAMPLE 1 ⎪x⎥ = √ 15 15 x = ± √ ALGEBRA If x = a and a is a nonnegative real number, then x = ± √ a. 2 Solving Equations by Using the Square Root Property Solve each equation. A 3x 2 - 4 = 68 3x 2 = 72 Add 4 to both sides. x 2 = 24 Divide both sides by 3 to isolate the squared term. x = ± √ 24 Take the square root of both sides. x = ±2 √ 6 Read ± √ a as “plus or minus square root of a.” Simplify. Check Use a graphing calculator. B x 2 - 10x + 25 = 27 (x - 5) 2 = 27 x - 5 = ± √ 27 Factor the perfect square trinomial. Take the square root of both sides. x = 5 ± √ 27 Add 5 to x = 5 ± 3 √ 3 Simplify. both sides. Check Use a graphing calculator. Solve each equation. 1a. 4x 2 - 20 = 5 1b. x 2 + 8x + 16 = 49 5- 4 Completing the Square 341 The methods in the previous examples can be used only for expressions that are perfect squares. However, you can use algebra to rewrite any quadratic expression as a perfect square. You can use algebra tiles to model a perfect square trinomial as a perfect square. The area of the square at right is x 2 + 2x + 1. Because each side of the square measures x + 1 units, the area is also (x + 1)(x + 1), or (x + 1) 2. This shows that (x + 1) 2 = x 2 + 2x + 1. If a quadratic expression of the form x 2 + bx cannot model a square, you can add a term to form a perfect square trinomial. This is called completing the square. Completing the Square WORDS NUMBERS ALGEBRA x + 6x + x + bx + 2 To complete the square of 2 b . x 2 + bx, add _ 2 () x 2 + 6x + 2 (_26) 2 x 2 + bx + (x + _b2 ) x 2 + 6x + 9 (x + 3) 2 Ý ÓÊ ÊÈÝ The model shows completing the square for x 2 + 6x by adding 9 unit tiles. The resulting perfect square trinomial is x 2 + 6x + 9. Note that completing the square does not produce an equivalent expression. EXAMPLE 2 2 (_b2 ) 2 Ý ÓÊ ÊÈÝÊ Ê LÊÊÈ Ú Ó Ú Ó L Ê Ê ÊÊÊÊ Ê ÊÊÊÊ ÊÈÊÊÊÊ Ê ÊÊ Ó Ó Completing the Square Complete the square for each expression. Write the resulting expression as a binomial squared. A x 2 - 2x + ( ) -2 _ 2 2 (2) B 2 b . = (-1) 2 = 1 Find _ x 2 - 2x + 1 Add. (x - 1) 2 Factor. Check Find the square of the binomial. x 2 + 5x + 25 5 2=_ _ 4 2 () () 25 x 2 + 5x + _ 4 2 5 x+_ 2 ( ) Add. Factor. Check Find the square of the binomial. (x + _52 ) = (x + _52 )(x + _52 ) 2 (x - 1) 2 = (x - 1)(x - 1) 25 = x 2 + 5x + _ 4 = x 2 - 2x + 1 Complete the square for each expression. Write the resulting expression as a binomial squared. 2a. x 2 + 4x + 342 Chapter 5 Quadratic Functions 2 b . Find _ 2 2b. x 2 - 4x + 2c. x 2 + 3x + You can complete the square to solve quadratic equations. Solving Quadratic Equations ax 2 + bx + c = 0 by Completing the Square 1. Collect variable terms on one side of the equation and constants on the other. 2. As needed, divide both sides by a to make the coefficient of the x 2-term 1. () b 3. Complete the square by adding __ 2 2 to both sides of the equation. 4. Factor the variable expression as a perfect square. 5. Take the square root of both sides of the equation. 6. Solve for the values of the variable. EXAMPLE 3 Solving a Quadratic Equation by Completing the Square Solve each equation by completing the square. A x 2 = 27 - 6x x 2 + 6x = 27 x 2 + 6x + To keep the equation balanced, you b 2 to must add __ 2 both sides of the equation. () x 2 + 6x + Collect variable terms on one side. = 27 + Set up to complete the square. (_26 ) = 27 + (_26 ) 2 2 x 2 + 6x + 9 = 27 + 9 () b Add _ 2 2 to both sides. Simplify. 2 (x + 3) = 36 Factor. x + 3 = ± √ 36 Take the square root of both sides. x + 3 = ±6 Simplify. x + 3 = 6 or x + 3 = -6 Solve for x. x = 3 or x = -9 B 2x 2 + 8x = 12 x 2 + 4x = 6 2 x + 4x + =6+ (_) x 2 + 4x + 4 2 Divide both sides by 2. 2 Set up to complete the square. (_) =6+ 4 2 2 x 2 + 4x + 4 = 6 + 4 2 (x + 2) = 10 x + 2 = ± √ 10 x = -2 ± √ 10 () b Add _ 2 2 to both sides. Simplify. Factor. Take the square root of both sides. Solve for x. Solve each equation by completing the square. 3a. x 2 - 2 = 9x 3b. 3x 2 - 24x = 27 Recall the vertex form of a quadratic function from Lesson 5-1: 2 f (x) = a(x - h) + k, where the vertex is (h, k). You can complete the square to rewrite any quadratic function in vertex form. 5- 4 Completing the Square 343 EXAMPLE 4 Writing a Quadratic Function in Vertex Form Write each function in vertex form, and identify its vertex. A f (x) = x 2 + 10x - 13 f (x) = (x 2 + 10x + In Example 3, the equation was balanced by adding _b_ 2 to both sides. 2 (2) and subtracting _b_ on one side. 2 Set up to complete the square. 10 (_) - 13 - (_ 2 ) 10 f (x) = x 2 + 10x + 2 () Here, the equation is balanced by adding ) - 13 2 2 f (x) = (x + 5) 2 - 38 () b 2 Add and subtract __ . 2 Simplify and factor. Because h = -5 and k = -38, the vertex is (-5, -38). Check Use the axis of symmetry formula to confirm the vertex. b = -_ 10 = -5 x = -_ y = f (-5) = (-5) 2 + 10(-5) - 13 = -38 ✔ 2a 2(1) B g(x) = 2x 2 - 8x + 3 g(x) = 2(x 2 - 4x) + 3 g(x) = 2(x 2 - 4x + ( Factor so the coefficient of x 2 is 1. )+3- Set up to complete the square. -4 (_) ) + 3 - 2(_ 2 ) g(x) = 2 x 2 - 4x + -4 2 2 2 () () b 2 b Add __ . Because __ 2 2 2 is multiplied by 2, you must () b 2 subtract 2 __ . 2 g(x) = 2(x 2 - 4x + 4) - 5 Simplify. g(x) = 2(x - 2) 2 - 5 Factor. Because h = 2 and k = -5, the vertex is (2, -5). Check A graph of the function on a graphing calculator supports your answer. Write each function in vertex form, and identify its vertex. 4a. f (x) = x 2 + 24x + 145 4b. g(x) = 5x 2 - 50x + 128 THINK AND DISCUSS 1. Explain two ways to solve x 2 = 25. 2. Describe how to change a quadratic function from standard form to vertex form by completing the square. 3. GET ORGANIZED Copy and complete the graphic organizer. Compare and contrast two methods of solving quadratic equations. 344 Chapter 5 Quadratic Functions 1Ã}Ê-µÕ>Ài,ÌÊ*À«iÀÌÞÊÛÃ° «iÌ}ÊÌ iÊ-µÕ>Ài ->ÀÌiÃ vviÀiViÃ 5-4 Exercises KEYWORD: MB7 5-4 KEYWORD: MB7 Parent GUIDED PRACTICE 1. Vocabulary What must you add to the expression x 2 + bx to complete the square? SEE EXAMPLE 1 2. (x - 2) 2 = 16 p. 341 SEE EXAMPLE Solve each equation. 2 p. 342 3. x 2 - 10x + 25 = 16 Complete the square for each expression. Write the resulting expression as a binomial squared. 5. x 2 + 14x + SEE EXAMPLE 3 p. 343 6. x 2 - 12x + p. 344 7. x 2 - 9x + Solve each equation by completing the square. 8. x 2 - 6x = -4 9. x 2 + 8 = 6x 11. x 2 = 24 - 4x SEE EXAMPLE 4 4. x 2 - 2x + 1 = 3 10. 2x 2 - 20x = 8 12. 10x + x 2 = 42 13. 2x 2 + 8x - 15 = 0 Write each function in vertex form, and identify its vertex. 14. f (x) = x 2 + 6x - 3 15. g(x) = x 2 - 10x + 11 16. h(x) = 3x 2 - 24x + 53 17. f (x) = x 2 + 8x - 10 18. g(x) = x 2 - 3x + 16 19. h(x) = 3x 2 - 12x - 4 PRACTICE AND PROBLEM SOLVING Independent Practice For See Exercises Example 20–22 23–25 26–31 32–37 1 2 3 4 Extra Practice Skills Practice p. S12 Application Practice p. S36 Solve each equation. 20. (x + 2) 2 = 36 22. (x - 3) 2 = 5 21. x 2 - 6x + 9 = 100 Complete the square for each expression. Write the resulting expression as a binomial squared. 1x + 23. x 2 - 18x + 24. x 2 + 10x + 25. x 2 - _ 2 Solve each equation by completing the square. 26. x 2 + 2x = 7 27. x 2 - 4x = -1 28. 2x 2 - 8x = 22 29. 8x = x 2 + 12 30. x 2 + 3x - 5 = 0 31. 3x 2 + 6x = 1 Write each function in vertex form, and identify its vertex. 32. f (x) = x 2 - 4x + 13 33. g(x) = x 2 + 14x + 71 34. h(x) = 9x 2 + 18x - 3 35. f (x) = x 2 + 4x - 7 36. g(x) = x 2 - 16x + 2 37. h(x) = 2x 2 + 6x + 25 38. Engineering The height h above the roadway of the main cable of the Golden Gate 7 1 x 2 - __ x + 500, where x is the Bridge can be modeled by the function h(x) = ____ 15 9000 distance in feet from the left tower. x h a. Complete the square, and write the function in vertex form. b. What is the vertex, and what does it represent? c. Multi-Step The left and right towers have the same height. What is the distance in feet between them? 5- 4 Completing the Square 345 Caracas Venezuela Angel Falls 39. Waterfalls Angel Falls in Venezuela is the tallest waterfall in the world. Water falls uninterrupted for 2421 feet before entering the river below. The height h above the river in feet of water going over the edge of the waterfall is modeled by h(t) = -16t 2 + 2421, where t is the time in seconds after the initial fall. a. Estimate the time it takes for the water to reach the river. b. Multi-Step Ribbon Falls in California has a height of 1612 ft. Approximately how much longer does it take water to reach the bottom when going over Angel Falls than when going over Ribbon Falls? 40. Sports A basketball is shot with an initial vertical velocity of 24 ft/s from 6 ft above the ground. The ball’s height h in feet is modeled by h(t) = -16t 2 + 24t + 6, where t is the time in seconds after the ball is shot. What is the maximum height of the ball, and when does the ball reach this height? Solve each equation using square roots. 41. x 2 - 1 = 2 42. 25x 2 = 0 43. 8x 2 - 200 = 0 44. -3x 2 + 6 = -1 45. (x + 13) 2 = 7 46. 48. x 2 + 14x + 49 = 64 49. 9x 2 + 18x + 9 = 5 25 (x + _32 ) = _ 2 2 47. 50. 9 =0 (x + _14 ) - _ 16 2 Two attempts to write f (x) = 2x 2 - 8x in vertex form are shown. Which is incorrect? Explain the error. /////ERROR ANALYSIS///// J \ \ ¦\ J \ \ ¦\ J \ \ ¦\ ¦ J \ \¦ ¦ J \ \ \ J \ \ \ J \ \ \ J \ \ Solve each equation by completing the square. Sports Acapulco, Mexico, is famous for its cliff-diving shows. Divers perform complicated acrobatic dives from heights of up to 80 feet. 346 51. x 2 + 8x = -15 52. x 2 + 22x = -21 53. 3x 2 + 4x = 1 54. 2x 2 = 5x + 12 55. x 2 - 7x - 2 = 0 56. x 2 = 4x + 11 57. x 2 + 6x + 4 = 0 58. 5x 2 + 10x - 7 = 0 59. x 2 - 8x = 24 60. Sports A diver’s height h in meters above the water is approximated by h(t) = h 0 - 5t 2, where h 0 is the initial height in meters, -5 is a constant based on the acceleration due to gravity in m/s 2, and t is the time in seconds that the diver falls through the air. a. Find the total time that the diver falls through Dive Heights the air for each type of dive in the table. Type Height (m) b. How high is a dive that keeps the diver in the Platform 5 air twice as long as a 5-meter dive? Platform 10 c. The speed of a diver entering the water can be approximated by s = 18t, where s is the speed Cliff 20 in kilometers per hour and t is the time in Cliff 30 seconds. Using your results from part a, find the speed of the diver entering the water for each dive height. d. How many times as high is a dive that results in a speed that is twice as fast? Chapter 5 Quadratic Functions 61. This problem will prepare you for the Multi-Step Test Prep on page 364. The height h in feet of a baseball hit from home plate can be modeled by the function h(t) = -16t 2 + 32t + 5.5, where t is the time in seconds since the ball was hit. The ball is descending when it passes 7.5 ft over the head of a 6 ft player standing on the ground. a. To the nearest tenth of a second, how long after the ball is hit does it pass over the player’s head? b. The horizontal distance between the player and home plate is 120 ft. Use your answer from part a to determine the horizontal speed of the ball to the nearest foot per second. 62. Estimation A bag of grass seed will cover 525 square feet. Twenty bags of seed are used to cover an area shaped like a square. Estimate the side length of the square. Check your answer with a calculator. 63. Critical Thinking The functions f and g are defined by f (x) = x 2 + 2x - 2 and g (x) = (x + 1) 2 - 3. Use algebra to prove that f and g represent the same function. 64. Sports A player bumps a volleyball with an initial vertical velocity of 20 ft/s. a. Write a function h in standard form for the ball’s height in feet in terms of the time t in seconds after the ball is hit. b. Complete the square to rewrite h in vertex form. c. What is the maximum height of the ball? d. What if...? Suppose the volleyball were hit under the same conditions, but with an initial velocity of 32 ft/s. How much higher would the ball go? ? ft 4 ft Graphing Calculator Use a graphing calculator to approximate the roots of each equation to the nearest thousandth. 65. x 2 - 15 = 40 66. x 2 = 2.85 67. 1.4x 2 = 24.6 68. (x + 0.6) 2 = 7.4 x2 = _ 1 69. _ 7 3 70. (x + _14 ) = _65 2 71. Critical Thinking Why do equations of the form x 2 = k have no real solution when k < 0? 72. Write About It Compare the methods of factoring and completing the square for solving quadratic equations. 73. Which gives the solution to 3x 2 = 33? ± √3 ± √11 11 74. Which equation represents the graph at right? Þ È y = (x - 2) 2 + 1 y = (x - 2) 2 - 1 y = (x + 2) 2 + 1 y = (x + 2) 2 - 1 121 { Ó Ý Ó ä Ó { 5- 4 Completing the Square 347 75. Which gives the vertex of the graph of y = 3(x - 1) 2 - 22? (1, -22) (3, -22) (-1, -22) (-3, -22) 76. Which number should be added to x 2 + 14x to make a perfect square trinomial? 196 7 14 49 77. Gridded Response What is the positive root of the equation 2x 2 - x = 10? 78. Extended Response Solve the quadratic equation x 2 - 6x = 16 by completing the square. Explain each step of the solution process, and check your answer. CHALLENGE AND EXTEND Find the value of b in each perfect square trinomial. 79. x 2 - bx + 144 80. 4x 2 - bx + 16 81. 3x 2 + bx + 27 82. ax 2 + bx + c Find the zeros of each function. 84. f (x) = x 2 + 6x √ 3 + 23 83. f (x) = x 2 - 4x √ 5 + 19 85. Farming To create a temporary grazing area, a farmer is using 1800 feet of electric fencing to enclose a rectangular field and then to subdivide the field into two plots. The fence that divides the field into two plots is parallel to the field’s shorter sides. a. What is the largest area of the field that the farmer can enclose? b. What are the dimensions of the field with the largest area? c. What if...? What would be the largest area of a square field that the farmer could enclose and divide into two plots? *ÌÊ£ *ÌÊÓ SPIRAL REVIEW Express each set of numbers using set-builder notation. (Lesson 1-1) 86. (72, ∞) 87. numbers within 10 units of 4 88. positive multiples of 4 89. Î Ó £ ä £ Ó Î { x È Use the table for Exercises 90–93. (Lesson 4-1) Monthly Budget Food Housing Auto Aboline family $352 $895 $426 Hernandez family $675 $1368 $642 Walker family $185 $615 $295 90. Display the data in the form of a matrix B. 91. What are the dimensions of the matrix? 92. What is the address of the entry that has the value 185? 93. What is the value of the matrix entry with the address b 22? What does it represent? Identify the axis of symmetry and the vertex of the graph of each function. (Lesson 5-2) 2 x2 - 1 94. f (x) = 3(x - 2) 2 95. g (x) = _ 96. h(x) = 6x 2 + 2.5 5 348 Chapter 5 Quadratic Functions Areas of Composite Figures Geometry Quadratic equations can be used to solve problems involving the areas of composite figures. Write an equation that represents the information given in the problem. Then solve the equation. Example The diagram shows a rectangular garden surrounded by a walkway. The garden measures 10 m by 34 m. The total area of the garden and walkway is 640 m 2. What is the width x of the walkway? The total area is equal to the total length multiplied by the total width. The total length is 2x + 34 m, and the total width is 2x + 10 m. A=×w Ý £äÊ Ý Ý Î{Ê Write the formula for total area. 640 = (2x + 34)(2x + 10) Substitute. 640 = 4x 2 + 88x + 340 Multiply the binomials. Ý 0 = 4x 2 + 88x - 300 Subtract 640 from both sides. 0 = x 2 + 22x - 75 Divide both sides by 4. 0 = (x - 3)(x + 25) Factor. x - 3 = 0 or x + 25 = 0 /Ì>Ê>Ài>ÊÊÈ{äÊÓ Use the Zero Product Property. x = 3 or x = -25 Solve for x. The width cannot be negative. Therefore, the width of the walkway is 3 m. Try This Write an equation that represents each problem. Then solve. 1. Use figure 1 below. A ring of grass with an area of 314 yd 2 surrounds a circular flower bed. Find the width x of the ring of grass. 2. Use figure 2 below. Sid cuts four congruent squares from the corners of a 30-in.-by-50-in. rectangular piece of cardboard so that it can be folded to make a box. Find the side length s of the squares, given that the area of the bottom of the box is 200 in 2. 3. Use figure 3 below. Harriet has 80 m of fencing materials to enclose three sides of a rectangular garden. She will use the side of her garage as a border for the fourth side. Find the width x of the garden if its area is to be 700 m 2. Figure 1 Ý £äÊÞ` Figure 2 Figure 3 xäÊ° >À>}i Ã Ý ÓääÊÓ À>ÃÃÊ>Ài>ÊÊÎ£{ÊÞ`Ó ÎäÊ° ÇääÊÓ Ý Þ iV}Ê>ÌiÀ>ÊÊnäÊ Ã Ã Ã Connecting Algebra to Geometry 349 5-5 IA-3.3 Carry out a that procedure solve quadratic algebraically A2.3.2 quadratic equations in the complexequations number system. 1.3a Solve Recognize to solvetocertain problems and equations, number (including systems factoring, the square, and applying the quadratic need to becompleting extended from real numbers to complex numbers.formula). Why learn this? Complex numbers can be used to describe the zeros of quadratic functions that have no real zeros. (See Example 4.) Objectives Define and use imaginary and complex numbers. Solve quadratic equations with complex roots. Vocabulary imaginary unit imaginary number complex number real part imaginary part complex conjugate Andrew Toos/CartoonResource.com */ Complex Numbers and Roots You can see in the graph of f (x) = x 2 + 1 below that f has no real zeros. If you solve the corresponding equation 0 = x 2 + 1, you find that x = ± √ -1 , which has no real solutions. However, you can find solutions if you define the square root of negative numbers, which is why imaginary numbers were invented. The imaginary unit i is defined as √ -1 . You can use the imaginary unit to write the square root of any negative number. Þ vÝ®ÊÊÝÓÊ Ê£ Ý ÊÝÌiÀVi«ÌÃ Imaginary Numbers WORDS An imaginary number is the square root of a negative number. Imaginary numbers can be written in the form bi, where b is a real number and i is the imaginary unit. NUMBERS √ -1 = i If b is a positive real number, = i √ √-2 = √ -1 √ 2 = i √ 2 then √-b b √ -4 = √ -1 √ 4 = 2i The square of an imaginary number is the original negative number. EXAMPLE 1 ALGEBRA and √ -b 2 = bi. 2 ( √ -1 ) = i 2 = -1 2 ( √ -b ) = -b Simplifying Square Roots of Negative Numbers Express each number in terms of i. A 3 √ -16 B - √ -75 (16)(-1) Factor out -1. 3 √ (75)(-1) - √ Factor out -1. 3 √ 16 √ -1 Product Property - √ 75 √ -1 Product Property 3 · 4 √ -1 Simplify. - √ 25 √ 3 √ -1 Product Property 12 √ -1 Multiply. -5 √ 3 √ -1 Simplify. 12i Express in terms of i. -5 √ 3 i = -5i √ 3 Express in terms Express each number in terms of i. 1a. √ -12 1b. 2 √ -36 350 Chapter 5 Quadratic Functions of i. 1 √ 1c. -_ -63 3 EXAMPLE 2 Solving a Quadratic Equation with Imaginary Solutions Solve each equation. A x 2 = -81 B 3x 2 + 75 = 0 -81 Take square x = ± √ x = ±9i 3x 2 = -75 roots. Express in terms of i. x 2 = -25 Divide both sides by 3. x = ± √ -25 Take square roots. Check x = ±5i Express in terms of i. Check −−−−−−−−−−−− 3x 2 + 75 = 0 2 3(±5i) + 75 0 3(25)i 2 + 75 0 75(-1) + 75 0 ✔ x 2 = -81 x 2 = -81 −−−−−−−−− −−−−−−−−− 2 (9i) -81 (-9i) 2 -81 81i 2 -81 81i 2 -81 81(-1) -81 ✔ 81(-1) -81 ✔ Solve each equation. 2a. x 2 = -36 2b. x 2 + 48 = 0 A complex number is a number that can be written in the form a + bi, where a and b are real numbers and i = √ -1 . The set of real numbers is a subset of the set of complex numbers . 2c. 9x 2 + 25 = 0 «iÝÊ ÕLiÀÃÊ ο ÎÊ ÊÇ ÊÊÊÊÎÊ ÊÊÚÚ ÊÓÊÊÊÊ ÊÊÊÊ{ÊÊ Î ,i> ÕLiÀÃÊ ώ £ ÚÚ ÊÊÊÊÊÊÊ£°ÇÎÊÊÊÊäÊÊÊÊû Ó Ü е °ÊÈÊÊÊÊÊÊÊȖÓÊ Ê Every complex number has a real part a and an imaginary part b. Real part Add -75 to both sides. >}>ÀÞ ÕLiÀÃ ÊÊÊÊÎÊÊÊÊx е ÊȖÇÊ еÊ Imaginary part Real numbers are complex numbers where b = 0. Imaginary numbers are complex numbers where a = 0 and b ≠ 0. These are sometimes called pure imaginary numbers. Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. EXAMPLE 3 Equating Two Complex Numbers Find the values of x and y that make the equation 3x - 5i = 6 - (10y)i true. Real parts 3x - 5i = 6 - (10y)i Imaginary parts 3x = 6 x=2 Equate the real parts. Solve for x. -5 = -(10y) Equate the imaginary parts. 1 =y _ Solve for y. 2 Find the values of x and y that make each equation true. 3a. 2x - 6i = -8 + (20y)i 3b. -8 + (6y)i = 5x - i √ 6 5- 5 Complex Numbers and Roots 351 EXAMPLE 4 Finding Complex Zeros of Quadratic Functions Find the zeros of each function. A f (x) = x 2 - 2x + 5 B x 2 - 2x + 5 = 0 x 2 - 2x + x 2 + 10x + 35 = 0 Set equal to 0. = -5 + x 2 + 10x + Rewrite. () g (x) = x 2 + 10x + 35 2 = -35 + x - 2x + 1 = -5 + 1 b Add __ . x + 10x + 25 = -35 + 25 (x - 1) = -4 Factor. (x + 5) 2 = -10 2 2 2 x - 1 = ± √ -4 2 x + 5 = ± √ -10 Take square roots. x = 1 ± 2i x = -5 ± i √ 10 Simplify. Find the zeros of each function. 4a. f (x) = x 2 + 4x + 13 4b. g(x) = x 2 - 8x + 18 When given one complex root, you can always find the other by finding its conjugate. EXAMPLE The solutions -5 + i √ 10 and -5 - i √ 10 in Example 4B are related. These solutions are a complex conjugate pair. Their real parts are equal and their imaginary parts are opposites. The complex conjugate of any complex number a + bi is the complex number a - bi. If a quadratic equation with real coefficients has nonreal roots, those roots are complex conjugates. 5 Finding Complex Conjugates Find each complex conjugate. A 2i - 15 -15 + 2i -15 - 2i B Write as a + bi. Find a - bi. Find each complex conjugate. 5a. 9 - i 5b. i + √ 3 -4i 0 + (-4)i 0 - (-4)i 4i Write as a + bi. Find a - bi. Simplify. 5c. -8i THINK AND DISCUSS 1. Given that one solution of a quadratic equation is 3 + i, explain how to determine the other solution. 2. Describe a number of the form a + bi in which a ≠ 0 and b = 0. Then describe a number in which a = 0 and b ≠ 0. Are both numbers complex? Explain. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box or oval, give a definition and examples of each type of number. 352 Chapter 5 Quadratic Functions «iÝÊ ÕLiÀÃ ,i>Ê ÕLiÀÃ >}>ÀÞÊ ÕLiÀÃ 5-5 Exercises KEYWORD: MB7 5-5 KEYWORD: MB7 Parent GUIDED PRACTICE 1. Vocabulary The number 7 is the ? part of the complex number √ 5 + 7i. (real or ̶̶̶ imaginary) SEE EXAMPLE 1 p. 350 SEE EXAMPLE 2 p. 351 SEE EXAMPLE 6. x 2 = -9 3 p. 351 7. 2x 2 + 72 = 0 p. 352 p. 352 5. √ -144 8. 4x 2 = -16 9. x 2 + 121 = 0 11. -4 + (y)i = -12x - i + 8 Find the zeros of each function. 12. f (x) = x 2 - 12x + 45 5 4. - √ -32 Find the values of x and y that make each equation true. 10. -2x + 6i = (-24y)i - 14 SEE EXAMPLE 4 SEE EXAMPLE Express each number in terms of i. 1 √ 2. 5 √ -100 3. _ -16 2 Solve each equation. 13. g(x) = x 2 + 6x + 34 Find each complex conjugate. 14. -9i 15. √ 5 + 5i 16. 8i - 3 17. 6 + i √ 2 PRACTICE AND PROBLEM SOLVING Independent Practice For See Exercises Example 18–21 22–25 26–27 28–31 32–35 1 2 3 4 5 Extra Practice Skills Practice p. S13 Application Practice p. S36 Express each number in terms of i. 1 √-90 18. 8 √ -4 19. -_ 3 20. 6 √ -12 21. √ -50 24. 3x 2 + 27 = 0 1 x 2 = -32 25. _ 2 Solve each equation. 22. x 2 + 49 = 0 23. 5x 2 = -80 Find the values of x and y that make each equation true. 26. 9x + (y)i - 5 = -12i + 4 27. 5(x - 1) + (3y)i = -15i - 20 Find the zeros of each function. 28. f (x) = x 2 + 2x + 3 29. g(x) = 4x 2 - 3x + 1 30. f (x) = x 2 + 4x + 8 31. g(x) = 3x 2 - 6x + 10 Find each complex conjugate. 32. i √ 3 33. -_ - 2i 2 34. -2.5i + 1 36. What if...? A carnival game asks participants to strike a spring with a hammer. The spring shoots a puck upward toward a bell. If the puck strikes the bell, the participant wins a prize. Suppose that a participant strikes the spring and shoots the puck according to the model d(t) = 16t 2 - 32t + 18, where d is the distance in feet between the puck and the bell and t is the time in seconds since the puck was struck. Is it possible for the participant to win a prize? Explain your answer. i -1 35. _ 10 18 ft 5- 5 Complex Numbers and Roots 353 Given each solution to a quadratic equation, find the other solution. 5i 37. 1 + 14i 38. _ 5 39. 4i - 2 √ 7 17i _ 40. -12 - i 41. 9 - i √ 2 42. 3 Find the values of c and d that make each equation true. 43. 2ci + 1 = -d + 6 - ci 44. c + 3ci = 4 + di 45. c 2 + 4i = d + di 48. 2x 2 + 12.5 = 0 Solve each equation. The Granger Collection, New York Math History 46. 8x 2 = -8 1 x 2 + 72 = 0 49. _ 2 52. x 2 - 4x + 8 = 0 1 x 2 = -27 47. _ 3 50. x 2 = -30 51. 2x 2 + 16 = 0 53. x 2 + 10x + 29 = 0 54. x 2 - 12x + 44 = 0 55. x 2 + 2x = -5 56. x 2 + 18 = -6x 57. -149 = x 2 - 24x Tell whether each statement is always, sometimes, or never true. If sometimes true, give examples to support your answer. 58. A real number is an imaginary number. 59. An imaginary number is a complex number. The Swiss mathematician Leonhard Euler (1707– 1783) was the first to use the notation i to -1 . He also represent √ introduced the notation f(x) to represent the value of a function f at x. 60. A rational number is a complex number. 61. A complex number is an imaginary number. 62. An integer is a complex number. 63. Quadratic equations have no real solutions. 64. Quadratic equations have roots that are real and complex. 65. Roots of quadratic equations are conjugate pairs. Find the zeros of each function. 66. f (x) = x 2 - 10x + 26 67. g(x) = x 2 + 2x + 17 68. h(x) = x 2 - 10x + 50 69. f (x) = x 2 + 16x + 73 70. g(x) = x 2 - 10x + 37 71. h(x) = x 2 - 16x + 68 72. Critical Thinking Can you determine the zeros of f (x) = x 2 + 64 by using a graph? Explain why or why not. 73. Critical Thinking What is the complex conjugate of a real number? 74. Write About It Explain the procedures you can use to solve for nonreal complex roots. 75. This problem will prepare you for the Multi-Step Test Prep on page 364. A player throws a ball straight up toward the roof of an indoor baseball stadium. The height h in feet of the ball after t seconds can be modeled by the function h(t) = -16t 2 + 112t. a. The height of the roof is 208 ft. Solve the equation 208 = -16t 2 + 112t. b. Based on your answer to part a, does the ball hit the roof? Explain your answer. c. Based on the function model, what is the maximum height that the ball will reach? 354 Chapter 5 Quadratic Functions 76. What is the complex conjugate of -2 + i ? 2-i 2+i i-2 -2 - i in terms of i. 77. Express √-225 15i -15i i √15 -i √15 78. Find the zeros of f(x) = x 2 - 2x + 17. 4±i 1 ± 4i -1 ± 4i -4 ± i 79. What value of c makes the equation 3 - 4i - 5 = (9 + ci) - 11 true? 4 -2 -4 2 80. Which of the following equations has roots of -6i and 6i? 1 x2 = 6 -_ 6 1 x2 = 9 _ 4 x 2 - 30 = 6 20 - x 2 = -16 81. Short Response Explain the types of solutions that equations of the form x 2 = a have when a < 0 and when a > 0. CHALLENGE AND EXTEND 82. Find the complex number a + bi such that 5a + 3b = 1 and -5b = 7 + 4a. 83. Can a quadratic equation have only one real number root? only one imaginary root? only one complex root? Explain. 84. Given the general form of a quadratic equation x 2 + bx + c = 0, determine the effect of each condition on the solutions. a. b = 0 b. c ≤ 0 c. c > 0 d. What is needed for the solutions to have imaginary parts? SPIRAL REVIEW Use the following matrices for Exercises 85–88. Evaluate, if possible. (Lesson 4-2) ⎡ 1 -5 ⎤ S =⎢ ⎣ -2 0 ⎦ 85. T 2 86. TV T= ⎡ 10 1 ⎤ ⎡ -4 1 -2 ⎤ V = 0 -3 1 0 -1 ⎣ -5 5 ⎦ ⎣ 2 -2 2 ⎦ ⎢ ⎢ 88. S 2 87. ST For each function, (a) determine whether the graph opens upward or downward, (b) find the axis of symmetry, (c) find the vertex, (d) find the y-intercept, and (e) graph the function. (Lesson 5-2) 1 x 2 + x - 10 90. f (x) = -x 2 + 3 89. f (x) = _ 5 1 x 2 + 3x + 1 91. f (x) = 2x 2 + 4x - 3 92. f (x) = -_ 2 Find the roots of each equation by factoring. (Lesson 5-3) 93. x 2 + 5x = 14 94. 6x 2 = -x + 2 95. 4x 2 + 9 = 15x 96. 4x 2 = 1 97. x 2 + 11x = -24 98. x 2 = -7x 5- 5 Complex Numbers and Roots 355 5-6 */ The Quadratic Formula IA-3.3 Carry a procedure to solve quadratic equations algebraically A2.3.2 quadratic equations in complex number completing system. 2.3a Solve Solve out quadratic equations bythe graphing, factoring, the (including square factoring, completing and quadratic formula.the square, and applying the quadratic formula). Objectives Solve quadratic equations using the Quadratic Formula. Classify roots using the discriminant. Vocabulary discriminant Who uses this? Firefighting pilots can use the Quadratic Formula to estimate when to release water on a fire. (See Example 4.) You have learned several methods for solving quadratic equations: graphing, making tables, factoring, using square roots, and completing the square. Another method is to use the Quadratic Formula, which allows you to solve a quadratic equation in standard form. By completing the square on the standard form of a quadratic equation, you can determine the Quadratic Formula. Numbers Algebra ax 2 + bx + c = 0 (a ≠ 0) 3x 2 + 5x + 1 = 0 5x + _ 1=0 x2 + _ 3 3 5 x = -_ 1 x2 + _ 3 3 _ ( ) 5x + 5 x2 + _ 3 2(3) 2 Divide by a. _ ( ) 1+ 5 = -_ 3 2(3) 2 (_) b b Complete x 2 + _ a x + 2a the square. 5=± x+_ 6 ( ) x=- b - 4ac _ 4a 2 2 Factor. 13 √_ 36 (_b ) c = -_ a+ 2 2a b b =_ c -_ (x + _ a 2a ) 4a b =± _ b - 4ac x+_ √ 2a 4a 2 2 2 Take square roots. √ 13 _5 ± _ 6 2 2 2 _ c b x2 + _ ax = - a c Subtract _ a. 25 _ -1 (x + _56 ) = _ 3 36 To subtract fractions, you need a common denominator. c b2 - _ _ a 4a 2 c _ 4a b2 - _ _ a 4a 4a 2 b c _ x2 + _ ax + a = 0 6 -5 ± √ 13 x= _ 6 b. Subtract _ 2a Simplify. 2 √ b - 4ac b _ ±_ x=- 2 2a 2a -b ± √ b 2 - 4ac x = __ 2a Ó The symmetry of a quadratic function is evident in the next √ b 2 - 4ac b to last step, x = -___ ± ________ . These two zeros are the 2a 2a √ b 2 - 4ac ________ , 2a same distance, away from the axis of symmetry, b x = -___ , with one zero on either side of the vertex. 2a £ Ý Ó The Quadratic Formula If ax 2 + bx + c = 0 (a ≠ 0), then the solutions, or roots, are b 2 - 4ac -b ± √ x = __. 2a 356 Chapter 5 Quadratic Functions £ £ You can use the Quadratic Formula to solve any quadratic equation that is written in standard form, including equations with real solutions or complex solutions. EXAMPLE 1 Quadratic Functions with Real Zeros Find the zeros of f (x) = x 2 + 10x + 2 by using the Quadratic Formula. x 2 + 10x + 2 = 0 Set f(x) = 0. -b ± √b 2 - 4ac x = __ 2a Write the Quadratic Formula. -10 ± √(10 ) 2 - 4(1)(2) x = ___ 2(1) Substitute 1 for a, 10 for b, and 2 for c. -10 ± √100 - 8 -10 ± √92 x = __ = __ 2 2 Simplify. -10 ± 2 √23 x = __ = -5 ± √23 Write in simplest form. 2 Check Solve by completing the square. x 2 + 10x + 2 = 0 x 2 + 10x = -2 x 2 + 10x + 25 = -2 + 25 (x + 5) 2 = 23 x = -5 ± √23 ✔ Find the zeros of each function by using the Quadratic Formula. 1a. f (x) = x 2 + 3x - 7 1b. g (x) = x 2 - 8x + 10 EXAMPLE 2 Quadratic Functions with Complex Zeros Find the zeros of f (x) = 2x 2 - x + 2 by using the Quadratic Formula. 2x 2 - x + 2 = 0 Set f(x) = 0. -b ± √b 2 - 4ac x = __ 2a Write the Quadratic Formula. -(-1) ± √(-1) 2 - 4(2)(2) x = ___ 2(2) Substitute 2 for a, -1 for b, and 2 for c. 1 ± √1 - 16 1 ± √-15 x = __ = _ 4 4 Simplify. √15 1 ± i √15 1 ±_ x=_=_ i 4 4 4 Write in terms of i. 2. Find the zeros of g (x) = 3x 2 - x + 8 by using the Quadratic Formula. The discriminant is part of the Quadratic Formula that you can use to determine the number of real roots of a quadratic equation. -b ± √b - 4ac x = __ 2a 2 Discriminant 5- 6 The Quadratic Formula 357 Discriminant The discriminant of the quadratic equation ax 2 + bx + c = 0 (a ≠ 0) is b 2 - 4ac. b 2 - 4ac > 0 b 2 - 4ac = 0 b 2 - 4ac < 0 two distinct real solutions one distinct real solution EXAMPLE 3 two distinct nonreal complex solutions Analyzing Quadratic Equations by Using the Discriminant Find the type and number of solutions for each equation. A x 2 - 6x = -7 Make sure the equation is in standard form before you evaluate the discriminant, b 2 - 4ac. B x 2 - 6x = -9 C x 2 - 6x = -11 x 2 - 6x + 7 = 0 x 2 - 6x + 9 = 0 x 2 - 6x + 11 = 0 b 2 - 4ac b 2 - 4ac b 2 - 4ac (-6) 2 - 4(1)(7) (-6) 2 - 4(1)(9) (-6) 2 - 4(1)(11) 36 - 28 = 8 36 - 36 = 0 36 - 44 = -8 b - 4ac > 0; the equation has two distinct real solutions. b - 4ac = 0; the equation has one distinct real solution. b 2 - 4ac < 0; the equation has two distinct nonreal complex solutions. 2 2 Find the type and number of solutions for each equation. 3a. x 2 - 4x = -4 3b. x 2 - 4x = -8 3c. x 2 - 4x = 2 The graph shows the related functions for Example 3. Notice that the number of real solutions for the equation can be changed by changing the value of the constant c. { Ó ä Ó Ý®ÊÊÝÓÊÊÈÝÊ Ê££ }Ý®ÊÊÝÓÊÊÈÝÊ Ê Ý È vÝ®ÊÊÝÓÊÊÈÝÊ ÊÇ Double-Checking Roots If I get integer roots when I use the Quadratic Formula, I know that I can quickly factor to check the roots. Look at my work for the equation x 2 - 7x + 10 = 0. Quadratic Formula: Factoring: 2 x= Terry Cannon, Carver High School 358 -(-7) ± √(-7) - 4(1)(10) ___ 2(1) 7 ± √9 10 = or 4 = 5 or 2 = 2 2 2 Chapter 5 Quadratic Functions _ _ _ x 2 - 7x + 10 = 0 (x - 5)(x - 2) = 0 x = 5 or x = 2 EXAMPLE 4 Aviation Application The pilot of a helicopter plans to release a bucket of water on a forest fire. The height y in feet of the water t seconds after its release is modeled by y = -16t 2 - 2t + 500. The horizontal distance x in feet between the water and its point of release is modeled by x = 91t. At what horizontal distance from the fire should the pilot start releasing the water in order to hit the target? Path of water Release point Target x ft Step 1 Use the first equation to determine how long it will take the water to hit the ground. Set the height of the water equal to 0 feet, and use the quadratic formula to solve for t. y = -16t 2 - 2t + 500 0 = -16t 2 - 2t + 500 Set y equal to 0. -b ± √b 2 - 4ac t = __ 2a Use the Quadratic Formula. -(-2) ± √(-2) 2 - 4(-16)(500) t = ___ 2(-16) Substitute for a, b, and c. 2 ± √32,004 t = __ -32 Simplify. t ≈ -5.65 or t ≈ 5.53 The time cannot be negative, so the water lands on the target about 5.5 seconds after it is released. Once you have found the value of t, you have solved only part of the problem. You will use this value to find the answer you are looking for. Step 2 Find the horizontal distance that the water will have traveled in this time. x = 91t x = 91(5.5) Substitute 5.5 for t. x = 500.5 Simplify. The water will have traveled a horizontal distance of about 500 feet. Therefore, the pilot should start releasing the water when the horizontal distance between the helicopter and the fire is 500 feet. Check Use substitution to check that the water hits the ground after about 5.53 seconds. y = -16t 2 - 2t + 500 y = -16(5.53) 2 - 2(5.53) + 500 y ≈ -0.3544 ✔ The height is approximately equal to 0 when t = 5.53. Use the information given above to answer the following. 4. The pilot’s altitude decreases, which changes the function describing the water’s height to y = -16t 2 - 2t + 400. To the nearest foot, at what horizontal distance from the target should the pilot begin releasing the water? 5- 6 The Quadratic Formula 359 Summary of Solving Quadratic Equations Method Graphing When to Use Examples 2x 2 + 5x - 14 = 0 Only approximate solutions or the number of real solutions is needed. x ≈ -4.2 or x ≈ 1.7 Factoring No matter which method you use to solve a quadratic equation, you should get the same answer. Square roots c = 0 or the expression is easily factorable. x 2 + 4x + 3 = 0 (x + 3)(x + 1) = 0 x = -3 or x = -1 (x - 5)2 = 24 The variable side of the equation is a perfect square. (x - 5) 2 = ± √ 24 √ x - 5 = ±2 √ 6 x = 5 ± 2 √ 6 Completing the square a = 1 and b is an even number. x 2 + 6x = 10 x 2 + 6x + = 10 + () 6 x 2 + 6x + _ 2 2 () 6 = 10 + _ 2 2 (x + 3) 2 = 19 x = -3 ± √19 Quadratic Formula Numbers are large or complicated, and the expression does not factor easily. 5x - 7x - 8 = 0 2 (-7) 2 - 4(5)(-8) -(-7) ± √ x = ___ 2(5) 7 ± √ 209 x=_ 10 THINK AND DISCUSS 1. Describe how the graphs of quadratic functions illustrate the type and number of zeros. 2. Describe the values of c for which the equation x 2 + 8x + c = 0 will have zero, one, or two distinct solutions. 3. GET ORGANIZED Copy and complete the graphic organizer. Describe the possible solution methods for each value of the discriminant. 360 Chapter 5 Quadratic Functions �������� ������������ �������� ���� �������� �������� ��������� ����������������� ������� 5-6 Exercises KEYWORD: MB7 5-6 KEYWORD: MB7 Parent GUIDED PRACTICE 1. Vocabulary What information does the value of the discriminant give about a quadratic equation? SEE EXAMPLE 1 p. 357 SEE EXAMPLE 2 p. 357 SEE EXAMPLE 3 p. 358 SEE EXAMPLE 4 p. 359 Find the zeros of each function by using the Quadratic Formula. 2. f (x) = x 2 + 7x + 10 3. g(x) = 3x 2 - 4x - 1 4. h(x) = 3x 2 - 5x 5. g(x) = -x 2 - 5x + 6 6. h(x) = 4x 2 - 5x - 6 7. f (x) = 2x 2 - 19 8. f (x) = 2x 2 - 2x + 3 9. r(x) = x 2 + 6x + 12 10. h(x) = 3x 2 + 4x + 3 11. p(x) = x 2 + 4x + 10 12. g(x) = -5x 2 + 7x - 3 13. f (x) = 10x 2 + 7x + 4 Find the type and number of solutions for each equation. 14. 4x 2 + 1 = 4x 15. x 2 + 2x = 10 16. 2x - x 2 = 4 17. Geometry One leg of a right triangle is 6 in. longer than the other leg. The hypotenuse of the triangle is 25 in. What is the length of each leg to the nearest inch? PRACTICE AND PROBLEM SOLVING Independent Practice For See Exercises Example 18–23 24–29 30–35 36 1 2 3 4 Extra Practice Skills Practice p. S13 Application Practice p. S36 Find the zeros of each function by using the Quadratic Formula. 18. f (x) = 3x 2 - 10x + 3 19. g(x) = x 2 + 6x 20. h(x) = x(x - 3) - 4 21. g(x) = -x 2 - 2x + 9 22. p(x) = 2x 2 - 7x - 8 23. f (x) = 7x 2 - 3 24. r (x) = x 2 + x + 1 25. h(x) = -x 2 - x - 1 26. f (x) = 2x 2 + 8 27. f (x) = 2x 2 + 7x - 13 28. g(x) = x 2 - x - 5 29. h(x) = -3x 2 + 4x - 4 Find the type and number of solutions for each equation. 30. 2x 2 + 5 = 2x 31. 2x 2 - 3x = 8 32. 2x 2 - 16x = -32 33. 4x 2 - 28x = -49 34. 3x 2 - 8x + 8 = 0 35. 3.2x 2 - 8.5x + 1.3 = 0 36. Safety If a tightrope walker falls, he will land on a safety net. His height h in feet after a fall can be modeled by h(t) = 60 - 16t 2, where t is the time in seconds. How many seconds will the tightrope walker fall before landing on the safety net? 60 ft 37. Physics A bicyclist is riding at a speed of 20 mi/h when she starts down a long hill. The distance d she travels in feet can be 11 ft modeled by the function d(t) = 5t 2 + 20t, where t is the time in seconds. a. The hill is 585 ft long. To the nearest second, how long will it take her to reach the bottom? b. What if...? Suppose the hill were only half as long. To the nearest second, how long would it take the bicyclist to reach the bottom? 5- 6 The Quadratic Formula 361 Find the zeros of each function. Then graph the function. Aerospace SpaceShipOne was the winner of the Ansari X Prize competition. The X Prize was awarded to the first nongovernmental spacecraft to reach an altitude of at least 100 km twice within a 2 week period. 38. f (x) = 3x 2 - 4x - 2 39. g(x) = 2x 2 - 2x - 1 40. h(x) = 2x 2 + 6x + 5 41. p(x) = 2x 2 + 3x - 1 42. h(x) = 3x 2 - 5x - 4 43. r (x) = x 2 - x + 22 44. Aerospace In 2004, the highest spaceplane flight was made by Brian Binnie in SpaceShipOne. A flight with this altitude can be modeled by the function h(t) = -0.17t 2 + 187t + 61,000, where h is the altitude in meters and t is flight time in seconds. a. Approximately how long did the Earth’s Atmosphere flight last? Layer Altitude (in km) b. What was the highest altitude to the Troposphere 0 to 10 nearest thousand meters? c. The table shows the altitudes of layers Stratosphere 10 to 50 of Earth’s atmosphere. According to Mesosphere 50 to 85 the model, which of these layers did Thermosphere 85 to 600 SpaceShipOne enter, and at what time(s) did the spaceplane enter them? Solve each equation by any method. 45. x 2 - 3x = 10 46. x 2 - 16 = 0 47. 4x 2 + 4x = 15 48. x 2 + 2x - 2 = 0 49. x 2 - 4x - 21 = 0 50. 4x 2 - 4x - 1 = 0 51. 6x 2 = 150 52. x 2 = 7 53. x 2 - 16x + 64 = 0 54. Critical Thinking If you are solving a real-world problem involving a quadratic equation, and the discriminant is negative, what can you conclude? 55. Multi-Step The outer dimensions of a picture frame are 25 inches by 20 inches. If the area inside the picture frame is 266 square inches, what is the width w of the frame? w Critical Thinking Find the values of c that make each equation have one real solution. 56. x 2 + 8x + c = 0 57. x 2 + 12x = c 58. x 2 + 2cx + 49 = 0 59. Write About It What method would you use to solve the equation -14x 2 + 6x = 2.7? Why would this method be easier to use than the other methods? 60. This problem will prepare you for the Multi-Step Test Prep on page 364. An outfielder throws a baseball to the player on third base. The height h of the ball in feet is modeled by the function h(t) = -16t 2 + 19t + 5, where t is time in seconds. The third baseman catches the ball when it is 4 ft above the ground. a. To the nearest tenth of a second, how long was the ball in the air before it was caught? b. A player on the opposing team starts running from second base to third base 1.2 s before the outfielder throws the ball. The distance between the bases is 90 ft, and the runner’s average speed is 27 ft/s. Will the runner reach third base before the ball does? Explain. 362 Chapter 5 Quadratic Functions 61. Which best describes the graph of a quadratic function with a discriminant of -3? Parabola with two x-intercepts Parabola with no x-intercepts Parabola that opens upward Parabola that opens downward 62. What is the discriminant of the equation 2x 2 - 8x = 14? 48 176 -176 -48 63. Which function has zeros of 3 ± i? f(x) = x 2 + 6x + 10 f(x) = x 2 + 6x - 10 g(x) = x 2 - 6x + 10 h(x) = x 2 - 6x - 10 64. Which best describes the discriminant of the function whose graph is shown? Positive Zero Þ Negative Undefined Ý CHALLENGE AND EXTEND 65. Geometry The perimeter of a right triangle is 40 cm, and its hypotenuse measures 17 cm. Find the length of each leg. 66. Geometry The perimeter of a rectangle is 88 cm. a. Find the least possible value of the length of the diagonal. Round to the nearest tenth of a centimeter. b. What are the dimensions of the rectangle with this diagonal? Write a quadratic equation whose solutions belong to the indicated sets. 67. integers 68. irrational real numbers 69. complex numbers 70. A quadratic equation has the form ax 2 + bx + c = 0 (a ≠ 0). a. What is the sum of the roots of the equation? the product of the roots? b. Determine the standard form of a quadratic equation whose roots have a sum of 2 and a product of -15. 71. Describe the solutions to a quadratic equation for which a = b = c. SPIRAL REVIEW 72. Biology The length of a human hair is a linear function of time. Juan’s hair grows 2.1 cm in 60 days. Express the growth in centimeters of Juan’s hair as a function of the number of days since his last haircut. (Lesson 2-4) Write the augmented matrix, and use row reduction to solve. (Lesson 4-6) 3y = 2x + 7 2x = -3y + 12 4x + 5y = -1 73. 74. 75. x - 6y = 1 x + y = 14 9 + 7y = 2x Solve each equation by completing the square. (Lesson 5-4) 76. x 2 - 5x = 1 77. 2x 2 = 16x - 4 78. 3x = 5x 2 - 12 5- 6 The Quadratic Formula 363 SECTION 5A Quadratic Functions and Complex Numbers Ballpark Figures When a baseball is thrown or hit into the air, its height h in feet after t seconds can be modeled by h(t) = -16t 2 + v yt + h 0, where v y is the initial vertical velocity of the ball in feet per second and h 0 is the ball’s initial height. The horizontal distance d in feet that the ball travels in t seconds can be modeled by d(t) = v xt, where v x is the ball’s initial horizontal velocity in feet per second. 1. A short stop makes an error by dropping the ball. As the ball drops, its height h in feet is modeled by h(t) = -16t 2 + 3. A slow-motion replay of the error shows the play at half speed. What function describes the height of the ball in the replay? 2. A player hits a foul ball with an initial 90 ft vertical velocity of 70 ft/s and an initial height of 5 ft. To the nearest foot, what is the maximum height reached by the ball? 3. A pitch will be a strike if its height is between 2.5 ft and 5 ft when it crosses home plate. The pitcher throws the ball from a height of 6 ft with an initial vertical velocity of 5 ft/s and a horizontal velocity of 116 ft/s. Could this pitch be a strike? Explain. 4. The next pitch crosses home plate 1 ft too high to be a strike. The pitch is thrown from a height of 6 ft with an initial vertical velocity of 8 ft/s. What is the initial horizontal velocity of this pitch? 5. A player throws the ball home from a height of 5.5 ft with an initial vertical velocity of 28 ft/s. The ball is caught at home plate at a height of 5 ft. Three seconds before the ball is thrown, a runner on third base starts toward home plate at an average speed of 25 ft/s. Does the runner reach home plate before the ball does? Explain. 364 Chapter 5 Quadratic Functions Pitcher’s mound 90 ft 90 ft 60 ft 6 in. 90 ft Home plate SECTION 5A Quiz for Lessons 5-1 Through 5-6 5-1 Using Transformations to Graph Quadratic Functions Using the graph of f (x) = x 2 as a guide, describe the transformations, and then graph each function. 1 x2 + 1 1. g (x) = (x + 2)2 - 4 2. g (x) = -4(x - 1)2 3. g (x) = _ 2 Use the description to write each quadratic function in vertex form. 4. f (x) = x 2 is vertically stretched by a factor of 9 and translated 2 units left to create g. 5. f (x) = x 2 is reflected across the x-axis and translated 4 units up to create g. 5-2 Properties of Quadratic Functions in Standard Form For each function, (a) determine whether the graph opens upward or downward, (b) find the axis of symmetry, (c) find the vertex, (d) find the y-intercept, and (e) graph the function. 6. f (x)= x 2 - 4x + 3 7. g (x) = -x 2 + 2x - 1 8. h (x) = x 2 - 6x 9. A football kick is modeled by the function h(x) = -0.0075x 2 + 0.5x + 5, where h is the height of the ball in feet and x is the horizontal distance in feet that the ball travels. Find the maximum height of the ball to the nearest foot. 5-3 Solving Quadratic Equations by Graphing and Factoring Find the roots of each equation by factoring. 10. x 2 - 100 = 0 11. x 2 + 5x = 24 12. 4x 2 + 8x = 0 5-4 Completing the Square Solve each equation by completing the square. 13. x 2 - 6x = 40 14. x 2 + 18x = 15 15. x 2 + 14x = 8 Write each function in vertex form, and identify its vertex. 16. f (x) = x 2 + 24x + 138 17. g(x) = x 2 - 12x + 39 18. h(x) = 5x 2 - 20x + 9 5-5 Complex Numbers and Roots Solve each equation. 19. 3x 2 = -48 20. x 2 - 20x = -125 21. x 2 - 8x + 30 = 0 5-6 The Quadratic Formula Find the zeros of each function by using the Quadratic Formula. 22. f (x) = (x + 6)2 + 2 23. g (x) = x 2 + 7x + 15 24. h (x) = 2x 2 - 5x + 3 25. A bicyclist is riding at a speed of 18 mi/h when she starts down a long hill. The distance d she travels in feet can be modeled by d (t) = 4t 2 + 18t, where t is the time in seconds. How long will it take her to reach the bottom of a 400-foot-long hill? Ready to Go On? 365 5-7 Solving Quadratic Inequalities Who uses this? Tour companies and other businesses use quadratic inequalities to make predictions of profits. (See Example 4.) Objectives Solve quadratic inequalities by using tables and graphs. Solve quadratic inequalities by using algebra. Vocabulary quadratic inequality in two variables Many business profits can be modeled by quadratic functions. To ensure that the profit is above a certain level, financial planners may need to graph and solve quadratic inequalities. A quadratic inequality in two variables can be written in one of the following forms, where a, b, and c are real numbers and a ≠ 0. Its solution set is a set of ordered pairs (x, y). y < ax 2 + bx + c y ≤ ax 2 + bx + c y > ax 2 + bx + c y ≥ ax 2 + bx + c In Lesson 2-5, you solved linear inequalities in two variables by graphing. You can use a similar procedure to graph quadratic inequalities. Graphing Quadratic Inequalities To graph a quadratic inequality T Þ 1. Graph the parabola that defines the boundary. 1 2. Use a solid parabola for y ≤ and y ≥ and a dashed parabola for y < and y >. Ý 3. Shade above the parabola for y > or ≥ and below the parabola for y ≤ or <. EXAMPLE 1 Graphing Quadratic Inequalities in Two Variables Graph y < -2x 2 - 4x + 6. Step 1 Graph the boundary of the related parabola y = -2x 2 - 4x + 6 with a dashed curve. Þ È Its y-intercept is 6, its vertex is (-1, 8), and its x-intercepts are -3 and 1. Step 2 Shade below the parabola because the solution consists of y-values less than those on the parabola for corresponding x-values. Check Use a test point to verify the solution region. y < -2x 2 - 4x + 6 0 < -2(0) 2 - 4(0) + 6 Try (0, 0). 0<6✔ 366 Chapter 5 Quadratic Functions { Ó { ä]Êä® Ó ä Ý { Graph each inequality. 1a. y ≥ 2x 2 - 5x - 2 1b. y < -3x 2 - 6x - 7 Quadratic inequalities in one variable, such as ax 2 + bx + c > 0 (a ≠ 0), have solutions in one variable that are graphed on a number line. EXAMPLE 2 Solving Quadratic Inequalities by Using Tables and Graphs Solve each inequality by using tables or graphs. A x 2 - 6x + 8 ≤ 3 Use a graphing calculator to graph each side of the inequality. Set Y1 equal to x 2 - 6x + 8 and Y2 equal to 3. Identify the values of x for which Y1 ≤ Y2. The parabola is at or below the line when x is between 1 and 5 inclusive. So, the solution set is 1 ≤ x ≤ 5, or 1, 5. The table supports your answer. The number line shows the solution set. Ó £ ä £ Ó Î { x È Ç n B x 2 - 6x + 8 > 3 Use a graphing calculator to graph each side of the inequality. Set Y1 equal to x 2 - 6x + 8 and Y2 equal to 3. Identify the values of x for which Y1 > Y2. For and statements, both of the conditions must be true. For or statements, at least one of the conditions must be true. The parabola is above the line y = 3 when x is less than 1 or greater than 5. So the solution set is x < 1 or x > 5, or (-∞, 1) (5, ∞). The number line shows the solution set. Ó £ ä £ Ó Î { x È Ç n Solve each inequality by using tables or graphs. 2a. x 2 - x + 5 < 7 2b. 2x 2 - 5x + 1 ≥ 1 The number lines showing the solution sets in Example 2 are divided into three distinct regions by the points 1 and 5. These points are called critical values. By finding the critical values, you can solve quadratic inequalities algebraically. 5- 7 Solving Quadratic Inequalities 367 EXAMPLE 3 Solving Quadratic Inequalities by Using Algebra Solve the inequality x 2 - 4x + 1 > 6 by using algebra. Step 1 Write the related equation. x 2 - 4x + 1 = 6 Step 2 Solve the equation for x to find the critical values. x 2 - 4x - 5 = 0 (x - 5)(x + 1) = 0 x - 5 = 0 or x + 1 = 0 x = 5 or x = -1 Write in standard form. Factor. Zero Product Property Solve for x. The critical values are 5 and -1. The critical values divide the number line into three intervals: x < -1, -1 < x < 5, and x > 5. Step 3 Test an x-value in each interval. #RITICAL VALUES x - 4x + 1 > 6 2 (-2) - 4(-2) + 1 > 6 ✔ Try x = -2. 2 Î Ó £ ä £ Ó Î { x È Ç { x È Ç (0) 2 - 4(0) + 1 > 6 ✘ Try x = 0. 4EST POINTS (6) 2 - 4(6) + 1 > 6 ✔ Try x = 6. Shade the solution regions on the number line. Use open circles for the critical values because the inequality does not contain or equal to. Î Ó £ ä £ Ó Î The solution is x < -1 or x > 5, or (-∞, -1) (5, ∞). Solve each inequality by using algebra. 3a. x 2 - 6x + 10 ≥ 2 3b. -2x 2 + 3x + 7 < 2 EXAMPLE 4 Problem-Solving Application 1 Chapter 5 Quadratic Functions Barcelos ver zon R i R s e i õ v Ama er Solim Manaus Tefé Understand the Problem The answer will be the number of people required for a profit that is greater than or equal to $5000. List the important information: • The profit must be at least $5000. • The function for the profit is P(x) = -25x 2 + 1000x -3000. 368 Río N egro Belém Xingu River A business offers tours to the Amazon. The profit P that the company earns for x number of tourists can be modeled by P(x) = -25x 2 + 1000x - 3000. How many people are needed for a profit of at least $5000? B R A Z I L Travel Brazil 2 Make a Plan Write an inequality showing profit greater than or equal to $5000. Then solve the inequality by using algebra. 3 Solve Write the inequality. -25x 2 + 1000x - 3000 ≥ 5000 Find the critical values by solving the related equation. -25x 2 + 1000x - 3000 = 5000 Write as an equation. -25x 2 + 1000x - 8000 = 0 Write in standard form. -25(x 2 - 40x + 320) = 0 Factor out -25 to simplify. Use the -(-40) ± √( -40) 2 - 4(1)(320) -b ± √ b 2 - 4ac __ ___ Quadratic x= = 2a 2(1) Formula. 40 ± √ 320 = _ 2 Simplify. x ≈ 28.94 or x ≈ 11.06 Test an x-value in each of the three regions formed by the critical x-values. x -25(10) 2 + 1000(10) - 3000 5000 £ä £x Óä Óx Îä Îx Try x = 10. 4500 ≥ 5000 ✘ -25(20) + 1000(20) - 3000 5000 2 Try x = 20. 7000 ≥ 5000 ✓ -25(30) + 1000(30) - 3000 5000 2 Try x = 30. 4500 ≥ 5000 ✘ A compound inequality such as 12 ≤ x ≤ 28 can be written as x | x ≥ 12 x ≤ 28 , or x ≥ 12 and x ≤ 28. (See Lesson 2-8.) Write the solution as an inequality. The solution is approximately 11.06 ≤ x ≤ 28.94. Because you cannot have a fraction of a person, round each critical value to the appropriate whole number. 12 ≤ x ≤ 28 For a profit of at least $5000, from 12 to 28 people are needed. 4 Look Back Enter y = -25x 2 + 1000x - 3000 into a graphing calculator, and create a table of values. The table shows that integer values of x between 12 and 28 inclusive result in y-values greater than or equal to 5000. 4. The business also offers educational tours to Patagonia, a region of South America that includes parts of Chile and Argentina. The profit P for x number of persons is P(x) = -25x 2 + 1250x - 5000. The trip will be rescheduled if the profit is less than $7500. How many people must have signed up if the trip is rescheduled? 5- 7 Solving Quadratic Inequalities 369 THINK AND DISCUSS 1. Compare graphing a quadratic inequality with graphing a linear inequality. 2. Explain how to determine if the intersection point(s) is/are included in the solution set when you solve a quadratic inequality by graphing. 3. GET ORGANIZED Copy and complete the graphic organizer. Compare the solutions of quadratic equations and inequalities. µÕ>Ì ® »iÃÃÊ/ >» iµÕ>ÌÞ ÀÊɠ® ºÀi>ÌiÀÊ/ >» iµÕ>ÌÞ ÀÊɡ® Ý>«i À>« -ÕÌÊ-iÌ 5-7 Exercises KEYWORD: MB7 5-7 KEYWORD: MB7 Parent GUIDED PRACTICE 1. Vocabulary Give an example of a quadratic inequality in two variables. SEE EXAMPLE 1 SEE EXAMPLE 2 3 p. 368 SEE EXAMPLE 4 p. 368 3. y ≤ 2x 2 - 4x - 1 4. y ≤ -3x 2 + x + 3 Solve each inequality by using tables or graphs. 5. x 2 - 5x + 3 ≤ 3 p. 367 SEE EXAMPLE Graph each inequality. 2. y > -(x + 1) 2 + 5 p. 366 6. 3x 2 - 3x - 1 > -1 7. 2x 2 - 9x + 5 ≤ -4 Solve each inequality by using algebra. 8. x 2 + 10x + 1 ≥ 12 9. x 2 + 13x + 45 < 5 10. -2x 2 + 3x + 12 > 10 11. Business A consultant advises the owners of a beauty salon that their profit p each month can be modeled by p(x) = -50x 2 + 3500x - 2500, where x is the average cost that a customer is charged. What range of costs will bring in a profit of at least $50,000? PRACTICE AND PROBLEM SOLVING Graph each inequality. 12. y < x 2 + 2x - 5 15. y ≥ x 2 + 6 1 x2 + 3 13. y > -_ 2 16. y < (x + 1)(x + 4) 14. y ≤ 2(x - 1) 2 - 3 17. y ≤ x 2 - 2x + 6 Solve each inequality by using tables or graphs. 370 18. x 2 - x + 5 < 11 19. 2x 2 + 3x + 6 ≥ 5 20. x 2 - 5x + 12 > 6 21. x 2 - 2x - 8 > 0 22. x 2 + 7x + 6 ≤ 6 23. x 2 - 12x + 32 < 12 Chapter 5 Quadratic Functions Independent Practice For See Exercises Example 12–17 18–23 24–26 27 1 2 3 4 Extra Practice Skills Practice p. S13 Application Practice p. S36 Solve each inequality by using algebra. 24. x 2 - 11x + 13 ≤ 25 25. -2x 2 + 3x + 4 ≥ -1 26. x 2 - 5x - 4 < -9 27. Sports A football thrown by a quarterback follows a path given by h(x) = -0.0095x 2 + x + 7, where h is the height of the ball in feet and x is the horizontal distance the ball has traveled in feet. If any height less than 10 feet can be caught or knocked down, at what distances from the quarterback can the ball be knocked down? Graph each quadratic inequality. 28. y ≤ 2x 2 + 4x - 3 29. y < 3x 2 - 12x - 4 31. y > -2(x + 3) 2 + 1 32. y > -x 2 - 2x - 1 34. Circus The human cannonball is an act where a performer is launched through the air. The height of the performer can be modeled by h(x) = -0.007x 2 + x + 20, where h is the height in feet and x is the horizontal distance traveled in feet. The circus act is considering a flight path directly over the main tent. 30. y ≥ -3x 2 + 4x 1 x 2 + 2x - 1 33. y ≤ _ 3 At least 5 ft a. If the performer wants at least 5 ft of vertical height clearance, how tall can the tent be? b. How far from the central pole should the “cannon” be placed? Solve each inequality by using any method. 35. x 2 - 5x - 24 ≤ 0 36. x 2 - 14 ≥ 2 37. -2x 2 - x + 8 > 6 38. x 2 - 4x - 5 ≤ -9 39. 3x 2 + 6x + 11 < 10 40. 4x 2 - 9 > 0 41. 3x 2 + 5x + 13 ≤ 16 42. -2x 2 + 3x + 17 ≥ 11 43. 5x 2 - 2x - 1 ≥ 0 44. (x - 2)(x + 11) ≥ 2 45. x 2 + 27 > 12x 46. -2x 2 + 3x + 6 > 0 47. Multi-Step A medical office has a rectangular parking lot that measures 120 ft by 200 ft. The owner wants to expand the size of the parking lot by adding an equal distance to two sides as shown. If zoning restrictions limit the total size of the parking lot to 35,000 ft 2, what range of distances can be added? Ý £ÓäÊvÌ Ý ÓääÊvÌ Match each graph with one of the following inequalities. A. y < x 2 + 2x - 3 48. B. y > -x 2 - 2x + 3 49. Þ Ý { Ó Ó 50. Þ È Þ Ý Ó ä Ó C. y < x 2 - 2x + 3 Ó { Ó ä Ó Ó Ó Ý Ó ä Ó { 5- 7 Solving Quadratic Inequalities 371 51. This problem will prepare you for the Multi-Step Test Prep on page 390. A small square tile is placed on top of a larger square tile as shown. This creates four congruent triangular regions. a. Write a function for the area A of one of the triangular regions in terms of x. Ý b. For what values of x, to the nearest tenth, is the area of each triangular region at least 30 cm 2? ÓäÊV c. For what values of x, to the nearest tenth, is the area of each triangular region less than 40 cm 2? 52. Music A manager estimates a band’s profit p for a concert by using the function p(t) = -200t 2 + 2500t - c, where t is the price per ticket and c is the band’s operating cost. The table shows the band’s operating cost at three different concert locations. What range of ticket prices should the band charge at each location in order to make a profit of at least $1000 at each concert? 53. Gardening Lindsey has 40 feet of metal fencing material to fence three sides of a rectangular garden. A tall wooden fence serves as her fourth side. a. Write a function for the area of the garden A in terms of x, the width in feet. b. What measures for the width will give an area of at least 150 square feet? c. What measures for the width will give an area of at least 200 square feet? Band’s Costs Location Operating Cost Freemont Park $900 Saltillo Plaza $1500 Riverside Walk $2500 x Graphing Calculator Use the intersect feature of a graphing calculator to solve each inequality to the nearest tenth. 54. x 2 + 6x - 13 > 4 55. x 2 - 15x + 20 ≤ 7 56. x 2 - 24 < 28 57. 2x 2 + 3x + 5 ≥ 8 58. Business A wholesaler sells snowboards to sporting-good stores. The price per snowboard varies based on the number purchased in each order. The function r(x) = -x 2 + 125x models the wholesaler’s revenue r in dollars for an order of x snowboards. a. To the nearest dollar, what is the maximum revenue per order? b. How many snowboards must the wholesaler sell to make at least $1500 in revenue in one order? 59. Critical Thinking Explain whether the solution to a quadratic inequality in one variable is always a compound inequality. 60. Critical Thinking Can a quadratic inequality have a solution set that is all real numbers? Give an example to support your answer. 61. Write About It Explain how the solutions of x 2 - 3x - 4 ≤ 6 differ from the solutions of x 2 - 3x - 4 = 6. 372 Chapter 5 Quadratic Functions 62. Which is the solution set of x 2 - 9 < 0? -3 < x < 3 -9 < x < 9 x < -3 or x > 3 x < -9 or x > 9 63. Which is the graph of the solution to x 2 - 7x + 10 ≥ 0? È x { Î Ó £ ä £ Ó Î { x È x { Î Ó £ ä ä È 64. Which is the solution set of x 2 - 7x ≤ 0? 0<x<7 0≤x≤7 £ Ó Î { x ä È x < 0 or x > 7 x ≤ 0 or x ≥ 7 65. Short Response Demonstrate the process for solving x 2 + 4x + 4 > 1 algebraically. Justify each step in the solution process. CHALLENGE AND EXTEND Graph each system of inequalities. 66. { y ≤ x2 y ≥ -x 2 + 5 67. { y ≥ x2 - 3 68. y ≤ -x 2 - 2x + 9 { y ≥ 2x 2 - 12x + 20 1 x 2 - 2x + 8 y≥_ 3 Geometry The area inside a parabola bounded from above or below by a horizontal line segment is __23 bh, where b is the length of the line segment and h is the vertical distance from the vertex of the parabola to the line segment. Find the area bounded by the graphs of each pair of inequalities. 69. y > x 2 + 5x - 6; y < 8 Þ Ý L 70. y < -2x 2 + 3x + 9; y > -5 SPIRAL REVIEW 71. Community Once a month, four teams of teens (lawn team, shopping team, cleaning team, and laundry team) spend a day assisting elderly residents of their neighborhood. Lynnette started the assignment chart for June but was interrupted. Complete the chart. Each home has only one team helping during each shift. (Previous course) Shifts Reed Home Brown Home Sondi Home Clem Home Lawn Cleaning ? ? 10:00 A.M.–12:30 P.M. ? Shopping ? Lawn 1:00 P.M.–3:30 P.M. ? ? Laundry ? 4:00 P.M.–6:30 P.M. Cleaning ? ? ? 7:00 A.M.–9:30 A.M. Graph each inequality by using intercepts. (Lesson 2-5) 72. 4x - 3y > 15 73. 6x - y ≤ 8 74. 8x + 5y < 40 Find the values of c that make each equation true. (Lesson 5-5) 75. 4 - 2c + 7i = 7i - 14 76. 4c + 2 - 3i + 2(i - 5) = 4(2i - 6) - 9i 5- 7 Solving Quadratic Inequalities 373 5-8 IA-3.5 Analyze given information models) to for solve contextual A2.3.5 problems that be(including modeled quadratic using quadratic equations functions, 3.1b Solve Identify whether thecan model/equation is a curve of best fit theand data, using problems. interpret the solutions and determine solutionscalculator. are reasonable. various methods and tools which maywhether include the a graphing Who uses this? Film preservationists use quadratic relationships to estimate film run times. (See Example 3.) Objectives Use quadratic functions to model data. Use quadratic models to analyze and predict. New York Recall that you can use differences to analyze patterns in data. For a set of ordered pairs with equally spaced x-values, a quadratic function has constant nonzero second differences, as shown below. r Collection, Vocabulary quadratic model quadratic regression The Grange */ Curve Fitting with Quadratic Models Equally spaced x-values ⎧ ⎨ ⎩ x -3 -2 -1 0 1 2 3 f (x) = x 2 9 4 1 0 1 4 9 -5 1st differences 2nd differences -3 2 -1 2 1 2 3 2 Þ n È 5 2 { Constant 2nd differences vÝ®ÊÊÊÝÊÓÊ Ó Ý Ó EXAMPLE 1 ä Ó Identifying Quadratic Data Determine whether each data set could represent a quadratic function. Explain. A x 0 2 4 6 8 y 12 10 9 9 10 Find the first and second differences. Equally spaced x-values B x y 2 4 8 16 12 10 9 9 10 y -2 -1 0 1 2 4 8 16 1 1st 2nd 2 1 2 1 4 2 ⎧ y x 8 ⎨ 6 4 ⎩ ⎧ ⎨ ⎩ 2 Quadratic function; second differences are constant for equally spaced x-values. 2 1 Equally spaced x-values 0 -2 -1 0 1 1 1 1 1 0 Find the first and second differences. x 1st 2nd -2 -1 8 4 Not a quadratic function; second differences are not constant for equally spaced x-values. Determine whether each data set could represent a quadratic function. Explain. 1a. x 3 4 5 6 7 1b. x 10 9 8 7 6 y 374 Chapter 5 Quadratic Functions 11 21 35 53 75 y 6 8 10 12 14 Just as two points define a linear function, three noncollinear points define a quadratic function. You can find the three coefficients, a, b, and c, of f (x) = ax 2 + bx + c by using a system of three equations, one for each point. The points do not need to have equally spaced x-values. EXAMPLE 2 Writing a Quadratic Function from Data Write a quadratic function that fits the points (0, 5), (2, 1), and (3, 2). Use each point to write a system of equations to find a, b, and c in f (x) = ax 2 + bx + c. Collinear points lie on the same line. Noncollinear points do not all lie on the same line. (x, y) f(x) = ax 2 + bx + c (0, 5) (2, 1) (3, 2) 5 = a(0) + b(0) + c 2 1 = a(2) + b(2) + c 2 2 = a(3) + b(3) + c 2 System in a, b, c c = 5 1 4a + 2b + c = 1 2 9a + 3b + c = 2 3 Substitute c = 5 from equation 1 into both equation 2 and equation 3. 2 4a + 2b + c = 1 9a + 3b + c = 2 3 4a + 2b + 5 = 1 9a + 3b + 5 = 2 4a + 2b = -4 4 9a + 3b = -3 5 Solve equation 4 and equation 5 for a and b using elimination. 4 5 3(4a + 2b) = 3(-4) → 12a + 6b = -12 -2(9a + 3b) = -2(-3) → -18a - 6b = 6 −−−−−−−−−−−− -6a = -6 a =1 Multiply by 3. Multiply by -2. Add the equations. Substitute 1 for a into equation 4 or equation 5 to find b. 4 4a + 2b = -4 → 4(1) + 2b = -4 2b = -8 b = -4 Write the function using a = 1, b = -4, and c = 5. f (x) = ax 2 + bx + c → f (x) = 1x 2 - 4x + 5, or f (x) = x 2 - 4x + 5 Check Substitute or create a table to verify that (0, 5), (2, 1), and (3, 2) satisfy the function rule. 2. Write a quadratic function that fits the points (0, -3), (1, 0), and (2, 1). You may use any method that you studied in Chapters 3 or 4 to solve the system of three equations in three variables. For example, you can use a matrix equation as shown. c=5 0 0 1a 5 a 1 → 4a + 2b + c = 1 4 2 1 b = 1 → b = -4 9a + 3b + c = 2 9 3 1 c 2 c 5 5- 8 Curve Fitting with Quadratic Models 375 A quadratic model is a quadratic function that represents a real data set. Models are useful for making estimates. In Chapter 2, you used a graphing calculator to perform a linear regression and make predictions. You can apply a similar statistical method to make a quadratic model for a given data set using quadratic regression. EXAMPLE 3 Film Application The table shows approximate run times for 16 mm films, given the diameter of the film on the reel. Find a quadratic model for the run time given the diameter. Use the model to estimate the run time for a reel of film with a diameter of 15 in. The coefficient of determination R 2 shows how well a quadratic model fits the data. The closer R 2 is to 1, the better the fit. In this model, R 2 ≈ 0.996, which is very close to 1, so the quadratic model is a good fit. Film Run Times (16 mm) Diameter (in.) Reel Length (ft) Run Time (min) 5 200 5.55 7 400 11.12 9.25 600 16.67 10.5 800 22.22 12.25 1200 33.33 13.75 1600 44.45 Step 1 Enter the data into two lists in a graphing calculator. Step 2 Use the quadratic regression feature. Step 3 Graph the data and function model to verify that the model fits the data. Step 4 Use the table feature to find the function value at x = 15. xä Î £x ä A quadratic model is T (d) ≈ 0.397d 2 - 3.12d + 11.94, where T is the run time in minutes and d is the film diameter in inches. For a 15 in. diameter, the model predicts a run time of about 54.5 min, or 54 min 30 s. Use the information given above to answer the following. 3. Find a quadratic model for the reel length given the diameter of the film. Use the model to estimate the reel length for an 8-inch-diameter film. 376 Chapter 5 Quadratic Functions THINK AND DISCUSS 1. Describe how to determine if a data set is quadratic. 2. Explain whether a quadratic function is a good model for the path of an airplane that ascends, descends, and rises again out of view. 3. GET ORGANIZED Copy and complete the graphic organizer. Compare the different quadratic models presented in the lesson. 5-8 +Õ>`À>ÌVÊ`i 7 iÊ««À«À>Ìi *ÀVi`ÕÀi Ý>VÌÊ`i ««ÀÝ>Ìi `i Exercises KEYWORD: MB7 5-8 KEYWORD: MB7 Parent GUIDED PRACTICE 1. Vocabulary How does a quadratic model differ from a linear model? SEE EXAMPLE 1 p. 374 SEE EXAMPLE 2. 2 p. 375 SEE EXAMPLE p. 376 Determine whether each data set could represent a quadratic function. Explain. 3 x -2 -1 0 1 2 y 0 -8 -16 16 8 3. x 1 2 3 4 5 y 1 3 9 27 81 4. x 2 y 4 4 6 8 10 -5 -8 -5 4 Write a quadratic function that fits each set of points. 5. (-2, 5), (0, -3), and (3, 0) 6. (0, 1), (2, -1), and (3, -8) 7. (-1, 8), (0, 4), and (2, 2) 8. (-4, 9), (0, -7), and (1, -1) 9. (2, 3), (6, 3), and (8, -3) 10. (-1, -12), (1, 0), and (2, 9) 11. Hobbies The cost of mounting different-sized photos is shown in the table. Find a quadratic model for the cost given the average side length. (For an 8 in. × 10 in. photo, the 8 + 10 average side length is _____ = 9 in.) 2 Estimate the cost of mounting a 24 in. × 36 in. photo. Costs of Mounting Photos Size (in.) Cost ($) 8 × 10 10 14 × 18 16 16 × 20 19 24 × 30 27 32 × 40 39 PRACTICE AND PROBLEM SOLVING Determine whether each data set could represent a quadratic function. Explain. 12. x 0 f (x) -1 2 4 6 8 2 11 26 47 13. x 0 1 2 3 4 f (x) 10 9 6 1 -6 14. x 1 f (x) -3 2 3 4 5 0 3 6 9 5- 8 Curve Fitting with Quadratic Models 377 Independent Practice For See Exercises Example 12–14 15–18 19 1 2 3 Extra Practice Skills Practice p. S13 Application Practice p. S36 Write a quadratic function that fits each set of points. 15. (-2, 5), (-1, 0), and (1, -2) 16. (1, 2), (2, -1), and (5, 2) 17. (-4, 12), (-2, 0), and (2, -12) 18. (-1, 2.6), (1, 4.2), and (2, 14) 19. Gardening The table shows the amount spent on water gardening in the United States between 1999 and 2003. Find a quadratic model for the annual amount in millions of dollars spent on water gardening based on number of years since 1999. Estimate the amount that people in the United States will spend on water gardening in 2015. Water Gardening Amount Spent (million $) Year Write a function rule for each situation, and identify each relationship as linear, quadratic, or neither. 1999 806 2000 943 2001 1205 2002 1441 2003 1565 20. the circumference C of a bicycle wheel, given its radius r 21. the area of a triangle A with a constant height, given its base length b 22. the population of bacteria P in a petri dish doubling every hour t 23. the area of carpet A needed for square rooms of length s 24. Physics In the past, different Relative Distance Fallen (units) mathematical descriptions of Time Interval Aristotle’s da Vinci’s Galileo’s falling objects were proposed. (s) Rule Rule Rule a. Which rule shows the 0 0 0 0 greatest increase in the distance fallen per second and 1 1 1 1 thus the greatest rate 2 2 3 4 of increase in speed? 3 3 6 9 b. Identify each rule as linear, 4 4 10 16 quadratic, or neither. c. Describe the differences in da Vinci’s rule, and compare it with the differences in Galileo’s. d. The most accurate rule is sometimes described as the odd-number law. Which rule shows an odd-number pattern of first differences and correctly describes the distance for falling objects? Find the missing value for each quadratic function. 25. x f (x) -1 0 0 1 1 0 2 3 -8 26. x -3 -2 -1 0 f (x) 12 2 0 1 8 27. x -2 f (x) -2 0 2 4 6 2 7 14 28. This problem will prepare you for the Multi-Step Test Prep on page 390. A home-improvement store sells several sizes of Length (in.) Area (in 2) rectangular tiles, as shown in the table. a. Find a quadratic model for the area of a tile based 4 28 on its length. 6 54 b. The store begins selling a new size of tile with a 8 88 length of 9 in. Based on your model, estimate the 10 130 area of a tile of this size. 378 Chapter 5 Quadratic Functions 29. Food The pizza prices for DeAngelo’s pizza parlor are shown at right. a. Find a quadratic model for the price of a pizza based upon the size (diameter). b. Use the quadratic model to find the price of a pizza with an 18 in. diameter. c. Graph the quadratic function. Does the function have a minimum or maximum point? What does this point represent? d. What if...? According to the model, how much should a 30 in. pizza cost? How much should an 8 in. pizza cost? e. Is the quadratic function a good model for the price of DeAngelo’s pizza? Explain your reasoning. Determine whether each data set could represent a quadratic function. If so, find a quadratic function rule. 30. x 0 1 2 3 4 31. y -1 0 -1 -4 -9 33. x -2 -1 0 1 2 y 0 7 24 16 3 x 1 2 3 4 5 32. y 10 20 40 60 80 34. x 0 1 2 3 4 y 9 5 3 1 0 36. Winter Sports The diagram shows the motion of a skier following a jump. Find a quadratic model of the skier’s height h in meters based on time t in seconds. Estimate the skier’s height after 2 s. x 2 4 6 8 10 0 1 3 5 x -2 -1 0 1 2 y 9 27 81 y -1 35. 0 3 t = 1.1 s h = 18.7 m t=0s h = 13.2 m 37. Data Collection Use a graphing calculator and a motion detector to measure the height of a basketball over time. Drop the ball from t = 3.0 s a height of 1 m, and let it bounce several h=0m times. Position the motion detector 0.5 m above the release point of the ball. a. What is the greatest height the ball reaches during its first bounce? b. Find an appropriate model for the height of the ball as a function of time during its first bounce. 38. Safety The light produced High-Pressure Sodium Vapor Streetlamps by high-pressure sodium Energy Use vapor streetlamps for different 35 50 70 100 150 (watts) energy usages is shown in the table. Light Output 2250 4000 5800 9500 16,000 (lumens) a. Find a quadratic model for the light output with respect to energy use. b. Find a linear model for the light output with respect to energy use. c. Apply each model to estimate the light output in lumens of a 200-watt bulb. d. Which model gives the better estimate? Explain. 5- 8 Curve Fitting with Quadratic Models 379 39. Sports The table lists the average distance that a normal shot travels for different golf clubs. 2 iron Average Distance for Normal Shot Club Iron (no.) 2 3 4 5 6 7 8 9 Loft Angle 16° 20° 24° 28° 32° 36° 40° 44° Distance (yd) 186 176 166 155 143 132 122 112 16º 9 iron a. Select three data values (club number, distance), and use a system of equations to find a quadratic model. Check your model by using a quadratic regression. b. Is there a quadratic relationship between club number and average distance of a normal shot? Explain. c. Is the relationship between club number and loft angle quadratic or linear? Find a model of this relationship. Math History Pythagoras made numerous contributions to mathematics, including the Pythagorean Theorem, which bears his name. 40. Multi-Step Use the table of alloy-steel chain data. a. Do each of the last two columns appear to be quadratic functions with respect to the nominal chain size? Explain. b. Verify your response in part a by finding each of the quadratic regression equations. Do the models fit the data well? Explain. c. Predict the values for the last two columns for a chain with a nominal size of __85 in. 44º Alloy-Steel Chain Specifications Nominal Size (in.) Maximum Length 100 Links (in.) Maximum Weight 100 Links (lb) 1 _ 4 1 _ 2 3 _ 4 98 84 156 288 208 655 1 277 1170 1 1_ 4 371 1765 41. Math History The Greek mathematician Pythagoras developed a formula for triangular numbers, the first four of which are shown. Write a quadratic function that determines a triangular number t in terms of its place in the sequence n. (Hint: The fourth triangular number has n = 4.) 42. Critical Thinking Two points define a unique line. How many points define a unique parabola, and what restriction applies to the points? 43. Critical Thinking Consider the following data set. x 10 8 13 9 11 14 6 4 12 7 5 y 9.14 8.14 8.74 8.77 9.29 8.1 6.13 3.1 9.13 7.26 4.74 a. b. c. d. Create a scatter plot of the data. Perform a linear regression on the data. Perform a quadratic regression on the data. Which model best describes the data set? Explain your answer. 44. Write About It What does it mean when the coefficient a in a quadratic regression model is zero? 380 Chapter 5 Quadratic Functions Loft angle 45. Which of the following would best be modeled by a quadratic function? Relationship between circumference and diameter Relationship between area of a square and side length Relationship between diagonal of a square and side length Relationship between volume of a cube and side length 46. If (7, 11) and (3, 11) are two points on a parabola, what is the x-value of the vertex of this parabola? 11 3 5 7 47. If y is a quadratic function of x, which value completes the table? x -2 0 2 4 y -8 0 12 28 6 48 44 20 12 48. The graph of a quadratic function having the form f (x) = ax 2 + bx + c passes through the points (0, -8), (3, 10), and (6, 34). What is the value of the function when x = -3? -32 -26 -20 10 49. Extended Response Write a quadratic function in standard form that fits the data points (0, -5), (1, -3), and (2, 3). Use a system of equations, and show all of your work. CHALLENGE AND EXTEND 50. Three points defining a quadratic function are (1, 2), (4, 6), and (7, w). a. If w = 9, what is the quadratic function? Does it have a maximum value or a minimum value? What is the vertex? b. If w = 11, what is the quadratic function? Does it have a maximum value or a minimum value? What is the vertex? c. If w = 10, what function best fits the points? 51. Explain how you can determine from three points whether the parabola that fits the points opens upward or downward. SPIRAL REVIEW Determine whether each data set could represent a linear function. (Lesson 2-3) 52. -2 1 4 f (x) -5 7 1 x 53. x -8 -6 f (x) -1 0 0 3 Find the inverse of the matrix, if it is defined. (Lesson 4-5) 54. 1 0 __ 3 -4 1 2 -2 55. 1 -1 56. -2 0 1 0 0 1 4 2 2 57. 3 -4 0 -__12 Find the zeros of each function by using the Quadratic Formula. (Lesson 5-6) 58. f (x) = 2x 2 - 4x + 1 59. f (x) = x 2 + 9 60. f (x) = -3x 2 + 10x + 12 5- 8 Curve Fitting with Quadratic Models 381 5-9 */ Operations with Complex Numbers A2.3.1 add, subtract, and divide complex numbers. Represent complex IA-3.2 Carry out a procedure todivide, perform with complex numbers 1.3b Define, Add, subtract, multiply,multiply andoperations simplify expressions involving numbers, and the addition, subtraction and absolute value of complex numbers, (including addition, subtraction, multiplication, and division). complex numbers. in the complex plane. Why learn this? Complex numbers can be used in formulas to create patterns called fractals. (See Exercise 84.) Objective Perform operations with complex numbers. Vocabulary complex plane absolute value of a complex number Just as you can represent real numbers graphically as points on a number line, you can represent complex numbers in a special coordinate plane. >}>ÀÞÊ>ÝÃ The complex plane is a set of coordinate axes in which the horizontal axis represents real numbers and the vertical axis represents imaginary numbers. Ó äÊ Êä Ó Ó ,i> >ÝÃ Ó EXAMPLE 1 Graphing Complex Numbers >}>ÀÞÊ>ÝÃ Graph each complex number. ÓÊ { A -3 + 0i The real axis corresponds to the x-axis, and the imaginary axis corresponds to the y-axis. Think of a + bi as x + yi. { {Ê Î Ó B -3i ÎÊ ä { C 4 + 3i Ó ä Ó { ,i> >ÝÃ Ó Î D -2 + 4i { Graph each complex number. 1a. 3 + 0i 1b. 2i 1c. -2 - i 1d. 3 + 2i Recall that the absolute value of a real number is its distance from 0 on the real axis, which is also a number line. Similarly, the absolute value of an imaginary number is its distance from 0 along the imaginary axis. Absolute Value of a Complex Number WORDS The absolute value of a complex number a + bi is the distance from the origin to the point (a, b) in the complex plane, and is denoted ⎪a + bi⎥. ALGEBRA EXAMPLE ⎪a + bi⎥ = √ a2 + b2 >}>ÀÞÊ>ÝÃ ÎÊ { { { ]ÎÊ {] Î ä Ó { ,i> >ÝÃ ⎪3 + 4i⎥ = √ 32 + 42 = √ 9 + 16 =5 382 Chapter 5 Quadratic Functions EXAMPLE 2 Determining the Absolute Value of Complex Numbers Find each absolute value. A ⎪-9 + i⎥ B ⎪6⎥ C ⎪-4i⎥ ⎪-9 + 1i⎥ ⎪6 + 0i⎥ ⎪0 + (-4)i⎥ (-9) 2 + 1 2 √ √ 62 + 02 0 2 + (-4) 2 √ √ 81 + 1 √ 36 √ 16 √ 82 6 4 Find each absolute value. 1 2a. ⎪1 - 2i⎥ 2b. -_ 2 ⎪ ⎥ 2c. ⎪23i⎥ Adding and subtracting complex numbers is similar to adding and subtracting variable expressions with like terms. Simply combine the real parts, and combine the imaginary parts. The set of complex numbers has all the properties of the set of real numbers. So you can use the Commutative, Associative, and Distributive Properties to simplify complex number expressions. EXAMPLE 3 Adding and Subtracting Complex Numbers Add or subtract. Write the result in the form a + bi. A (-2 + 4i) + (3 - 11i) (-2 + 3) + (4i - 11i) 1 - 7i B (4 - i) - (5 + 8i) (4 - i) - 5 - 8i (4 - 5) + (-i - 8i) -1 - 9i Complex numbers also have additive inverses. The additive inverse of a + bi is -(a + bi), or -a - bi. C (6 - 2i) + (-6 + 2i) (6 - 6) + (-2i + 2i) 0 + 0i Associative and Commutative Properties Add real parts and imaginary parts. Distributive Property Associative and Commutative Properties Add real parts and imaginary parts. Associative and Commutative Properties Add real parts and imaginary parts. 0 D (10 + 3i) - (10 - 4i) (10 + 3i) - 10 - (-4i) (10 - 10) + (3i + 4i) 0 + 7i Distributive Property Associative and Commutative Properties Add real parts and imaginary parts. 7i Add or subtract. Write the result in the form a + bi. 3a. (-3 + 5i) + (-6i) 3b. 2i - (3 + 5i) 3c. (4 + 3i) + (4 - 3i) You can also add complex numbers by using coordinate geometry. 5- 9 Operations with Complex Numbers 383 EXAMPLE 4 Adding Complex Numbers on the Complex Plane Find (4 + 3i) + (-2 + i) by graphing on the complex plane. Step 1 Graph 4 + 3i and -2 + i on the complex plane. Connect each of these numbers to the origin with a line segment. >}>ÀÞÊ>ÝÃ ÓÊ { { {Ê Î ÓÊ { Step 2 Draw a parallelogram that has these two line segments as sides. The vertex that is opposite the origin represents the sum of the two complex numbers, 2 + 4i. Therefore, (4 + 3i) + (-2 + i) = 2 + 4i. ä Ó Ó { ,i> >ÝÃ Ó { Check Add by combining the real parts and combining the imaginary parts. (4 + 3i) + (-2 + i) = 4 + (-2) + (3i + i) = 2 + 4i Find each sum by graphing on the complex plane. 4a. (3 + 4i) + (1 - 3i) 4b. (-4 - i) + (2 - 2i) You can multiply complex numbers by using the Distributive Property and treating the imaginary parts as like terms. Simplify by using the fact i 2 = -1. EXAMPLE 5 Multiplying Complex Numbers Multiply. Write the result in the form a + bi. A 2i(3 - 5i) 6i - 10i 2 B (5 - 6i)(4 - 3i) 20 - 15i - 24i + 18i 2 Multiply. Distribute. 6i - 10(-1) 20 - 39i + 18(-1) Use i 2 = -1. 10 + 6i Write in a + bi form. C (7 + 2i)(7 - 2i) Use i 2 = -1. 2 - 39i D (6i)(6i) 49 - 14i + 14i - 4i 2 Multiply. 36i 2 49 - 4(-1) Use i 2 = -1. 36(-1 ) Use i 2 = -1. 53 -36 Multiply. Write the result in the form a + bi. 5a. 2i(3 - 5i) 5b. (4 - 4i)(6 - i) 5c. (3 + 2i)(3 - 2i) The imaginary unit i can be raised to higher powers as shown below. Notice the repeating pattern in each row of the table. The pattern allows you to express any power of i as one of four possible values: i, -1, -i, or 1. 384 Powers of i i =i i5 = i4 · i = 1 · i = i i9 = i i 2 = -1 i 6 = i 4 · i 2 = 1 · (-1) = -1 i 10 = -1 i 3 = i 2 · i = -1 · i = -i i 7 = i 4 · i 3 = 1 · (-i) = -i i 11 = -i i 4 = i 2 · i 2 = -1 · (-1) = 1 i8 = i4 · i4 = 1 · 1 = 1 i 12 = 1 1 Chapter 5 Quadratic Functions EXAMPLE 6 Evaluating Powers of i A Simplify -3i 12. -3i 12 = -3(i 2) 6 Rewrite i 12 as a power of i 2. = -3(-1) 6 = -3(1) = -3 Simplify. B Simplify i 25. i 25 = i · i 24 = i · (i 2) Rewrite as a product of i and an even power of i. 12 Rewrite i 24 as a power of i 2. = i · (-1) 12 = i · 1 = i Simplify. 1 i 7. 6a. Simplify _ 2 The complex conjugate of a complex number a + bi is a - bi. (Lesson 5-5) EXAMPLE 6b. Simplify i 42. Recall that expressions in simplest form cannot have square roots in the denominator (Lesson 1-3). Because the imaginary unit represents a square root, you must rationalize any denominator that contains an imaginary unit. To do this, multiply the numerator and denominator by the complex conjugate of the denominator. 7 Dividing Complex Numbers A Simplify _. 3 + 7i 8i 3 + 7i -8i _ 8i -8i (_) -24i - 56i 2 __ -64i 2 -24i + 56 _ 64 -3i + 7 _ 3i _ = 7 -_ 8 8 8 B Simplify _ . Multiply by the conjugate. Distribute. Use i 2 = -1. Simplify. 3 + 8i 7a. Simplify _. -i 5+i 2 - 4i 5 + i 2 + 4i _ 2 - 4i 2 + 4i (_) 10 + 20i + 2i + 4i 2 __ 4 + 8i - 8i - 16i 2 10 + 22i - 4 __ 4 + 16 6 + 22i _ 11 i _ = 3 +_ 10 10 20 3 - i. 7b. Simplify _ 2-i THINK AND DISCUSS 1. Explain when a complex number a + bi and its conjugate are equal. 2. Find the product (a + bi)(c + di), and identify which terms in the product are real and which are imaginary. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, give an example. LÃÕÌi Û>Õi ``} «iÝ ÕLiÀÃ ÕÌ«Þ} Õ}>ÌiÃ 5- 9 Operations with Complex Numbers 385 5-9 Exercises KEYWORD: MB7 5-9 KEYWORD: MB7 Parent GUIDED PRACTICE 1. Vocabulary In the complex number plane, the horizontal axis represents ? numbers, and the vertical axis represents ? numbers. (real, irrational, −−− −−− or imaginary) SEE EXAMPLE 1 p. 382 SEE EXAMPLE 2 3 p. 383 SEE EXAMPLE 4 5 p. 384 SEE EXAMPLE 6 p. 385 SEE EXAMPLE 7 p. 385 4. 3 + 2i 5. -2 - 3i Find each absolute value. 6. 4 - 5i 7. -33.3 8. -9i 9. 5 + 12i 10. -1 + i 11. 15i Add or subtract. Write the result in the form a + bi. 12. (2 + 5i) + (-2 + 5i) 13. (-1 - 8i) + (4 + 3i) 14. (1 - 3i) - (7 + i) 15. (4 - 8i) + (-13 + 23i) 16. (6 + 17i) - (18 - 9i) 17. (-30 + i) - (-2 + 20i) Find each sum by graphing on the complex plane. 18. (3 + 4i) + (-2 - 4i) p. 384 SEE EXAMPLE 3. -i 2. 4 p. 383 SEE EXAMPLE Graph each complex number. 19. (-2 - 5i) + (-1 + 4i) 20. (-4 - 4i) + (4 + 2i) Multiply. Write the result in the form a + bi. 21. (1 - 2i)(1 + 2i) 22. 3i(5 + 2i) 23. (9 + i)(4 - i) 24. (6 + 8i)(5 - 4i) 2 25. (3 + i) 26. (-4 - 5i)(2 + 10i ) 27. -i 9 28. 2i 15 29. i 30 5 - 4i 30. _ i 17 33. _ 4+i 11 - 5i 31. _ 2 - 4i 45 - 3i 34. _ 7 - 8i 8 + 2i 32. _ 5+i -3 - 12i 35. _ 6i Simplify. PRACTICE AND PROBLEM SOLVING Independent Practice For See Exercises Example 36–39 40–45 46–51 52–54 55–60 61–63 64–69 1 2 3 4 5 6 7 Extra Practice Skills Practice p. S13 Graph each complex number. 36. -3 37. -2.5i 38. 1 + i Find each absolute value. 39. 4 - 3i 40. 2 + 3i 41. -18 4i 42. _ 5 43. 6 - 8i 44. -0.5i 45. 10 - 4i Add or subtract. Write the result in the form a + bi. 46. (8 - 9i) - (-2 - i) 47. 4i - (11 - 3i) 48. (4 - 2i) + (-9 - 5i) 49. (13 + 6i) + (15 + 35i) 50. (3 - i) - (-3 + i) 51. -16 + (12 + 9i) Application Practice p. S36 Find each sum by graphing on the complex plane. 52. (4 + i) + (-3i) 386 Chapter 5 Quadratic Functions 53. (5 + 4i) + (-1 + 2i) 54. (-3 - 3i) + (4 - 3i) Multiply. Write the result in the form a + bi. 55. -12i (-1 + 4i) 56. (3 - 5i)(2 + 9i) 57. (7 + 2i)(7 - 2i) 2 58. (5 + 6i) 59. (7 - 5i)(-3 + 9i) 60. -4(8 + 12i) 62. -i 11 5 - 2i 65. _ 3+i 8 + 4i _ 68. 7+i 63. 5i 10 3 66. _ -1 - 5i 6 + 3i 69. _ 2 - 2i Simplify. 61. i 27 2 - 3i 64. _ i 19 + 9i _ 67. 5+i Write the complex number represented by each point on the graph. >}>ÀÞÊ>ÝÃ 70. A { Ó 71. B { 72. C ä Ó { ,i> >ÝÃ Ó 73. D { 74. E Find the absolute value of each complex number. Fractals Fractals are self-similar, which means that smaller parts of a fractal are similar to the fractal as a whole. Many objects in nature, such as the veins of leaves and snow crystals, also exhibit selfsimilarity. As a result, scientists can use fractals to model these objects. 75. 3 - i 76. 7i 77. -2 - 6i 78. -1 - 8i 3 -_ 1i 81. _ 2 2 79. 0 80. 5 + 4i 82. 5 - i √ 3 83. 2 √ 2 - i √ 3 84. Fractals Fractals are patterns produced using complex numbers and the repetition of a mathematical formula. Substitute the first number into the formula. Then take the result, put it back into the formula, and so on. Each complex number produced by the formula can be used to assign a color to a pixel on a computer screen. The result is an image such as the one at right. Many common fractals are based on the Julia Set, whose formula is Z n + 1 = (Z n) 2 + c, where c is a constant. a. Find Z 2 using Z 2 = (Z 1) 2 + 0.25. Let Z 1 = 0.5 + 0.6i. b. Find Z 3 using Z 3 = (Z 2) 2 + 0.25. Use Z 2 that you obtained in part a. c. Find Z 4 using Z 4 = (Z 3) 2 + 0.25. Use Z 3 that you obtained in part b. Simplify. Write the result in the form a + bi. 85. (3.5 + 5.2i) + (6 - 2.3i) 86. 6i - (4 + 5i) 87. (-2.3 + i) - (7.4 - 0.3i) 88. (-8 - 11i) + (-1 + i) 89. i(4 + i) 2 90. (6 - 5i) 2 91. (-2 - 3i) 92. (5 + 7i)(5 - 7i) 93. (2 - i)(2 + i)(2 - i) 94. 3 - i 11 12 + i 97. _ i 1+i _ 100. -2 + 4i 95. i 52 - i 48 18 - 3i 98. _ i 4 _ 101. 2 - 3i 96. i 35 - i 24 + i 18 4 + 2i 99. _ 6+i 6 _ 102. √ 2-i 5- 9 Operations with Complex Numbers 387 Multi-Step Impedance is a measure of the opposition of a circuit to an electric current. Electrical engineers find it convenient to model impedance Z with complex numbers. In a parallel AC circuit with two impedances Z 1 and Z 2, the equivalent or total impedance in ohms can be determined by using the Z 1Z 2 formula Z eq = ______ . Z +Z 1 �� �� 2 ���� ������������������� 103. Find the equivalent impedance Z eq for Z 1 = 3 + 2i and Z 2 = 1 - 2i arranged in a parallel AC circuit. 104. Find the equivalent impedance Z eq for Z 1 = 2 + 2i and Z 2 = 4 - i arranged in a parallel AC circuit. Tell whether each statement is sometimes, always, or never true. If the statement is sometimes true, give an example and a counterexample. If the statement is never true, give a counterexample. 105. The sum of any complex number a + bi and its conjugate is a real number. 106. The difference between any complex number a + bi (b ≠ 0) and its conjugate is a real number. 107. The product of any complex number a + bi (a ≠ 0) and its conjugate is a positive real number. 108. The product of any two imaginary numbers bi (b ≠ 0) and di (d ≠ 0) is a positive real number. 3 are shown. Which is 109. /////ERROR ANALYSIS///// Two attempts to simplify _ 2+i incorrect? Explain the error. � � � � ���������� � ���������� ������� ���������� ������ ������ ������� ������ � ������������ ������� ������ � � � ������������ � � ��� �� � � � � ���������� � ���������� ������� ���������� ������ ������ ������� ������ � ������������ ������� ������ � ������������ � 110. Critical Thinking Why are the absolute value of a complex number and the absolute value of its conjugate equal? Use a graph to justify your answer. 111. Write About It Discuss how the difference of two squares, a 2 - b 2 = (a + b)(a - b), relates to the product of a complex number and its conjugate. 112. This problem will prepare you for the Multi-Step Test Prep on page 390. You have seen how to graph sums of complex numbers on the complex plane. a. Find three pairs of complex numbers whose sum is 4 + 4i. b. Graph each of the sums on the same complex plane. c. Describe the results of your graph. �������������� �� � 388 Chapter 5 Quadratic Functions ������ �� � � ���� ���� Use the graph for Exercises 113–114. 113. Which point on the graph represents 1 - 2i? C A D B >}>ÀÞÊ>ÝÃ { Ó 114. What is the value of the complex number represented in the graph by E? -2 -2i ä { ,i> >ÝÃ { Ó { 2i 2 Ó 115. Which expression is equivalent to (2 - 5i) - (2 + 5i) ? 4 + 10i 10i -10i 4 - 10i 116. Which expression is equivalent to (-5 + 3i) ? 16 - 15i 16 - 30i 34 - 15i 34 - 30i 2 CHALLENGE AND EXTEND 117. Consider the powers of i. a. Complete the table, and look for a pattern. i1 = i -1 = i0 = i -2 = i -3 = i -4 = i -5 = b. Explain the pattern that you observed for i raised to negative powers. What are the only possible values of i raised to a negative integer power? c. Simplify i -12, i -37, and i -90. Find the general form of the result for each complex operation. a + bi 118. (a + bi)(c + di) 119. _ c + di SPIRAL REVIEW 120. Money The table shows the amount that James spent for lunches each week over an eight-week period. Make a scatter plot of the data. Sketch a line of best fit, and find its equation. (Lesson 2-7) Lunches Purchased 5 7 3 5 6 2 4 5 Weekly Cost ($) 10 13 8 9 8 5 10 11 Solve each inequality by using algebra. (Lesson 5-7) 121. 0 ≥ 3x 2 - 6x 122. 10 < x 2 - 4x - 11 123. -6 ≥ 2x + 7x - 21 124. 3 - x 2 < 7 - 5x 2 Determine whether each data set could represent a quadratic function. Explain. (Lesson 5-8) 125. x -2 -1 0 1 y 5 -1 -3 -1 126. x 0 2 4 6 y 18 10 2 -6 5- 9 Operations with Complex Numbers 389 SECTION 5B Applying Quadratic Functions Tilted Tiles Mitch and Jacob are making mosaics in an art class. To make one mosaic, Mitch first divides a wall into a grid made up of squares with a side length of 20 cm. Then Jacob glues a tile on each square, making sure that each corner of the tile touches a side of the grid square. They measure the side length of each tile as well as the distance x from the upper right corner of the grid square to a corner of the tile. They find that for each tile there are two possible values of x, as shown. 1. Complete the table by finding the area of each tile and the ratio y of the area of each tile to the area of the grid square. 2. Make a scatterplot of the ordered pairs (x, y). Find and graph a quadratic model for the data. Is the model a reasonable representation of the data? Explain. 3. Describe the domain for the problem situation. Explain why the domain of the problem situation is different from the domain of the model. 4. Use your model to determine the value of y when x = 3.8. Explain the meaning of your answer in the context of the problem. 5. For what values of x does a tile cover at least 75% of the grid square? Round to the nearest tenth. 390 Chapter 5 Side Length of Tile (cm) x x 20 cm Area of x (cm) Tile (cm 2) 15 6.4 15 13.6 15.5 5.5 15.5 14.5 16 4.7 16 15.3 17 3.3 17 16.7 18 2.1 18 17.9 19 1.1 19 18.9 20 0 y SECTION 5B Quiz for Lessons 5-7 Through 5-9 5-7 Solving Quadratic Inequalities Graph each inequality. 2 1. y > -x 2 + 6x 2. y ≤ -x - x + 2 Solve each inequality by using tables or graphs. 3. x 2 - 4x + 1 > 6 4. 2x 2 + 2x -10 ≤ 2 Solve each inequality by using algebra. 5. x 2 + 4x - 7 ≥ 5 6. x 2 - 8x < 0 7. The function p(r) = -1000r 2 + 6400r - 4400 models the monthly profit p of a small DVD-rental store, where r is the rental price of a DVD. For what range of rental prices does the store earn a monthly profit of at least $5000? 5-8 Curve Fitting with Quadratic Models Determine whether each data set could represent a quadratic function. Explain. 8. x 5 6 7 8 9 y 13 11 7 1 -7 9. x -4 -2 0 2 4 y 10 8 4 8 10 Write a quadratic function that fits each set of points. 10. (0, 4), (2, 0), and (3, 1) 11. (1, 3), (2, 5), and (4, 3) For Exercises 12–14, use the table of maximum load allowances for various heights of spruce columns. Maximum Load Allowance No. 1 Common Spruce 12. Find a quadratic regression equation to model the maximum load given the height. 13. Use your model to predict the maximum load allowed for a 6.5 ft spruce column. Height of Column (ft) Maximum Load (lb) 4 7280 5 7100 6 6650 7 5960 14. Use your model to predict the maximum load allowed for an 8 ft spruce column. 5-9 Operations with Complex Numbers Find each absolute value. 15. -6i 16. 3 + 4i 17. 2 - i Perform each indicated operation, and write the result in the form a + bi. 18. (3 - 5i) - (6 - i) 19. (-6 + 4i) + (7 - 2i) 20. 3i (4 + i) 21. (3 + i)(5 - i) 22. (1 - 4i)(1 + 4i) 2 - 7i 24. _ -i 23. 3i 15 3-i 25. _ 4 - 2i Ready to Go On? 391 Vocabulary absolute value of a complex number . . . . . . . . . . . . . . . . . 382 imaginary number . . . . . . . . . . 350 quadratic model . . . . . . . . . . . . 376 imaginary part . . . . . . . . . . . . . 351 quadratic regression . . . . . . . . 376 axis of symmetry . . . . . . . . . . . 323 imaginary unit . . . . . . . . . . . . . 350 real part . . . . . . . . . . . . . . . . . . . 351 binomial . . . . . . . . . . . . . . . . . . . 336 maximum value . . . . . . . . . . . . 326 root of an equation . . . . . . . . . 334 completing the square . . . . . . 342 minimum value . . . . . . . . . . . . 326 standard form . . . . . . . . . . . . . . 324 complex conjugate . . . . . . . . . . 352 parabola . . . . . . . . . . . . . . . . . . . 315 trinomial. . . . . . . . . . . . . . . . . . . 336 complex number . . . . . . . . . . . 351 quadratic function . . . . . . . . . . 315 vertex form . . . . . . . . . . . . . . . . . 318 complex plane . . . . . . . . . . . . . 382 quadratic inequality in two variables . . . . . . . . . . . . . . . . 366 vertex of a parabola . . . . . . . . . 318 discriminant . . . . . . . . . . . . . . . 357 zero of a function . . . . . . . . . . . 333 Complete the sentences below with vocabulary words from the list above. 1. The number 5i can be classified as both a(n) ? and a ? . −−−−−− −−−−−− 2. The value of the input x that makes the output f (x) equal zero is called the 3. The ? . −−−−−− is the point at which the parabola intersects the axis of symmetry. ? −−−−−− 4. The type and number of solutions to a quadratic equation can be determined by finding the ? . −−−−−− 5. When a parabola opens upward, the y-value of the vertex is the ? of a −−−−−− quadratic function. 5-1 Using Transformations to Graph Quadratic Functions (pp. 315–322) EXERCISES EXAMPLES ■ 2 Using the graph of f (x) = x as a guide, describe the transformations, and then graph 1 2 g (x) = __ x + 3. Þ 2 n v } È g (x) = __12 x 2 + 3 is f vertically compressed by a factor of __12 and translated 3 units up. ■ { ä]ÊÎ® Ý Ó ä Ó Use the description to write a quadratic function in vertex form. The function f (x) = x 2 is translated 1 unit right to create g. translation 1 unit right: h=1 2 g (x) = a(x - h) + k → g (x) = (x - 1)2 392 Chapter 5 Quadratic Functions Graph each function by using a table. 1 x 2 + 3x - 4 6. f (x) = -x 2 - 2x 7. f (x) = _ 2 Using the graph of f (x) = x 2 as a guide, describe the transformations, and then graph each function. 8. g (x) = 4(x - 2)2 9. g (x) = -2(x + 1)2 1 x2 - 3 11. g (x) = -(x + 2)2 + 6 10. g (x) = _ 3 Use the description to write each quadratic function in vertex form. 12. f (x) = x 2 is reflected across the x-axis and translated 3 units down to create g. 13. f (x) = x 2 is vertically stretched by a factor of 2 and translated 4 units right to create g. 14. f (x) = x 2 is vertically compressed by a factor of __14 and translated 1 unit left to create g. 5-2 Properties of Quadratic Functions in Standard Form (pp. 323–330) EXERCISES EXAMPLE ■ For f (x) = -x 2 + 2x + 3, (a) determine whether the graph opens upward or downward, (b) find the axis of symmetry, (c) find the vertex, (d) find the y-intercept, and (e) graph the function. e. a. Because a < 0, the parabola opens downward. � ������ � ������ b. axis of symmetry: b = -_ 2 =1 x = -_ 2a 2(-1) c. f (1) = -1 2 + 2(1) + 3 = 4 The vertex is (1, 4). ������ � � � ����� For each function, (a) determine whether the graph opens upward or downward, (b) find the axis of symmetry, (c) find the vertex, (d) find the y-intercept, and (e) graph the function. 15. f (x) = x 2 - 4x + 3 16. g (x) = x 2 + 2x + 3 1 x 2 - 2x + 4 17. h(x) = x 2 - 3x 18. j(x) = _ 2 Find the minimum or maximum value of each function. 19. f (x) = x 2 + 2x + 6 20. g (x) = 6x - 2x 2 21. f (x) = x 2 - 5x + 1 22. g (x) = -2x 2 - 8x + 10 23. f (x) = -x 2 - 4x + 8 24. g (x) = 3x 2 + 7 d. Because c = 3, the y-intercept is 3. 5-3 Solving Quadratic Equations by Graphing and Factoring (pp. 333–340) EXERCISES EXAMPLES ■ ■ Find the roots of x 2 + x = 30 by factoring. x 2 + x - 30 = 0 Rewrite in standard form. (x - 5)(x + 6) = 0 Factor. x - 5 = 0 or x + 6 = 0 Zero Product Property. Solve each equation. x = 5 or x = -6 Write a quadratic function with zeros 8 and -8. Write zeros as solutions. x = 8 or x = -8 x - 8 = 0 or x + 8 = 0 Set equations equal to 0. (x - 8)(x + 8) = 0 Converse Zero Product Property f (x) = x 2 - 64 Replace 0 with f (x). Find the roots of each equation by factoring. 25. x 2 - 7x - 8 = 0 26. x 2 - 5x + 6 = 0 27. x 2 = 144 28. x 2 - 21x = 0 29. 4x 2 - 16x + 16 = 0 30. 2x 2 + 8x + 6 = 0 31. x 2 + 14x = 32 32. 9x 2 + 6x + 1 = 0 Write a quadratic function in standard form for each given set of zeros. 33. 2 and -3 34. 1 and -1 35. 4 and 5 36. -2 and -3 37. -5 and -5 38. 9 and 0 5-4 Completing the Square (pp. 342–349) EXERCISES EXAMPLE ■ Solve x 2 - 8x = 12 by completing the square. 2 x - 8x + = 12 + x 2 - 8x + 16 = 12 + 16 (x - 4)2 = 28 x - 4 = ± √ 28 x = 4 ± 2 √ 7 Set up equation. Solve each equation by completing the square. 39. x 2 - 16x + 48 = 0 40. x 2 + 20x + 84 = 0 b 2 . Add __ 2 41. x 2 - 6x = 16 () Factor. Take square roots. Solve for x. 42. x 2 - 14x = 13 Write each function in vertex form, and identify its vertex. 44. g(x) = x 2 + 2x - 7 43. f (x) = x 2 - 4x + 9 Study Guide: Review 393 5-5 Complex Numbers and Roots (pp. 350–355) EXERCISES EXAMPLE ■ Solve x 2 - 22x + 133 = 0. x 2 - 22x + = -133 + x 2 - 22x + 121 = -133 + 121 (x - 11 )2 = -12 -12 x - 11 = ± √ x = 11 ± 2i √ 3 Rewrite. b 2 Add __ . 2 () Factor. Take square roots. Solve. Solve each equation. 45. x 2 = -81 46. 6x 2 + 150 = 0 47. x 2 + 6x + 10 = 0 48. x 2 + 12x + 45 = 0 49. x 2 - 14x + 75 = 0 50. x 2 - 22x + 133 = 0 Find each complex conjugate. 51. 5i - 4 52. 3 + i √5 5-6 The Quadratic Formula (pp. 356–363) EXERCISES EXAMPLES ■ Find the zeros of f (x) = 3x 2 - 5x + 3 by using the Quadratic Formula. 2 - 4ac -b ± √b Quadratic x = __ 2a Find the zeros of each function by using the Quadratic Formula. ( -5 )2 - 4( 3 )( 3 ) -(-5 ) ± √ x = ___ 2(3 ) 55. f (x) = 2x 2 - 10x + 18 Formula √11 5 ± √ -11 5 ±i_ =_=_ 6 6 6 ■ Substitute. 53. f (x) = x 2 - 3x - 8 54. h(x) = (x - 5)2 + 12 56. g(x) = x 2 + 3x + 3 Simplify. Find the type and number of solutions for x 2 + 9x + 20 = 0. b 2 - 4ac = 9 2 - 4(1)(20) = 81 - 80 = 1 There are two distinct real roots because the discriminant is positive. 57. h(x) = x 2 - 5x + 10 Find the type and number of solutions for each equation. 58. 2x 2 - 16x + 32 = 0 59. x 2 - 6x = -5 60. x2 + 3x + 8 = 0 61. x 2 - 246x = -144 62. x 2 + 5x = -12 63. 3x 2 - 5x + 3 = 0 5-7 Solving Quadratic Inequalities (pp. 366–373) EXAMPLE ■ Solve x 2 - 4x - 9 ≥ 3 by using algebra. Graph each inequality. Write and solve the related equation. Write in standard form. x 2 - 4x - 12 = 0 Factor. (x + 2)(x - 6) = 0 Solve. x = -2 or x = 6 64. y > x 2 + 3x + 4 The critical values are -2 and 6. These values divide the number line into three intervals: x ≤ -2, -2 ≤ x ≤ 6, and x ≥ 6. Testing an x-value in each interval gives the solution of x ≤ -2 or x ≥ 6. 394 EXERCISES Chapter 5 Quadratic Functions 65. y ≤ 2x 2 - x - 5 Solve each inequality by using tables or graphs. 66. x 2 + 2x - 4 ≥ -1 67. -x 2 - 5x > 4 Solve each inequality by using algebra. 68. -x 2 + 6x < 5 69. 3x 2 - 25 ≤ 2 70. x 2 - 3 < 0 71. 3x 2 + 4x - 3 ≤ 1 5-8 Curve Fitting with Quadratic Models (pp. 374–381) EXERCISES EXAMPLE ■ Find a quadratic model for the wattage of fluorescent bulbs F given the comparable incandescent bulb wattage I. Use the model to estimate the wattage of a fluorescent bulb that produces the same amount of light as a 120-watt incandescent bulb. Write a quadratic function that fits each set of points. 72. (-1, 8), (0, 6), and (1, 2) 73. (0, 0), (1, -1), and (2, -6) Construction For Exercises 74–77, use the table of copper wire gauges. Wattage Comparison Incandescent (watts) 40 60 75 90 100 Fluorescent (watts) 11 15 20 23 28 Enter the data into two lists in a graphing calculator. Use the quadratic regression feature. Common U.S. Copper Wire Gauges Resistance per 1000 ft (ohms) Gauge Diameter (in.) 24 0.0201 25.67 22 0.0254 16.14 20 0.0320 10.15 18 0.0403 6.385 74. Find a quadratic regression equation to model the diameter given the wire gauge. 75. Use your model to predict the diameter for a 12-gauge copper wire. The model is F(I ) ≈ 0.0016I 2 + 0.0481I + 6.48. A 36-watt fluorescent bulb produces about the same amount of light as a 120-watt incandescent bulb. 76. Find a quadratic regression equation to model the resistance given the wire gauge. 77. Use your model to predict the resistance for a 26-gauge copper wire. 5-9 Operations with Complex Numbers (pp. 382–389) EXERCISES EXAMPLES Perform each indicated operation, and write the result in the form a + bi. ■ ⎜-2 + 4i⎟ (-2)2 + 4 2 = √ 4 + 16 = √ ■ √ 20 = 2 √ 5 (3 + 2i)(4 - 5i) 2 12 - 15i + 8i - 10i 12 - 7i - 10(-1 ) = 22 - 7i ■ -5 + 3i _ 1 - 2i 80. ⎜12 - 16i⎟ 81. ⎜7i⎟ 82. (1 + 5i) + (6 - i) 83. (9 + 4i) - (3 + 2i) 84. (5 - i) - (11 - i) 85. -5i (3 - 4i) 86. (5 - 2i)(6 + 8i) 87. (3 + 2i)(3 - 2i) 88. (4 + i)(1 - 5i) 89. (-7 + 4i)(3 + 9i) 32 -5 + 3i _ -5 - 7i + 6i _ ( 1 + 2i ) = __ 2 1 - 2i Perform each indicated operation, and write the result in the form a + bi. 78. ⎜-3i⎟ 79. ⎜4 - 2i⎟ 1 + 2i 1 - 4i 2 7i -11 - 7i = -_ 11 - _ =_ 5 5 1+4 90. i 2 + 9i 92. _ -2i 8 _ 94. - 4i 1+i 91. -5i 21 5 + 2i 93. _ 3 - 4i -12 + 26i _ 95. 2 + 4i Study Guide: Review 395 Using the graph of f (x) = x 2 as a guide, describe the transformations, and then graph each function. 1 x2 + 2 1. g (x) = (x + 1)2 - 2 2. h (x) = -_ 2 3. Use the following description to write a quadratic function in vertex form: f(x) = x 2 is vertically compressed by a factor of __12 and translated 6 units right to create g. For each function, (a) determine whether the graph opens upward or downward, (b) find the axis of symmetry, (c) find the vertex, (d) find the y-intercept, and (e) graph the function. 4. f (x) = -x 2 + 4x + 1 5. g (x) = x 2 - 2x + 3 6. The area A of a rectangle with a perimeter of 32 cm is modeled by the function A(x) = -x 2 + 16x, where x is the width of the rectangle in centimeters. What is the maximum area of the rectangle? Find the roots of each equation by using factoring. 7. x 2 - 2x + 1 = 0 8. x 2 + 10x = -21 Solve each equation. 9. x 2 + 4x = 12 10. x 2 - 12x = 25 11. x 2 + 25 = 0 12. x 2 + 12x = -40 Write each function in vertex form, and identify its vertex. 13. f (x) = x 2 - 4x + 9 14. g (x) = x 2 - 18x + 92 Find the zeros of each function by using the Quadratic Formula. 15. f (x) = (x - 1)2 + 7 16. g (x) = 2x 2 - x + 5 17. The height h in feet of a person on a waterslide is modeled by the function h (t) = -0.025t 2 - 0.5t + 50, where t is the time in seconds. At the bottom of the slide, the person lands in a swimming pool. To the nearest tenth of a second, how long does the ride last? 18. Graph the inequality y < x 2 - 3x - 4. Solve each inequality. 19. -x 2 + 3x + 5 ≥ 7 20. x 2 - 4x + 1 > 1 For Exercises 21 and 22, use the table showing the average cost of LCD televisions at one store. 21. Find a quadratic model for the cost of a television given its size. Costs of LCD Televisions Size (in.) 15 17 23 30 Cost ($) 550 700 1500 2500 22. Use the model to estimate the cost of a 42 in. LCD television. Perform the indicated operation, and write the result in the form a + bi. 23. (12 - i) - (5 + 2i) 396 24. (6 - 2i)(2 - 2i) Chapter 5 Quadratic Functions 25. -2i 18 1 - 8i 26. _ 4i FOCUS ON SAT MATHEMATICS SUBJECT TESTS The SAT Mathematics Subject Tests assess knowledge from course work rather than ability to learn. The Level 1 test is meant to be taken by students who have completed two years of algebra and one year of geometry, and it tests more elementary topics than the Level 2 test. You will need to use a calculator for some of the problems on the SAT Mathematics Subject Tests. Before test day, make sure that you are familiar with the features of the calculator that you will be using. You may want to time yourself as you take this practice test. It should take you about 8 minutes to complete. 1. For what value of c will 3x 2 - 2x + c = 0 have exactly one distinct real root? 2 (A) -_ 3 4. What is the solution set of y 2 - 2y ≤ 3y + 14? (A) y ≥ -2 (B) y ≤ 7 (C) y ≤ -2 or y ≥ 7 1 (B) -_ 3 (C) 0 (D) -7 ≤ y ≤ 2 (E) -2 ≤ y ≤ 7 1 (D) _ 3 2 (E) _ 3 5. Which of the following is a factor of (a - 1)2 - b 2 ? 2. If m and n are real numbers, i 2 = -1, and (m - n) - 4i = 7 + ni, what is the value of m? (A) a + b - 1 (B) a - b (A) -4 (C) a - 1 (B) -3 (D) a - b + 1 (C) 1 (E) 1 - b (D) 3 (E) 4 6. If z = 5 - 4i and i 2 = -1, what is z ? (A) 1 3. If x 2 - 5x + 6 = (x - h) + k, what is the value of k? 25 (A) -_ 4 2 5 (B) -_ 2 (B) 3 (C) 9 (D) √41 (E) √42 1 (C) -_ 4 (D) 0 (E) 6 College Entrance Exam Practice 397 Multiple Choice: Work Backward When taking a multiple-choice test, you can sometimes work backward to determine which answer is correct. Because this method can be time consuming, it is best used only when you cannot solve a problem in any other way. Which expression is equivalent to 2x 2 - 3x - 14? (2x + 7)(x + 2) (2x - 7)(x + 2) (2x - 7)(x - 2) (2x + 7)(x - 2) If you have trouble factoring the quadratic expression given in the question, you can multiply the binomials in the answer choices to find the product that is the same as 2x 2 - 3x - 14. Try Choice A: (2x + 7)(x + 2) = 2x 2 + 11x + 14 Try Choice B: (2x - 7)(x - 2) = 2x 2 - 11x + 14 Try Choice C: (2x - 7)(x + 2) = 2x 2 - 3x - 14 Choice C is the answer. Note: Trying choice D can help you check your work. What is the solution set of x 2 - 36 < 0? x < -6 or x > 6 -36 < x < 36 -6 < x < 6 x < -36 or x > 36 If you have trouble determining the solution set, substitute values of x into the inequality. Based on whether the values make the inequality true or false, you may be able to eliminate one or more of the answer choices. Substitute 0 for x: x 2 - 36 < 0 → (0)2 - 36 0 → -36 < 0 ✔ When x = 0, the inequality is true. Therefore, the solution set must include x = 0. Because choices F and J do not include x = 0, they can be eliminated. Substitute 10 for x: x 2 - 36 < 0 → (10)2 - 36 0 → 64 0 ✘ When x = 10, the inequality is false. Therefore, the solution set does not include x = 10. Because choice H includes x = 10, it can be eliminated. The only remaining choice is choice G. Therefore, choice G must be correct. 398 Chapter 5 Quadratic Functions You can also work backward to check whether the answer you found by another method is correct or reasonable. Read each test item, and answer the questions that follow. Item A What are the zeros of the function g (x) = 6x 2 - 8x - 4, rounded to the nearest hundredth? -10.32 and 2.32 1.72 and -0.39 -1.72 and 0.39 10.32 and -2.32 1. Rachel cannot remember how to determine the zeros of a quadratic function, so she plans to pick one of the answer choices at random. What could Rachel do to make a more educated guess? 2. Describe how to find the correct answer by working backward. Item B A portable television has a screen with a diagonal of 4 inches. The length of the screen is 1 inch greater than its width. What are the dimensions of the screen to the nearest hundredth? Item C Which of the following is a solution of (x + 4)2 = 25? x = -9 x=0 x = -1 x=9 5. Explain how to use substitution to determine the correct answer. 6. Check whether choice A is correct by working backward. Explain your findings. What should you do next? Item D The height h of a golf ball in feet t seconds after it is hit into the air is modeled by h(t) = -16t 2 + 64t. How long is the ball in the air? 2 seconds 12 seconds 4 seconds 16 seconds 7. The measurements given in the answer choices represent possible values of which variable in the function? 8. Describe how you can work backward to determine that choice F is not correct. Item E Ü { ÜÊ Ê£ The base of a triangle is 4 in. longer than twice its height. If the triangle has an area of 24 in 2, what is its height? 1.28 inches by 2.28 inches 1.28 inches by 3.28 inches LÊÓ Ê Ê{ 2.28 inches by 2.28 inches 2.28 inches by 3.28 inches 3. Can any of the answer choices be eliminated immediately? If so, which choices and why? 4. Describe how you can determine the correct answer by using the Pythagorean Theorem and working backward. 2 in. 6 in. 4 in. 8 in. 9. What equation do you need to solve to find the value of h? 10. Try choice A by working backward. Explain your findings. What should you do next? Test Tackler 399 KEYWORD: MB7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1– 5 Multiple Choice 5. Which graph represents the function 1 (x - 3) - 4? f(x) = -_ 2 ⎡-1 8 2⎤ N=⎢ ⎣ 0 1 6⎦ ⎡6 -2⎤ 1. M = ⎢ ⎣3 7⎦ 8 What is the matrix product 2MN? ⎡ -24 184 0⎤ ⎢ ⎣ -12 124 192 ⎦ -8 ⎡ -12 92 0⎤ ⎢ ⎣ -6 62 96⎦ ⎡-24 -12⎤ -4 4 8 4 8 y x -4 ⎡-12 -6 ⎤ 92 62 ⎣ 0 96⎦ 0 (1, -6) (-3, -4) -8 2. Which of these functions does NOT have zeros at -1 and 4? 4 (-5, 0) f(x) = x 2 - 3x - 4 -8 f(x) = 2x + 6x - 8 2 -4 y x 0 4 8 -4 f(x) = -x 2 + 3x + 4 f(x) = 2x 2 - 6x - 8 (5, -5) -8 3. Dawn and Julia are running on a jogging trail. Dawn starts running 5 minutes after Julia does. If Julia runs at an average speed of 8 ft/s and Dawn runs at an average speed of 9 ft/s, how many minutes after Dawn starts running will she catch up with Julia? 5 minutes 40 minutes 27 minutes 45 minutes 4. Which equation has intercepts at (20, 0, 0), (0, 40, 0), and (0, 0, 5)? (7, 2) 0 4 124 0 192⎦ ⎢ 4 -4 ⎢ 184 ⎣ (-1, 6) y 8 y (5, 0) x (-1, 3) -8 -4 0 4 -4 6. What is the equation of the function graphed below? y 20x + 40y + 5z = 0 (-1, 3) 20x + 40y + 5z = 1 2 4x + 8y + z = 5 2x + y + 8z = 40 400 Chapter 5 Quadratic Functions x -6 0 -4 (-3, -1) -2 2 y = (x - 3)2 - 1 y = (x - 1)2 - 3 y = (x + 3)2 - 1 y = (x + 1)2 - 3 7. If the relationship between x and y is quadratic, which value of y completes the table? x -3 -1 y 21 7 1 3 5 27 61 Short Response ⎧-4x + 8y - 2z = 8 14. ⎨ 4x - 4y + 2z = -5 ⎩ x + 4y - 2z = 15 a. Write the augmented matrix that could be used 3 9 7 17 to solve the system of equations given above. b. Find the solution of the system, and explain 5(6 - 8i ) 8. Which is equivalent to the expression _ ? -20 + 10i 15 - 40i 15 - 8i 20 - 10i 2-i how you determined your answer. 15. The graph below shows a feasible region for a set of constraints. 8 9. What is the inverse of the following matrix? ⎡-2 -4⎤ ⎢ ⎣ 4 2⎦ y 6 4 ⎡ _ 1⎤ 1 _ 3 6 1 _ _ - -1 6⎦ ⎣ 3 ⎡ _ 1⎤ - 1 -_ 6 3 1 1 _ _ ⎣ 3 6⎦ ⎢ ⎢ ⎡ _ 1⎤ - 1 -_ 2 4 1 1 _ _ ⎣ 4 2⎦ ⎢ 2 x 0 2 4 6 8 a. Write the constraints for the feasible region. b. Maximize the objective function P = 3x - 4y ⎡ 2 4⎤ ⎢ ⎣-4 -2⎦ under these constraints. In nearly all standardized tests, you cannot enter a negative value as the answer to a griddedresponse question. If you get a negative value as an answer to one of these questions, you have probably made a mistake in your calculations. Gridded Response 16. Consider the function f(x) = x 2 - 2x - 48. a. Determine the roots of the function. Show your work. b. The function f is translated to produce the function g. The vertex of g is the point (3, 30). Write the function rule for g in vertex form, and explain how you determined your answer. Extended Response 10. What value of x makes the equation 17. A small alteration store charges $15.00 per hour x 2 + 64 = 16x true? 11. The table shows the fees that are charged at an airport parking lot for various lengths of time. What is the slope of the linear function that models the parking fee f in dollars for h number of hours? plus a $12.50 consulting fee for alterations. A competing store charges $20.00 per hour but does not charge a consulting fee. a. For each store, write a linear function c that can be used to find the total cost of an alteration that takes h hours. b. For which values of h is the small alteration Time (h) Parking Fee ($) 1 3 5 7 3.35 5.05 6.75 8.45 12. What is the x-value of the vertex of f(x) = 2x - 15x + 5? 2 13. What is the value of c given that the following store less expensive than the competing store? Explain how you determined your answer. c. The small store wants to adjust its pricing so that it is less expensive than the competing store for any alteration job that takes an hour or more. By how much should the small store lower its consulting fee in order to make this adjustment? system is dependent? ⎧ 2y - x + 10 = 0 ⎨ ⎩ 3x - 6y - c = 16 Cumulative Assessment, Chapters 1–5 401

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