2. Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240

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4. Contents Lesson 6-1Graphing Quadratic Functions Lesson 6-2Solving Quadratic Equations by Graphing Lesson 6-3Solving Quadratic Equations by Factoring Lesson 6-4Completing the Square Lesson 6-5The Quadratic Formula and the Discriminant Lesson 6-6Analyzing Graphs of Quadratic Functions Lesson 6-7Graphing and Solving Quadratic Inequalities

5. Lesson 1 Contents Example 1Graph a Quadratic Function Example 2Axis of Symmetry, y-Intercept, and Vertex Example 3Maximum or Minimum Value Example 4Find a Maximum Value

6. Example 1-1a First, choose integer values for x. Then, evaluate the function for each x value. Graph the resulting coordinate pairs and connect the points with a smooth curve. Graph by making a table of values. Answer: (1, 3) 31 (0, –1) –10 (–1, –3) –3–1 (–2, –3) –3–2 (–3, –1) –1–3 (x, y) f (x)x

7. Example 1-1a

8. Example 1-1b 167214 f(x)f(x) 210–1–2 x Answer: Graph by making a table of values.

9. Example 1-2a Consider the quadratic function Find the y -intercept, the equation of the axis of symmetry, and the x -coordinate of the vertex. Begin by rearranging the terms of the function so that the quadratic term is first, the linear term is second and the constant term is last. Then identify a, b, and c. So,and The y -intercept is 2.

10. Example 1-2a You can find the equation of the axis of symmetry using a and b. Answer:The y -intercept is 2. The equation of the axis of symmetry is x = 2.Therefore, the x -coordinate of the vertex is 2. Equation of the axis of symmetry Simplify.

11. Example 1-2a Make a table of values that includes the vertex. Choose some values for x that are less than 2 and some that are greater than 2. This ensures that points on either side of the axis of symmetry are graphed. Vertex Answer: (4, 2) 24 (3, –1) –13 (2, –2) –22 (1, –1 ) –11 (0, 2) 20 (x, f(x)) f(x)f(x)x

12. Answer: Then graph the points from your table connecting them with a smooth curve. Graph the vertex and the y -intercept. Example 1-2a Use this information to graph the function. As a check, draw the axis of symmetry,, as a dashed line. (0, 2) The graph of the function should be symmetric about this line. (2, –2)

13. Consider the quadratic function a. Find the y -intercept, the equation of the axis of symmetry, and the x -coordinate of the vertex. b. Make a table of values that includes the vertex. Example 1-2b Answer: y -intercept: 3 ; axis of symmetry: x -coordinate: 3 –2–5–6–5–23f(x)f(x) 543210x Answer:

14. Example 1-2b c. Use this information to graph the function. Answer:

15. Example 1-3a Consider the function Determine whether the function has a maximum or a minimum value. Answer:Since the graph opens down and the function has a maximum value. For this function,

16. Example 1-3a State the maximum or minimum value of the function. The maximum value of this function is the y -coordinate of the vertex. Answer:The maximum value of the function is 4. The x -coordinate of the vertex is Find the y -coordinate of the vertex by evaluating the function for Original function

17. Consider the function a. Determine whether the function has a maximum or a minimum value. b. State the maximum or minimum value of the function. Example 1-3b Answer: minimum Answer: –5

18. Example 1-4a Economics A souvenir shop sells about 200 coffee mugs each month for $6 each. The shop owner estimates that for each $0.50 increase in the price, he will sell about 10 fewer coffee mugs per month. How much should the owner charge for each mug in order to maximize the monthly income from their sales? Words The income is the number of mugs multiplied by the price per mug. Variables Letthe number of $0.50 price increases. Then the price per mug and the number of mugs sold.

19. Example 1-4a Let I(x) equal the income as a function of x. The income is the number of mugs sold times the price per mug. Rewrite in form. Multiply. Simplify. Equation I(x)=

20. Example 1-4a I(x) is a quadratic function with and Since the function has a maximum value at the vertex of the graph. Use the formula to find the x -coordinate of the vertex. x -coordinate of the vertex Formula for the x -coordinate of the vertex Simplify.

21. Example 1-4a This means that the shop should make 4 price increases of $0.50 to maximize their income. Answer: The mug price should be

22. Example 1-4a What is the maximum monthly income the owner can expect to make from these items? Answer: Thus, the maximum income is $1280. To determine the maximum income, find the maximum value of the function by evaluating Income function Use a calculator.

23. Example 1-4a Check Graph this function on a graphing calculator, and use the CALC menu to confirm this solution. ENTER2nd [CALC] 4 0 Keystrokes: 10 ENTER At the bottom of the display are the coordinates of the maximum point on the graph of The y value of these coordinates is the maximum value of the function, or 1280.

24. Economics A sports team sells about 100 coupon books for $30 each during their annual fund-raiser. They estimate that for each $0.50 decrease in the price, they will sell about 10 more coupon books. a. How much should they charge for each book in order to maximize the income from their sales? b. What is the maximum monthly income the team can expect to make from these items? Example 1-4b Answer: $17.50 Answer: $6125

25. End of Lesson 1

26. Lesson 2 Contents Example 1Two Real Solutions Example 2One Real Solution Example 3No Real Solution Example 4Estimate Roots Example 5Write and Solve an Equation

27. Example 2-1a Solveby graphing. Graph the related quadratic function The equation of the axis of symmetry is Make a table using x values around Then graph each point. x–101234 f (x)f (x)0–4–6 –40

28. Example 2-1a From the table and the graph, we can see that the zeroes of the function are –1 and 4. Answer: The solutions of the equation are –1 and 4.

29. Example 2-1a CheckCheck the solutions by substituting each solution into the original equation to see if it is satisfied.

30. Example 2-1b Answer: –3 and 1 Solveby graphing.

31. Example 2-2a Solveby graphing. Write the equation in form. Add 4 to each side. Graph the related quadratic function x01234 f(x)f(x)41014

32. Example 2-2a Notice that the graph has only one x -intercept, 2. Answer: The equation’s only solution is 2.

33. Example 2-2b Solveby graphing. Answer: 3

34. Example 2-3a Number Theory Find two real numbers whose sum is 4 and whose product is 5 or show that no such numbers exist. ExploreLet one of the numbers. Then PlanSince the product of the two numbers is 5, you know that Original equation Distributive Property Add x 2 and subtract 4x from each side.

35. Example 2-3a x01234 f (x)52125 Solve You can solve by graphing the related function.

36. Example 2-3a Notice that the graph has no x -intercepts. This means that the original equation has no real solution. Answer:It is not possible for two numbers to have a sum of 4 and a product of 5. Examine Try finding the product of several numbers whose sum is 4.

37. Example 2-3b Number Theory Find two real numbers whose sum is 7 and whose product is 14 or show that no such numbers exist. Answer: no such numbers exist

38. Example 2-4a Solveby graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. The equation of the axis of symmetry of the related function is x0123456 f(x)3–2–5–6–5–23

39. Example 2-4a The x -intercepts of the graph are between 0 and 1 and between 5 and 6. Answer:One solution is between 0 and 1 and the other is between 5 and 6.

40. Example 2-4b Solveby graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. Answer:between 0 and 1 and between 3 and 4

41. Example 2-5a Royal Gorge Bridge The highest bridge in the United States is the Royal Gorge Bridge in Colorado. The deck of the bridge is 1053 feet above the river below. Suppose a marble is dropped over the railing from a height of 3 feet above the bridge deck. How long will it take the marble to reach the surface of the water, assuming there is no air resistance? Use the formula where t is the time in seconds and h 0 is the initial height above the water in feet. We need to find t whenand Original equation

42. Example 2-5a Graph the related function using a graphing calculator. Adjust your window so that the x -intercepts are visible. Use the zero feature, [CALC], to find the positive zero of the function, since time cannot be negative. Use the arrow keys to locate a left bound for the zero and press. 2nd ENTER Then locate a right bound and presstwice. ENTER

43. Example 2-5a Answer: The positive zero of the function is approximately 8. It should take about 8 seconds for the marble to reach the surface of the water.

44. Example 2-5b Hoover Dam One of the largest dams in the United States is the Hoover Dam on the Colorado River, which was built during the Great Depression. The dam is 726.4 feet tall. Suppose a marble is dropped over the railing from a height of 6 feet above the top of the dam. How long will it take the marble to reach the surface of the water, assuming there is no air resistance? Use the formula where t is the time in seconds and h 0 is the initial height above the water in feet.

45. Example 2-5b Answer: about 7 seconds

46. End of Lesson 2

47. Lesson 3 Contents Example 1Two Roots Example 2Double Root Example 3Greatest Common Factor Example 4Write an Equation Given Roots

48. Example 3-1a Solveby factoring. Answer: The solution set is {0, –4}. Original equation Add 4x to each side. Factor the binomial. Solve the second equation. Zero Product Property or

49. Example 3-1a Check Substitute 0 and –4 in for x in the original equation.

50. Example 3-1a Solveby factoring. Original equation Subtract 5x and 2 from each side. Factor the trinomial. Zero Product Property or Solve each equation. Answer: The solution set is Check each solution.

51. Solve each equation by factoring. a. b. Example 3-1b Answer: {0, 3} Answer:

52. Example 3-2a Answer: The solution set is {3}. Solveby factoring. Original equation Add 9 to each side. Factor. Zero Product Property or Solve each equation.

53. Example 3-2a Check The graph of the related function, intersects the x -axis only once. Since the zero of the function is 3, the solution of the related equation is 3.

54. Example 3-2b Answer: {–5} Solveby factoring.

55. Example 3-3a Multiple-Choice Test Item What is the positive solution of the equation ? A –3 B 5 C 6 D 7 Read the Test Item You are asked to find the positive solution of the given quadratic equation. This implies that the equation also has a solution that is not positive. Since a quadratic equation can either have one, two, or no solutions, we should expect this equation to have two solutions.

56. Example 3-3a Solve the Test Item Answer: D Both solutions, –3 and 7, are listed among the answer choices. However, the question asks for the positive solution, 7. Original equation Factor. Divide each side by 2. Factor. or Zero Product Property Solve each equation.

57. Example 3-3b Multiple-Choice Test Item What is the positive solution of the equation ? A 5 B – 5 C 2 D 6 Answer: C

58. Example 3-4a Write a quadratic equation with and 6 as its roots. Write the equation in the form where a, b, and c are integers. Write the pattern. Simplify. Replace p with and q with 6.

59. Example 3-4a Use FOIL. Multiply each side by 3 so that b is an integer. Answer: A quadratic equation with roots and 6 and integral coefficients is You can check this result by graphing the related function.

60. Example 3-4b Write a quadratic equation with and 5 as its roots. Write the equation in the form where a, b, and c are integers. Answer:

61. End of Lesson 3

62. Lesson 4 Contents Example 1Equation with Rational Roots Example 2Equation with Irrational Roots Example 3Complete the Square Example 4Solve an Equation by Completing the Square Example 5Equation with a  1 Example 6Equation with Complex Solutions

63. Example 4-1a Solve by using the Square Root Property. Original equation Factor the perfect square trinomial. Square Root Property Subtract 7 from each side. Solve each equation. Answer: The solution set is {–15, 1}. Write as two equations. or

64. Example 4-1b Solve by using the Square Root Property. Answer: {3, 13}

65. Example 4-2a Solve by using the Square Root Property. Original equation Factor the perfect square trinomial. Square Root Property Add 5 to each side. Use a calculator. Write as two equations. or

66. Example 4-2a Answer: The exact solutions of this equation are and The approximate solutions are 1.5 and 8.5. Check these results by finding and graphing the related quadratic function. Original equation Subtract 12 from each side. Related quadratic function

67. Example 4-2a Check Use the ZERO function of a graphing calculator. The approximate zeros of the related function are 1.5 and 8.5.

68. Example 4-2b Solve by using the Square Root Property. Answer:

69. Example 4-3a Find the value of c that makesa perfect square. Then write the trinomial as a perfect square. Step 1Find one half of 16. Step 2Square the result of Step 1. Step 3Add the result of Step 2 to Answer:The trinomial can be written as

70. Example 4-3b Answer: 9 ; (x + 3) 2 Find the value of c that makes a perfect square. Then write the trinomial as a perfect square.

71. Example 4-4a Solve by completing the square. Write the left side as a perfect square by factoring. Since add 4 to each side. Rewrite so the left side is of the form Notice that is not a perfect square.

72. Example 4-4a Answer: The solution set is {–6, 2}. Square Root Property Subtract 2 from each side. Solve each equation. Write as two equations. or

73. Example 4-4b Answer: {–6, 1} Solve by completing the square.

74. Example 4-5a Solve by completing the square. Divide by the coefficient of the quadratic term, 3. Notice that is not a perfect square. Add to each side. Since add to each side.

75. Example 4-5a Write the left side as a perfect square by factoring. Simplify the right side. Square Root Property Add to each side.

76. Example 4-5a Solve each equation. Answer: The solution set is Write as two equations. or

77. Example 4-5b Solve by completing the square. Answer:

78. Example 4-6a Solve by completing the square. Notice that is not a perfect square. Rewrite so the left side is of the form Since add 1 to each side. Write the left side as a perfect square by factoring.

79. Example 4-6a Square Root Property Subtract 1 from each side. Answer: The solution set is Notice that these are imaginary solutions.

80. Example 4-6a Check A graph of the related function shows that the equation has no real solutions since the graph has no x -intercepts. Imaginary solutions must be checked algebraically by substituting them in the original equation.

81. Example 4-6b Solve by completing the square. Answer:

82. End of Lesson 4

83. Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

84. Example 5-1a Solve by using the Quadratic Formula. First, write the equation in the form and identify a, b, and c.

85. Example 5-1a Replace a with 1, b with –8, and c with –33. Simplify. Then, substitute these values into the Quadratic Formula. Quadratic Formula

86. Example 5-1a Answer: The solutions are 11 and –3. Write as two equations. or Simplify.

87. Example 5-1b Answer: 2, –15 Solve by using the Quadratic Formula.

88. Replace a with 1, b with –34, and c with 289. Identify a, b, and c. Then, substitute these values into the Quadratic Formula. Example 5-2a Solve by using the Quadratic Formula. Quadratic Formula Simplify.

89. Example 5-2a Answer: The solution is 17. Check A graph of the related function shows that there is one solution at

90. Solve by using the Quadratic Formula. Example 5-2b Answer: 11

91. Example 5-3a Solve by using the Quadratic Formula. Quadratic Formula Replace a with 1, b with –6, and c with 2. Simplify.

92. Example 5-3a Answer: The exact solutions are and The approximate solutions are 0.4 and 5.6. or

93. Example 5-3a CheckCheck these results by graphing the related quadratic function, Using the ZERO function of a graphing calculator, the approximate zeros of the related function are –2.9 and 0.9.

94. Example 5-3b Solve by using the Quadratic Formula. Answer: or approximately 0.7 and 4.3

95. Example 5-4a Now use the Quadratic Formula. Solve by using the Quadratic Formula. Write the equation in the form Simplify. Replace a with 1, b with –6, and c with 13. Quadratic Formula

96. Answer: The solutions are the complex numbers and Example 5-4a Simplify.

97. Example 5-4a A graph of the function shows that the solutions are complex, but it cannot help you find them.

98. Example 5-4a Original equation CheckTo check complex solutions, you must substitute them into the original equations. The check for is shown below. Sum of a square; Distributive Property Simplify.

99. Example 5-4b Solve by using the Quadratic Formula. Answer:

100. Example 5-5a Answer: The discriminant is 0, so there is one rational root. Find the value of the discriminant for. Then describe the number and type of roots for the equation.

101. Example 5-5a Answer: The discriminant is negative, so there are two complex roots. Find the value of the discriminant for. Then describe the number and type of roots for the equation.

102. Example 5-5a Answer: The discriminant is 80, which is not a perfect square. Therefore, there are two irrational roots. Find the value of the discriminant for. Then describe the number and type of roots for the equation.

103. Example 5-5a Answer: The discriminant is 81, which is a perfect square. Therefore, there are two rational roots. Find the value of the discriminant for. Then describe the number and type of roots for the equation.

104. Find the value of the discriminant for each quadratic equation. Then describe the number and type of roots for the equation. a. b. c. d. Example 5-5b Answer: 0 ; 1 rational root Answer: –24 ; 2 complex roots Answer: 5 ; 2 irrational roots Answer: 64 ; 2 rational roots

105. End of Lesson 5

106. Lesson 6 Contents Example 1Graph a Quadratic Function in Vertex Form Example 2Write y = x 2 + bx + c in Vertex Form Example 3Write y = ax 2 + bx + c in Vertex Form, a  1 Example 4Write an Equation Given Points

107. Example 6-1a Now use this information to draw the graph. Analyze Then draw its graph. h = 3 and k = 2 Answer: The vertex is at (h, k) or (3, 2) and the axis of symmetry is The graph has the same shape as the graph of but is translated 3 units right and 2 units up.

108. Example 6-1a Step 1 Plot the vertex, (3, 2). Step 3 Find and plot two points on one side of the axis of symmetry, such as (2, 3) and (1, 6). Step 4 Use symmetry to complete the graph. Step 2 Draw the axis of symmetry, (1, 6)(5, 6) (2, 3)(4, 3) (3, 2)

109. Example 6-1b Answer: Analyze Then draw its graph. The vertex is at (–2, – 4), and the axis of symmetry is The graph has the same shape as the graph of ; it is translated 2 units left and 4 units down.

110. Example 6-2a Write in vertex form. Then analyze the function. Notice that is not a perfect square. Balance this addition by subtracting 1. Complete the square by adding

111. Example 6-2a This function can be rewritten as So, and Answer: The vertex is at (–1, 3), and the axis of symmetry is Since the graph opens up and has the same shape as but is translated 1 unit left and 3 units up. Write as a perfect square.

112. Example 6-2b Write in vertex form. Then analyze the function. Answer: vertex: (–3, –4) ; axis of symmetry: opens up; the graph has the same shape as the graph of but it is translated 3 units left and 4 units down.

113. Example 6-3a Write in vertex form. Then analyze and graph the function. Original equation Group and factor, dividing by a. Complete the square by adding 1 inside the parentheses. Balance this addition by subtracting –2(1).

114. Example 6-3a Write as a perfect square. Now graph the function. Two points on the graph to the right of are (0, 2) and (0.5, –0.5). Use symmetry to complete the graph. Answer: The vertex form is So, andThe vertex is at (–1, 4) and the axis of symmetry is Since the graph opens down and is narrower than It is also translated 1 unit left and 4 units up.

115. Example 6-3a

116. Example 6-3b Write in vertex form. Then analyze and graph the function. Answer: vertex: (–1, 7) ; axis of symmetry: x = –1 ; opens down; the graph is narrower than the graph of y = x 2,and it is translated 1 unit left and 7 units up.

117. Example 6-4a Write an equation for the parabola whose vertex is at (1, 2) and passes through (3, 4). The vertex of the parabola is at (1, 2) so and Since (3, 4) is a point on the graph of the parabola, and Substitute these values into the vertex form of the equation and solve for a. Vertex form Substitute 1 for h, 2 for k, 3 for x, and 4 for y. Simplify.

118. Example 6-4a Subtract 2 from each side. Divide each side by 4. Answer:The equation of the parabola in vertex form is

119. Example 6-4a Check A graph ofverifies that the parabola passes through the point at (3, 4).

120. Answer: Example 6-4b Write an equation for the parabola whose vertex is at (2, 3) and passes through (–2, 1).

121. End of Lesson 6

122. Lesson 7 Contents Example 1Graph a Quadratic Inequality Example 2Solve ax 2 + bx + c  0 Example 3Solve ax 2 + bx + c  0 Example 4Write an Inequality Example 5Solve a Quadratic Inequality

123. Example 7-1a Graph Step 1 Graph the related quadratic equation, Since the inequality symbol is >, the parabola should be dashed.

124. Example 7-1a Graph Step 2 Test a point inside the parabola, such as (1, 2). So, (1, 2) is a solution of the inequality. (1, 2)

125. Example 7-1a Step 3 Shade the region inside the parabola. Graph

126. Example 7-1b Answer: Graph

127. Example 7-2a The solution consists of the x values for which the graph of the related quadratic function lies above the x -axis. Begin by finding the roots of the related equation. Solve by graphing. Related equation Factor. Solve each equation. Zero Product Property or

128. Example 7-2a Sketch the graph of the parabola that has x -intercepts at 3 and 1. The graph lies above the x -axis to the left of and to the right of Answer: The solution set is

129. Example 7-2b Solve by graphing. Answer:

130. Example 7-3a Solve by graphing. This inequality can be rewritten as The solution consists of the x -values for which the graph of the related quadratic equation lies on and above the x -axis. Begin by finding roots of the related equation. Related equation Use the Quadratic Formula. Replace a with –2, b with –6 and c with 1.

131. Example 7-3a Simplify and write as two equations. or Simplify. Sketch the graph of the parabola that has x -intercepts of –3.16 and 0.16. The graph should open down since a < 0.

132. Example 7-3a The graph lies on and above the x -axis at and and between these two values. The solution set of the inequality is approximately Answer:

133. Example 7-3a Check Test one value of x less than – 3.16, one between –3.16 and 0.16, and one greater than 0.16 in the original inequality.

134. Example 7-3b Solve by by graphing. Answer:

135. Example 7-4a Sports The height of a ball above the ground after it is thrown upwards at 40 feet per second can be modeled by the function where the height h(x) is given in feet and the time x is in seconds. At what time in its flight is the ball within 15 feet of the ground? The function h(x) describes the height of the ball. Therefore, you want to find values of x for which Original inequality Subtract 15 from each side.

136. Example 7-4a Graph the related function using a graphing calculator. The zeros are about 0.46 and 2.04. The graph lies below the x -axis whenor

137. Example 7-4a Answer: Thus, the ball is within 15 feet of the ground for the first 0.46 second of its flight and again after 2.04 seconds until the ball hits the ground at 2.5 seconds.

138. Example 7-4b Answer: The ball is within 10 feet of the ground for the first 0.5 second of its flight and again after 1.25 seconds until the ball hits the ground. Sports The height of a ball above the ground after it is thrown upwards at 28 feet per second can be modeled by the function where the height h(x ) is given in feet and the time x is given in seconds. At what time in its flight is the ball within 10 feet of the ground?

139. Example 7-5a Solve algebraically. First, solve the related equation. Related quadratic equation Subtract 2 from each side. Factor. Solve each equation. Zero Product Property or

140. Example 7-5a Plot –2 and 1 on a number line. Use closed circles since these solutions are included. Notice that the number line is separated into 3 intervals.

141. Example 7-5a Test a value in each interval to see if it satisfies the original inequality.

142. Example 7-5a Answer: The solution set is This is shown on the number line below.

143. Example 7-5b Solve algebraically. Answer:

144. End of Lesson 7

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