Warm-Up Oct. 22. Daily Agenda:

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1 Evaluate y = 2x 3x + 5 when x = 1, 0, and 2. Daily Agenda: Grade Assignment Go over Ch 3 Test; Retakes must be done by next Tuesday 5.1 notes / assignment Graphing Quadratic Functions 5.2 notes / assignment Solving Quadratic Equations by Factoring Warm-Up Oct. 22

2 Chapter 5 Quadratic Function

3 Four ways to solve quadratic equations How to graph quadratic functions and inequalities What you ll learn

4 Chapter 5.1 Graphing Quadratic Functions

5 Goals What you should learn How to graph a quadratic function. How to use quadratic functions to solve real-life problems. Why you should learn it To model real-life objects, such as the cables of the Golden Gate Bridge.

6 A quadratic function has the form y = ax + bx + c Where a 0 The graph of a quadratic is a u-shape called a parabola The turning point is called the vertex The axis of symmetry is the vertical line through the vertex The parabola is symmetric about that line Goal: Learn how to graph a quadratic function. Graphing a Quadratic Function

7 P249 Answer questions #3 and #4 in notes. Goal: Learn how to graph a quadratic function. Graphing Calculator activity

8 The graph of y = ax + bx + c(called standard form) is a parabola with these characteristics: Opens up if a > 0 Opens down if a < 0 Wider than y = x if a < 1 Narrower than y = x if a > 1 X-coordinate of vertex is Axis of Symmetry is line x = Goal: Learn how to graph a quadratic function. The graph of a Quadratic Equation

9 Goal: Learn how to graph a quadratic function. Graph y = x + x + 12 Example

10 Goal: Learn how to graph a quadratic function. Graph y = x x 6 Example

11 Vertex form: y = a x h + k Intercept form: y = a x p x q Goal: Learn how to graph a quadratic function. The vertex is (h, k) The axis of symmetry is x = h The x-intercepts are p and q The axis of symmetry is halfway between (p, 0) and (q, 0). Opens up if a > 0 Opens down if a < 0 Additional Forms of a Quadratic Function

12 Goal: Learn how to graph a quadratic function. Graph y = 2 x Example

13 Goal: Learn how to graph a quadratic function. Graph y = x Example

14 Goal: Learn how to graph a quadratic function. Graph y = 2 x 3 x + 1 Example

15 Goal: Learn how to graph a quadratic function. Graph y = 2x x 4 Example

16 Goal: Learn how to graph a quadratic function. Write the quadratic function in standard form. y = x 6 x 4 y = 4 x Example

17 Write the quadratic function in standard form. y = x Goal: Learn how to graph a quadratic function. y = 3 x + 1 x 5 Example

18 Goal: Learn how to use quadratic functions to solve real-life problems. Suppose that a group of high school students conducted an experiment to determine the number of hours of study that leads to the highest score on a comprehensive year-end exam. The exam score y for each student who studied for x hours can be modeled by y = x x Which amount of studying produced the highest score on the exam? What is the highest percent score the model predicts? Example

19 Goal: Learn how to use quadratic functions to solve real-life problems. The path of a ball thrown by a baseball player forms a parabola with equation: y = x , where x is the horizontal distance in feet of the ball from the player and y is the height in feet of the ball. How far does the ball travel before it again reaches the same height from which it was thrown? How high was the ball at its highest point? Example

20 P253: 17 19, x 3s, 50, 52, 54, 65, 72 #50 requires a graphing calculator! Assignment

21 Solve the equation 2 x 3 = 6 Daily Agenda: Grade assignment 5.2 notes / assignment Warm-Up Oct. 24

22 Chapter 5.2 Solving Quadratic Equations by Factoring

23 Goals What you should learn How to solve quadratic expressions by factoring How to find zeros of quadratic functions Why you should learn it To solve real-life problems, such as finding appropriate dimensions for a painted mural.

24 Factoring Review: Trinomial x2 + 8x + 15 can be factored into two binomials, (x + 3) and (x + 5) To FACTOR means to write as multiplication So (x + 3)(x + 5) To check to see if you have factored correctly, multiply out answer Or FOIL Goal: Learn how to solve quadratic expressions by factoring. From ax2 + bx + c, looking for factors of c that add to be b. Factoring Quadratic Expressions

25 Goal: Learn how to solve quadratic expressions by factoring. Example Factor the following: x2 2x 48 x2 21x + 80 x2 + 5x + 6 x2 3x 4 x2 + 9x 36 x2 10x + 21

26 Goal: Learn how to solve quadratic expressions by factoring. Example Factor the following: 4y2 4y 3 5x2 + 17x + 14

27 Difference of Two Squares: a2 b2 = (a + b)(a b) Goal: Learn how to solve quadratic expressions by factoring. Perfect Square Trinomial: a2 + 2ab + b2 = (a + b)2 a2 2ab + b2 = (a b)2 GCF: Greatest Common Factor ALWAYS look to factor it out first! Special Factoring Patterns

28 Goal: Learn how to solve quadratic expressions by factoring. Example Factor the following: 16y x2 + 2x 12 3v2 18v 4z2 12z x2 + 3x w2 + 60w u2 36

29 Goal: Learn how to solve quadratic expressions by factoring. Example Factor the following: 64x2 9 30u2 57u x2 + 8x + 1

30 Write equation in standard form ax2 + bx + c = 0 The y has changed to a zero. Because when y is zero, where are we? We re looking for the x-intercepts or roots or zeros Factor left-hand side Set each factor equal to zero Zero Product Property: If AB = 0, then A = 0, or B = 0, or they both = 0. Solve each simple equation. Goal: Learn how to solve quadratic expressions by factoring. How is this different from standard form in 5.1? Solving Quadratic Equations by Factoring

31 Goal: Learn how to solve quadratic expressions by factoring. Example Solve by factoring: 9t2 12t + 4 = 0 3x 6 = x2 10

32 Solve by factoring: 2w2 10w = 23w w2 Goal: Learn how to solve quadratic expressions by factoring. Example

33 Goal: Learn how to solve quadratic expressions by factoring. A painter is making a rectangular canvas for her next painting. She wants the length of the canvas to be 4 ft more than twice the width of the canvas. The area of the canvas must be 30 ft2. What should the dimensions of the canvas be? Example

34 Remember intercept form of a quadratic y = a(x p)(x q) Goal: Learn how to find zeros of quadratic functions We can find the zeros by factoring! Find Zeros of Quadratic Functions

35 Find the zeros of y = 3x2 + 14x 5. Goal: Learn how to find zeros of quadratic functions Find the zeros of y = x2 + 8x Example

36 Goal: Learn how to find zeros of quadratic functions You own an amusement park that averages 75,000 visitors per year who each pay a $12 admission charge. You plan to lower the admission price to attract new customers. It has been shown that each $1 decrease in price results in 15,000 new visitors. What admission should you charge to maximize your annual revenue? What is the maximum revenue? Example

37 P260: x 3s, 47, 50, 53, x 3s, 100, 128, 131, 135 Assignment

38 Solve the equations: 5x 3 = 17 0 = t Daily Agenda: Grade assignment 5.3 notes / assignment 1st half of 5.4 notes / assignment Warm-Up Oct. 29

39 Chapter 5.3 Solving Quadratic Equations by Finding Square Roots

40 Goals What you should learn How to solve quadratic equations by finding square roots. How to use quadratic equations to solve real-life problems. Why should you learn it To be able to model real-life quantities, such as the height of a rock dropped off a tower.

41 Goal: Learn how to solve quadratic equations by finding square roots. Square root: one of the two identical factors of a number What are two identical factors of 16? Which means that the square root of 16 is A positive number has two square roots, s and s Radical sign: Radicand: the number beneath the radical sign Radical: any expression in the form s Solving Quadratic Equations

42 Activity on P264 Goal: Learn how to solve quadratic equations by finding square roots. Write answers to all 6 problems plus answer the two questions in your notes. Investigating Properties of Square Roots

43 Goal: Learn how to solve quadratic equations by finding square roots. Product Property: ab = a b Quotient Property: = A square-root expression is considered simplified if: No radicand has a perfect-square factor other than 1 There is no radical in the denominator Properties of Square Roots (a > 0, b > 0)

44 Simplify the expression. 500 Goal: Learn how to solve quadratic equations by finding square roots Example

45 Goal: Learn how to solve quadratic equations by finding square roots. Rationalizing the denominator is the process of multiplying both the numerator and denominator by the same radical expression The purpose is to eliminate the radical from the denominator Rationalizing the Denominator

46 Goal: Learn how to solve quadratic equations by finding square roots. Simplify the expression. Example

47 Goal: Learn how to solve quadratic equations by finding square roots. You can solve equations in the form x = s by taking the square root of both sides of the equation. If s > 0, x = s has two real-number solutions Written x = ± s Solving Quadratic Equations

48 Solve 3 5x = 9 Goal: Learn how to solve quadratic equations by finding square roots. Example

49 Goal: Learn how to solve quadratic equations by finding square roots. Solve 3 x 2 = 21 Example

50 Solve the following equations: 4x 6 = 42 Goal: Learn how to solve quadratic equations by finding square roots. x 4 = 6 Example

51 We can model the height/time relationship for any item that is dropped on Earth with the following equation: h = 16t + h h = current height Goal: Learn how to use quadratic equations to solve real-life problems. t = after t seconds in time h0 = initial height the object was dropped from Using Quadratic Models in Real-Life

52 Goal: Learn how to use quadratic equations to solve real-life problems. The tallest building in the United States is in Chicago, Ill. It is 1450 ft. tall. How long would it take a penny to drop from the top of this building? How fast would the penny be traveling when it hits the ground if the speed is given by s = 32t where t is the number of seconds since the penny was dropped? Example

53 P267: x 3s, 70, 71, 78, 86 Assignment

54 Chapter 5.4 Complex Numbers

55 Goals What you should learn How to solve quadratic equations with complex solutions and perform operations with complex numbers. Apply complex numbers to fractal geometry. Why you should learn it To solve problems involving complex solutions

56 What happens when you take the square root of a negative number? The answer is not a real number! To compensate, mathematicians created the imaginary number,i i = 1 Goal: How to solve quadratic equations with complex solutions. Properties of the Square Root of a Negative Number If r is a positive real number, then r = i r Then it follows that i r = r Or i = 1 Imaginary Numbers

57 Solve 2x + 26 = 10 Goal: How to solve quadratic equations with complex solutions. Example

58 Solve x + 1 = 5 Goal: How to solve quadratic equations with complex solutions. Example

59 Goal: How to perform operations with complex numbers. Standard form for a complex number is a + bi a is the real part bi is the imaginary part Complex Numbers

60 Every complex number corresponds to a point in the complex plane Horizontal axis is the real axis Vertical axis is the imaginary axis Goal: How to perform operations with complex numbers. Complex Plane

61 Goal: How to perform operations with complex numbers. Plot the complex numbers in the complex plane. 4 i i Example

62 To add or subtract complex numbers, Add/subtract the real parts Add/subtract the imaginary parts Goal: How to perform operations with complex numbers. Operations with Complex Numbers

63 Write the expression as a complex number in standard form i i Goal: How to perform operations with complex numbers. 2 3i 3 7i 2i 3 + i + 2 3i Example

64 Distribute or FOIL just as you would when multiplying real numbers. Remember that i = 1 Goal: How to perform operations with complex numbers. Multiplying Complex Numbers

65 Write the expression as a complex number in standard form. i 3 + i Goal: How to perform operations with complex numbers i 6 2i 1 + 2i 1 2i Example

66 Goal: How to perform operations with complex numbers. To divide complex numbers, multiply by the complex conjugate The complex conjugate is the same expression as the denominator with the opposite operation in the middle Dividing Complex Numbers

67 Goal: How to perform operations with complex numbers. Write the quotient in standard form. Example

68 Write the expressions as a complex number in standard form. 3i 9 i Goal: How to perform operations with complex numbers i 3 6i Example

69 P277: x 3s Assignment #1

70 No warm-up Be ready to grade homework when the bell rings Daily Agenda: Explore the Mendelbrot Set Oct. 31

71 Get our a piece of paper During video, answer the following List 5 characteristics, properties, or items that you learned about fractals Write at least two complete sentences summarizing what you learned. lated Hunting the Hidden Dimension

72 Graph 3 + 4i How can we find the distance between the number and the origin? z = a + b Absolute Value of a Complex Number

73 Find the absolute value of each complex number. Which is closest to the origin in a complex plane? 2 + 5i 6i 5 3i Example

74 To determine if a complex number c belongs in the Mandelbrot set, plug it into the following equation and work it out as follows: f z = z + c z = 0, z = f z, z = f z, z = f z, If the absolute values of this list of numbers is less than some fixed number, than c belongs in the set If the absolute values of this list of numbers becomes infinitely large, then c does no belong in the set Complex Numbers in the Mandelbrot Set

75 f z = z + c z = 0, z = f z, z = f z, z = f z, Tell whether the complex number c belongs to the Mandelbrot set. c = 0.5i c = 3 c = 2 + i Example

76 P278: Assignment

77 Solve the equation 3 x + 5 = 24 Daily Agenda: Grade assignment 5.5 notes / assignment 5.6 notes / assignment Quiz Tuesday! Warm-Up Nov. 2

78 Chapter 5.5 Completing the Square

79 Goals What you should learn How to solve quadratic equations by completing the square How to use completing the square to write quadratic functions in vertex form. Why you should learn it To solve real-life problems.

80 Goal: How to solve quadratic equation by completing the square. Perfect Square Trinomial: a + 2ab + b = a + b or a 2ab + b = a b To solve by completing the square, make a trinomial be a perfect square trinomial Then can factor into one binomial squared. Steps: Move constant to other side Add number to each side to make a perfect square trinomial Half of b squared Factor left hand side Square root both sides Add plus/minus sign Solve Completing the Square

81 Goal: How to solve quadratic equation by completing the square. Find the value of c that makes x 3x + c a perfect square trinomial. Then write the expression as the square of a binomial. Example

82 Goal: How to solve quadratic equation by completing the square. Solve x 6x 8 = 0 by completing the square. Example

83 Goal: How to solve quadratic equation by completing the square. Solve x + 4x 1 = 0 by completing the square. Example

84 Goal: How to solve quadratic equation by completing the square. Solve 5x 10x + 30 = 0 by completing the square. Example

85 Goal: How to solve quadratic equation by completing the square. Solve 3x 12x + 16 = 0 by completing the square. Example

86 Goal: How to solve quadratic equation by completing the square. Under certain road conditions, the formula for a car s stopping distance is given by d = 0.1s + 1.1s. If a driver leaves 5 car lengths, approximately 75 ft, between him and the driver in front of him, what is the maximum speed he can drive and still stop safely? Example

87 Goal: How to solve quadratic equation by completing the square. You have 30 ft of chain link fence to make a rectangular enclosure for your dog. A pet store owner recommended that an enclosure for one dog be at least 48 ft2 in area. What should the dimensions of the enclosure be to make the area 48 ft2. Example

88 Goal: How to use completing the square to write quadratic functions in vertex form. Can use completing the square to write in the form y = a x h + k Writing in Vertex Form

89 Goal: How to use completing the square to write quadratic functions in vertex form. Write the quadratic function y = x + 6x + 16 in vertex form. What is the vertex of the function s graph? Example

90 Goal: How to use completing the square to write quadratic functions in vertex form. Write the quadratic function y = x + 3x + 3 in vertex form. What is the vertex of the function s graph? Example

91 Goal: How to use completing the square to write quadratic functions in vertex form. An agricultural researcher finds that the height h (in inches) of one type of pepper plant can be modeled by the function h = 0.88r + 8.8r + 20 where r is the amount of rainfall (in inches) that fell during the growing season. How much rain would maximize the height of the pepper plants? What is the maximum height? Example

92 P286: x 3s, 94, 96, 98, 116 Assignment

93 Chapter 5.6 The Quadratic Formula and the Discriminant

94 Goals What you should learn How to solve quadratic equations using the quadratic formula. How to use the quadratic formula in real-life situations. Why you should learn it To solve real-life problems.

95 Goal: How to solve quadratic equations using the quadratic formula. To solve ax + bx + c = 0 x = ± The Quadratic Formula

96 Goal: How to solve quadratic equations using the quadratic formula. Solve 3x + 8x = 35 Example

97 Goal: How to solve quadratic equations using the quadratic formula. Solve 12x 5 = 2x + 13 Example

98 Goal: How to solve quadratic equations using the quadratic formula. Solve 2x = 2x + 3 Example

99 Goal: How to solve quadratic equations using the quadratic formula. Solve 2x + x = x 2x + 4 Example

100 Solve x = 2x 5 Goal: How to solve quadratic equations using the quadratic formula. Example

101 Evaluate the expression b 4ac for a = 2, b = -2, and c = 0. Daily Agenda: Grade assignment Quiz Finish 5.6 notes / assignment 5.7 notes / assignment Warm-Up Nov. 6

102 Goal: How to solve quadratic equations using the quadratic formula. The discriminant is the expression b 4ac The expression under the radical sign Tells the types of solutions If positive, two real solutions If zero, one real solution If negative, two imaginary solutions The Discriminant

103 Goal: How to solve quadratic equations using the quadratic formula. Find the discriminant of the quadratic equations and give the number and types of solutions of the equations. 9x + 6x + 1 = 0 9x + 6x 4 = 0 9x + 6x + 5 = 0 Example

104 Goal: How to use the quadratic formula in real-life situations. The water in a large fountain leaves the spout with a vertical velocity of 30 ft per second. After going up in the air it lands in a basin 6 ft below the spout. If the spout is 10 ft above the ground, how long does it take a single drop of water to travel from the spout to the basin? Use the model h = 16t + v t + h. Example

105 P295: x 4s, 80, 99 Assignment

106 Solve and graph (on a number line) 3 6 x 1 > 9 Daily Agenda: Grade assignment 5.7 notes / assignment 5.8 notes assignment Warm-Up Nov. 8

107 Chapter 5.7 Graphing and Solving Quadratic Inequalities

108 Goals What you should learn How to graph quadratic inequalities in two variables How to solve quadratic inequalities in one variable Why you should learn it To solve real-life problems like how much weight a rope can support.

109 Goal: How to graph quadratic inequalities in two variables. This combines graphing quadratics and graphing inequalities Steps to graphing quadratic inequalities Graph the equation y = ax + bx + c Find axis of symmetry, vertex, intercepts from solving Pick a test point and check in the original inequality Shade the appropriate region Graphing a Quadratic Inequality

110 Graph y 2x 5x 3 Goal: How to graph quadratic inequalities in two variables. Example

111 Goal: How to graph quadratic inequalities in two variables. You are making a photo album. Each album page needs to be able to hold 6 square pictures. If the length of one side of each picture is x, then A 6x is the area of one album page. Graph this function. If you have an album page that has an area of 70 square inches, will it be able to accommodate 6 pictures with 3-inch sides? Example

112 Two or more inequalities graphed on the same coordinate plane. Shade each graph separately Goal: How to graph quadratic inequalities in two variables. Shade darker the overlapping region This region has the solutions in common to both inequalities Systems of Quadratic Inequalities

113 Graph the system of inequalities y x + 9 y x + 5x 6 Goal: How to graph quadratic inequalities in two variables. Example

114 One way to solve quadratic inequalities in one variable is by graphing After graphed: Goal: How to solve quadratic inequalities in one variables. If graphed ax + bx + c < 0, looking at x values below x-axis If graphed ax + bx + c > 0, looking at x values above x-axis Solving Quadratic Inequalities in One Variable

115 Solve x 5x Goal: How to solve quadratic inequalities in one variables. Example

116 Solve x 11x Goal: How to solve quadratic inequalities in one variables. Example

117 Solve x 9x + 36 > 0. Goal: How to solve quadratic inequalities in one variables. Example

118 Solve 3x + x + 7 < 0. Goal: How to solve quadratic inequalities in one variables. Example

119 Goal: How to solve quadratic inequalities in one variables. Another way to solve quadratic inequalities in one variable is algebraically. Use factoring, square root property, completing the square, or quadratic formula Graph the critical x-values on a number line. This creates 3 intervals or regions, check a value in each of the regions to determine the inequalities that satisfy the problem. Solving Quadratic Inequalities in One Variable

120 Solve 2x x > 3 Goal: How to solve quadratic inequalities in one variables. Example

121 Solve 3x + 11x 4 Goal: How to solve quadratic inequalities in one variables. Example

122 Goal: How to solve quadratic inequalities in one variables. Suppose a study was conducted to test the average reading comprehension of a person x years of age. The study found that the number of points P(x) scored on a reading comprehension test could be modeled by: P x = 0.017x + 1.9x + 31, 5 x 95. At what ages does the average person score greater than 60 points on this test? Example

123 P303: by 3s, 49, Assignment

124 Solve the system of linear equations: x + y = 2 x + 3y z = 5 2x y + z = 6 Daily Agenda: Grade assignment Go over quizzes 5.8 notes / assignment Wednesday: Quiz, Review, Project Friday: Test, assignment Tuesday: Finish Project Warm-Up Nov. 12

125 Chapter 5.8 Modeling with Quadratic Functions

126 Goals What you should learn How to write quadratic functions given characteristics of their graphs How to use technology to find quadratic models for data Why you should learn it To solve real-life problems such as determining the effect of wind on a runner s performance

127 Goal: How to write quadratic functions given characteristics of their graphs. Write a quadratic function for the parabola shown. Example

128 Goal: How to write quadratic functions given characteristics of their graphs. Write a quadratic function for the parabola shown. Example

129 Goal: How to write quadratic functions given characteristics of their graphs. Given three points Plug each point individually into y = ax + bx + c You now have three equations with three variables (a, b, and c) Solve the system for a, b, and c Write an equation in standard form. Writing a Quadratic in Standard Form

130 Goal: How to write quadratic functions given characteristics of their graphs. A group of students dropped a rubber ball and measured the height in inches of the ball for each of its successive bounces. The results are shown in the table. Find the quadratic model in standard form for the data using the first three points. Example

131 Goal: How to use technology to find quadratic models for data. Quadratic Regression: a quadratic equation that best models the data On graphing calculator: STAT, CALC, QuadReg VARS, 5, EQ, 1 to put in Y= Best-Fitting Quadratic Model

132 Goal: How to use technology to find quadratic models for data. A bank adjusts its interest rates for new certificates of deposits daily. The table shows the interest rates on the first of the month for January through May. Find the best-fitting quadratic model for the data. According to the model, during which month did the certificates of deposit have the highest interest rate. What was that rate? Example

133 Goal: How to use technology to find quadratic models for data. Find the best-fitting quadratic model for the data in the table. Example

134 P309: 8 32 even, 38 Assignment

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