Chapter 3: The Parabola

Transcription

1 Chapter 3: The Parabola SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza

2 Chapter 3: The Parabola Lecture 7: Introduction to Parabola Lecture 8: Converting General Form of a Parabola to Its Standard Form and Vice-Versa Lecture 9: The Graph of Parabola with Vertex at the Origin Lecture 10: The Graph of Parabola with Vertex at (h, k) Lecture 11: The Parabola and the Tangent Line Lecture 12: Some Applications of Parabola

3 TED Ed Video Number 3: The Math Behind Michael Jordan s Legendary Hang Time by Andy Peterson and Zack Patterson

4 Lecture 7: Introduction to Parabola SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza

5 A Short Recap From our previous lesson about conic section, how does a parabola formed?

6 A Short Recap A parabola is graph of what kind of function?

7 Classroom Task Number 4: Using the figure on the next slide, give your own definition of a parabola.

8 Parabola to You: Look at Me, How Do You Define Me?

9 The Definition of Parabola A parabola is the set of all points in a plane equidistant from a fixed point and a fixed line.

10 The Focus and the Directrix The fixed point is called the focus and the fixed line is called the directrix.

11 The Graph of the Parabola

12 Something to think about What do you think are the roles of the focus and the directrix in the graph of a parabola?

13 Type 1: Parabola Opening Upward Whenever the directrix is below the focus, then the parabola opens upward.

14 Type 2: Parabola Opening Downward Whenever the directrix is above the focus, then the parabola opens downward.

15 Type 3: Parabola Opening to the Right Whenever the directrix is on the left side of the focus, then the parabola opens to the right.

16 Type 4: Parabola Opening to the Left Whenever the directrix is on the right side of the focus, then the parabola opens to the left.

17 The Vertex and the Axis of Symmetry The vertex of the parabola is the midpoint of the perpendicular segment from the focus to the directrix, while the line that passes through the vertex and focus is the axis of symmetry.

18 The Graph of the Parabola

19 Something to think about What condition will guarantee that the vertex is exactly the midpoint of the perpendicular segment from focus to the directrix?

20 Parabola to You: Look at Me, How Do You Define Me?

21 Something to think about When we say an image/ an object is symmetric, what do we mean by that?

22 The Graph of the Parabola

23 The Latus Rectum The line segment through the focus perpendicular to the axis of symmetry is called the latus rectum whose length is 4c.

24 Did you know? The word latus rectum (singular, plural: latera recta) came from a Latin word latus which means line or side and rectum which means straight. So, the its English translation is straight line or straight side.

25 The Graph of the Parabola

26 Something to think about What do you think is the relationship of latus rectum to the directrix?

27 Something to think about Why is it called a "latus rectum"? What do you think its purpose?

28 The Focal Length The focal length is the undirected distance between the vertex and focus in this case c.

29 The Graph of the Parabola

30 Classroom Task Number 5: Derive the formula of the Parabola Opening to the Right having the Vertex at the Origin

31 The Definition of Parabola A parabola is the set of all points in a plane equidistant from a fixed point and a fixed line.

32 Parabola Opening to the Right having the Vertex at the Origin (0, 0)

33 Our Equation: Using the definition, we will use the equation below to derive the formula: d 1 d 2

34 Formula of the Parabola Opening to the Right having the Vertex at the Origin Formula of the Parabola Opening to the Right having the Vertex at the Origin: 2 y 4cx

35 Classroom Task Number 6: Derive the formula of the Parabola Opening Upward having the Vertex at the Origin

36 The Definition of Parabola A parabola is the set of all points in a plane equidistant from a fixed point and a fixed line.

37 Parabola Opening Upward having the Vertex at the Origin (0, 0)

38 Our Equation: Using the definition, we will use the equation below to derive the formula: d 1 d 2

39 Formula of the Parabola Opening Upward having the Vertex at the Origin Formula of the Parabola Opening Upward having the Vertex at the Origin 2 x 4cy

40 Performance Task 6: Using the definition of the parabola, its parts and its graph, derive the formulae for the following: o Parabola Opening Downward having the Vertex at the Origin; and o Parabola Opening to the Left having the Vertex at the Origin.

41 Lecture 8: Converting General Form to Standard Form of a Parabola SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza

42 Learning Expectations: This section presents how to convert form of a parabola to its standard form and viceversa.

43 The General and Standard Equations of the Parabola Vertex General Form Standard Form (0, 0) (h, k) 0 2 F Dx y cx y F Ey x y 4cx 2 x 4cy 2 x 4cy F Ey Dx y ) ( 4 ) ( 2 h x c k y ) ( 4 ) ( 2 h x c k y 0 2 F Ey Dx x ) ( 4 ) ( 2 k y c h x ) ( 4 ) ( 2 k y c h x

44 Example 21: Convert the general equations to standard form: y 2 12 x 2y x 2 12 x y 16 0

45 Final Answer: The standard forms of the given general equations are: ( 2 y 1) 12( x 2) 1 ( x 3) 2 ( y 2 2)

46 Example 22: Convert the standard equations to general form: ( 2 y 3) 7( x 8) 2 ( x 2) 8( y 5)

47 Final Answer: The general forms of the given standard equations are: y 2 7 x 6 y 65 0 x 2 4x 8 y 44 0

48 Performance Task 7: Please download, print and answer the Let s Practice 7. Kindly work independently.

49 Lecture 9: The Graph of the Parabola with Vertex at the Origin SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza

50 The Parts of the Graph of the Parabola with Vertex at the Origin Standard Equation Focus Directrix Axis of Symmetry Endpoints of Latus Rectum Graph y 2 4cx (c, 0) x = -c x-axis E 1 (c, 2c) E 2 (c, -2c) Opening to the Right 2 y 4cx (-c, 0) x = c x-axis E 1 (-c, 2c) E 2 (-c, -2c) Opening to the Left x 2 4cy (0, c) y = -c y-axis E 1 (2c, c) E 2 (-2c, c) Opening Upward 2 x 4cy (0, -c) y = c y-axis E 1 (2c, -c) E 2 (-2c, -c) Opening Downward

51 Classroom Task Number 8: To understand where the coordinates of each part of the graph of the parabola with vertex at the origin V (0, 0) derived from, show it to the class using the graph of the parabola opening upward.

52 Example 22: Sketch and discuss: 2 x 8y

53 Supply the Table: Vertex: Focus: Axis of Symmetry: Directrix: Length of Latus Rectum: Endpoints of Latus Rectum:

54

55 Example 23: Find the equation of the parabola with vertex at (0, 0) and focus at ( 0, 3 4).

56 Supply the Table: Vertex: Focus: Axis of Symmetry: Directrix: Length of Latus Rectum: Endpoints of Latus Rectum:

57

58 Performance Task 8: Please download, print and answer the Let s Practice 8. Kindly work independently.

59 Lecture 10: The Graph of the Parabola with Vertex at (h, k) SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza

60 The Parts of the Graph of the Parabola with Vertex at the (h, k) Standard Equation Focus Directrix Axis of Symmetry Endpoints of Latus Rectum Graph ( y k) ( y k) 2 ( x h) ( x h) c( x h) 4c( x h) 4c( y k) 4c( y k) (h+c, k) x = h-c y = k (h-c, k) x = h+c y = k E 1 (h+c, k+2c) E 2 (h+c, k-2c) E 1 (h-c, k+2c) E 2 (h-c, k-2c) (h, k+c) y = k-c x = h E 1 (h+2c, k+c) E 2 (h-2c, k+c) (h, k-c) y = k+c x = h E 1 (h+2c, k-c) E 2 (h-2c, k-c) Opening to the Right Opening to the Left Opening Upward Opening Downward

61 Classroom Task Number 9: To understand where the coordinates of each part of the graph of the parabola with vertex at V (h, k) derived from, show it to the class using the graph of the parabola opening to the right.

62 Example 24: Reduce the general equation y 2 4x 6y 11 0 standard form. Determine the vertex, focus, directrix, axis of symmetry, length of latus rectum and its endpoints and sketch the graph.

63 Supply the Table: Vertex: Focus: Axis of Symmetry: Directrix: Length of Latus Rectum: Endpoints of Latus Rectum:

64

65 Example 25: Find the general equation of the parabola with vertex at (3, 6) and focus at (3, 4). Sketch the graph.

66

67 Final Answer: Thus, general equation of the parabola is x 2 6x 8 y 37 0.

68 Example 26: Find the general equation of the parabola with vertex at (2, 1) and directrix x = 5. Sketch the graph.

69

70 Final Answer: Thus, general equation of the parabola is y 2 12 x 2 y 23 0.

71 Performance Task 9: Please download, print and answer the Let s Practice 9. Kindly work independently.

72 Lecture 11: The Parabola and the Tangent Line SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza

73 Learning Expectations: This section illustrates the tangent line to a parabola and a line that pass through two points of the parabola.

74 A Review: Aside from what we have discussed about the introduction to parabola, what is another way of writing the general equation of a parabola?

75 The General Equation of a Parabola The General Equation of a Parabola is: Ax 2 Bx C 0

76 Did you know? A quadratic equation denoted by Ax Bx C 0, contains one and only one solution if B 4AC

77 The Parabola and the Tangent Line

78 Example 27: Find the equation of the tangent line to y 2 12 x at the point (-3, 6).

79 Something to think about Based on our previous discussion about a line tangent to the circle, what relationship exists between the tangent line, the point of tangency PT (- 3, 6) and the equation of the parabola 2 y 12 x.

80 What we knew: We can use our prior knowledge about our lesson about tangent to a circle. That is, if a line is tangent to a parabola, then they shared a common point which is the point of tangency denoted by PT (-3, 6). This common point is one of the infinitely many points of the tangent line and the parabola, y 2 12 x which can make both of their equations true once substituted.

81 Something to think about Since we all knew that the point PT (-3, 6) is one of the infinitely many points of the tangent line, it is associated with linear equations, and we were asked to look for its equation, which among the five forms of linear equations can we use in determining its equation?

82 What we know: We can use the point slope form to determine the equation of the tangent line denoted by: y y m x x 1 1 where x 1 is -3 and y 1 is 6.

83 Something to think about Since the value of slope denoted by m is unknown, what can we do to find its value to finally complete the equation of the tangent line?

84 Something to think about What method can we use to simplify: mx 3m x?

85 Something to think about How to simplify mx 3m 6 2 using square of trinomial?

86 Something to think about mx 3m x To simplify, we can use FOIL method given 2 three terms denoted by: 2 2 a b c 2ab 2ac 2bc

87 Something to think about What idea/ theorem can we use to ensure that we can only have one value of slope denoted by m?

88 To Ensure We Can Have One and Only One Slope: A quadratic equation denoted by Ax Bx C 0, contains one and only one solution if B 4AC

89 Final Answer: Thus, the tangent line is x y 3 0.

90 Example 28: Find the points of intersection between the parabola y 2 4x 8y 16 0 and the line x y 4 0. Sketch the graph.

91 Take Note: These points of intersection we are about to compute are two of infinitely many points of both the parabola and the line. Moreover, they are solutions that will make the two equations true. When we say we are to find the points of intersection, it is same as to say we are to find the solutions to the given equations. To determine these two points of intersection, it is more convenient to use substitution method.

92 Final Answer: Thus, the points of intersection are (-4, 0) and (0, 4).

93 Performance Task 10: Please download, print and answer the Let s Practice 10. Kindly work independently.

94 Classroom Task 10: Please prepare a five-minute presentation about the applications of ellipse in the real-world by family. The presentation can be in a form of a family report, skit, video presentation, etc. This will be presented on our next meeting.

95 Lecture 12: Some Applications of Parabola SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza

96 Parabolas in Real Life: Arch Bridges; Microphones; Suspension Car Headlights; Bridges; Mirror in Radar Equipment; Reflecting Telescope; Solar Furnaces; Television; and Search Lights; Radio Antennas.

97 Example 29: How high is the parabolic arch of span 20 feet and height of 16 feet, at a distance 5 feet from the center?

98 Unlocking Terms in the Given Problem: To help us easily solve the given problem, we shall unlock the following terms: 1. PARABOLIC ARCH 2. SPAN What do you know about these?

99 Parabolic Arch It is usually a curved part of a structure that is over an opening and that supports a wall or other weight above the opening.

100 The Span The span is the distance from the end of the thumb to the end of the little finger of a spread hand.

101 The Parabolic Arc:

102 Something to think about How do you understand the problem?

103 Our Understanding of the Problem: We can deduce from the problem that the height of the parabolic arch from the center of its span is 16 feet. However, the problem is asking us to find the height of the parabola when the distance, denoted by span, is 5 feet away from the center.

104 Our Strategy in Solving the Problem: As we have observed, a parabolic arch is modeled by a parabola opens downward with its vertex as the maximum value. Thus, to solve the given problem, we will use the equation of the parabola opens downward graphed on the Cartesian Coordinate Plane, where the x-axis serves as its span while the y-axis is the its height.

105 Our Representations: Let: x-axis be the span of the parabolic arch with 20 feet width y-axis be the height of the parabolic arch with 16 feet height

106 Our Task: Our Task: To find the height (y) when the width or the span (x) is 5 feet away from the center. Thus: 5 feet, y

107 Please Take Note: The y-axis doesn t only represent the height of our parabolic arch. But, by definition, it also serves as the axis of symmetry which divides our parabolic arch into two congruent parts. Hence, our span in the x-axis is also divided into two, resulting to 10 feet each side from the center. Thus, we will have S 1 (-10, 0) and S 2 (10, 0) to denote the endpoints of the span of our parabolic arc.

108 Please Take Note: Remember, when the parabola is opening downward the highest/ maximum point of it is its vertex. Thus, the vertex is V (0, 16) since the height denoted by y-axis is 16 feet and the span denoted by x-axis is zero. Moreover, its vertex is of the form V (h, k), to find the height of the parabolic arch when the distance is 5 feet away from the center, we will use the equation of a parabola opening downward with vertex at V (h, k): x h 2 4c y k

109 Something to think about As we can observed, to find the height of the parabolic arch when its span is 5 feet away from its center, denoted by y, we need to find first the its focal length denoted by c. What will be your strategy? What will you do to find the focal length?

110 Helpful Illustration:

111 Final Answer The height is 12 feet.

112 Example 30: The cable of a suspension bridge has supporting towers which are 70 feet high and 500 feet apart. If the lowest point of the cable is 20 feet above the floor of the bridge, find the length of a supporting rod 80 feet from the center of the span.

113 The Suspension Bridge:

114 Helpful Illustration:

115 Final Answer: Thus, the length of the supporting rod is feet.

116 Example 31: A truck has to pass under an overhead parabolic arch bridge which has a span of 20 meters and is 16 meters high. If the tank is 14 meters wide, is placed in the truck with its sides vertical, and the top of the tank is 7.5 meters above the street level, what is the smallest clearance from the top of the tank so that the truck can pass under the bridge?

117 The Overhead Parabolic Arch Bridge/ Tunnel:

118 Helpful Illustration:

119 Final Answer: Thus, the clearance is 0.66 meters.

120 Example 32: The cross section of a large parabolic microphone can be molded by the equation 2 y 116 x. What is the length of the feedhorn?

121 The Large Parabolic Microphone

122 The Large Parabolic Microphone

123 Helpful Illustration:

124 Final Answer: Thus, the feedhorn should be 29 inches.

125 Performance Task 11: Please download, print and answer the Let s Practice 11. Kindly work independently.