Mathematics
Parabola
A parabola is a U-shaped curve that can be represented by a quadratic equation. It is symmetrical and has a vertex, which is the highest or lowest point on the curve. Parabolas are commonly found in algebra and geometry and are used to model various real-world phenomena, such as the path of a thrown object or the shape of a satellite dish.
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10 Key excerpts on "Parabola"
No longer available |Learn more- James Stewart, Lothar Redlin, Saleem Watson(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
782 CHAPTER 11 ■ Conic Sections 11.1 ParabolaS ■ Geometric Definition of a Parabola ■ Equations and Graphs of Parabolas ■ Applications ■ Geometric Definition of a Parabola We saw in Section 3.1 that the graph of the equation y ax 2 bx c is a U-shaped curve called a Parabola that opens either upward or downward, depend- ing on whether the number a is positive or negative. In this section we study Parabolas from a geometric, rather than an algebraic, point of view. We begin with the geometric definition of a Parabola and show how this leads to the algebraic formula that we are already familiar with. GEOMETRIC DEFINITION OF A Parabola A Parabola is the set of all points in the plane that are equidistant from a fixed point F (called the focus) and a fixed line l (called the directrix). This definition is illustrated in Figure 1. The vertex V of the Parabola lies halfway between the focus and the directrix, and the axis of symmetry is the line that runs through the focus perpendicular to the directrix. Parabola l Axis Focus Vertex Directrix F V FIGURE 1 In this section we restrict our attention to Parabolas that are situated with the vertex at the origin and that have a vertical or horizontal axis of symmetry. (Parabolas in more general positions will be considered in Section 11.4.) If the focus of such a Parabola is the point F1 0, p 2 , then the axis of symmetry must be vertical, and the directrix has the equation y p. Figure 2 illustrates the case p 0. Deriving the Equation of a Parabola If P1 x, y 2 is any point on the Parabola, then the distance from P to the focus F (using the Distance Formula) is "x 2 1 y p 2 2 The distance from P to the directrix is 0 y 1 p 2 0 0 y p 0 y=_p F(0, p) P(x, y) y x y 0 p p FIGURE 2 Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
eBook - PDF- James Stewart, Lothar Redlin, Saleem Watson, , James Stewart, Lothar Redlin, Saleem Watson(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
In this section we study Parabolas from a geometric, rather than an algebraic, point of view. We begin with the geometric definition of a Parabola and show how this leads to the algebraic formula that we are already familiar with. GEOMETRIC DEFINITION OF A Parabola A Parabola is the set of all points in the plane that are equidistant from a fixed point F (called the focus ) and a fixed line l (called the directrix ). This definition is illustrated in Figure 1. The vertex V of the Parabola lies halfway between the focus and the directrix, and the axis of symmetry is the line that runs through the focus perpendicular to the directrix. Parabola l Axis Focus Vertex Directrix F V FIGURE 1 In this section we restrict our attention to Parabolas that are situated with the vertex at the origin and that have a vertical or horizontal axis of symmetry. (Parabolas in more general positions will be considered in Section 7.4.) If the focus of such a Parabola is the point F 1 0, p 2 , then the axis of symmetry must be vertical, and the directrix has the equation y p . Figure 2 illustrates the case p 0. Deriving the Equation of a Parabola If P 1 x , y 2 is any point on the Parabola, then the distance from P to the focus F (using the Distance Formula) is x 2 1 y p 2 2 The distance from P to the directrix is 0 y 1 p 2 0 0 y p 0 y=_p F(0, p) P(x, y) y x y 0 p p FIGURE 2 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. SECTION 7.1 ■ Parabolas 549 By the definition of a Parabola these two distances must be equal.
eBook - PDF- Paul A. Calter, Michael A. Calter(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
It is used in the design of optical devices, highway curves, and much more. Definition of a Parabola We said that the Parabola results when we cut a cone by a plane that is parallel to an element of the cone. Here is another definition. A Parabola is the set of points in a plane, each of which is equidistant from a fixed point, the focus, 221 and a fixed line, the directrix. ■ Exploration: Try this. Use the preceding definition to construct a Parabola whose distance from focus to directrix is 2.0 in. (a) Draw a line to represent the directrix as shown in Fig. 22–48, and indicate a focus F 2.0 in. from that line. (b) Then draw a line L parallel to the directrix at some arbitrary distance, say, 3.0 in. (c) With the same (3.0-in.) distance as radius and F as center, use a compass to draw arcs intersecting L at and Each of these points is now at the same distance (3.0 in.) from F and from the directrix and is hence a point on the Parabola. (d) Repeat the construction with distances other than 3.0 in. to get more points on the Parabola. ■ Figure 22–49 shows the typical shape of a Parabola. The Parabola has an axis of symmetry which intersects it at the vertex. The distance p from directrix to vertex is equal to the directed distance from the vertex to the focus. Standard Equation of a Parabola: Vertex at the Origin Let us place the Parabola on coordinate axes with the vertex at the origin and with the axis of symmetry along the x axis, as shown in Fig. 22–50. Choose any point P on the Parabola. Then, by the definition of a Parabola, But in right triangle FBP, and But, since Squaring both sides yields Collecting terms, we get the standard equation of a Parabola with vertex at the origin. 222 y 2 4px Standard Equation of a Parabola: Vertex at Origin, Axis Horizontal x 2 2px p 2 y 2 p 2 2px x 2 (x p) 2 y 2 p 2 2px x 2 4 (x p) 2 y 2 p x FP AP, AP p x FP 4 (x p) 2 y 2 FP AP.
eBook - PDFPrecalculus
Functions and Graphs, Enhanced Edition
- Earl Swokowski, Jeffery Cole(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
Conic sections were studied extensively by the ancient Greeks, who discovered properties that enable us to state their definitions in terms of points and lines, as we do in our discussion. From our work in Section 2.6, if , the graph of is a Parabola with a vertical axis. We shall next state a general definition of a Parabola and derive equations for Parabolas that have either a vertical axis or a horizontal axis. We shall assume that F is not on l , for this would result in a line. If P is a point in the plane and is the point on l determined by a line through P that is perpendicular to l (see Figure 2), then, by the preceding definition, P is on the Parabola if and only if the distances and are equal. The axis of the Parabola is the line through F that is perpendicular to the directrix. The vertex of the Parabola is the point V on the axis halfway from F to l . The vertex is the point on the Parabola that is closest to the directrix. To obtain a simple equation for a Parabola, place the y -axis along the axis of the Parabola, with the origin at the vertex V , as shown in Figure 3. In this case, the focus F has coordinates for some real number , and the equation of the directrix is . (The figure shows the case .) By the p 0 y p p 0 0, p d P , P d P , F P y ax 2 bx c a 0 l Definition of a Parabola A Parabola is the set of all points in a plane equidistant from a fixed point F (the focus ) and a fixed line l (the directrix ) that lie in the plane. FIGURE 2 P V Axis Directrix F P l Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
eBook - PDFPrecalculus
Building Concepts and Connections 2E
- Revathi Narasimhan(Author)
- 2016(Publication Date)
- XYZ Textbooks(Publisher)
If the curve is closed, the conic is an ellipse; otherwise, it is a Parabola. ■ If the plane is parallel to the axis of the cone and does not pass through the vertex, it intersects the cone in a pair of non-intersecting open curves, called a hyperbola. Each curve is called a branch of the hyperbola. In this section, we will define Parabolas differently than we did in Chapter 2. Later on, we will make connections between the two definitions. Definition of a Parabola A Parabola is the set of all points ( x , y ) in a plane such that the distance of ( x , y ) from a fixed line is equal to the distance of ( x , y ) from a fixed point that is not on the fixed line. The fixed line and the fixed point, which lie in the plane of the Parabola, are called the directrix and the focus , respectively. The axis of symmetry of a Parabola is the line that is perpendicular to the directrix and passes through the focus of the Parabola. The vertex of a Parabola is on the axis of symmetry, midway between the focus and the directrix. See Figure 2. The only Parabolas discussed in this section are those whose axis of symmetry is either vertical or horizontal. Figure 2 Directrix Vertex Focus Axis of symmetry Objectives ■ Define a Parabola. ■ Find the focus, directrix, and axis of symmetry of a Parabola. ■ Determine the equation of a Parabola and write it in standard form. ■ Translate a Parabola in the xy -plane. ■ Sketch a Parabola. ■ Understand the reflective property of a Parabola. 738 Chapter 9 Conic Sections Parabolas with Vertex at the Origin Suppose a Parabola has its vertex at the origin and the y -axis as its axis of symmetry. Suppose also that the equation of its directrix is y = − p , for some p > 0. The coordinates of the focus are (0, p ), since the vertex is midway between the focus and directrix. Let P be any point on the Parabola with coordinates ( x , y ), and let dist( PF ) denote the distance from P to the focus F .
eBook - PDFCollege Algebra
Building Concepts and Connections 2E
- Revathi Narasimhan(Author)
- 2019(Publication Date)
- XYZ Textbooks(Publisher)
■ If the plane is parallel to the axis of the cone and does not pass through the vertex, it intersects the cone in a pair of non-intersecting open curves, called a hyperbola. Each curve is called a branch of the hyperbola. In this section, we will define Parabolas differently than we did in Chapter 2. Later on, we will make connections between the two definitions. Definition of a Parabola A Parabola is the set of all points ( x , y ) in a plane such that the distance of ( x , y ) from a fixed line is equal to the distance of ( x , y ) from a fixed point that is not on the fixed line. The fixed line and the fixed point, which lie in the plane of the Parabola, are called the directrix and the focus , respectively. The axis of symmetry of a Parabola is the line that is perpendicular to the directrix and passes through the focus of the Parabola. The vertex of a Parabola is on the axis of symmetry, midway between the focus and the directrix. See Figure 2. The only Parabolas discussed in this section are those whose axis of symmetry is either vertical or horizontal. Figure 2 Directri x Vertex Focus Axis of symmetry Objectives ■ Define a Parabola. ■ Find the focus, directrix, and axis of symmetry of a Parabola. ■ Determine the equation of a Parabola and write it in standard form. ■ Translate a Parabola in the xy -plane. ■ Sketch a Parabola. ■ Understand the reflective property of a Parabola. 6.1 The Parabola 497 Parabolas with Vertex at the Origin Suppose a Parabola has its vertex at the origin and the y -axis as its axis of symmetry. Suppose also that the equation of its directrix is y = − p , for some p > 0. The coordinates of the focus are (0, p ), since the vertex is midway between the focus and directrix. Let P be any point on the Parabola with coordinates ( x , y ), and let dist( PF ) denote the distance from P to the focus F . To find the distance from P to the directrix, we draw a perpendicular line segment from P to the line y = − p (the directrix).
- Arun Datta(Author)
- 2013(Publication Date)
- CRC Press(Publisher)
Parabola As in Figure 1.9, let AA ′ be a straight line and e be a point (e is not on line AA ′ ). The locus of all points B such that the distance of B from line AA ′ is equal to the distance of B from point e will form a Parabola. The point e is the focus and the line AA ′ is the directrix. The general equation of a Parabola is y 2 = 4 ax (1.77) Table 1.4 Conic Section b 2 − 4 ac < 0 b 2 − 4 ac = 0 b 2 − 4 ac > 0 Δ ≠ 0 a Δ < 0 Parabola Hyperbola a ≠ c , an ellipse a = c , a circle a Δ > 0, no locus Δ = 0 Point Two parallel lines if Q = d 2 + e 2 − 4( a + c ) f > 0; one straight line if Q = 0; no locus if Q < 0 Two intersecting straightlines A A uni2032 B e x y Figure 1.9 Parabola. 37 Chapter one: Basic mathematics © 2010 Taylor & Francis Group, LLC Parametric equations of a Parabola Let a and b be two lines in a plane with equations x = 4 at 2 (1.78a) y = 4 at (1.78b) respectively. The real number t is a parameter. Now, to obtain an equation of the curve, we eliminate the parameter t from the two equations. Eliminating t from Equations 1.78a and 1.78b, we get x a y a = 4 4 2 or y 2 = 4 ax which is the equation of the Parabola. Equation of tangent of a Parabola The slope of a tangent line can be obtained as 2 yy ′ = 4 a or slope y ′ = 2 a / y The slope of the tangent line at a point ( x 0 , y 0 ) = 2 a / y 0 . The equation of the tangent line will be y y a y x x -= -0 0 0 2 ( ) (1.79a) or yy 0 − 4 ax 0 = 2 ax − 2 ax 0 (1.79b) or yy 0 = 2 a ( x + x 0 ) (1.79c) Equation 1.79c is the equation of the tangent of the Parabola y 2 = 4 ax at point ( x 0 , y 0 ).
eBook - PDF- David Cohen, Theodore Lee, David Sklar, , David Cohen, Theodore Lee, David Sklar(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
Let B be the point where the line drawn from P perpendicular to the axis of the Parabola meets the axis. Show that AB 2 p . 17. Let P ( x 0 , y 0 ) be a point on the Parabola y 2 4 px . Let A be the point where the normal line to the Parabola at P meets the axis of the Parabola. Let F be the focus of the Parabola. If a line is drawn from A perpendicular to meeting at Z , show that ZP 2 p . FP FP , PQ . PQ y 2 p y 0 x y 0 2 864 CHAPTER 12 The Conic Sections 12.4 THE ELLIPSE We have here apparently [in the work of Anthemius of Tralles (a sixth-century Greek architect and mathematician)] the first mention of the construction of an ellipse by means of a string stretched tight round the foci. —Sir Thomas Heath in A History of Greek Mathematics, Vol. II (Oxford: The Clarendon Press, 1921) The heavenly motions are nothing but a continuous song for several voices, to be per-ceived by the intellect, not by the ear. —Johannes Kepler (1571–1630) In this section we discuss the symmetric, oval-shaped curve known as the ellipse. As Kepler discovered and Newton later proved, this is the curve described by the plan-ets in their motions around the sun. The true orbit of Mars was even less of a circle than the Earth’s. It took almost two years for Kepler to realize that its orbital shape is that of an ellipse. An ellipse is the shape of a circle when viewed at an angle. —Phillip Flower in Understanding the Universe (St. Paul: West Publishing Co., 1990) Copyright 201 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
eBook - PDFFunctions Modeling Change
A Preparation for Calculus
- Eric Connally, Deborah Hughes-Hallett, Andrew M. Gleason(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
11. The maximum or minimum point of a Parabola is called its vertex. 12. If a Parabola is concave up, its vertex is a maximum point. 13. If the equation of a Parabola is written as = ( − ℎ) 2 + , then the vertex is located at the point (−ℎ, ). 14. If the equation of a Parabola is written as = ( − ℎ) 2 + , then the axis of symmetry is found at = ℎ. 15. If the equation of a Parabola is = 2 + + and < 0, then the Parabola opens downward.
eBook - PDF- Ron Larson(Author)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
The falling stream of water has the shape of a Parabola whose vertex (0, 48) is at the end of the pipe (see figure). The stream of water strikes the ocean at the point (10radical.alt23 , 0). Write an equation for the path of the water. x 48 ft 40 30 20 10 10 20 30 40 y 4 ft 8 ft y x Figure for 67 Figure for 68 68. Window Design A church window is bounded above by a Parabola (see figure). Write an equation of the Parabola. 69. Archway A parabolic archway is 12 meters high at the vertex. At a height of 10 meters, the width of the archway is 8 meters (see figure). How wide is the archway at ground level? x y (0, 12) (- 4, 10) (4, 10) - 4 - 8 4 8 2 12 x (0, 8) (- 2, 4) (2, 4) y Figure for 69 Figure for 70 70. Lattice Arch A parabolic lattice arch is 8 feet high at the vertex. At a height of 4 feet, the width of the lattice arch is 4 feet (see figure). How wide is the lattice arch at ground level? 71. Suspension Bridge Each cable of a suspension bridge is suspended (in the shape of a Parabola) between two towers (see figure). 10 - 20 20 40 60 - 40 - 60 - 10 30 20 x y (- 60, 20) (60, 20) (a) Find the coordinates of the focus. (b) Write an equation that models the cables. 73. Satellite Orbit A satellite in a 100-mile-high circular orbit around Earth has a velocity of approximately 17,500 miles per hour. When this velocity is multiplied by radical.alt22 , the satellite has the minimum velocity necessary to escape Earth’s gravity and follow a parabolic path with the center of Earth as the focus (see figure). Parabolic path 4100 miles x y Not drawn to scale Circular orbit (a) Find the escape velocity of the satellite. (b) Write an equation for the parabolic path of the satellite. (Assume that the radius of Earth is 4000 miles.) Each cable of the Golden Gate Bridge is suspended (in the shape of a Parabola) between two towers that are 1280 meters apart. The top of each tower is 152 meters above the roadway. The cables touch the roadway at the midpoint between the towers.
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