11.5 Conic Sections. Objective A. To graph a parabola

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1 Section 11.5 / Conic Sections 11.5/ Conic Sections Objective A To graph a parabola VIDEO & DVD CD TUTOR WEB SSM Point of Interest Hpatia (c. 3 15) is considered the first prominent woman mathematician. She lectured in mathematics and philosoph at the Museum in Aleandria, the most distinguished place of learning in the world. One of the topics on which Hpatia lectured was conic sections. One historian has claimed that with the death (actuall the murder) of Hpatia, the long and glorious histor of Greek mathematics was at an end. The conic sections are curves that can be constructed from the intersection of a plane and a right circular cone. The parabola, which was introduced earlier, is one of these curves. Here we will review some of that previous discussion and look at equations of parabolas that were not discussed before. Ever parabola has an ais of smmetr and a verte that is on the ais of smmetr. To understand the ais of smmetr, think of folding the paper along that ais. The two halves of the curve will match up. Verte Ais of smmetr The graph of the equation a b c, a, is a parabola with the ais of smmetr parallel to the -ais. The parabola opens up when a and opens down when a. When the parabola opens up, the verte is the lowest point on the parabola. When the parabola opens down, the verte is the highest point on the parabola. The coordinates of the verte can be found b completing the square. Copright Houghton Mifflin Compan. All rights reserved. HOW TO Find the verte of the parabola whose equation is 5. 5 ( ) 5 ( ) 5 ( ) 1 The coefficient of is positive, so the parabola opens up. The verte is the lowest point on the parabola, or the point that has the least -coordinate. Because ( ) for all, the least -coordinate occurs when ( ), which occurs when. This means the -coordinate of the verte is. To find the -coordinate of the verte, replace in ( ) 1 b and solve for. ( ) 1 ( ) 1 1 The verte is (, 1). Group the terms involving. Complete the square on. Note that is added and subtracted. Because 5, the equation is not changed. Factor the trinomial and combine like terms. Verte

2 11.5/ Chapter 11 / Functions and Relations Point of Interest The suspension cables for some bridges, such as the Golden Gate bridge, hang in the shape of a parabola. Parabolic shapes are also used for mirrors in telescopes and in certain antenna designs. B following the procedure of the last eample and completing the square on the equation a b c, we find that the -coordinate of the verte is b. a The -coordinate of the verte can then be determined b substituting this value of into a b c and solving for. Because the ais of smmetr is parallel to the -ais and passes through the verte, the equation of the ais of smmetr is 5. b a HOW TO Find the verte and ais of smmetr of the parabola whose equation is Then sketch its graph. -coordinate: b a 6 (3) 1 Find the -coordinate of the verte and the ais of smmetr, a 53, b 5 6. The -coordinate of the verte is 1. The ais of smmetr is the line 1. To find the -coordinate of the verte, replace b 1 and solve for (1) 6(1) 1 The verte is (1, ). Because a is negative, the parabola opens down. Find a few ordered pairs and use smmetr to sketch the graph. HOW TO Find the verte and ais of smmetr of the parabola whose equation is. Then sketch its graph. -coordinate: b a (1) Find the -coordinate of the verte and the ais of smmetr. a 5 1, b 5 The -coordinate of the verte is. The ais of smmetr is the line. To find the -coordinate of the verte, replace b and solve for. The verte is (, ). Because a is positive, the parabola opens up. Find a few ordered pairs and use smmetr to sketch the graph. Copright Houghton Mifflin Compan. All rights reserved.

3 Section 11.5 / Conic Sections 11.5/3 The graph of an equation of the form a b c, a, is also a parabola. In this case, the parabola opens to the right when a is positive and opens to the left when a is negative. For a parabola of this form, the -coordinate of the b verte is. The ais of smmetr is the line a b 5. a Ais of smmetr Verte Using the vertical line test, the graph of a parabola of this form is not the graph of a function. The graph of a b c is a relation. HOW TO Find the verte and ais of smmetr of the parabola whose equation is 5. Then sketch its graph. -coordinate: b a () Find the -coordinate of the verte and the ais of smmetr. a 5, b 5 The -coordinate of the verte is. The ais of smmetr is the line. To find the -coordinate of the verte, replace b and solve for. 5 () () 5 3 The verte is (3, ). Since a is positive, the parabola opens to the right. Find a few ordered pairs and use smmetr to sketch the graph. HOW TO Find the verte and ais of smmetr of the parabola whose equation is 3. Then sketch its graph. -coordinate: b a () 1 Find the -coordinate of the verte and the ais of smmetr. a 5, b 5 Copright Houghton Mifflin Compan. All rights reserved. The -coordinate of the verte is 1. The ais of smmetr is the line 1. To find the -coordinate of the verte, replace b 1 and solve for. 3 (1) (1) 3 1 The verte is (1, 1). Because a is negative, the parabola opens to the left. Find a few ordered pairs and use smmetr to sketch the graph.

4 11.5/ Chapter 11 / Functions and Relations Eample 1 Find the verte and ais of smmetr of the parabola whose equation is 3. Then sketch its graph. You Tr It 1 Find the verte and ais of smmetr of the parabola whose equation is 1. Then sketch its graph. b a (1) Ais of smmetr: () 3 1 Verte: (, 1) Eample Find the verte and ais of smmetr of the parabola whose equation is 1. Then sketch its graph. You Tr It Find the verte and ais of smmetr of the parabola whose equation is. Then sketch its graph. b a () 1 Ais of smmetr: 1 (1) (1) 1 1 Verte: (1, 1) Eample 3 Find the verte and ais of smmetr of the parabola whose equation is 1. Then sketch its graph. You Tr It 3 Find the verte and ais of smmetr of the parabola whose equation is 1. Then sketch its graph. b a (1) Ais of smmetr: 1 1 Verte: (, 1) s on p. 11.5/S1 Copright Houghton Mifflin Compan. All rights reserved.

5 Section 11.5 / Conic Sections 11.5/5 Objective B To find the equation of a circle and to graph a circle VIDEO & DVD CD TUTOR WEB SSM TAKE NOTE As the angle of the plane that intersects the cone changes, different conic sections are formed. For a parabola, the plane was parallel to the side of the cone. For a circle, the plane is parallel to the base of the cone. A circle is a conic section formed b the intersection of a cone and a plane parallel to the base of the cone. r ( h, k) radius center A circle can be defined as all points (, ) in the plane that are a fied distance from a given point (h, k) called the center. The fied distance is the radius of the circle. The Standard Form of the Equation of a Circle Let r be the radius of a circle and let (h, k) be the coordinates of the center of the circle. Then the equation of the circle is given b ( h) ( k) r HOW TO Sketch a graph of ( 1) ( ) 9. ( 1) [ ()] 3 Center: (1, ) Radius: 3 Rewrite the equation in standard form. Copright Houghton Mifflin Compan. All rights reserved. HOW TO Find the equation of the circle with radius and center (1, ). Then sketch its graph. ( h) ( k) r [ (1)] ( ) ( 1) ( ) 16 Use the standard form of the equation of a circle. Replace r b, h b 1, and k b. Sketch the graph b drawing a circle with center (1, ) and radius.

6 11.5/6 Chapter 11 / Functions and Relations Appling the vertical-line test reveals that the graph of a circle is not the graph of a function. The graph of a circle is the graph of a relation. Eample Sketch a graph of ( ) ( 1). ( h) ( k) r [ ()] ( 1) Center: (h, k) (, 1) Radius: r You Tr It Sketch a graph of ( ) ( 3) 9. Eample 5 Find the equation of the circle with radius 5 and center (1, 3). Then sketch its graph. ( h) ( k) r [ (1)] ( 3) 5 ( 1) ( 3) 5 You Tr It 5 Find the equation of the circle with radius and center (, 3). Then sketch its graph. 6 s on p. 11.5/S1 Copright Houghton Mifflin Compan. All rights reserved.

7 Section 11.5 / Conic Sections 11.5/7 Objective C To graph an ellipse with center at the origin VIDEO & DVD CD TUTOR WEB SSM The orbits of the planets around the sun are oval shaped. This oval shape can be described as an ellipse, which is another of the conic sections. There are two aes of smmetr for an ellipse. The intersection of these two aes is the center of the ellipse. An ellipse with center at the origin is shown at the right. Note that there are two -intercepts and two -intercepts. (, b) ( a, ) (, b) (a, ) Point of Interest The word ellipse comes from the Greek word ellipsis, which means deficient. The method b which the earl Greeks analzed the conics caused a certain area in the construction of the ellipse to be less than another area (deficient). The word ellipsis in English, which means omission, has the same Greek root as the word ellipse. The Standard Form of the Equation of an Ellipse with Center at the Origin The equation of an ellipse with center at the origin is. a b 1 The -intercepts are (a, ) and (a, ). The -intercepts are (, b) and (, b). B finding the - and -intercepts of an ellipse and using the fact that the ellipse is oval shaped, we can sketch a graph of the ellipse. HOW TO Sketch the graph of the ellipse whose equation is 1. 9 Copright Houghton Mifflin Compan. All rights reserved. Comparing 1 with 1, we have a 9 and b. 9 a b Therefore, a 3 and b. The -intercepts are (3, ) and (3, ). The -intercepts are (, ) and (, ). Use the intercepts to sketch a graph of the ellipse. Using the vertical-line test, we find that the graph of an ellipse is not the graph of a function. The graph of an ellipse is the graph of a relation.

8 11.5/ Chapter 11 / Functions and Relations HOW TO Sketch a graph of the ellipse whose equation is Point of Interest For a circle, a b and a thus. Earl Greek b 1 astronomers thought that each planet had a circular orbit. Toda we know that the planets have elliptical orbits. However, in most cases the ellipse is ver nearl a circle. a For Earth, 1.1. The b most elliptical orbit is Pluto s, a for which 1.3. b The -intercepts are (, ) and (, ). The -intercepts are (, ) and (, ). a 5 16, b 5 16 Use the intercepts and smmetr to sketch the graph of the ellipse. The graph in this eample is the graph of a circle. A circle is a special case of an ellipse. It occurs when a b in the equation 1. a b Eample 6 Sketch a graph of the ellipse whose equation is You Tr It 6 Sketch a graph of the ellipse whose equation is intercepts: (3, ) and (3, ) -intercepts: (, ) and (, ) Eample 7 You Tr It 7 Sketch a graph of the ellipse whose equation is intercepts: (, ) and (, ) -intercepts: (, 3 ) and (, 3 ) ( 3 3.5) Sketch a graph of the ellipse whose equation is s on p. 11.5/S1 Copright Houghton Mifflin Compan. All rights reserved.

9 Section 11.5 / Conic Sections 11.5/9 Objective D To graph a hperbola with center at the origin VIDEO & DVD CD TUTOR WEB SSM A hperbola is a conic section that is formed b the intersection of a cone and a plane perpendicular to the base of the cone. The hperbola has two vertices and an ais of smmetr that passes through the vertices. The center of a hperbola is the point halfwa between the two vertices. The graphs at the right show two possible graphs of a hperbola with center at the origin. In the first graph, an ais of smmetr is the -ais and the vertices are -intercepts. In the second graph, an ais of smmetr is the -ais and the vertices are -intercepts. Note that in either case, the graph of a hperbola is not the graph of a function. The graph of a hperbola is the graph of a relation. Copright Houghton Mifflin Compan. All rights reserved. Point of Interest The word hperbola comes from the Greek word perboli, which means eceeding. The method b which the earl Greeks analzed the conics caused a certain area in the construction of the hperbola to be greater than (to eceed) another area. The word hperbole in English, meaning eaggeration, has the same Greek root as the word hperbola. The word asmptote comes from the Greek word asmptotos, which means not capable of meeting. The Standard Form of the Equation of a Hperbola with Center at the Origin The equation of a hperbola for which an ais of smmetr is the -ais is. The vertices are (a, ) and (a, ). a b 1 The equation of a hperbola for which an ais of smmetr is the -ais is. The vertices are (, b) and (, b). b a 1 To sketch a hperbola, it is helpful to draw two lines that are approached b the hperbola. These two lines are called asmptotes. As a point on the hperbola gets farther from the origin, the hperbola gets closer to the asmptotes. Because the asmptotes are straight lines, their equations are linear equations. The equations of the asmptotes for a hperbola with center at b the origin are and b. a a

10 11.5/1 Chapter 11 / Functions and Relations Point of Interest Hperbolas are used in LORAN (LOng RAnge Navigation) as a method b which a ship s navigator can determine the position of the ship, as shown in the figure below. The are also used as mirrors in some telescopes to focus incoming light. T 1 T HOW TO Sketch a graph of the hperbola whose equation is 1. 9 An ais of smmetr is the -ais. b 9, a The vertices are (, 3) and (, 3). 3 The asmptotes are and 3. The vertices are (, b) and (, b). b The asmptotes are 5 and a b 5. a Sketch the asmptotes. Use smmetr and the fact that the hperbola will approach the asmptotes to sketch its graph. T 3 Eample Sketch a graph of the hperbola whose equation is You Tr It Sketch a graph of the hperbola whose equation is Ais of smmetr: -ais Vertices: (, ) and (, ) Asmptotes: 1 and 1 Eample 9 You Tr It 9 Sketch a graph of the hperbola whose equation is Ais of smmetr: -ais Vertices: (, ) and (, ) Asmptotes: and 5 5 Sketch a graph of the hperbola whose equation is s on p. 11.5/S1 Copright Houghton Mifflin Compan. All rights reserved.

11 Section 11.5 / Conic Sections 11.5/ Eercises Objective A To graph a parabola State (a) whether the ais of smmetr is a vertical or a horizontal line and (b) in what direction the parabola opens Find the verte and ais of smmetr of the parabola given b the equation. Then sketch its graph Copright Houghton Mifflin Compan. All rights reserved

12 11.5/1 Chapter 11 / Functions and Relations Objective B To find the equation of a circle and to graph a circle Sketch a graph of the circle given b the equation. 16. ( ) ( ) ( ) ( 3) ( 3) ( 1) ( ) ( 3). ( ) ( ) 1. ( 1) ( ) 5. Find the equation of the circle with radius and center (, 1). Then sketch its graph. 3. Find the equation of the circle with radius 3 and center (1, ). Then sketch its graph.. Find the equation of the circle with radius 5 and center (1, 1). Then sketch its graph. 5. Find the equation of the circle with radius 5 and center (, 1). Then sketch its graph. Copright Houghton Mifflin Compan. All rights reserved.

13 Section 11.5 / Conic Sections 11.5/13 Objective C To graph an ellipse with center at the origin Sketch a graph of the ellipse given b the equation Objective D To graph a hperbola with center at the origin Copright Houghton Mifflin Compan. All rights reserved. Sketch a graph of the hperbola given b the equation

14 11.5/1 Chapter 11 / Functions and Relations APPLYING THE CONCEPTS Write the equation in standard form. Identif the graph, and then graph the equation Copright Houghton Mifflin Compan. All rights reserved.

15 s to You Tr It 11.5/S1 SECTION 11.5 You Tr It 1 1 b a (1) 1 You Tr It 5 ( h) ( k) r ( ) [ (3)] ( ) ( 3) 16 Ais of smmetr: 1 (1) (1) 1 Verte: (1, ) You Tr It b a (1) 1 Ais of smmetr: 1 (1) (1) 3 Verte: (3, 1) You Tr It 3 1 b a (1) 1 Ais of smmetr: 1 1 (1) 1 Verte: (1, ) You Tr It 6 -intercepts: (, ) and (, ) -intercepts: (, 5) and (, 5) You Tr It 7 -intercepts (3, ) and (3, ) -intercepts: (, 3) and (, 3) 3 1 You Tr It Ais of smmetr: -ais Vertices: (3, ) and (3, ) Asmptotes: 5 and Copright Houghton Mifflin Compan. All rights reserved. You Tr It ( h) ( k) r ( ) [ (3)] 3 Center: (h, k) (, 3) Radius: r 3 You Tr It 9 Ais of smmetr: -ais Vertices: (, 3) and (, 3) Asmptotes: and

16 Answers to Selected Eercises 11.5/A1 SECTION a. A vertical line b. Opens up 3a. A horizontal line b. Opens right 5a. A horizontal line b. Opens left , 3 Verte: Verte: (, ) Verte: (1, ) Ais of smmetr: 3 Ais of smmetr: Ais of smmetr: Verte: (1, ) Verte: 1, 7 Ais of smmetr: Ais of smmetr: ( 1) ( ) 9 5. ( ) ( 1) Copright Houghton Mifflin Compan. All rights reserved Ellipse Hperbola Hperbola