Chapter 7 Page 1 of 16 Lecture Guide Math 105 - College Algebra Chapter 7 to accompan College Algebra b Julie Miller Corresponding Lecture Videos can be found at Prepared b Stephen Toner & Nichole DuBal Victor Valle College Last updated: 5/9/13
Chapter 7 Page 2 of 16 Chapter 7 Conic Sections Overview 7.1 The Ellipse Definition: An ellipse is the set of points, P, in a plane such that the of the distances from P to two fied points is constant. Derrivation Case #1: P is at "high noon" Circle Review... Definition: A circle is the set of points, P, in a plane which are a fied distance (radius) from a fied point (center).
Chapter 7 Page 3 of 16 Derrivation Case #2: P is arbitrar General Formula for an ellipse, centered at ( ): To find the foci, use. 1. Graph. Label the foci. From our definition, ( ) ( ) ( ) ( ( ) ) ( ) ( ) ( ) ( ) ( ) ( ) Substitute in
7.1 #24 Chapter 7 Page 4 of 16 7.1 #36 a. Identif the center of the ellipse. a. Identif the center of the ellipse. b. Determine the value of. b. Identif the vertices. c. Determine the value of. c. Identif the endpoints on the minor ais. d. Identif the vertices. e. Identif the endpoints on the minor ais. d. Identif the foci. f. Identif the foci. g. Determine the length of the major ais. e. Graph the ellipse: h. Determine the length of the minor ais. i. Graph the ellipse:
Chapter 7 Page 5 of 16 7.1 #44 Find the equation of the ellipse from its graph: a. Write the equation of the ellipse in standard form. 3 2 1-1 1 2 3 4 5 6 7-1 b. Identif the center, vertices, endpoints of the minor ais, and foci. 7.1 #62 Write the standard form of an equation of an ellipse subject to the given conditions. Vertices: ( ) and ( ) Foci: ( ) and ( ) What happens if ou get fractions?
Chapter 7 Page 6 of 16 7.1 #60 Write the standard form of an equation of an ellipse subject to the given conditions. Vertices: ( ) and ( ) Passes Through ( ) 7.2 The Hperbola Definition: A hperbola is the set of points, P, in a plane such that the of the distances from P to two fied points is constant. General Formulas for a hperbola, centered at ( ): To find the foci, use. 1. Graph and analze: The eccentricit of an ellipse:
7.2 #24 Chapter 7 Page 7 of 16 7.2 #36 a. Identif the center. b. Identif the vertices. c. Identif the foci. a. Identif the center. b. Identif the vertices. c. Identif the foci. d. Write the equation of the asmptotes. d. Write the equation of the asmptotes. e. Graph the hperbola: e. Graph the hperbola: 7.2 #46 a. Write the equation of the hperbola in standard form.
Chapter 7 Page 8 of 16 b. Identif the center, vertices and foci. 7.2 #56 Vertices: ( ) and ( ) Slope of the asmptotes: For eercises 51-60, write the standard form of an equation of the hperbola subject to the given conditions. 7.2 #52 Vertices: ( ) and ( ) Foci: ( ) and ( ) 7.2 #60 Corners of the reference rectangle: ( ),, ( ) Horizontal transverse ais
Chapter 7 Page 9 of 16 7.3 The Parabola Recall the "old wa" of graphing a parabola: ( ) Connection between the "old wa" and the "conic wa"... verte: ma/min value: ais of smmetr: Definition: A parabola is the set of points, P, in a plane which are from a fied point (focus) and a fied line (called the ). From the definition, General Formulas:
Chapter 7 Page 10 of 16 For eercises 34-36, an equation of a parabola or is given. a. Identif the verte, value of, focus, and focal diameter of the parabola. b. Identif the endpoints of the latus rectum. c. Graph the parabola. d. Write an equation for the directri and ais of smmetr. 7.3 #34 For eercise 46, an equation of a parabola or is given. a. Identif the verte, value of, focus, and focal diameter of the parabola. b. Identif the endpoints of the latus rectum. c. Graph the parabola. d. Write an equation for the directri and ais of smmetr. 7.3 #46 7.3 #36 7.3 #70 Find the standard form of an equation of a parabola with focus ( ) and verte ( ).
Chapter 7 Page 11 of 16 For #58, an equation of a parabola is given. a. Write the equation of the parabola in standard form. b. Identif the verte, focus, and directri. Similar to 7.3 #74 Find the standard form of an equation of a parabola with verte ( ) and passing through ( ) (Hint: There are two possible answers.) 7.3 #58 7.3 #72 Determine the standard form of an equation of the parabola subject to the given conditions. Focus: ( ); Directri:
Chapter 7 Page 12 of 16 7.PRE Problem Recognition Eercises 7.PRE #12 Determine whether the equation represents a circle, an ellipse, a hperbola, or a parabola. Write the equation in standard form. 7.PRE #9 7.PRE #13 7.PRE #10 7.PRE #14 7.PRE #11
Chapter 7 Page 13 of 16 7.PRE #15 Shape Formulas Comments Circle Ellipse 7.PRE #16 Hperbola Parabola
Chapter 7 Page 14 of 16 Chapter 7 Review 3. Find the vertices and foci. Graph. 1. Graph and analze. 2. Graph and analze.
Chapter 7 Page 15 of 16 4. Find the vertices, foci and asmptotes. Graph. 5. Find the equation of the having vertices at ( ) and ( ) and foci at ( ) and ( ). 6. Find the equation of the ellipse having a major ais of length 10 and foci at ( ) and ( ).
Chapter 7 Page 16 of 16 7. Find the equation of the ellipse having its center at ( ), a minor ais of length 10, and a verte at ( ). 9. Graph. Identif the verte, focus, directri and -intercepts. 8. Graph