8.0 Introduction to Conic Sections PreCalculus INTRODUCTION TO CONIC SECTIONS Lesson Targets for Intro: 1. Know and be able to eplain the definition of a conic section.. Identif the general form of a quadratic function (in two variables) as a parabola, circle, ellipse or hperbola. 3. Determine the tpe of conic using the discriminant when given the general form equation. 4. Complete the square to write the equation of a conic in transformational form. Conic Section: the shape formed b the intersection of a right circular cone and a plane. The basic conic sections (also called non-degenerative conic sections) are shown below. Some tetbooks refer to the circle as a specialized form of an ellipse. We will stud the geometric definitions that make each shape AND the algebraic equation for each shape this unit. All conic sections can be modeled b using the general form of a quadratic equation in two variables: where A, B and C are not all zero. You ma recall from previous work with solving quadratic functions, the discriminant is the value under the square root of the quadratic formula ( b 4ac ). The value of the discriminant tells us the number and tpe of zeros. Likewise, the general form for a conic has a discriminant ( B 4AC ). The value of the discriminant tells us the tpe of conic section we have. > 0 If B 4AC is = 0 < 0 Remember a circle is a special ellipse where Eample 1: Use the discriminant to identif the tpe of conic section. a) 3 4 3 0 + = b) = 1 Unit 4 -
8.0 Introduction to Conic Sections PreCalculus In addition to identifing the tpe of conic section, we will want each equation written in transformational form that is written so that we can easil identif the transformations that have been applied from the parent function. To write in change how the equation is written, we. Notice we ma need to for or or both! Eample : Use the discriminant to identif the tpe of conic section. Then, rewrite each general form quadratic into a transformational form. a) + 4 8 + 1= 0 b) + 6 + 1 + 36 = 0 c) 9 4 + 90 16 115 = 0 Closing questions Which term from the general form equation is MISSING from all of the equations in eample? What do ou notice about the equation of the parabola as compared to the other conics for eample? Unit 4-3
8.1 Parabolas Pre-Calculus 8.1 PARABOLAS Learning Targets for 8.1 1. Write the equation of a parabola when given at least two of the important features of the graph.. Identif the ke features of a parabola when given an equation. 3. Sketch the graph of a parabola b hand including verte, directri, focus and curve. 4. Prove a general form equation is a parabola. Parabola: The set of all points equidistant from a line called the directri and a fied point called the focus. Vertical Parabolas: 1 ( ) k = h Horizontal Parabolas: 4 p 1 ( ) h= k 4 p Notice the verte is equidistant from the focus and directri just like ever other point on the parabola. This distance is noted b in the equation above. Remember: the parabola is the focus and never crosses the. Eample 1: Find the equation for the parabola that satisfies the given conditions. A sketch is helpful! a) Focus: (, 3), Directri 5 = b) Focus( 5,3), Verte: ( 5,6) Eample : Sketch the graph of ( ) 8( 1) = + b hand. Unit 4 4
8.1 Parabolas Pre-Calculus The table below summarizes what we have learned Opening Vertical Up: p > 0 Down: p < 0 Horizontal Right: p > 0 Left: p < 0 Equation Verte Focus Directri Ais of Smmetr Eample 3: Prove that the graph of + 4 11 = 0 is a parabola. Then find its verte, focus, and directri. Unit 4 5
PreCalculus Eploration: 8. Ellipses An ellipse is different from a circle because it is longer from the center to the edge in one direction than it is in the other direction. Answer the questions below. 1. Eamine the ellipses at the right. a) What is the same about them? A B b) What is different about them?. The equation for ellipse A is + = 1. The equation for ellipse B is 9 36 + = 1. 36 9 a) How can ou determine from the equation whether an ellipse will be longer in the horizontal direction or longer in the vertical direction? b) How can ou determine the distance from the center of the ellipse to the edge in the horizontal direction? c) How can ou determine the distance from the center of the ellipse to the edge in the vertical direction? 3. Use the graph of ( 1) ( + 4) + = 1 provided at the right to help ou answer the questions below. 5 4 a) Based on the translation rules from the equation for a circle and the sample given, how would ou translate an ellipse? b) Write the equation for the translated form of an ellipse (using variables like,, h, k, a and b). Identif the center. Center: (1, 4) Unit 4 6
8. Ellipses Pre-Calculus 8. ELLIPSE Learning Targets 8. 1. Write the equation of an ellipse when given at least two of the important features of the graph.. Identif the ke features of an ellipse when given an equation. 3. Sketch the graph of an ellipse b hand including center, vertices and co-vertices. 4. Prove a general form equation is an ellipse. Before watching the Video for Lesson 8., complete the Eploration for 8.! Ellipse: The set of all points in a plane whose sum of distances from two fied points is a constant. The fied points are called. For the equations below, and. Major Ais has length and contains the center, vertices and foci. Minor Ais has length is perpendicular to the transverse ais. Vertical Ellipse: ( h) ( k) + = 1 Horizontal Ellipse: b a ( h) ( k) + = 1 a b (, ) (, ) Vertical Ellipse Horizontal Ellipse ALL Ellipses Vertices Co-Vertices Foci a units awa from center along major ais. b units awa from center along minor ais c units awa from center along major ais Unit 4 7
8. Ellipses Pre-Calculus ( + 3 ) ( 1 ) Eample 1: Find the center, vertices, and foci of + = 1. 16 5 Eample : Find the equation for the ellipse that satisfies the given conditions. Foci: (,1) and (,5), the major ais endpoints are (, 1) and (, 7) Eample 3: Prove that the graph of 4 8 4 4 0 + + + = is an ellipse. Then, find its center and foci. Unit 4 8
8.3 Hperbolas Pre-Calculus 8.3 HYPERBOLAS Learning Targets 8.3 1. Write the equation of a hperbola when given at least two of the important features of the graph.. Identif ke features of a hperbola when given an equation. 3. Sketch the graph of a hperbola b hand including center and asmptotes. 4. Prove a general form equation is a hperbola 5. Write the equations of the asmptotes of a hperbola. Hperbola: The set of all points in a plane whose difference of distances from two fied points is constant. The fied points are called. For the equations below, and. Transverse Ais contains the center, vertices and foci. Conjugate Ais is perpendicular to the transverse ais. Vertical Hperbola: ( k) ( h) = 1 Horizontal Hperbola: a b (, ) ( h) ( k) = 1 a b (, ) Eample1: List the center, vertices and foci of each hperbola. Then, sketch the graph b hand. ( 1 ) ( + 3 ) a) = 1 b) 9 5 ( ) + = 1 16 4 Unit 4 9
8.3 Hperbolas Pre-Calculus In summar Vertices Vertical Transverse Ais Horizontal Transverse Ais ALL Hperbolas a units awa from center along transverse ais. Foci Asmptotes c units awa from center along transverse ais. Drawn through the corners of bo with dimensions a b b. Eample : Find the equation for the hperbola that satisfies the given conditions. Center at ( 1, 4), focus ( 1, 10) and verte ( 1, 1). Eample 3: Prove the graph of equations of the asmptotes. 5 9 50 50 81 0 = is a hperbola. Then, find its center and the Unit 4 10