10 6 Identifying Conic Sections Warm Up Lesson

10 -6 Identifying. Conic. Sections Warm Up Lesson Presentation Lesson Quiz Holt Algebra 22

10 -6 Identifying Conic Sections Warm Up Solve by completing the square. 1. x 2 + 6 x = 91 2. 2 x 2 + 8 x – 90 = 0 Holt Algebra 2

10 -6 Identifying Conic Sections Objectives Identify and transform conic functions. Use the method of completing the square to identify and graph conic sections. Holt Algebra 2

10 -6 Identifying Conic Sections In Lesson 10 -2 through 10 -5, you learned about the four conic sections. Recall the equations of conic sections in standard form. In these forms, the characteristics of the conic sections can be identified. Holt Algebra 2

10 -6 Identifying Conic Sections Holt Algebra 2

10 -6 Identifying Conic Sections Example 1: Identifying Conic Sections in Standard Form Identify the conic section that each equation represents. A. x+4 = (y – 2)2 10 This equation is of the same form as a parabola with a horizontal axis of symmetry. B. This equation is of the same form as a hyperbola with a horizontal transverse axis. Holt Algebra 2

10 -6 Identifying Conic Sections Example 1: Identifying Conic Sections in Standard Form Identify the conic section that each equation represents. C. This equation is of the same form as a circle. Holt Algebra 2

10 -6 Identifying Conic Sections Check It Out! Example 1 Identify the conic section that each equation represents. a. x 2 + (y + 14)2 = 112 b. (y – 6)2 22 Holt Algebra 2 – (x – 1)2 212 = 1

10 -6 Identifying Conic Sections All conic sections can be written in the general form Ax 2 + Bxy + Cy 2 + Dx + Ey+ F = 0. The conic section represented by an equation in general form can be determined by the coefficients. Holt Algebra 2

10 -6 Identifying Conic Sections Example 2 A: Identifying Conic Sections in General Form Identify the conic section that the equation represents. 4 x 2 – 10 xy + 5 y 2 + 12 x + 20 y = 0 A = 4, B = – 10, C = 5 Identify the values for A, B, and C. B 2 – 4 AC 2 (– 10) – 4(4)(5) Substitute into B 2 – 4 AC. 20 Simplify. Because B 2 – 4 AC > 0, the equation represents a hyperbola. Holt Algebra 2

10 -6 Identifying Conic Sections Example 2 B: Identifying Conic Sections in General Form Identify the conic section that the equation represents. 9 x 2 – 12 xy + 4 y 2 + 6 x – 8 y = 0. A = 9, B = – 12, C = 4 Identify the values for A, B, and C. B 2 – 4 AC 2 (– 12) – 4(9)(4) Substitute into B 2 – 4 AC. 0 Simplify. Because B 2 – 4 AC = 0, the equation represents a parabola. Holt Algebra 2

10 -6 Identifying Conic Sections Example 2 C: Identifying Conic Sections in General Form Identify the conic section that the equation represents. 8 x 2 – 15 xy + 6 y 2 + x – 8 y + 12 = 0 A = 8, B = – 15, C = 6 Identify the values for A, B, and C. B 2 – 4 AC (– 15)2 – 4(8)(6) Substitute into B 2 – 4 AC. 33 Simplify. Because B 2 – 4 AC > 0, the equation represents a hyperbola. Holt Algebra 2

10 -6 Identifying Conic Sections Check It Out! Example 2 a Identify the conic section that the equation represents. 9 x 2 + 9 y 2 – 18 x – 12 y – 50 = 0 Holt Algebra 2

10 -6 Identifying Conic Sections Check It Out! Example 2 b Identify the conic section that the equation represents. 12 x 2 + 24 xy + 12 y 2 + 25 y = 0 Holt Algebra 2

10 -6 Identifying Conic Sections If you are given the equation of a conic in standard form, you can write the equation in general form by expanding the binomials. If you are given the general form of a conic section, you can use the method of completing the square from Lesson 5 -4 to write the equation in standard form. Remember! You must factor out the leading coefficient of x 2 and y 2 before completing the square. Holt Algebra 2

10 -6 Identifying Conic Sections Example 3 A: Finding the Standard Form of the Equation for a Conic Section Find the standard form of the equation by completing the square. Then identify and graph each conic. x 2 + y 2 + 8 x – 10 y – 8 = 0 Rearrange to prepare for completing the square in x and y. x 2 + 8 x + + y 2 – 10 y + Complete both squares. 2 Holt Algebra 2 =8+ +

10 -6 Identifying Conic Sections Example 3 A Continued (x + 4)2 + (y – 5)2 = 49 Factor and simplify. Because the conic is of the form (x – h)2 + (y – k)2 = r 2, it is a circle with center (– 4, 5) and radius 7. Holt Algebra 2

10 -6 Identifying Conic Sections Example 3 B: Finding the Standard Form of the Equation for a Conic Section Find the standard form of the equation by completing the square. Then identify and graph each conic. 5 x 2 + 20 y 2 + 30 x + 40 y – 15 = 0 Rearrange to prepare for completing the square in x and y. 5 x 2 + 30 x + + 20 y 2 + 40 y + = 15 + + Factor 5 from the x terms, and factor 20 from the y terms. 5(x 2 + 6 x + Holt Algebra 2 )+ 20(y 2 + 2 y + ) = 15 + +

10 -6 Identifying Conic Sections Example 3 B Continued Complete both squares. 2 2 é é ù 6ö 2ö æ æ 2 2 5 ê x + 6 x + ç ÷ ú + 20 ê y + 2 y + ç ÷ è 2 ø úû êë êë è 2ø 5(x + 3)2 + 20(y + 1)2 = 80 (x + 3)2 + (y +1 )2 = 16 Holt Algebra 2 4 1 2 ù 6 æ ö æ 2ö ú = 15 + 5 ç ÷ + 20 ç ÷ è 2ø úû è 2ø Factor and simplify. Divide both sides by 80.

10 -6 Identifying Conic Sections Example 3 B Continued Because the conic is of the form (x – h)2 (y – k)2 + = 1, 2 2 a b it is an ellipse with center (– 3, – 1), horizontal major axis length 8, and minor axis length 4. The covertices are (– 3, – 3) and (– 3, 1), and the vertices are (– 7, – 1) and (1, – 1). Holt Algebra 2

10 -6 Identifying Conic Sections Check It Out! Example 3 a Find the standard form of the equation by completing the square. Then identify and graph each conic. y 2 – 9 x + 16 y + 64 = 0 Holt Algebra 2

10 -6 Identifying Conic Sections Check It Out! Example 3 a Continued Holt Algebra 2

10 -6 Identifying Conic Sections Check It Out! Example 3 b Find the standard form of the equation by completing the square. Then identify and graph each conic. 16 x 2 + 9 y 2 – 128 x + 108 y + 436 = 0 Holt Algebra 2

10 -6 Identifying Conic Sections Check It Out! Example 3 b Continued Holt Algebra 2