Date: 9.1(a) Notes: The Ellipse Lesson Objective: Graph and write the equa- tions of ellipses in standard form. CCSS: G.GPE.3 You will need: colored pens Real-World App: Will your truck clear the opening of the archway? This is Jeopardy!!!: This is the center of a circle whose equation is: x 2 + y 2 + 4x – 8y – 16 = 0

Lesson 1: Conic Sections Conic Sections: Curves that result from the intersection of a right cone and a plane.

Lesson 1: Conic Sections Here are some pictures of conic sections: Parabola on Golden Gate Bridge, San Francisco

Lesson 1: Conic Sections Hyperbolic buildings, San Francisco

Lesson 1: Conic Sections Hyperbolic buildings, San Francisco

Lesson 1: Conic Sections Semiellipse in Chinatown, San Francisco

Lesson 2: The Anatomy of an Ellipse Ellipse: The set of all points, P, in a plane the sum of whose distances from 2 foci, F 1 and F 2, is constant. Graph the ellipse whose equation is: (x – 0) 2 + (y – 0) 2 = Plot foci F 1 at (-4, 0) and F 2 at (4, 0).

Lesson 2: The Anatomy of an Ellipse (x – 0) 2 + (y – 0) 2 = Plot foci F 1 at (-4, 0) and F 2 at (4, 0).

Lesson 2: The Anatomy of an Ellipse Ellipse Equation in Standard Form: (x – h) 2 + (y – k) 2 = 1 a 2 b 2 Always 1

Lesson 2: The Anatomy of an Ellipse (x – h) 2 + (y – k) 2 = 1 b 2 a 2

Lesson 2: The Anatomy of an Ellipse Ellipse Equation in Standard Form: (x – h) 2 + (y – k) 2 = 1 a 2 b 2 b 2 a 2 Center: (h, k); midpoint of major and minor axes Major Axis: The longer axis; length = 2a Minor Axis: The shorter axis; length = 2b Vertices: +a units from the center; the end- points of the major axis Foci: +c units from the center on the major axis; a 2 = b 2 + c 2 or c 2 = a 2 b 2 –

Lesson 2: The Anatomy of an Ellipse Ellipse Summary: (x – h) 2 + (y – k) 2 = 1 a 2 b 2 (x – h) 2 + (y – k) 2 = 1 b 2 a 2 Center (h, k) Major Axis Parallel to x-axis, a is under (x – h) 2 Parallel to y-axis, a is under (y – k) 2 Vertices (h + a, k)(h, k + a) Foci (h + c, k) c 2 = a 2 – b 2 (h, k + c) c 2 = a 2 – b 2

Lesson 3: Graphing Ellipses at the Origin Graph and locate the foci: 4x 2 + y 2 = 36

Lesson 4: Graphing Ellipses at (h, k) Graph and locate the foci: 4x 2 + 9y 2 + 8x – 36y + 4 = 0

9.1(a): Do I Get It? Yes or No Graph and locate the foci. 1. x 2 + y 2 = x y 2 = x 2 + 4y 2 – 18x + 16y – 11 = 0