April 18, 2012 Conic Sections Intro - Parabola Warm-up: Write the quadratic equation in vertex form and identify the vertex. 1.y = x 2 – 6x y = 3x 2 + 6x – 4

Check HW 8.1 If you did not check your answers in the back of the book, check your answers with someone in your group who did.

Conic Sections are images that are formed by slicing a plane through through a double cone.

The first section we will study is the parabola.

A parabola can be defined as the set of all points in a plane that are the same distance from a given point called the focus and a given line called the directrix. The standard equation (aka vertex form) of a parabola is: y = a(x – h) 2 + k If a > 0, then it opens up If a < 0, then It opens down Vertex: (h, k) Axis of symmetry: x = h Focus: Directrix:

Example 1: Identify the parts of the parabola by looking at the equation y = a(x – h) 2 + k If a > 0, then it opens up If a < 0, then It opens down Vertex: (h, k) Axis of symmetry: x = h Focus: Directrix: y = -2(x + 4) 2 – 1 Direction of Opening: Vertex: Axis of Symmetry Focus: down (-4, -1) x = 4 (-4, -9/8) Directrix:y = -7/8 Identify a = h = k = -2-4 “Is it in standard form?”

Practice: Identify coordinates of the vertex and focus, the equations of the axis of symmetry and directrix, and the direction of opening of the parabola with the given equation. y = (x – 4) a = h = k = Opens up Vertex: (4, 3) Focus: (4, 13/4) Axis of sym: x = 4 Directrix: y = 11/4 “Is it in standard form?”

What if it’s not in Standard Form/Vertex Form? Example 2a: Write in standard form, then identify each part of the parabola. y = x 2 + 6x – 4 y = (x 2 + 6x + ) - 4 – y = (x + 3) 2 – 13 You try: y = x 2 + 4x – 15 Use completing the square 99

What if it’s not in Standard Form/Vertex Form? Example 2b: Write in standard form, then identify each part of the parabola. y = 3x 2 + 6x – 4 y = 3(x 2 + 2x + ) − 4 – y = 3(x + 1) 2 – 7 You try: y = -2x 2 + 4x – 15 Use completing the square 1 3