Homework Answers (Completing the Square #2-10 even) 2) 4) 6) 8) 10)

Objectives: *Analyze and graph Parabolas *Write equations of parabolas https://www.youtube.com/watch?v=ky5Q6hEtjKk watch until 3:22 Conics 7.1: Parabolas Objectives: *Analyze and graph Parabolas *Write equations of parabolas

VOCABULARY Conic section: a figure formed when a plane intersects a double-napped right cone. Locus: the set of all points that fulfill a geometric property. Parabola: the locus of points in a plane that are equidistant from a fixed point and a specific line.

Parabola: the locus of points in a plane that are equidistant from a fixed point and a specific line. Focus: the fixed point from which the locus of points on a parabola are equidistant. Directrix: the line from which points on a parabola are equidistant. Vertex: the intersection of a parabola and its axis of symmetry. Axis of symmetry: a line perpendicular to the directrix through the focus of a parabola.

You can use the standard form of the equation for a parabola to determine the characteristics of the parabola such as the vertex, focus, and directrix.

Example 1: A) For the graph (y – 3)2 = –8 (x + 1), identify the vertex, focus, axis of symmetry, and directrix. Then graph the parabola.

Example 1: B) For the graph (x + 1)2 = –4(y – 2) identify the vertex, focus, axis of symmetry, and directrix. Then graph the parabola.

Example 2: The parabolic mirror for the California Institute of Technology’s Hale telescope at Mount Palomar has a shape modeled by y2 = 2668x, where x and y are measured in inches. What is the focal length of the mirror?

To determine the characteristics of a parabola, you may sometimes need to write an equation in standard form. In some cases, you can simply rearrange the equation, but other times it may be necessary to use mathematical skills such as completing the square.

Example 3: A) Write x2 − 8x − y = −18 in standard form Example 3: A) Write x2 − 8x − y = −18 in standard form. Identify the vertex, focus, axis of symmetry, and directrix. Then graph the parabola.

Example 3: B) Write y = − 1 2 x2 + 3x + 6 in standard form Example 3: B) Write y = − 1 2 x2 + 3x + 6 in standard form. Identify the vertex, focus, axis of symmetry, and directrix. Then graph the parabola.

Example 3: C) Write y 2 + 16x = 55 – 6y in standard form Example 3: C) Write y 2 + 16x = 55 – 6y in standard form. Identify the vertex, focus, axis of symmetry, and directrix. Then graph the parabola.

Warm-Up Use completing the square to write the equation in standard form for a parabola. Then identify the vertex, focus, axis of symmetry, and directrix. 1)

Homework Answers 1) vertex: (1, 2) focus: (1, 4) A of S: x = 1 directrix: y = 0 2) Eqn: (y + 3)2 = -12(x – 1) vertex: (1, -3) focus: (-2, -3) A of S: y = -3 directrix: x = 4

3) Eqn: (x + 4)2 = -4(y – 2) vertex: (-4, 2) focus: (-4, 1) A of S: x = - 4 directrix: y = 3 6) x2 = 8y 7) 4 in

PG. 14 HW #19: Write the equation for the given parabola.

A) focus (2, 1) and vertex (−5, 1) Example 4: Write an equation for and graph a parabola with the given characteristics. A) focus (2, 1) and vertex (−5, 1)

B) vertex (3, −2), directrix y = −1 Example 4: Write an equation for and graph a parabola with the given characteristics. B) vertex (3, −2), directrix y = −1 

C) focus (−1, 7), opens up, contains (3, 7) Example 4: Write an equation for and graph a parabola with the given characteristics. C) focus (−1, 7), opens up, contains (3, 7)  

Complete the 2 problems on the front and add this to the back: Exit Slip: Complete the 2 problems on the front and add this to the back: Write an equation for the graph of a parabola with the given information: Focus (11,4); opens right; contains (20, 16)

Line Tangent to a Parabola A line ℓ that is tangent to a parabola at a point P forms an isosceles triangle such that: *The segment from P to the focus forms one leg of the triangle. *The segment along the axis of symmetry from the focus to another point on the tangent line forms the other leg.

Example 6: Write an equation for the line tangent to y = x2 − 2 at (2, 2).

Example 7: Write an equation for the line tangent to