U5D2 Assignment, pencil, red pen, highlighter, calculator, notebook Have out: Bellwork: 1. Find the standard form of the equation by completing the square then graph the circle. Identify the center and radius. 2. Find the equation of the circle in standard form. 5 -5 x y y 5 x -5 5 total: -5

We need to complete the square “twice” for both the x’s and y’s. 1. Find the standard form of the equation by completing the square then graph the circle. We need to complete the square “twice” for both the x’s and y’s. a) 1 4 1 4 +2 5 –5 x y +1 +1 +1 center: (1, –2) +1 r = 1 +1 +1

2. Find the equation of the circle in standard form. 5 -5 x y center: (–1,1) r = 3 +1 +1 +1 total:

Copy into the worksheet: Ellipse sum Ellipse: The set of all points in a plane such that the ______ of the distances from two points in the plane is ___________. sum constant

Get out a piece of blank paper Steps to draw an ellipse: 1. Put the blank paper on the cardboard, and push the two pins through the paper and the board. 2. Draw the top half of an ellipse using a pencil and the string. Then draw the bottom half. Draw several different ellipses: Move the pins further apart. Move the pins closer. Consider the following questions:  What happens to the shape when the pins are moved apart? Closer?  Does the total length of the string ever change?

The two fixed points are called the ______. foci The two fixed points are called the ______. Every ellipse has ____ axes of ____________. The _________axis has a length ______ and the _________ axis has a length of _____. The ________ intersect at the _________ of the ellipse. The _____ always lie on the _________ axis. ____ is always larger than ____! (plural of a focus) 2 symmetry major 2a minor 2b axes center foci major a b y (0, b) x major (-a, 0) (-c, 0) (c, 0) (a, 0) (0, -b) minor

By Pythagorean theorem, x (-a, 0) (a, 0) (0, b) (0, -b) (-c, 0) (c, 0) d1 + d2 = 2a d1 d2 The sum of the distances from the _____ to any point on the ellipse is the same as the length of the ______ axis, or ___ units. This distance from the center to either _____ is c units. By the ___________ Theorem, a, b, and c are related by the equation _________, where a > b. foci major 2a By Pythagorean theorem, c2+b2 = d22 c2+b2 = d12 foci since here d1 = d2 and d1+d2 = 2a 2d1 = 2a d1 = 2a 2 d1 = d2 = a c2 + b2 = a2 c2 = a2 – b2 Pythagorean c2 = a2 – b2

Summary Chart of Ellipses with Center (h, k) The largest denominator is ____ Standard Form of Equation Center Direction of Major Axis Foci Length of Major Axis Length of Minor Axis (h, k) (h, k) horizontal vertical notice a is under x notice a is under y (h ± c, k) (h, k ± c) 2a 2a 2b 2b

I. Graph each ellipse. Label the coordinates of the center, the foci, and the endpoints of the major and minor axes. (0,5) 5 y –5 x a) Center: (0, 0) Which denominator is bigger? (–2,0) (2,0) major axis: Y, so this is a vertical ellipse. a = 5 c2 = a2 – b2 b = 2 c2 = 25 – 4 c = 4.58 c2 = 21 (0,–5) foci: (0, 0 ± 4.58) (0, 4.58) c ≈ 4.58 (0, -4.58)

I. Graph each ellipse. Label the coordinates of the center, the foci, and the endpoints of the major and minor axes. 10 x y –10 b) (2,7) Which denominator is bigger? Center: (2, 5) (–5,5) (9,5) major axis: X, so this is a horizontal ellipse. (2,3) a = 7 b = 2 c2 = a2 – b2 c = 6.71 c2 = 49 – 4 foci: (2 ± 6.71, 5) c2 = 45 (8.71, 5) (–4.71, 5) c ≈ 6.71

Finish parts d – f on your own. I. Graph each ellipse. Label the coordinates of the center, the foci, and the endpoints of the major and minor axes. 10 x y –10 c) (0,6) Center: (0, 0) major axis: X, horizontal ellipse (–10,0) a = 10 b = 6 (10,0) c2 = a2 – b2 c = 8 c2 = 100 – 36 foci: (0 ± 8, 0) (0,–6) c2 = 64 (±8, 0) c = 8 Finish parts d – f on your own.

II. Write the standard form equation for each ellipse and identify the coordinates of the foci. 5 -5 x y a) Center: (–3,–1) major axis: Y 4 a = 4 1 b = 1 Where does a2 go? c = 3.87 (x + 3)2 (y + 1)2 Equation: + = 1 c2 = a2 – b2 12 42 c2 = 42 – 12 c2 = 16 – 1 c2 = 15 foci: (–3, –1 ± 3.87) c ≈ 3.87 (–3, 2.87) and (–3, –4.87)

II. Write the standard form equation for each ellipse and identify the coordinates of the foci. x -7 7 F1 F2 b) Center: (0, 0) uh oh! no b? major axis: Y a = 6 b = ≈ 5.20 Where does a2 go? c = 3 x2 y2 Equation: + = 1 62 c2 = a2 – b2 32 = 62 – b2 9 = 36 – b2 –27 = –b2 foci: (0, 0 ± 3) b2 = 27 (0, 3) and (0, –3) b ≈ 5.20

Finish parts d – f on your own. II. Write the standard form equation for each ellipse and identify the coordinates of the foci. 5 -5 x y c) Finish parts d – f on your own. Center: (0, 0) major axis: X 3 2 a = 3 b = 2 Where does a2 go? c = 2.24 x2 y2 + = 1 Equation: c2 = a2 – b2 32 22 c2 = 32 – 22 c2 = 9 – 4 c2 = 5 foci: (0 ± 2.24, 0) c ≈ 2.24 (–2.24, 0) and (2.24, 0)

Complete the packet

pencil, red pen, highlighter, GP notebook, calculator 1/5/2012 Have out: Compute / identify all of the following parts of the parabola. Then make a complete sketch using the information. Bellwork: y x 8 14 –2 –8 vertex: axis of symmetry: focus: directrix: total: latus rectum:

Name the vertex, axis of symmetry, focus, and directrix of the parabola whose equation is given. Then find the length of the latus rectum and sketch a graph of the parabola. a) Compute first to help find the focus and directrix. +1 y x 8 14 –2 –8 vertex: (7, -5) +1 axis of symmetry: x = 7 +1 +1 graph +1 focus: (7, –5 + 1) = (7, – 4) y = –5 – 1 directrix: +1 y = –6 +1 F +1 latus rectum: v 4 y = –6 +1 +1 +1 total: Graph half of the L.R. on each side of the focus.