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Learn about ellipses and how to draw them using a string and pins. Understand the equation of ellipses in standard form and identify the center, foci, and endpoints of major and minor axes.
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y 5 x -5 5 -5 total: 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. y 5 x -5 5 -5
1. Find the standard form of the equation by completing the square then graph the circle. y 5 x 5 –5 –5 We need to complete the square “twice” for both the x’s and y’s. a) 1 4 1 4 +2 +1 +1 +1 center: (1, –2) +1 r = 1 +1 +1
y 5 x -5 5 -5 total: 2. Find the equation of the circle in standard form. center: (–1,1) r = 3 +1 +1 +1
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 ___________. 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?
foci (plural of a focus) 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 ____! 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
y (0, b) x (-a, 0) (-c, 0) (c, 0) (a, 0) (0, -b) d1 + d2 = 2a d2 d1 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 2a major 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) a The largest denominator is ____ (h, k) (h, k) vertical horizontal notice a is under y notice a is under x (h ± c, k) (h, k ± c) 2a 2a 2b 2b
y 5 x –5 5 –5 I. Graph each ellipse. Label the coordinates of the center, the foci, and the endpoints of the major and minor axes. (0,5) a) Center: (0, 0) Which denominator is bigger? (–2,0) (2,0) major axis: Y, so this is a vertical ellipse. 5 a = c2 = a2 – b2 2 b = c2 = 25 – 4 4.58 c = c2 = 21 (0,–5) foci: (0, 0± 4.58) (0, 4.58) c ≈ 4.58 (0, -4.58)
y 10 x –10 10 –10 I. Graph each ellipse. Label the coordinates of the center, the foci, and the endpoints of the major and minor axes. b) (2,7) Which denominator is bigger? (2, 5) Center: (–5,5) (9,5) major axis: X, so this is a horizontal ellipse. (2,3) a = 7 2 b = 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
y 10 x –10 10 –10 I. Graph each ellipse. Label the coordinates of the center, the foci, and the endpoints of the major and minor axes. c) (0,6) (0, 0) Center: major axis: X, horizontal ellipse (–10,0) a = 10 (10,0) 6 b = 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.
y 5 x -5 5 -5 II. Write the standard form equation for each ellipse and identify the coordinates of the foci. a) Center: (–3,–1) 4 Y major axis: 4 a = 1 1 b = 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)
y 7 F1 x -7 7 F2 -7 II. Write the standard form equation for each ellipse and identify the coordinates of the foci. b) (0, 0) Center: uh oh! no b? major axis: Y 6 a = 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) (0, –3) and b ≈ 5.20
y 5 x -5 5 -5 II. Write the standard form equation for each ellipse and identify the coordinates of the foci. c) Finish parts d – f on your own. Center: (0, 0) X major axis: 2 3 3 a = 2 b = 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)
y 8 x 14 –2 –8 total: 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: vertex: axis of symmetry: focus: directrix: latus rectum:
y 8 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. x 14 –2 –8 total: Compute first a) +1 to help find the focus and directrix. vertex: (7, -5) +1 +1 axis of symmetry: x = 7 +1 graph +1 focus: (7, –5 + 1) = (7, – 4) y = –5 – 1 directrix: +1 +1 y = –6 F +1 v latus rectum: 4 y = –6 +1 +1 +1 Graph half of the L.R. on each side of the focus.
