U5D5 Have out: Bellwork: Answer the following for the equation: red pen, highlighter, GP notebook, calculator, ruler U5D5 Have out: Bellwork: Answer the following for the equation: (a) Write the equation for the ellipse in standard from. (b) Identify: center a, b, & c foci (c) Sketch a graph of the ellipse and the foci. total:

Bellwork: 4 4 100 16 +1 +1 +1 +1 1 1 1 +1 1 25 4 +3

Bellwork: (2, 3) Center: (2, –2) (0, -2) (4,-2) a = 5 b = 2 c = 4.58 10 y x –10 Bellwork: (2, 3) +1 Center: (2, –2) +1 (0, -2) (4,-2) a = 5 +1 +1 +1 b = 2 c = 4.58 (2, -7) foci: (2, –2 ± 4.58) +1 (2, 2.58) and (2, –6.58) c2 = a2 – b2 +1 +1 +1 c2 = 25 – 4 +1 graphed foci c2 = 21 total: Which denominator is bigger? Y, so this is a vertical ellipse c ≈ 4.58

Hyperbolas Let’s review the four types of conic sections: Parabola Circle Ellipse Hyperbola Hyperbolas There is only one left to study:

Hyperbola: The set of all points in a plane such that the ________________ of the ___________ of the distances from two fixed points, called the _____, is constant. absolute value difference foci positive d2 – d1 is a _________ constant. Recall:  For an ellipse, the sum of the distances is a positive constant.  But for the hyperbola, the difference of the distances is a positive constant.

A hyperbola has some similarities to an ellipse.  The distance from the center to a vertex is ___ units. a  The distance from the center to a focus is ___ units. c y x vertices (0, b) focus focus (-c, 0) (-a, 0) (a, 0) (c, 0) (0, -b)

2 symmetry transverse 2a vertices conjugate 2b perpendicular  There are _____ axes of ___________: 2 symmetry transverse 2a The ___________ axis is a line segment of length ____ units whose endpoints are _________ of the hyperbola. vertices The ___________ axis is a line segment of length ____units that is ________________ to the ____________ axis at the _________. conjugate 2b perpendicular y x conjugate axis transverse center (0, b) transverse axis (-c, 0) (-a, 0) (a, 0) (c, 0) (0, -b)

c2 = a2 + b2 asymptote asymptote center branches asymptotes The values of a, b, and c relate differently for a hyperbola than for an ellipse. For a hyperbola, ___________. c2 = a2 + b2 Hmm… the ellipse is a “minus” (c2 = a2 – b2) and the hyperbola is a “plus” (c2 = a2 + b2). That looks like the Pythagorean Theorem!?! y asymptote (0, b) asymptote As a hyperbola recedes from its _______, the ___________ approach the lines called ______________. center branches x asymptotes (-c, 0) (-a, 0) (a, 0) (c, 0) (0, -b)

Summary Chart of Hyperbolas with Center (0, 0) Standard Form of Equation Direction of Transverse Axis Foci Vertices Length of Transverse Axis Length of Conjugate Axis Equations of Asymptotes horizontal vertical (±c, 0) (0, ±c) (±a, 0) (0, ±a) 2a 2a 2b 2b

Note: The equations for hyperbolas and ellipses are identical except for a _______ sign. In graphing either, the __________ axis variable, x or y, is the axis which contains the _________ and _______. minus positive vertices foci Okay, so what does all of this mean? With a couple of exceptions that we will talk about, graph hyperbolas like you would an ellipse. But instead of drawing an ellipse, we will draw a rectangle and some lines.

Which variable is positive? x So, it’s a horizontal hyperbola a) I. Graph each hyperbola. Label the vertices and foci. Identify the asymptotes. Which variable is positive? x So, it’s a horizontal hyperbola a) –10 x y 10 Center: (0, 0) c2 = a2 + b2 a = (0,3) 5 c2 = 25 + 9 b = 3 c2 = 34 c = 5.83 (-5, 0) (5, 0) foci: (± 5.83, 0) c ≈ 5.83 vertices: (±5, 0) (0,-3) asymptotes: vertices

Which variable is positive? y So, it’s a vertical hyperbola b) I. Graph each hyperbola. Label the vertices and foci. Identify the asymptotes. Which variable is positive? y So, it’s a vertical hyperbola b) –10 x y 10 Center: (0, 0) c2 = a2 + b2 a = 4 c2 = 16 + 49 (0, 4) b = 7 c2 = 65 (–7,0) (7,0) c = 8.06 foci: (0, ± 8.06) c ≈ 8.06 (0,–4) vertices: (0, ±4) asymptotes:

Take a few minutes to complete Part I.

Which variable is positive? x So, it’s a horizontal hyperbola c) I. Graph each hyperbola. Label the vertices and foci. Identify the asymptotes. Which variable is positive? x So, it’s a horizontal hyperbola c) –10 x y 10 Center: (0, 0) c2 = a2 + b2 (0, 4) a = 2 c2 = 4 + 16 b = 4 c2 = 20 c = 4.47 (–2, 0) (2, 0) foci: (± 4.47, 0) c ≈ 4.47 vertices: (±2, 0) (0,–4) asymptotes:

Which variable is positive? So, it’s a vertical hyperbola I. Graph each hyperbola. Label the vertices and foci. Identify the asymptotes. y Which variable is positive? So, it’s a vertical hyperbola –10 x y 10 Center: (0, 0) (0,6) a = 6 b = 8 (–8,0) (8,0) c = 10 c2 = a2 + b2 foci: (0, ±10) c2 = 64 +36 vertices: (0, ±6) asymptotes: c2 = 100 (0,–6) c = 10

a = 3 b = 2 c = equation: x2 y2 32 22 general form II. Write the standard form equation for each hyperbola. a) y x -7 7 The hyperbola opens left and right. Therefore, x is positive. “a” will be under x. b a a = 3 b = 2 c = equation: x2 y2 32 22 general form

a = 1 equation: b = 2 y2 x2 c = 12 22 general form II. Write the standard form equation for each hyperbola. b) The hyperbola opens up and down. y x -5 5 Therefore, y is positive. “a” will be under y. To determine “a” and “b”, we need to draw in the rectangle. a b Look at the places where the asymptotes cross the grid. Draw a rectangle that goes through the vertices and intersections on the grid. a = 1 equation: b = 2 y2 x2 c = 12 22 general form

a c a = 2 b = equation: x2 y2 c = 4 22 general form II. Write the standard form equation for each hyperbola. 5 -5 x y d) The hyperbola opens left and right. Therefore, x is positive. “a” will be under x. We don’t know b, and drawing a rectangle won’t help since the asymptotes are not given. a c However, we know the foci. a = 2 b = equation: x2 y2 c = 4 22 general form

Finish today’s worksheets.