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Answer the following for the equation:

Understand the standard form, graphing, and properties of ellipses and hyperbolas with detailed explanations and examples.

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Answer the following for the equation:

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  1. total: 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.

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

  3. y 10 x –10 10 –10 total: Bellwork: (2, 3) +1 (2, –2) +1 Center: (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 Which denominator is bigger? Y, so this is a vertical ellipse c ≈ 4.58

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

  5. 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.

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

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

  8. The values of a, b, and c relate differently for a hyperbola than for an ellipse. For a hyperbola, ___________. c2 = a2 + b2 y asymptote (0, b) asymptote center branches x asymptotes (a, 0) (-a, 0) (c, 0) (-c, 0) (0, -b)

  9. Summary Chart of Hyperbolas with Center (0, 0) horizontal vertical (±c, 0) (0, ±c) (±a, 0) (0, ±a) 2a 2a 2b 2b

  10. 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.

  11. y 10 x –10 10 –10 I. Graph each hyperbola. Label the vertices and foci. Identify the asymptotes. Which variable is positive? x So, it’s a horizontal hyperbola a) (0, 0) Center: c2 = a2 + b2 (0,3) a = 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

  12. y 10 x –10 10 –10 I. Graph each hyperbola. Label the vertices and foci. Identify the asymptotes. Which variable is positive? y So, it’s a vertical hyperbola b) (0, 0) Center: 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:

  13. Take a few minutes to complete Part I.

  14. y 10 x –10 10 –10 I. Graph each hyperbola. Label the vertices and foci. Identify the asymptotes. Which variable is positive? x So, it’s a horizontal hyperbola c) (0, 0) Center: 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:

  15. y 10 x –10 10 –10 I. Graph each hyperbola. Label the vertices and foci. Identify the asymptotes. y Which variable is positive? So, it’s a vertical hyperbola (0,6) (0, 0) Center: 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

  16. y 7 x 7 -7 -7 II. Write the standard form equation for each hyperbola. a) 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

  17. y 5 x 5 -5 -5 II. Write the standard form equation for each hyperbola. b) The hyperbola opens up and down. 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

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

  19. Finish today’s worksheets.

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