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Conics

Conics. Chapter 7. Parabolas. Definition - a parabola is the set of all points equal distance from a point (called the focus) and a line (called the directrix). Parabolas are shaped like a U or C. Parabolas. Equations - y = a(x - h) 2 + k opens up if a > 0, opens down if a < 0.

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Conics

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  1. Conics Chapter 7

  2. Parabolas • Definition - a parabola is the set of all points equal distance from a point (called the focus) and a line (called the directrix). • Parabolas are shaped like a U or C

  3. Parabolas • Equations - • y = a(x - h)2 + k • opens up if a > 0, opens down if a < 0. • x = a(y - k)2 + h • opens right if a > 0, opens left if a < 0.

  4. Parabolas • y = a(x - h)2 + k • x = a(y - k)2 + h • Vertex - the bottom of the curve that makes up a parabola. Represented by the point (h, k).

  5. Parabolas • Given the following equations for a parabola, give the direction of opening and the vertex. • y = (x - 6)2 - 4 • opens up • vertex is at (6, -4)

  6. Parabolas • x = (y + 5)2 + 4 • opens right. • vertex = (4, -5) • y = -5(x + 2)2 • opens down • vertex = (-2, 0)

  7. Parabolas • x = -y2 - 1 • opens left • vertex = (-1, 0)

  8. Parabolas • How are we going to graph these? • Calculator of course!!! • We will be using the conics menu (#9). • Typing it will be KEY!!!!!

  9. Parabolas • Notice that you have four choices for parabolas. Two for x = and two for the y = types. • How would we graph y = (x - 6)2 - 4?

  10. Parabolas • y = (x - 6)2 - 4 • Which form would we use? • The third one. • A = 1 • H = 6 • K = -4

  11. Parabolas • y = (x - 6)2 - 4 • We already know that the vertex is at (6, 4), but the calculator will tell us if we hit G-Solv and then VTX (F5, then F4).

  12. Parabolas • Steps to graph a parabola (cause you gotta put in on graph paper for me to see). • 1) choose the general equation that you will be working with.

  13. Parabolas • 2) Enter your variables. • 3) Draw (F6) • 4) Find the vertex (G-solve, then VRX => F5 then F4). • 5) Plot the vertex on your graph paper.

  14. Parabolas • Now we need to plot a point on each side of the vertex. • 6) if it is a y = equation, use the x value of the vertex as your reference. Plug in a value larger and smaller into the equation to get your y.

  15. Parabolas • 6) if it is a x = equation, use the y value of the vertex as your reference. Plug in a value larger and smaller into the equation to get your x. • 7) Plot these two points on your graph paper.

  16. Parabolas • 8) connect your three points in a C or U shape. • You’re done!!!

  17. Parabolas • Let’s try to graph some together. • x = (y + 5)2 + 4 • y = -5(x + 2)2 • x = -y2 - 1

  18. Parabolas • Assignment: • wkst 58 • pg. 420 • #’s 21 - 25

  19. Circles • Definition: the set of all points that are equidistant from a given point (the center). The distance between the center and any point is called the radius.

  20. Circles • Equation - (x - h)2 + (y - k)2 = r2 • the center is at (h, k) • the radius is r (notice that in the equation r is squared)

  21. Circles • Give the center and the radius of each equation. • (x - 1)2 + (y + 3)2 = 9 • center = (2, -3) • radius = 3

  22. Circles • (x - 2)2 + (y + 4)2 = 16 • center = (2, 4) radius = 4 • (x - 3)2 + y2 = 9 • center = (3, 0) radius = 3 • x2 + (y + 5)2 = 4 • center = (0, -5) radius = 2

  23. Circles • Of course the calculator will do this for us. Let’s look at the circles in the conics menu. • The 5th and 6th choices are circles. We will be using the 5th choice most often.

  24. Circles • Let’s graph (x - 1)2 + (y + 3)2 = 9 using the calculator. • Select the correct equation and plug in h, k and r. • h = 1, k = -3, and r = 3

  25. Circles • Draw it. • By hitting G-Solv we can get the center and radius. • Check it with what we found earlier.

  26. Circles • Graphing on paper • 1) plot the center. • 2) make 4 points, one up, down, left and right from the center. The distance between the points and the center is the radius.

  27. Circles • 3) Connect the points in a circular fashion. DO NOT create a square. This will take practice.

  28. x y Circles • (x - 1)2 + (y + 3)2 = 9 • Center = (1, -3) Radius = 3

  29. Circles - graph these • (x - 2)2 + (y + 4)2 = 16 • center = (2, 4) radius = 4 • (x - 3)2 + y2 = 9 • center = (3, 0) radius = 3 • x2 + (y + 5)2 = 4 • center = (0, -5) radius = 2

  30. Circles • Assignment: • Circles wkst 60/61

  31. Ellipses • Do not call ellipses ovals, even though they have the same shape. • Equation:

  32. Ellipses • The center is at (h, k). • a is the horizontal distance from the center to the edge of the oval. • b is the vertical distance from the center to the edge of the oval

  33. Ellipses • Give the center of the ellipse, a, and b. • center is (0, 2), a = 4, b = 2

  34. Ellipses • center is (0, 2), a = 4, b = 2 • to graph we will plot the center, then use a to create points on each side of the center and use b to create points above and below the center.

  35. x y Ellipses • center is (0, 2), a = 4, b = 2

  36. Ellipses • The calculator will be useful in confirming your answer, but will not give you the center or any of the distances. We use the next to the last option for ellipses.

  37. Ellipses • Graph - find the center and a & b. Check using the calc.

  38. Ellipses • Assignment: • wkst 63/64

  39. Hyperbola • Hyperbola look like two parabolas facing out from each other. • I am not going to make you graph them by hand. Just use the calculator.

  40. Hyperbola • the equation is just like that of an ellipse except that the fractions are being subtracted.

  41. Hyperbola • Enter the following equation into the calculator. • h = -3, k = 5, a = 3, b = 2

  42. Hyperbola • hit G-Solve, then VTX (F4). • this will give you one of the two vertices, use the arrow keys to get the other. • graph these points.

  43. Hyperbola • use the x or y intercepts (which will be given to you using G-Solv) to sketch the graph.

  44. Hyperbola • Assignment • wkst 66/67

  45. Conics • Points to use to distinguish between the conics sections. • the equation of a parabola is the ONLY equation where only one variable is being squared.

  46. Conics • for circles, both x and y are being squared, it is usually not set equal to 1 and there are no fractions.

  47. Conics • for ellipses, both variables are squared, and the equation is the sum of fractions set equal to 1

  48. Conics • for hyperbola, both variables are squared, and the equation is the difference of fractions set equal to 1

  49. Conics • Assignment • Conics worksheet

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