Documents
Resources
Learning Center
Upload
Plans & pricing Sign in
Sign Out

Conics

VIEWS: 5 PAGES: 44

									                         Conics
0011 0010 1010 1101 0001 0100 1011

                                     1
                                         2
                                     4
                       Parabolas
• Definition - a parabola is the
0011 0010 1010 1101 0001 0100 1011


  set of all points equal
  distance from a point (called
  the focus) and a line (called      1
                                         2
  the directrix).
• Parabolas are shaped like a
  U or C
                                     4
                       Parabolas
• Equations -
0011 0010 1010 1101 0001 0100 1011

• y = a(x - h) 2+k

   –opens up if a > 0, opens down
    if a < 0.                        1
                                         2
• x = a(y -

    if a < 0.
                        k) 2    +h
   –opens right if a > 0, opens left
                                     4
                       Parabolas
• y = a(x -     +k      h) 2
0011 0010 1010 1101 0001 0100 1011

• x = a(y - k) 2+h

• Vertex - the bottom of the
  curve that makes up a              1
                                         2
  parabola. Represented by
  the point (h, k).
                                     4
                       Parabolas
• Given the following equations
0011 0010 1010 1101 0001 0100 1011
  for a parabola, give the
  direction of opening and the
  vertex.
• y = (x -
                                     1
                                         2
                                     4
              -4     6) 2

• opens up
• vertex is at (6, -4)
                       Parabolas
• x = (y +    +4      5) 2
0011 0010 1010 1101 0001 0100 1011

• opens right.
• vertex = (4, -5)
• y = -5(x + 2) 2                    1
                                         2
• opens down
• vertex = (-2, 0)                   4
                       Parabolas
•x =     -1-y 2
0011 0010 1010 1101 0001 0100 1011

• opens left
• vertex = (-1, 0)
                                     1
                                         2
                                     4
                       Parabolas
• How are we going to graph
0011 0010 1010 1101 0001 0100 1011

  these?
• Calculator of course!!!
• We will be using the conics        1
                                         2
  menu (#9).
• Typing it will be KEY!!!!!
                                     4
                       Parabolas
• Notice that you have four
0011 0010 1010 1101 0001 0100 1011

  choices for parabolas. Two
  for x = and two for the y =
  types.                             1
                                         2
• How would we graph
    y = (x - 6) 2 - 4?
                                     4
                       Parabolas
• y = (x -   -4      6) 2
0011 0010 1010 1101 0001 0100 1011

• Which form would we use?
• The third one.
•A = 1                               1
                                         2
•H = 6
• K = -4                             4
                       Parabolas
• y = (x -    -4     6) 2
0011 0010 1010 1101 0001 0100 1011

• We already know that the
  vertex is at (6, -4), but the
  calculator will tell us if we      1
                                         2
  hit G-Solv and then VTX
  (F5, then F4).
                                     4
                       Parabolas
• Steps to graph a parabola
0011 0010 1010 1101 0001 0100 1011

  (cause you gotta put in on
  graph paper for me to see).
• 1) choose the general              1
                                         2
  equation that you will be
  working with.
                                     4
                       Parabolas
• 2) Enter your variables.
0011 0010 1010 1101 0001 0100 1011

• 3) Draw (F6)
• 4) Find the vertex (G-solve,
  then VRX => F5 then F4).           1
                                         2
• 5) Plot the vertex on your
  graph paper.
                                     4
                       Parabolas
• Now we need to plot a point
0011 0010 1010 1101 0001 0100 1011

  on each side of the vertex.
• 6) if it is a y = equation, use
  the x value of the vertex as       1
                                         2
  your reference. Plug in a
  value larger and smaller into
  the equation to get your y.        4
                       Parabolas
• 6) if it is a x = equation, use
0011 0010 1010 1101 0001 0100 1011

  the y value of the vertex as
  your reference. Plug in a
  value larger and smaller into      1
                                         2
  the equation to get your x.
• 7) Plot these two points on
  your graph paper.                  4
                       Parabolas
• 8) connect your three points
0011 0010 1010 1101 0001 0100 1011

  in a C or U shape.
• You’re done!!!
                                     1
                                         2
                                     4
                       Parabolas
• Let’s try to graph some
0011 0010 1010 1101 0001 0100 1011

  together.
• x = (y + 5) 2+4

• y = -5(x + 2) 2                    1
                                         2
• x = -y 2-1


                                     4
                           Circles
• Definition: the set of all
0011 0010 1010 1101 0001 0100 1011

  points that are equidistant
  from a given point (the
  center). The distance              1
                                         2
  between the center and any
  point is called the radius.
                                     4
                           Circles
• Equation -
0011 0010 1010 1101 0001 0100 1011

    (x - h) 2 + (y - k)2 = r2

• the center is at (h, k)
• the radius is r (notice that in    1
                                         2
  the equation r is squared)
                                     4
                           Circles
• Give the center and the
0011 0010 1010 1101 0001 0100 1011

  radius of each equation.
• (x - 1) 2 + (y + 3)2 = 9

• center = (1, -3)                   1
                                         2
• radius = 3
                                     4
                           Circles
• (x -      2) 2
            + (y +   = 16            4) 2
0011 0010 1010 1101 0001 0100 1011

• center = (2, -4) radius = 4
• (x - 3) 2 + y2 = 9

• center = (3, 0) radius = 3                1
                                                2
•x 2 + (y + 5)2 = 4

• center = (0, -5) radius = 2               4
                           Circles
• Of course the calculator will
  do this for us. Let’s look at
0011 0010 1010 1101 0001 0100 1011




  the circles in the conics
  menu.                              1
                                         2
• The 5th and 6th choices are
  circles. We will be using the
  5th choice most often.             4
                           Circles
• Let’s graph
0011 0010 1010 1101 0001 0100 1011

  (x - 1) 2 + (y + 3)2 = 9 using

  the calculator.
• Select the correct equation        1
                                         2
  and plug in h, k and r.
• h = 1, k = -3, and r = 3
                                     4
                           Circles
• Draw it.
0011 0010 1010 1101 0001 0100 1011

• By hitting G-Solv we can get
  the center and radius.
• Check it with what we found        1
                                         2
  earlier.
                                     4
                           Circles
• Graphing on paper
0011 0010 1010 1101 0001 0100 1011

• 1) plot the center.
• 2) make 4 points, one up,
  down, left and right from          1
                                         2
  the center. The distance
  between the points and the
  center is the radius.              4
                           Circles
• 3) Connect the points in a
0011 0010 1010 1101 0001 0100 1011

  circular fashion. DO NOT
  create a square. This will
  take practice.                     1
                                         2
                                     4
                           Circles
• (x -   + (y +
            1) 2   =9                3) 2
0011 0010 1010 1101 0001 0100 1011

• Center = (1, -3) Radius = 3

                                                1
                                                    2
                                            4
                                            x




                                     y
          Circles - graph these
• (x -      2) 2
            + (y +   = 16            4) 2
0011 0010 1010 1101 0001 0100 1011

• center = (2, 4) radius = 4
• (x - 3) 2 + y2 = 9

• center = (3, 0) radius = 3                1
                                                2
•x 2 + (y + 5)2 = 4

• center = (0, -5) radius = 2               4
                           Ellipses
• Do not call ellipses ovals,
0011 0010 1010 1101 0001 0100 1011

  even though they have the
  same shape.
• Equation:                           1
                                          2
(x  h) (y  k)
   a 2 
           b2   1
                       2

                                      4
                                      2
                          Ellipses
• The center is at (h, k).
0011 0010 1010 1101 0001 0100 1011

• a is the horizontal distance
  from the center to the edge
  of the oval.                       1
                                         2
• b is the vertical distance
  from the center to the edge
  of the oval                        4
                          Ellipses
• Give the center of the ellipse,
0011 0010 1010 1101 0001 0100 1011

  a, and b.
                 x   2
                      (y  2)        2


                 16
                    
                         4
                              1
                                         1
                                             2
• center is (0, 2), a = 4, b = 2
                                         4
                          Ellipses
• center is (0, 2), a = 4, b = 2
0011 0010 1010 1101 0001 0100 1011

• to graph we will plot the
  center, then use a to create
  points on each side of the         1
                                         2
  center and use b to create
  points above and below the
  center.                            4
                          Ellipses
• center is (0, 2), a = 4, b = 2
0011 0010 1010 1101 0001 0100 1011




                                     x
                                         1
                                             2
                               y     4
                          Ellipses
• The calculator will be useful
0011 0010 1010 1101 0001 0100 1011

  in confirming your answer,
  but will not give you the
  center or any of the               1
                                         2
  distances. We use the next to
  the last option for ellipses.
                                     4
                          Ellipses
• Graph - find the center and
0011 0010 1010 1101 0001 0100 1011

  a & b. Check using the calc.
            2             2
         x     y
                  1
          9 25                       1
                                         2
         (x  3)
            4
                 2
                            2
                    (y  1)  1
                                     4
                      Hyperbola
• Hyperbola look like two
0011 0010 1010 1101 0001 0100 1011

  parabolas facing out from
  each other.
• I am not going to make you         1
                                         2
  graph them by hand. Just
  use the calculator.
                                     4
                      Hyperbola
• the equation is just like that
0011 0010 1010 1101 0001 0100 1011

  of an ellipse except that the
  fractions are being
  subtracted.
(x  h)               2
                                     (y  k)
                                            1
                                            2
                                                2
   a 2  
                                       b2
                                            4   1
                      Hyperbola
• Enter the following equation
0011 0010 1010 1101 0001 0100 1011

  into the calculator.
(x  3) (y  5)      2               2
               1                   1
                                         2
   9       4
• h = -3, k = 5, a = 3, b = 2        4
                      Hyperbola
• hit G-Solve, then VTX (F4).
0011 0010 1010 1101 0001 0100 1011

• this will give you one of the
  two vertices, use the arrow
  keys to get the other.             1
                                         2
• graph these points.
                                     4
                      Hyperbola
• use the x or y intercepts
0011 0010 1010 1101 0001 0100 1011

  (which will be given to you
  using G-Solv) to sketch the
  graph.                             1
                                         2
                                     4
                           Conics
• Points to use to distinguish
0011 0010 1010 1101 0001 0100 1011

  between the conics sections.
• the equation of a parabola is
  the ONLY equation where            1
                                         2
  only one variable is being
  squared.
                                     4
                           Conics
• for circles, both x and y are
0011 0010 1010 1101 0001 0100 1011

  being squared, it is usually
  not set equal to 1 and there
  are no fractions.                  1
                                         2
                                     4
                           Conics
• for ellipses, both variables
0011 0010 1010 1101 0001 0100 1011

  are squared, and the
  equation is the sum of
  fractions set equal to 1           1
                                         2
                                     4
                           Conics
• for hyperbola, both
0011 0010 1010 1101 0001 0100 1011

  variables are squared, and
  the equation is the difference
  of fractions set equal to 1        1
                                         2
                                     4

								
To top