Graphing Quadratic Functions(2)

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					Graphing Quadratic Functions

   A “Shortcut” and A “Summary”
   All the slides in this presentation are timed.

    Trying to advance the slides before you are asked to do so
will result in skipping part of the presentation on that slide.

   When each slide is finished a box will appear to let you
know there is nothing left on that slide.




                                                                  DONE
                  Before we begin …
Things you should know/understand before we begin.


     1. What is a parabola?
     2. What is the vertex of a parabola?
     3. How do you find the vertex of a parabola?
               a) In Standard Form … y = ax2 + bx + c
               b) In Vertex Form … y = a (x – h)2 + k
               c) In Intercept Form … y = a (x – p)(x – q)
     4. What is the line of symmetry of a parabola?


     5. The symbol, , means “change”. Thus x, means “the change in x”.


                                                                           DONE
       A Little Exploration
Fill out the following table. Click the screen when you are done.


 x                   x                y = x2               y
                      0                   0
  1                   1                   1                   1
  1                   2                   4                   3
  1                   3                   9                   5
  1                   4                  16                   7
  1                   5                  25                   9
  1                   6                  36                  11
  1                   7                  49                  13
  1                   8                  64                  15

                                                                    DONE
       A Little Exploration
Fill out the following table. Click the screen when you are done.


 x                   x                y = 2x2              y
                      0                   0
  1                   1                   2               2 = 2(1)
  1                   2                   8               6 = 2(3)
  1                   3                  18              10 = 2(5)
  1                   4                  32              14 = 2(7)
  1                   5                  50              18 = 2(9)
  1                   6                  72               22 = 2(11)
  1                   7                  98               26 = 2(13)
  1                   8                 128               30 = 2(15)

                                                                       DONE
       A Little Exploration
Fill out the following table. Click the screen when you are done.


 x                   x                y = 3x2              y
                      0                   0
  1                   1                   3               3 = 3(1)
  1                   2                  12               9 = 3(3)
  1                   3                  27              15 = 3(5)
  1                   4                  48              21 = 3(7)
  1                   5                  75              27 = 3(9)
  1                   6                 108               33 = 3(11)
  1                   7                 147               39 = 3(13)
  1                   8                 192               45 = 3(15)

                                                                       DONE
       A Little Exploration
Fill out the following table. Click the screen when you are done.


 x                   x              y = 0.5 x2             y
                      0                   0
  1                   1                  0.5            0.5 = 0.5(1)
  1                   2                   2             1.5 = 0.5(3)
  1                   3                  4.5            2.5 = 0.5(5)
  1                   4                   8             3.5 = 0.5(7)
  1                   5                 12.5            4.5 = 0.5(9)
  1                   6                  18             5.5 = 0.5(11)
  1                   7                 24.5            6.5 = 0.5(13)
  1                   8                  32             7.5 = 0.5(15)

                                                                        DONE
       A Little Exploration
Fill out the following table. Click the screen when you are done.


 x                   x                y = ax2              y
                      0                   0
  1                   1                   a                  1a
  1                   2                  4a                  3a
  1                   3                  9a                  5a
  1                   4                 16a                  7a
  1                   5                 25a                  9a
  1                   6                 36a                 11a
  1                   7                 49a                 13a
  1                   8                 64a                 15a

                                                                    DONE
                               So What?
First notice that in every example we just did the vertex was at (0, 0).
    The vertex of a parabola is not always at (0, 0), but the patterns we are
    going to employ will still work IF you start at the vertex.
Second, as we moved right on the x-axis, how far did we go every time? 1
Third, as we moved up the y-axis, we followed a pattern. Did you see that pattern?
                              Up a, 3a, 5a, 7a, 9a, etc.
   Notice that in every example so far, the a value was positive. That is why
   we moved UP. If a is negative, then the pattern of odd multiples of a still
   holds, but you would move DOWN.
To put it together, if we want to graph a parabola by hand …
FIRST: Find the vertex
SECOND: To find other points on the parabola without making a table of
values, start at the vertex and move right 1, up a … then move right 1, up 3a
… then move right 1 , up 5a … etc.
THIRD: To finish graphing the parabola, reflect the points you graphed in
the last step over the line of symmetry.
                                                                                 DONE
                   Let’s Try a Few
Graph each parabola below. Click on either the answer to see just the graph or
the “HELP” button to see a step by step explanation.

#1: y = 2x2 – 4x + 1

                           ANSWER                     HELP

#2: y = –½(x – 5)2 + 4

                           ANSWER                     HELP

#3: y = –3(x + 2)(x + 4)

                           ANSWER                     HELP
#1: y = 2x2 – 4x + 1       STANDARD FORM


                       y




                                               I’VE GOT IT! TRY
                                                ANOTHER ONE.




                                           x      Click one




                                               HOW DO YOU DO
                                               THIS PROBLEM?
#1: y = 2x2 – 4x + 1       STANDARD FORM       First, find the line of symmetry:

                                                       -b   +4
                       y                          x=      =    =1
                                                       2a 2(2)
                                               Second, find the vertex by
                                               plugging in the line of
                                               symmetry for the x-value.
                                                          2
                                               y = 2 (1) - 4 (1)+ 1 = - 1
                                                So, the vertex is (1, –1).

                                           x
                                                Since a = 2 …
                                                Move Right 1, Up 2 … (1a)

                                                Move Right 1, Up 6 … (3a)

                                                Reflect the points across the
                                                line of symmetry and draw the
                                                parabola.
                                           Click here to go back to problems
                                                                     DONE
#2: y = –½(x – 5)2 + 4       VERTEX FORM


                         y




                                               I’VE GOT IT! TRY
                                                ANOTHER ONE.




                                           x      Click one




                                               HOW DO YOU DO
                                               THIS PROBLEM?
#2: y = –½(x – 5)2 + 4       VERTEX FORM       First, plot the vertex at (5, 4).

                                               Since a = – ½ …
                         y
                                           Move Right 1, Down ½ … (1a)

                                           Move Right 1, Down 3 2 … (3a)

                                               Move Right 1, Down 5 2 … (5a)




                                           x

                                               The line of symmetry of a
                                               parabola is a vertical line
                                               through the vertex.

                                               Reflect the points across the
                                               line of symmetry and draw the
                                               parabola.
                                           Click here to go back to problems
                                                                     DONE
#3: y = –3(x + 2)(x + 4)       INTERCEPT FORM


                           y




                                                    I’VE GOT IT! TRY
                                                     ANOTHER ONE.




                                                x      Click one




                                                    HOW DO YOU DO
                                                    THIS PROBLEM?
#3: y = –3(x + 2)(x + 4)       INTERCEPT FORM     First, the x-intercepts are the
                                                  values of x such that each factor
                                                  equals 0. So, the x-intercepts
                           y                      are x = –2 and x = –4.
                                                 The line of symmetry is halfway
                                                 between these two values. So,
                                                 the line of symmetry is x = –3.
                                                 Plug this value of x into the
                                                 original equation to find the y-
                                                 value of the vertex.
                                                    y = - 3(- 3 + 2)(- 3 + 4) = 3
                                                x So the vertex is (–3, 3)

                                                  Since a = –3…
                                                  Move Right 1, Down 3 … (1a)
                                                  Move Right 1, Down 9 … (3a)
                                                  Reflect the points across the
                                                  line of symmetry and draw the
                                                  parabola.
                                                Click here to go back to problems
                                                                          DONE

				
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posted:10/5/2011
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Jun Wang Jun Wang Dr
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