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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

#2: y = –½(x – 5)2 + 4

#3: y = –3(x + 2)(x + 4)

#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.
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.
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.