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2y x
This is the parent graph of all quadratic functions.
The graph of a quadratic function is called a parabola.
The parent function is given as
2y xA table of values can be constructed from the graph as given to the right.
x y
-3 9 -2 4 -1 1 0 0 1 1 2 4 3 9
(-3,9)
(-2,4)
(-1,1)
(0,0)
(1,1)
(2,4)
(3,9)
2y x(-3,9)
(-2,4)
(-1,1)
(0,0)
(1,1)
(2,4)
(3,9)
All other quadratic functions can be expressed in the form:
This is called the standard form.
The general form is given as:
2( )y a x h k
2y ax bx c
y a x h 2 k+=
In standard form,
(h,k) identifies the vertex of the parabola.
h k
(h,k)
y a x h 2 k+=
In standard form,
a affects the direction the parabola opens and how wide or narrow it will open.
a
Since a=2 and it is positive, the parabola opens up and
the y-values are all 2 times larger than on the parent graph.
(1,8)
(2,2)
(3,0)
(4,2)
(5,8)
y 2 x 3 2 0+= 2
y a x h 2 k+=
In standard form,
If a is negative, the parabola will open down.
a
Since a=-2 and it is negative, the parabola opens down
andthe y-values are all 2 times larger than on the parent graph.
(1,-8)
(2,-2)
(3,0)
(4,-2)
(5,-8)
y 2 x 3 2 0+= - 2
- 2- 2
- 2- 2- 2- 2- 2- 2
The points where the parabola intersects the x-axis are called theRoots or Zeros of the function.
These roots occur when the y-value is equal to zero.
Solving for x we get the values:(1,0)
(2,6)
(3,8)
(4,6)
(5,0)X= X=1 51 51 51 51
5
1 51 551
22( 3) 8y x
20 2( 3) 8x
Example: Graph 23( 5) 2y x
The vertex is (5, -2)
The graph opens upward because 3 is positive.The y-values are multiplied by 3.
(5, -2)(5, -2)(5, -2)(5, -2)(5, -2)(5, -2)(5, -2)(5, -2)(5, -2)(5, -2)(5, -2)(5, -2)(5, -2)(5, -2)(5, -2)(5, -2)(5, -2)(5, -2)(5, -2)(5, -2)(5, -2)(5, -2)
Over 1
up 3
The zeros of the function can be found by setting y=0.
20 3( 5) 2x
Now solve for x.
2
2
2
2
2
2
0 3( 5) 2
0 2 3( 5) 2 2
2 3( 5)
1 12 3( 5)
3 32
( 5)3
2( 5)
3
25
3
25 5 5
3
25 5 5
3
25
35.66
4.33
x
x
x
x
x
x
x
x
x
x
x
x
The roots or zeros are:
(4.33, 0) and (5.66,0)
