Sec. 3.2 –
Quadratic Functions, and Graphs
A quadratic function is a polynomial function of degree two.
Therefore, there will be an
2
x
term. The graph of a quadratic function can be found from
the squaring function, i.e.
2
( )
f x
x
=
. It will be a parabola!
Given a quadratic function in its normal polynomial form, we can change it to one with
reflections, shifts and/or stretches/contractions.
For ex.
if
2
( )
2
2
3
f x
x
x
=
+
+
then we can use the complete the square process !
2
( )
3
2
2
f x
x
x
=
+
+
divide both sides by 2
2
( )
3
____
2
2
f x
x
x

=
+
+
subtract 3/2 from both sides
2
( )
3
1
1
2
2
4
4
f x
x
x

+
=
+
+
add ¼ to both sides
2
( )
5
1
2
4
2
f x
x

=
+
simplify and factor
2
( )
1
5
2
2
4
f x
x
=
+
+
add 5/4 to both sides, now multiply thru by 2
2
1
1
( )
2(
)
2
2
2
f x
x
=
+
+
so we stretch the
2
x
graph by 2 vertically,
shift it
1
2
a unit to the left, and up
1
2
2
.
Notice that the “origin” point of
2
x
, shifts to
1
1
(
,2
)
2
2

. This is the low point of the
graph and is called the
vertex
of the parabola! It is the high/low point of the parabola,
depending upon which way the parabola opens.
We
could
do this with
ANY
quadratic function to see how the graph relates to our basic
squaring function.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 stean
 Optimization

Click to edit the document details