Basic graphs
Graphs of functions of the form
=
(
)
f
x
x
r
.
•
First we consider the graph of the equation
=
y
x
2
.
If
=
(
)
f
x
x
2
, then
=
(
)
f

x
(
)
f
x
for all real numbers
x
, that is, changing the sign of an input number
x
for the function f does not change the
value of f.
For example,
=
f

3
2
f
3
2
=
9
4
.
This leads to the fact that the graph of
=
y
x
2
is
symmetrical about the
y
axis
.
In general, a function that satisfies the condition that
=
(
)
f

x
(
)
f
x
for all numbers
x
in the domain of f is called an
even function
.
The graph of an even function is always symmetrical about the
y
axis.
The graph of
=
y
x
2
is an example of a
parabola
. In general, a parabola is a curve consisting of all points that lie at an equal distance from
a specified point ( called the
focus
of the parabola ) and a specified line ( called the
directrix
of the parabola ).
The focus of the parabola given by
=
y
x
2
is the point
,
0
1
4
and the directrix is the horizontal line with equation
=
y

1
4
.
As an example, we can check that the point (
)
,
1
1
is equidistant from
,
0
1
4
and
=
y

1
4
.
The point (
)
,
1
1
clearly lies at a distance of
5
4
vertically above the line
=
y

1
4
.
The difference in the
x
coordinates of the two points
,
0
1
4
and (
)
,
1
1
is 1 unit while the difference
between the
y
coordinates is
3
4
.
The distance between the two points can then be calculated using Pythagoras' theorem and is
=
+
1
2
3
4
2
+
1
9
16
=
25
16
=
5
4
.
•
Now consider the graph of the equation
=
y
x
3
.
If
=
(
)
f
x
x
3
, then
=
(
)
f

x

(
)
f
x
for all real numbers
x
, that is, changing the sign of an input number
x
for the function f just changes the
sign of
(
)
f
x
. For example,
=
f
5
4
125
64
=
1.953125
and
=
f

5
4

125
64
=

1.953125.
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 Spring '08
 Uri
 Real Numbers, Web page, Conic section, 45 degrees, Peter Stone

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