Reflections of graphs in the x and y axes
Introductory example
.
Suppose that we have already constructed a table of values for the graph of the equation
=
y
+
x
4 and that we wish to draw the graph of
=
y

+
x
4.
For each point (
)
,
x
1
y
1
on the graph of
=
y
+
x
4
we can obtain a
corresponding point (
)
,
x
1

y
1
on the graph of
=
y

+
x
4 by just
changing the sign of the
y
coordinate of the first point.
x


4

15
4

3

7
4
0
9
4
5
33
4
12
=
y
+
x
4

0
1
2
1
3
2
2
5
2
3
7
2
4
=
y

+
x
4

0

1
2

1

3
2
2

5
2

3

7
2

4
The vertical line joining two points (
)
,
x
1
y
1
and
(
)
,
x
1

y
1
is
bisected
by the
x
axis.
We may say that the second point
(
)
,
x
1

y
1
is the
reflection
of the first point (
)
,
x
1
y
1
in the
x
axis as a "
mirror line
". It should be thought of as a two sided mirror so that also the first point
(
)
,
x
1
y
1
is the reflection of the second point (
)
,
x
1

y
1
.
In the following picture the graph of
=
y

+
x
4
(
blue
graph)
is the reflection in the
x
axis of the graph of
=
y
+
x
4
(
red
graph).
Suppose that we have already constructed a table of values for the graph of the equation
=
y
+
x
4 and that we wish to draw the graph of
=
y
 +
x
4.
The second equation is obtained from the first by replacing
x
by its additive inverse

x
.
For each point (
)
,
x
1
y
1
on the graph of
=
y
+
x
4
we can obtain a
corresponding point (
)
,

x
1
y
1
on the graph of
=
y
 +
x
4
by just
changing the sign of the
x
coordinate of the first point. Thus we can obtain a table of values for plotting
=
y
 +
x
4
by changing the sign
of the
x
coordinates in the table used for the first graph.
The horizontal line joining two points (
)
,
x
1
y
1
and
(
)
,
x
1

y
1
is
bisected
by the
y
axis.
The second point
(
)
,

x
1
y
1
is the
reflection
of
the first point (
)
,
x
1
y
1
in the
y
axis as a "
mirror line
".
Thus the table . . .
x


4

15
4

3

7
4
0
9
4
5
33
4
12
=
y
+
x
4

0
1
2
1
3
2
2
5
2
3
7
2
4
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Documentgives rise to the table . . .
x

4
15
4
3
7
4
0

9
4

5

33
4

12
=
y
 +
x
4

0
1
2
1
3
2
2
5
2
3
7
2
4
Rearranging the values in the new table in reverse order so that the
x
coordinates are increasing from left to right gives the following table.
x
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 Spring '08
 Uri
 Euclidean geometry, Peter Stone

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