(1) Circles, and
(2) Symmetry
Transform graphs, and observe Symmetry
Identify this graph
What is its equation?
Identify this graph
What is its equation?
Part (2) : Symmetry
 First, define a function of two variables called
F1, as follows:
F[x_, y_] = x y + x2 – y3 - 2
Now the equation “x y + x2 - y3 == 2” can be
written
F[x, y] == 0
 Plot the graph of the equation, using
ContourPlot, as you did with circles. Use
the window [-5, 5] x [-5, 5]
Plotting a typical equation
Reflecting in the X-axis
 To do this, we replace y in the original
equation F[x, y] == 0, with –y. In other
words, we plot F[x, -y] == 0. Keep the
same window for both x and y.
4
4
2
2
0
0
2
2
4
The original graph
4
2
0
2
4
The transformed graph
4
4
2
0
2
4
Reflection in the Y - Axis
4
4
2
2
0
0
2
2
4
4
The original graph
4
2
0
2
4
The transformed graph
4
2
0
2
4
Reflection in the Origin
4
4
2
2
0
0
2
2
4
4
The original graph
4
2
0
2
4
The transformed graph
4
2
0
2
4
Just in case you don’t believe it, here
are both graphs on the same axes:
And now for something completely different:
Reflection on the diagonal x = y
4
2
0
2
4
The original graph
4
2
0
2
4
Verifying Symmetry
 The curve we have been using was
selected because of its lack of symmetry.
 If a curve has symmetry corresponding to
one of these transformations, then the
transformation will leave it unchanged.
 In the following graphs and equations,
take a good look at the graph, and then
make one transformation that leaves it
looking the same as before. Choose your
transformation carefully.
+
+2 +2 −1=0
 Examine the graph,
and decide which
symmetries it
possesses, if any.
 This one has only
one.
+
2.
−
−
=
 Define the function F2 as follows:
, =
+ −2 −2 .
4
2
0
2
4
4
2
0
2
4
Indentify the symmetries of
this graph
Indentify the symmetries of
this graph

This is a member of the family of Elliptic Curves, which were
recently (1997) used to prove Fermat’s Last Theorem.