Symmetry is more of a geometrical
than an algebraic concept, but the subject of symmetry does come up in
a couple of algebraic contexts. For instance, when you're graphing
quadratics, you may
be asked for the parabola's axis of symmetry. This is usually just the
vertical line x
= h, where
"h"
is the x-coordinate
of the vertex, (h,
k). That is, a
parabola's axis of symmetry is usually just the vertical line through
its vertex.

The other customary context
for symmetry is judging from a graph whether a function is even
or odd. Warning:
By definition, no function can be symmetric about the x-axis
(or any other horizontal line), since anything that is mirrored around
an horizontal line will violate the Vertical
Line Test. On the
other hand, a function can be symmetric about a vertical line or
about a point. In particular, a function that is symmetric about the y-axis
is also an "even" function, and a function that is symmetric
about the origin is also an "odd" function. Because of this
correspondence between the symmetry of the graph and the evenness or oddness
of the function, "symmetry" in algebra is usually going to apply
to the y-axis
and the origin.

For each of the following
graphs, list any symmetries, and state whether the graph shows a function.

Graph B:
This graph is symmetric about the axes x
= 0 (the y-axis)
and y
= 0 (the x-axis),
and also about the origin. Since there exist vertical lines (such as
the line x
= 2) which will cross
this graph twice, it does not show a function.

Graph C:
This graph is symmetric about the axes x
= 1 and y
= –2, and symmetric
about the point (1,
–2). Since a vertical
line can be drawn to cross the ellipse twice, this is not a function.

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Graph D:
This graph is symmetric about slanty lines: y
= x and y
= –x. It is
also symmetric about the origin. Because this hyperbola is angled correctly
(so that no vertical line can cross the graph more than once), the graph
shows a function.

Graph E:
This graph (of a square-root function) shows no symmetry whatsoever,
but it is a function.

Graph F:
This graph (of a cubic function) is symmetric about the point (–4,
–1), but not around
any lines. This graph does show a function.

Graph G:
This parabola is lying on its side. It is symmetric about the line y
= 2. It is not a function.

Graph H:
This parabola is vertical, and is symmetric about the y-axis.
It is a function; in fact, it is an even function.

As mentioned earlier, you'll
mostly only be looking for symmetry about the y-axis
(that is, for whether the function is even) or about the origin (that
is, for whether the function is odd).

Determine from the
graphs whether the displayed functions are even, odd, or neither.

Graph A:
This linear graph goes through the origin. If I rotate the graph 180°
around the origin, I'll get the same picture. So this graph is odd.
(The function would not be odd if this line didn't go through
the origin.)

Graph B:
This parabola's vertex is on the y-axis,
so the axis of symmetry is the y-axis.
That means that the function is even.

Graph C: This
cubic is centered on the origin. If I rotate the graph 180° around the
origin, I'll get the same picture. So this graph is odd.

Graph D:
This cubic is centered at the point (0,
–3). This graph is
symmetric, but not about the origin or the y-axis.
So this function is neither
even nor odd.

Graph E:
This cube root is centered on the origin, so this function is odd.

Graph F:
This square root has no symmetry. The function is neither
even nor odd.

Graph G:
This graph looks like a bell-shaped curve. Since it is mirrored around
the y-axis,
the function is even.

Graph H:
This hyperbola is symmetric about the lines y
= x and y
= –x, but this
tells me nothing about evenness or oddness. But the graph is also symmetric
about the origin, so this function is odd.

When looking for symmetry,
you don't have to just sit there thinking the thing through. Instead,
take the paper and your pencil, and see if there is a spot where you can
plant the pencil's eraser, and then spin the paper on the table. When
it's spun halfway around, do you get the same picture as you had before?
Then your eraser marks a point of symmetry. Grab a ruler and stand it
on its edge in the middle of the graph. Look down onto the paper, and
eye-ball the two "sides" of the picture. Do the two portions
of the graph, one on either side of the ruler, look like mirror images?
Then the ruler marks a line of symmetry. Don't be shy about putting your
hands into the work; it can really help in getting a "feel"
for symmetry.