Please help with pre cal?

Question

An equation is being tested for symmetry with respect to the x-axis, the y-axis, and the origin. Explain why, if two of these symmetries are present, the remaining one must
also be present.

Thank you

Answer 1

if something is symmetric it contains a mirror image across the reference point and/or axis. if the eqn was symmetric about the x-axis, it would have similar positive and negative x values and similar y values, for example: (2, 2) and (-2, 2).
the same thing follows for the y-axis. and b/c the origin is the intersection of these two axii, the following statement is true.

Answer 2

All you really have to do is look closely at the definition of symmetry. If a graph is symmetric about something it means that you can fold the graph in some way and the different parts of the graph will overlap. A graph that is symmetric about the y-axis will overlap if you fold the graph along the y-axis. Similarly, a graph that is symmetric about the x-axis will overlap if you fold the graph along the x-axis. Finally, a graph that is symmetric about the origin will overlap if you fold the graph along the x-axis and the y-axis.

So if symmetric about x and y it will also be symmetric with respect to the origin, etc.

Answer 3

You would need to show three "proofs":

1. If symmetric by x-axis and y-axis, then symmetric by origin.

2. If symmetric by origin and y-axis, then symmetric by x-axis.

3. If symmetric by x-axis and origin, then symmetric by y-axis.


1. (proof) If symmetric by y-axis then (x,y) = (-x,y) for all x. If symmetric by x-axis then (x,y) = (x,-y) for all x.
Therefore, (x,-y) = (-x,y) for all x which implies (x,y) = (-x,-y) for all x. Therefore, symmetric with respect to origin.

2. (proof) If symmetric w.r.t origin, (x,y) = (-x,-y) all x. If symmetric w.r.t y-axis, (x,y) = (-x,y) all x. So, (-x,-y) = (-x,y) all x or (x,-y)=(x,y) all x which means symmetric about x-axis.

3. (proof) If symmetric w.r.t origin, (x,y) = (-x,-y) all x. If symmetric w.r.t x-axis, (x,y) = (x,-y) all x. So, (-x,-y) = (x,-y) all x or (x,y)=(-x,y) all x which means symmetric about y-axis.

Therefore, if two of these symmetries are present, the remaining one must also be present.

There you go, a little repetitive but you've shown all three cases!