Note that your expression is only real for
Underneath the radical it's the difference of two squares, which iirc, are always symmetrical in the y axis. Plus your graph is a semi-circle.
No doubt someone can put it more elegantly than I though
i am trying to determine if this equation is symmetrical to the x-axis and/or the y-axis. or both. the equation is...
which i changed to...
the book states that this equation is ONLY symmetrical to the y-axis. i agree it is symmetrical to the y-axis, but it also seems to me to be symmetrical to the x-axis.
Because a square root has both a positive and negative value, doesn't this make sense?
what am i missing/misunderstanding?
p.s. is this algebra or pre-calc? i wasn't entirely sure
ok lets say a=.5
is this totally wrong?
my graphing calculator does show that it is a semi-circle.
edit: ok, so you can never have a negative number as the answer for a square root. so is this just something that one has to remember? i guess i'm struggling with the fact that the method i learned from the book i'm reading explained how to determine symmetry using specific rules.
it states that you can substitute in -y or -b in place of y or b respectively when trying to see if a graph or function displays symmetry with the x-axis. (y=b). If the resulting equation is the same as the original, then it is symmetrical with respect to the x-axis.
so when i plug in -y i get
so is it as simple as saying that ?
this inequality isn't always true though, since y could be zero.
In your case if you wanted to solve for y then you'd have and the negative solution is extraneous.
Your equation will hold for
Using your example of a = 0.5
Say if you were only given the second equation then you'd deduce that - but where has that negative solution come from?
Squaring both sides of an equation in general is a "non-reversible" step. When you "undo" the squaring operation you don't get back what you started with. For example, if , and you square both sides you get . Reversing this gives . An extra solution has been generated that wasn't there before.