I thought I understood this subject, but apparently not. Here goes.
The first part of the problem says graph the equation and determine visually whether it is symmetric with respect to the x,y and origin axes.
The equation is as follows
x^2+y^2=4. So y=(4-x^2)^(1/2), which visually comes out to be a half circle on the top half of the graph with x>-2 and x<2. Obviously anyone can see that it is symmetric across the Y axis. And I believe, because of the fact that is symmetric across the Y and it touches the X axis on both sides it would be Symmetric across the X also. Continuing on, I guess since the graph would then be a Circle, then it would be symmetric across the origin also. However my confussion comes when proving it.
For symmetry across the Y axis, we are given the equation f(x)=f(-x). Working this out, I can see that the equation is quite obviously symmetric across the Y.
However, for symmetry across the X axis we have the equation f(-x)= -f(x). Now, looking at this equation and the fact that everything lies under a square root sign, in my mind no matter what you do to whats under the square root sign the outside is always going to come out positive. So that means that f(-x) can't be equal to - f(x).
I know that my error has to be something very basic and I dare say stupid, but I can't figure it out. The answers once again have to prove that x^2+y^2=4 is symmetrical across all three axes. (THat is to say in other words that the answer has to show symmetry across X, Y and the origin.)
The first part of the problem says graph the equation and determine visually whether it is symmetric with respect to the x,y and origin axes.
The equation is as follows
x^2+y^2=4. So y=(4-x^2)^(1/2), which visually comes out to be a half circle on the top half of the graph with x>-2 and x<2. Obviously anyone can see that it is symmetric across the Y axis. And I believe, because of the fact that is symmetric across the Y and it touches the X axis on both sides it would be Symmetric across the X also. Continuing on, I guess since the graph would then be a Circle, then it would be symmetric across the origin also. However my confussion comes when proving it.
For symmetry across the Y axis, we are given the equation f(x)=f(-x). Working this out, I can see that the equation is quite obviously symmetric across the Y.
However, for symmetry across the X axis we have the equation f(-x)= -f(x). Now, looking at this equation and the fact that everything lies under a square root sign, in my mind no matter what you do to whats under the square root sign the outside is always going to come out positive. So that means that f(-x) can't be equal to - f(x).
I know that my error has to be something very basic and I dare say stupid, but I can't figure it out. The answers once again have to prove that x^2+y^2=4 is symmetrical across all three axes. (THat is to say in other words that the answer has to show symmetry across X, Y and the origin.)
