# Thread: Determining Lines of Symmetry from Scratch?

1. ## Determining Lines of Symmetry from Scratch?

Hi all ---

When I tried the following question in the black oval --- the answer is in red --- I was completely stuck on the logic/thought process/intuition in the first part. I'm fine with the algebra --- it's NOT what's troubling me for the past hours.

I know how to check for symmetry about the origin and y-axis. A function, f(x), is symmetric about the origin if $f(-x) = -f(x)$ and about y-axis if $f(-x) = f(x)$.

But here, how on earth would you even know to check for symmetry about $y = \pm x$? How would you even suspect or be aware of this in the first place?

My work: For $xy = v \Longleftrightarrow y = \frac{v}{x}$, I sort of see this because this is just the reciprocal function with the constant $v$.
So here, I'd expect symmetry about $y = \pm x$ because I've seen the graph of the reciprocal function before. But I'm completely lost for $x^4 + y^4 = u$?

Also, what would lead you to check for symmetry about x- and y-axis here? Question doesn't give any information that would lead you to do this?

In general, if I see this kind of question that only asks me to state all lines of symmetry (WITHOUT telling me what the lines are), how would I know
--- which lines I should check for symmetry?
--- how many lines there are?

Maybe because I don't understand it, but the answer looks like a bit of guesswork to me...

Thank you ---

2. ## Re: Determining Lines of Symmetry from Scratch?

The fact that the exponents on x and y are even so that $(-x)^4= x^4$ and $(-y)^4= y^4$ would lead me to say that the graph is symmetric about the axes: the point "symmetric" to (x, y) about the y-axis is (-x, y) and about the x-axis is (x, -y). The fact that the formula is "symmetric" with respect to x and y: replacing x with y and y with x changes $x^4+ y^4= u$ to $(y)^4+ (x^4)= u$ which is the same equation, tells me that it is symmetric about the lines y= x and y= -x.

3. ## Re: Determining Lines of Symmetry from Scratch?

Hi HallsofIvy ---

Thanks a lot for your answer. It really helped with why one'd think $y = \pm x$ and $x, y$ axes give symmetry for the functions.

Just one last question that was maybe unclear in my post --- How would you know/infer that there are no other lines of symmetry to check or test for either curve?

4. ## Re: Determining Lines of Symmetry from Scratch?

After thinking about this for the past week, I still do not see how and why there are no other lines of symmetry? Has someone gotten farther than me?

5. ## Re: Determining Lines of Symmetry from Scratch?

Hi all again --- I should point out that the red box is the solution, so we are NOT told or hinted that both graphs are symmetric about $y = \pm x$ and $x = 0, y = 0$.

After two more weeks of banging my head on this problem, I still do not see how you can tell or guess that there are no other lines of symmetry? Does anyone have more insight?

Thank you ---