Proof about symmetry (simple)

hms.tech

1. The problem statement, all variables and given/known data

Show that the curve is symmetrical about the x axis (without drawing the graph)
eq of the curve is : x^2 + 4y^2 = 1

also show that the curve is symmetric about the y axis
2. Relevant equations


3. The attempt at a solution

To prove that the curve was symmetric abou the x axis, i made x the subject of the equation of the curve:

x=(1-4y^2)^0.5 (can be positive or negative)

Then i used simple intuition:
let a particular value of y be "k" and the corresponding value of x be "c".
by simple calculation, we can conclude that for y=-k , x will still be equal to "c"

Can anyone guide me if this proof is enough or it lacks something, for the latter case, please provide an alternative but suitable proof.

SammyS

Generally, simply use the original equation. If you replace y with -y, and the resulting equation is equivalent to the original equation, then the graph is symmetric w.r.t. the x-axis.

For symmetry w,r,t, the y-axis, replace x with -x, in the original, and check to see that the result is equivalent to the original.

Mentallic

It is enough proof to show that f(x)=f(-x) for symmetry about the y-axis, and f(y)=f(-y) for symmetry about the x-axis. Can you see why?
Basically, this just means you just need to show that the function doesn't change when you swap x for -x and y for -y.

edit: If I bothered to refresh the page to see if a reply was already made, we wouldn't be here right now...

hms.tech

alright, from your replies i think the method i used is correct.

So thnx guys !
cheers

HallsofIvy

Yes, correct, but do you understand that everyone was telling you that you don't have to solve for one variable?