line symmetry help its greatly appreciated thanks for all the help?

Question

using algebraic tests to check for symmetry with respect to x and y axis and the origin

x-y^2=0
y= x^4-x^2+33
y= 1/x^2+1
xy=4

Answer 1

A curve is symmetrical around the x axis if replacing y by -y gives you the same equation; around the y axis if replacing x by -x gives the same equation; and about the origin if replacing both x and y by -x and -y simultaneously gives the same equation.

x - y^2 = 0:
x-axis: replace y by -y: x - (-y)^2 = 0 <=> x - y^2 = 0, so it is symmetrical about the x-axis.
y-axis: replace x by -x: -x - y^2 = 0, this is not the same, so not symmetrical about the y-axis.
From this we can also conclude that it is not symmetrical about the origin.

(Note that if a curve is symmetrical about both the x and y axes, it is also symmetrical about the origin; if it is symmetrical about one axis but not the other, it is not symmetrical about the origin; and if it is symmetrical about neither axis, it may or may not be symmetrical about the origin.)

You should be able to prove the following results:

y = x^4 - x^2 + 33: symmetrical about the y-axis but not the x-axis or the origin.

y = 1/x^2 + 1: same

xy = 4: Not symmetrical about the x or y axes but is symmetrical about the origin.

Answer 2

If it's symmetrical about the x-axis, then x(-y) = x(y).
Similarly, for y-axis symmetry, y(-x) = y(x).

1)
x - y² = 0
x(y) = y²
x(-y) = (-y)² = y² = x(y) ──► symmetrical about x-axis
y² = x
y(x) = ±√(x)
y(-x) = ±√(-x) ≠ y(x) ──► not symmetrical about y-axis

2)
y = x^4 - x² + 33
I don't know how to solve this for x, so I don't know how to find if it's symmetrical about the x-axis.
y(x) = x^4 - x² + 33
y(-x) = (-x)^4 - (-x)² + 33 = x^4 - x² + 33 = y(x)
──► Symmetrical about y-axis

3)
y = 1/x² + 1
1/x² = y - 1
x² = 1 / (y - 1)
x(y) = ±√(1 / (y-1))
x(-y) = ±√(1 / (-y-1)) ≠ x(y) ──► not symmetrical about x-axis
y(x) = 1/x² + 1
y(-x) = 1/(-x)² + 1 = 1/x² + 1 = y(x)
──► symmetrical about y-axis

4)
xy = 4
x(y) = 4 / y
x(-y) = 4 / -y ≠ x(y) ──► not symmetrical about x-axis
Identically, this is not symmetrical about the y-axis