Question

How do you determine whether the graph of `x^3+y^3=4` is symmetric with respect to the x, y axis, the line y=x, the line y=-x, or none of these?

Answer 1

See explanation and the Socratic graphs. The graph of `x^3+y^3=4` is symmetrical with respect to `x-y=0`.
graph{x^3+y^3-4=02 [-10, 10, -5, 5]}

Explanation:

If (x, y) is is on the graph and so is `(x, -y)`, then the graph is

symmetric, with respect to the x-axis.

Example: Vertical cosine wave `x = cos (y) = cos (-y)`

graph{x-cos y =0 [-10, 10, -5, 5]}

If (x, y) is is on the graph and so is `(-x, y)`, then the graph is

symmetric, with respect to the y-axis.

Example : Parabola @`x^2=(-x)^2=4y`

graph{x^2-4y=0 [-10, 10, -5, 5]}

If (x, y) is is on the graph and so are (x, -y) and (-x, y), then the graph

symmetric, with respect to both the axes.

Example: The circle `x^2+y^2=(-x)^2+y^2=x^2+(-y)^2=1`

graph{x^2+y^2=1 [-10, 10, -5, 5]}

If (x, y) is is on the graph and so is ((-x,- y), then the graph is

symmetric, with respect to the origin.

Example The cubic graph `f(x, y) =y- x^3=0` and so is #f(-x,-y)=-y-

x^3=-f(x, y) =0#

graph{x^3 [-10, 10, -5, 5]}

If the equation is of the form

`f(x, y) = g(x+y, x-y) = g(X, Y)`, where #X = x+y and Y =x-y, all above

apply with respect to new X=axis and Y-axis.

Example for symmetry with respect to `x-y =X =0`

|x+y)|=1

graph{(|x+y|-1)(x-y)=0 [-10, 10, -5, 5]}

The given graph is yet another.

Upon the transformation `x= X-Y and y = X +Y`, the equation

becomes g(X, Y) = 2X*3+6XY^2-4=g(X, -Y).All-exclusive example :

Look for symmetry about x-y =Y =0. The graph appears, in the answer

space.

Exponential growth curve `y=e^x`

graph{e^x [-10, 10, -5, 5]}