# 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?

Jan 3, 2017

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 $\left(x , - y\right)$, then the graph is

symmetric, with respect to the x-axis.

Example: Vertical cosine wave $x = \cos \left(y\right) = \cos \left(- y\right)$

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

If (x, y) is is on the graph and so is $\left(- x , y\right)$, then the graph is

symmetric, with respect to the y-axis.

Example : Parabola @${x}^{2} = {\left(- x\right)}^{2} = 4 y$

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 ${x}^{2} + {y}^{2} = {\left(- x\right)}^{2} + {y}^{2} = {x}^{2} + {\left(- y\right)}^{2} = 1$

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 $f \left(x , y\right) = y - {x}^{3} = 0$ 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

$f \left(x , y\right) = g \left(x + y , x - y\right) = g \left(X , Y\right)$, 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 $x = X - Y \mathmr{and} y = X + Y$, 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]}