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?

1 Answer
Jan 3, 2017

Answer:

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]}