How do you determine whether the graph of #y^2=(4x^2)/9-4# is symmetric with respect to the x axis, y axis, the line y=x or y=-x, or none of these?

Question

Cora Alexander · Accepted Answer

The graph is symmetrical, with respect to the axes. There is no symmetry, with respect to the bisectors #y=+-x#. See the illustrative Socratic graph of this hyperbola.

graph{x^2/8-y^2.4-1=0 [-10, 10, -5, 5]} The equation is #f(x, y)=(4x^2)/9-y^2-4=0#. Here, #f(+-x, +-y)=f(x, y)#. So, if (x, y) is a point on the graph, then (x, -y), (-x, y) and (-x, -y) lie on the graph. And so, the graph is symmetrical about both the axes. #y=+-x# become the new axes upon rotation of the axes about the origin, through #45^o#. The ad hoc transformations are #x =(X-Y)/sqrt2 and y=(X+Y)/sqrt2#. Referred to the new X and Y axes, the equation f(x, y) = 0 becomes #g(X, Y)=4/9(X-Y)^2/2-(X+Y)^2/2-4=0#. Now, only #g(-X, -Y)=g(X, Y)#, revealing, as expected, polar symmetry about the ( same ) origin. There is no symmetry about the new axes. So, there is no symmetry about #y = +-x#.

Liam Brennan · Answer

undefined To determine the symmetry of the graph of $ y^2 = \frac{4x^2}{9} - 4 $ with respect to the x-axis, y-axis, y = x, or y = -x, you can analyze the equation's properties.

For symmetry with respect to the x-axis, replace y with -y and check if the equation remains unchanged.

For symmetry with respect to the y-axis, replace x with -x and check if the equation remains unchanged.

For symmetry with respect to the line y = x, interchange x and y and check if the equation remains unchanged.

For symmetry with respect to the line y = -x, interchange x with -y and y with -x and check if the equation remains unchanged.

Once you've performed these substitutions, if the resulting equation remains unchanged, the graph is symmetric with respect to the corresponding axis or line. If it changes, then it is not symmetric with respect to that axis or line.