# 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?

The graph is symmetrical, with respect to the axes. There is no symmetry, with respect to the bisectors

graph{x^2/8-y^2.4-1=0 [-10, 10, -5, 5]}

The equation is

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.

Referred to the new X and Y axes, the equation f(x, y) = 0

becomes

symmetry about the ( same ) origin.

There is no symmetry about the new axes.

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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.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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