Determine if the graph of this equation is symmetric with respect to the x-axis, y-axis, y=x or y=-x. y=1/x^2

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flbyrne | (Level 3) Assistant Educator

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In the equation `y=1/x^2` , because x is squared, there is a pair of x values that have the same corresponding y value.

For x=1 `y=1/(-1)^2=1`

Thus the equation is symmetric with respect to the middle point between -1 and 1, which is 0.

Therefore the graph is symmetric with respect to the y axis.

Because square numbers can only be positive, y can only be positive, thus the graph is not symmetric with respect to the x axis.

Graph:

To determine if it's symmetric with respect to y=x, determine the horizontal and vertical distance from y=x to the curve. Choose a point different than (1,1) so the distance to the curve is different than 0.

Try a point (4,4)

For x=4 `y=1/4^2=1/16=0.0625` and the distance from y=4 to y=0.0625 is 3.9375

For y=4 `x=sqrt(1/y)=sqrt(1/4)=1/2=0.5` and the distance from x=4 to x=0.5 is 3.5

Thus the graph is not symmetric with respect to y=x. Because it is symmetric with respect to the y axis, then it is not symmetric with respect to y=-x either.

 

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