When you think of a reflection think of a mirror. The mirror imgae can be reflected over the y or x axis or through the origin.The left image is over the x axis, center thru the origin and the right over the y axis

A related concept of reflection is symmetry. Reflection is thought more often with transformation of a graph, the actual movement of the graph.

Symmetry is a more important concept. Lets look at some examples.

Symmetrical to the y axis:
y = x2 If you substitute -x in for x you have y = (-x)2 which is equal to y = x2
Therefore since y = (-x)2 equals y = x2 the function is symmetrical to the y axis.

Using numbers if x = 2→ y = x2 → y = 2 2 → y = 4; If you substitute x = -2 into the equation y = x2→ y = (-2)2 → y = 4; Notice the y value remains the same even though the x values are the negative of the other.

Therefore for y axis symmetry if the coordinate (x,y) is on the graph then (-x,y) is also on the graph. To use numbers if (3,9) is on the graph (hence a solution to the equation) then (-3,9) must also be on the graph. If it isn't then the equation is not symmetrical to the y axis.

Is y = -x2 symmetrical to the y axis? answer

This is the graph of y = x3   If x = 2  then y = 23 = 8   For the function to be symmetrical to the origin when x = -2   y must = -8.  To express it another way if the coordinate (x,y) is on the graph then (-x, -y) must also be on the graph. They are and hence y = x3 is symmetrical to the origin.

Symmetry with respect to the x axis.

An equation is symmetrical to the x axis if the coordinates (x,y) are on the graph and the coordinates (x,-y) are also on the graph.
For example:   x = y2     For symmetry if the coordinates (9,3) is on the graph then (9,-3) must also be on the graph.
So lets substitute:
        x = 32 = 9   x = (-3)2 = 9
The answers are equal and will be for any combination of (x,y) and (x,-y) and therefore the equation is symmetrical with the x axis.

Testing equations for symmetry:

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