# Reflection

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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 = x^{2} If you substitute -x in for x you have y = (-x)^{2} which is equal to y = x^{2}

Therefore since y = (-x)^{2} equals y = x^{2} the function is symmetrical to the y axis.

Using numbers if x = 2→ y = x^{2} → y = 2 ^{2} → y = 4; If you substitute x = -2 into the equation y = x^{2}→ 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 = -x^{2} symmetrical to the y axis?
answer

This is the graph of y = x^{3} If x = 2 then y = 2^{3} = 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 = x^{3} 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 = y^{2} For symmetry if the coordinates (9,3) is on the graph then (9,-3) must also be on the graph.

So lets substitute:

x = 3^{2} = 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:

- For y axis symmetry substitute -x for every x in the equation. If the result is an equivalent

equation the equation is symmetrical to the y axis.

example; y = x^{2} and y = (-x)^{2} = x^{2} Therefore y = x^{2} is symmetrical to the y axis.
- For x axis symmetry substitute -y for every y in the equation. If the result is an equivalent
equation the equation is symmetrical to the x axis.

example; x = y^{2} and x = (-y)^{2} = y^{2} Therefore x = y^{2} is symmetrical to the x axis.
- For origin symmetry substitute -y for every y in the equation and -x for every x in the equation.

If the result is an equivalent
equation the equation is symmetrical to the origin.

example; y = x^{3} and -y = (-x)^{3} → -y = -x^{3} If you multiply both sides by -1 you have y = x^{3}

Therefore y = x^{3} is symmetrical to the origin.

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