Reflection

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:

• 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 = x² and y = (-x)² = x²   Therefore y = x² 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 = y² and x = (-y)² = y²   Therefore x = y² 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 = x³ and -y = (-x)³ → -y = -x³  If you multiply both sides by -1 you have y = x³
  Therefore y = x³ is symmetrical to the origin.

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