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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