** Mon Jan 31 (symmetries)**
For the webwork you will need to know the following:
(1) To test for symmetry about the y-axis:
Replace x with (-x).
If you get an equivalent equation
then the graph is symmetric about the y-axis.
(2) To test for symmetry about the x-axis:
Replace y with (-y).
If you get an equivalent equation
then the graph is symmetric about the x-axis.
(3) To test for symmetry about the x-axis:
Replace x with (-x) and replace y with (-y).
If you get an equivalent equation
then the graph is symmetric about the origin.
Important points:
(A) When you substitute, to be safe you must use parentheses.
example: the equation y = x^2 has symmetry about the y-axis
because when you replace x with (-x) you get the same equation.
(B) Symmetry means that you get an equivalent equation,
not that you get the same equation.
example: the equation y = x^3 has symmetry about the origin
because when you replace y with (-y) and x with (-x) you
get the equation -y = -x^3 which is equivalent to
the equation y = x^3.
(C) To prove that a symmetry does NOT exist you only
need to find one point on the graph for which the
reflection is not on the graph. (But to prove that
symmetry does exist it is not enough to consider
individual points.)
example 1: y = x^3 is NOT symmetric about the y-axis
because the point (1,1) is on the graph but
the point (-1,1) is not.
(example 2: The points (1,0) and (-1,0) are both on
the graph of y=x(1-x^2), but the graph is not even.)