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3-1 Symmetry & Coordinate GraphsObjective: 1. To determine symmetry of a graph using algebraic tests.2. To determine if a function is even or odd.
Point Symmetry
Rotating a figure that has point symmetry 180o should look the same.
Point Symmetry2 points are symmetric with
respect to (wrt) a 3rd point “M” if & only if M is the midpoint of the 2 points.
M
Point Symmetry
The graph of a relation S is symmetric WRT the origin if and only if (a,b) S implies that (-a, -b) S.
Point Symmetry
Steps for determining point symmetry about the origin:1. Find f(-x) and –f(x)2. If f(-x) =-f(x) then the graph
has point symmetry.
Point Symmetry Example: Determine whether the graph
is symmetric WRT the origin. . 2)( xxf
Determine whether each graph is symmetric WRT the origin
6)( xxf
xxxg 53)( 3
a)
b)
Line Symmetry Two points P and P’ are symmetric wrt a
line if and only if that line is the perpendicular bisector of PP’. A point P is symmetric to itself wrt a line if and only if P is on the line.
Four cases we will look at: wrt x-axis wrt y-axis wrt y=x Wrt y=-x
Line Symmetry wrt x-axis If (a, -b) S and (a, b) S Substituting (a, b) and (a, -b) into the
equation produces the same result.
Line Symmetry wrt x-axis Example: x = y2
(4,2), (4,-2)are both members of this set
Line Symmetry- wrt y-axis If (-a, b) S and (a, b) S Substituting (-a, b) and (a, b) into the
equation produces the same result.
Line Symmetry- wrt y-axis Example: y= x2-2
(-2, 2), (2, 2)are both members of this set
Line Symmetry- wrt y=x If (b,a) S and (a, b) S Substituting (b,a) and (a, b) into the
equation produces the same result.
Line Symmetry- wrt y=x Example: xy= 12
(2,6), (6,2)are both members of this set
Line Symmetry- wrt y=-x If (-b,-a) S and (a, b) S Substituting (-b,-a) and (a, b) into the
equation produces the same result.
Line Symmetry- wrt y=-x Example: xy= -12
(2,-6), (6,-2)are both members of this set
Practice Determine whether the graph of x2 + y
= 3 is symmetric with respect to the x-axis, the y-axis the line y = x, the line y = -x.
Practice: Can the graph of an equation be
symmetric to more than one of these lines at a time? If yes, give an example, if no, give an explanation.
Practice Determine whether the graph of |y|=|x|
+1 is symmetric with respect to the x-axis, the y-axis, both or neither. Use the information about the equation’s symmetry to graph the relation.
Even and Odd FunctionsEven functions have
symmetry wrt the y-axisOdd functions have symmetry
wrt the origin
Practice: Look at the previous examples from today and
determine which are even or odd.
2)( xxf 6)( xxf
xxxg 53)( 3 2yx
2- xy 212xy
12xy
even
even
odd
neither
even
odd
odd
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