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