Symmetry Section 2.7 Notes (part 2)

Symmetry

Two Types of Symmetry:

Point Symmetry Two distinct points P and P’ are symmetric with respect to point M, if and only if M is the midpoint of P and P’ Example: P(3, -4) and P’ are symmetric with respect to M(2, 2). What is P’?

A good example of a graph being symmetric to a point is: This graph is symmetric to the ORIGIN M By the way, what is the domain and range of this graph? P’ D: (-∞, ∞) R: (-∞, ∞)

Interval Notation A parenthesis ( ) shows an open (not included) endpoint A bracket [ ] shows a closed [included] endpoint Examples: Set A with endpoints 1 and 3, neither endpoint included (1,3) Set B with endpoints 6 and 10, not including 10 [6,10) Set C with endpoints 20 and 25, including both endpoints [20,25] Set D with endpoints 28 and infinity, not including 28 (28, )

Line symmetry What does the this graph look like: y = x2 Two distinct points P and P’ are symmetric with respect with respect to line “l” if “l” is the perpendicular bisector of the line PP’ Line symmetry What does the this graph look like: y = x2 By the way, what is the domain and range of this graph? This graph is symmetric with respect to the y-axis or x = 0 D: (-∞, ∞) R: (0, ∞)

Infinite amount of lines x-axis y-axis origin What is each graph symmetric with respect to? And tell me the domain and range. x = 2 x-axis Finish putting in domain and ranges in this PPT x = 2 y = -5 P(2, -5) Infinite amount of lines x-axis y-axis origin

What is each graph symmetric with respect to? x = -1 y = 2 P(-1, 2) x = 1 y = 2 origin

Real quick… Circle equations look like: Ellipse equations look like: Same coefficients, both x and y squared Different POSITIVE coefficients, both x and y squared

A function is odd if f( -x) = - f(x) for every number x in the domain. A function is even if f( -x) = f(x) for every number x in the domain. So if you plug a –x into the function and you get the negative of the function back again (all terms change signs) it is odd. EVEN ODD

If a function is not even or odd we just say neither (meaning neither even nor odd) Determine if the following functions are even, odd or neither. Not the original and all terms didn’t change signs, so NEITHER. Got f(x) back so EVEN.

Challenge: Is it even, odd, or neither:

Ex. 1 Even, Odd or Neither? Graphically Algebraically EVEN

Ex. 2 Even, Odd or Neither? Graphically Algebraically ODD

Ex. 3 Even, Odd or Neither? Graphically Algebraically EVEN

Ex. 4 Even, Odd or Neither? Graphically Algebraically Neither

Even, Odd or Neither? EVEN ODD

Even functions are symmetric about the y-axis What do you notice about the graphs of even functions? Even functions are symmetric about the y-axis

Odd functions are symmetric about the origin What do you notice about the graphs of odd functions? Odd functions are symmetric about the origin

Even, Odd or Neither ?

The graph below is a portion of a complete graph The graph below is a portion of a complete graph. Sketch a complete graph for each of the following symmetries. With respect to:   The x-axis The y-axis The line y = x The line y = -x

Does (1, -1) exist? NO Not the x-axis (1, 1) exists Does (-1, 1) exist? YES Symmetric to y-axis