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Symmetry Section 3.1

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Symmetry

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Two Types of Symmetry:

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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’?

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

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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, )

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Line symmetry 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 ’ This graph is symmetric with respect to the y-axis or x = 0 What does the this graph look like: y = x 2 By the way, what is the domain and range of this graph? D: (-∞, ∞) R: (0, ∞)

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

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

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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

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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

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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.

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Challenge: Is it even, odd, or neither:

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Even, Odd or Neither? Ex. 1 GraphicallyAlgebraically

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Even, Odd or Neither? Ex. 2 GraphicallyAlgebraically

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Even, Odd or Neither? GraphicallyAlgebraically Ex. 3

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Even, Odd or Neither? GraphicallyAlgebraically Ex. 4

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Even, Odd or Neither?

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

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

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Even, Odd or Neither ?

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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

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(1, 1) exists Does (1, -1) exist?NONot the x-axis Does (-1, 1) exist?YESSymmetric to y-axis

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