9.5 HONORS Determine Odd and Even Functions Graphically and Algebraically

Transcription

1 9.5 HONORS Determine Odd and Even Functions Graphically and Algebraically Use this blank page to compile the most important things you want to remember for cycle 9.5: 181

2 Even and Odd Functions Even Functions: A function is even if f f for all in the domain of the Geometrically, the graph of an even function is symmetric with respect to the y-ais. That means that the graph of the function remains unchanged after reflection about the y-ais. 2 h is an eample of an even You can see in the graph at the right that it is symmetric about the y-ais. 2 h 2 2 h Note: Linear functions are even only if they are horizontal lines. Eponential functions are never even. Eamples: Notice that f is symmetric about the y-ais but g is not. f 4 g 4 f 4 f 4 g 4 1 g

3 Odd Functions: A function is odd if f f for all in the domain of the Geometrically, the graph of an odd function has rotational symmetry with respect to the origin. That means that the graph of the function will remain unchanged after a rotation of 180 about the origin. h is an eample of an odd Notice that if you rotate the graph 180 around the origin, it will match up with itself. h h h Note: Linear functions are odd only if they pass through the origin. Eponential functions are never odd therefore, they are neither odd nor even. Eamples: Notice that if you rotate the graph of f() 180 about the origin, it will match up again. The graph of g() is not symmetric with respect to the origin, even though it does pass through it. f 4 1 g f 4 f 4 f 4 4 g 1 g g

4 9.5a (build-refine) Even and Odd Functions Part I: Determine odd or even functions using symmetry in graphs From a graph, a function is EVEN if: it is symmetric about the y-ais. For eample, the function graphed at the right is even because 2,4 is a point on the graph and 2,4 is also a point on the graph. Notice that -2 is the opposite of 2, but both inputs give the same output. So f f, i.e., opposite inputs generate the same output. From a graph, a function is ODD if: it is symmetric about the origin. For eample, the function graphed at the right is odd because 4,4 is a point on the graph and 4, 4 is also a point on the graph. Notice that the input -4 is the opposite of 4, and gives the opposite output from 4. So f f, i.e., opposite inputs generate the outputs that are opposites of each other. For each of the functions graphed below: a) determine if they have even, odd, or no symmetry b) if they are linear, eponential, or neither

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6 Part II: Determine Odd or Even Functions Algebraically In part I, we saw that a. if a function is even, then the inputs and should produce the same outputs or f f. b. if a function is odd, then the inputs and should produce opposite outputs or f f. To test for even or odd functions, substitute for. Eample 1: f is an odd function because f f Test for even functions. Test for odd functions f f f simplify substitute for f so this function is not an even f f f simplify substitute for f so this function is an odd Eample 2: g 2 is neither even nor odd because f f and f f Test for even functions Test for odd functions. 2 2 g g substitute for 2 g 2 so this function is not an even 2 2 g g substitute for 2 g 2 so this function is not an odd Part III: Practice determining odd and even functions Test the following functions algebraically. Determine whether each function is even, odd, or neither then check by graphing the functions on a graphing calculator. 1. f ( ) f ( ) f ( ) f ( ) f ( ) f ( ) f ( ) f( ) f ( ) f ( ) f ( ) 2. f ( ) 5 186