NOTES Section 2.3 Day 1 Polynomial Functions and Their Graphs Polynomial Functions –graphs are continuous without breaks, gaps, sharp corners, or overlap Smooth and Continuous… Polynomial Not continuous; has a gap… NOT Polynomial Continuous but has sharp corners… NOT Polynomial Zeros and Multiplicity: Let f be a polynomial function, and let a be a real number… If 0 ) ( = a f , then 1. a is a solution of the polynomial equation 0 ) ( = x f 2. a is a root or zero of the function f 3. ) 0 , ( a is an x-intercept of the graph of f 4. ) ( a x - is a factor of the polynomial If ) ( a x - is a factor that occurs k times, then a is called a zero with multiplicity k. If k is odd, then the graph crosses the x- axis at a. If k is even, then the graph touches the x- axis at a. Example 1: Check out 3 4 ) 2 ( ) 3 )( 1 ( + - - x x x End behavior & Leading Coefficient Test: The end behavior of the graph depends on the sign of the leading coefficient and the overall degree of the polynomial. Odd Degree Even Degree Sign of Leading Coefficient Positive Negative Positive Negative End Behavior Example 2: Describe (or sketch) right and left end behavior of the graphs of each of the following equations. A. 6 2 4 3 1 ) ( x x x f - - = B. x x x f 9 ) ( 3 - = Intercepts: The graph of a polynomial function of degree n has one y-intercept and at most n x-intercepts. Turning Points/Local Extrema/peaks & valleys: A polynomial function of degree n has at most n – 1 turning points (one less than its degree). o f t t ye xd 6 8 a Ewe o 4 I a 3 factors X I 11 3 X 2 Zeros y x2 V 2 y x A 3 off 9 3 J f y X quad V cubic µ quartic W N