Graphing Polynomial Functions By the end of today’s class, I will be able to: þ understand the significance of the zeros þ graph from factored form þ graph.

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Graphing Polynomial Functions By the end of today’s class, I will be able to: þ understand the significance of the zeros þ graph from factored form þ graph from “factorable” standard form

What do the characteristics indicate? The degree of the polynomial and Recall: The degree of the polynomial and and the sign of the leading coefficient indicate the end-behaviour of the function Consider: y = 2x 3 + 4x x - 40 Since Degree = 3 is odd Therefore the graph has opposite ends Since Leading Coefficient = 2 is positive Therefore the graph generally rises right E.B.(End Behaviour): As x  -oo, y  -oo As x  +oo, y  +oo Y-Intercept = -40

Possible Cubic Graphs # of Turning Points  # of Zeros  Zero Turning Points 2 Turning Points 1 zero 2 zeros 3 zeros Y-Intercept = - 40

What do the zeros indicate? y = 2x 3 + 4x x - 40 may be written as y = 2( x + 5 )( x + 1 )( x – 4 ) Since it is a cubic polynomial function with 3 zeros Therefore the graph must have 2 turning points! x-ints = - 5, -1, 4 y-int = - 40

What does the order of the roots indicate? Consider y = ( x – 5 ) 2 “Bounces” A “Double” Root means that the graph “Bounces” at that x-intercept! =============================== Consider y = ( x – 5 ) 3 “Goes Through” A “Triple” Root means that the graph “Goes Through” that x-intercept!

Now try these… #1] y = ( x – 2 ) 3 ( x + 5 ) #2] y = - 2 / 3 ( x – 3 ) 2 ( x + 1 ) #3] y = 3 ( 2x + 1 ) 2 ( 1 – x ) ( 3x - 5 ) #4] y = - x 5 - 6x 4 - 9x 3 #5] y = 3x 3 + 9x 2 - 3x - 9 READ: p.139(right column) & p & “In Summary” on p.145 DO:p.146 #1,2b,6acef,10b,12b