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Polynomial Functions 2.1 (M3) Make sure you have book and working calculator EVERY day!!!
EXAMPLE 1 Identify polynomial functions 4 Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient. a. h (x) = x 4 – x 2 + 3 1 a.Yes its a Polynomial. It is in standard form. Degree 4 – Quartic Trinomial Its leading coefficient is 1. b. Yes its a Polynomial. Standard form is Degree 2 – Quadratic Trinomial Leading Coefficient is
EXAMPLE 1 Identify polynomial functions Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient. c. f (x) = 5x 2 + 3x –1 – x c. Not a polynomial function d. k (x) = x + 2 x – 0.6x 5 d. Not a polynomial function e. f (x) = 13 – 2x e. polynomial function; f (x) = –2x + 13; degree 1 linear binomial, leading coefficient: –2
Graph Trend based on Degree Even degree - end behavior going the same direction Odd degree – end behavior (tails) going in opposite directions Leading Coefficient
Symmetry: Even/Odd/Neither First look at degree Even if it is symmetric respect to y-axis –When you substitute -1 in for x, all signs stay the SAME Odd if it is symmetric with respect to the origin –When you substitute -1 in for x, all of the signs CHANGE Neither if it is NOT symmetric around the y-axis or origin
Tell whether it is even/odd/neither 1.f(x)= x 2 + 2 2.f(x)= x 2 - 4x 3.f(x)= x 3 4.f(x)= x 3 + x 5.f(x)= x 3 + 5x +1
Additional Vocabulary to Review End Behavior: Left side x –, f(x) ____ Right side x +, f(x) ____
Additional Vocabulary to Review Domain: set of all possible x values Range: set of all possible y values Symmetry: even (across y), odd (around origin), or neither Interval of increase (where graph goes up to the right) Interval of decrease (where the graph goes down to the right) End Behavior: Left side x –, f(x) ____ Right side x +, f(x) ____
Polynomial Functions and Their Graphs There are several different elements to examine on the graphs of polynomial functions: Local minima and maxima:
On the graph above:A local maximum: f(x) = A local minimum: f(x) = Give the Local Maxima and Minima Must use y to describe High and Low
Finding a local max and/or local min is EASY with the calculator! Graph each of the following and find all local maxima or minima: Now describe their end behavior.
Describe the Interval of Increasing and Decreasing Increasing when ___________ Decreasing when _____________ Increasing when ___________ Must use x to describe Left to Right x y (Left to Right) The graph is:
Assignment Page 69 #1-3 Classify #8-10 End Behavior #11-13 Symmetry (Even/Odd/Neither) #14-16 Max/Min, Domain/Range, Intervals of Increasing and Decreasing