# Example 2 identify each of the following as a graph

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Example 2 Identify each of the following as a graph of a polynomial (P) or not a polynomial (N). The simplest form of polynomial functions of various degrees are the single-termed polynomials, or monomials, of the form ( ) n f x x . You can think of these as the parent functions for all polynomials of degree n. For these monomials, there is a consistent pattern in the shapes of the graphs. In general, as the degree increases, the graphs become flatter between 1 x µ and steeper for 1 x ! . Why is that?? Inde p N 3X N 434W fat 2 3 3 2 N 7x x2 p p 24W N 3 42 7 43 X X 40 N x 2 4 1 k w x O N P N p Parents ODD ODD ODD even even

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Example 3 Sketch the following transformations of the parent quartic function. Determine the end behavior of each graph, as well as the number of x-intercepts and relative extrema. (a) ³ ´ 4 ( ) 3 2 f x x ± ± (b) 4 ( ) ( 2) 3 h x x ± ² ± (c) 4 2 ( ) 2 8 f x x x ± (d) ³ ´³ ´ 2 2 ( ) 3 1 16 d x x x ± ± ± There is an easy way to determine the end behavior of a polynomial equation in standard form, f(x), simply by looking at the leading term, n n a x . The sign of the leading coefficient will give you the right-end behavior. Do you know why this works? If 0 n a ! , then lim ( ) x f x of f If 0 n a µ , then lim ( ) x f x of ±f Once the right-end behavior is known, knowing whether the degree is even or odd will give you the left-end behavior. If n is even, lim ( ) lim ( ) x x f x f x o±f of If n is odd, lim ( ) lim ( ) x x f x f x o±f of ± Here s a visual standard transformation form fcxt AGCBCx.cm v2 y a.gg snmoex9indttEhaIiTocaimax same end behavior A localmin 2x i Otheydo figure outx.int sgmmetry End Behavior 0 36 11 131 4 Cx 4 Sym x.int lety o x int 1,1 4,1 4 meforall Eren ftx 2GxY 86 7 unfix 0 2 4 271 27 x a x 0,2 2 2 4 8 2 hmfcxt xflxkfc xxsr.TW mEEtipTioityy aEIiesnsymxt'If y n 3localmaxlm.ms 1 local max 26cal mins 3x int pos REBispos neg REBisneg
Example 4 Determine the end behavior of the following polynomials (a) 4 ( ) 2 f x x ± (b) 3 4 ( ) 5 3 8 2 P x x x x ± ± ± (c) 7 ( ) 2 h t t (d) 7 6 3 ( ) 2 4 6 11 8 N t t t t t ² ± ² ± In Example 3, we noticed that different equations of quartic polynomials had different number of relative extrema as well as a different number of x-intercepts. Information about both relative extrema and roots (zeros/x-intercepts) can be obtained from the degree of a polynomial equation. Theorem This theorem gives you upper bounds on both possibilities. Any polynomial can have fewer than the maximum, and we have to figure out what those other possibilities are. Example 5 Draw several possibilities for the shape of a polynomial of odd degree. Use a degree 5 polynomial as one such function. Analyze the possibilities for zeros and relative extrema.

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• Fall '19
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