Polynomial Functions and Models. Learning Objectives. Polynomials. P (x) = a n x n + a n 1 x n a 1 x + a 0, a n 0

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1 Polyomial Fuctios ad Models 1 Learig Objectives 1. Idetify polyomial fuctios ad their degree 2. Graph polyomial fuctios usig trasformatios 3. Idetify the real zeros of a polyomial fuctio ad their multiplicity 4. Aalyze the graph of a polyomial fuctio 5. Build cubic models from data 2 Polyomials A polyomial is a expressio that is costructed from oe or more variables ad costats, usig oly the operatios of additio, subtractio, multiplicatio, ad costat positive whole umber expoets. P (x) = a x + a 1 x a 1 x + a 0, a 0 Is the stadard form of a polyomial where is a oegative iteger ad a is called the leadig coefficiet. For example, 2 2x 4x 3 is a polyomial x 4 x 3x is ot a polyomial because it ivolves divisio by a variable ad also because it has a expoet that is ot a positive whole umber. 3 1

2 Polyomials P (x) = a x + a 1 x a 1 x + a 0, a 0 The stadard form of a polyomial that is show above is writte i descedig order. This meas that the term that has the highest degree is writte first, the term with the ext highest degree is writte ext, ad so forth A polyomial ca have missig terms Polyomials of degree higher tha 2 are referred to as higher-degree polyomials 4 Degree Polyomials Name of Polyomial Fuctio First Liear f(x) = 2x + 5 Secod Quadratic f(x) = 3x 2 5x + 2 Third Cubic f(x) = x 3 2x 1 Fourth Quartic f(x) = x 4 3x 3 +7x-6 Fifth Quitic f(x) = 2x 5 + 3x 4 x 3 +x 2 Domai: all real umbers. There are o holes or breaks i the graph. 5 Eve If is eve the y = x is eve ad symmetric about the y-axis passes through (0,0), (1,1) ad (-1,1) As x y As x - y As icreases, fuctio flattes for x <1 ad icreases more quickly as x icreases 2

3 f x 3 x 1 x 3 x The degree is 2+1+3=6 The leadig coefficiet is 3>0 The ed behavior is y=3x 6 with shape Odd If is odd the y = x is odd ad Symmetric with the origi passes through (0,0), (1,1) ad (-1,-1) As x y As x - y - As icreases, fuctio flattes for x <1 ad icreases more quickly as x icreases Which is larger x 2 or x 4? It depeds whe x > 1 the x 4 is larger but whe x <1 the x 2 is larger 3

4 If m Multiplicity x r is a factor of f ( x) the r is a root (zero) with a multiplicity of m m eve m odd f x touches x-axis at x r f x crosses x-axis at x r Multiplicity Suppose P( x) is a polyomial with factors x r with r a real umber m If m is odd, the graph crosses the x-axis if m is eve, the graph touches but does ot cross the x-axis s 2 f ( x) x 4 x 1 Zeroes are x = 1 ad x = 4 (multiplicity of 2) 2 f ( x) x 2 x 3 Zeroes are x = - 2 (multiplicity of 2) ad x = 3 4

5 Eve Multiplicity As the multiplicity icreases, we touch more flatly (x-2) 2 (x-2)4 Flatter Odd Multiplicity As the multiplicity icreases, we cross more flatly (x-2) (x-2)3 Flatter Remarks If x = r is a zero of the polyomial with multiplicity m the: If m is odd the the x-itercept correspodig to x = r will cross the x-axis If m is eve the the x-itercept correspodig to x = r will oly touch the x-axis ad ot actually cross it If m > 0 the the graph will flatte out at x = r 5

6 f x x x x At x=1, we touch but do ot cross x-axis At x=2, we cross the x-axis flatly At x=3, we clealy cross the x-axis f x x x x 2 3 At x=1, we clealy cross the x-axis At x=2, we touch but do ot cross x-axis At x=3, we cross the x-axis flatly Ed Behavior We wat to examie what happes to the output (yvalues) whe the iput (x-values) to a polyomial are very large We will do this by examiig two fuctios y = 2x 3 + 4x - 8 ad y = 2x 3 with icreasig x-values 6

7 Explore x = 500 x = 1000 Ed Behavior Polyomials ted to the value of the leadig term for large iputs (x-values) f ( x) a x a x... a x a x a As x we have f ( x) a x For large x values of x (positive or egative) a x a x... a x a x a a x This is a importat cocept to remember whe takig Calculus Ed Behavior 2 ad eve, a 0 3 ad odd, a 0 7

8 Ed Behavior 2 ad eve, a 0 3 ad odd, a 0 Ed Behavior Aother way to justify usig the leadig term to predict ed behavior ca be see by factorig the leadig term out 1 p x a x a x a x a a 1 a1 a 0 p x ax 1 1 ax ax ax As we have a a a x a x a x a x p x a x a x Turig Poit The graph of a fuctio turs at a local maximum or miimum of the fuctio Turig poit ad local maxima Turig poit ad local miima 8

9 Explore Two zeroes Oe tur Four zeroes Three turs Six zeroes Five turs Turs If f ( x) has degree, the the curve has at most 1 turs Remarks Whe we eed to graph a polyomial, we ofte begi by examiig the graph usig our graphig calculator Free graphig calculator 9

10 Process for Graphig a Polyomial Determie all the zeroes of the polyomial ad their multiplicity. Use the zeros to determie the x - itercepts Use the multiplicity to determie if the zeros cross the x - axis or just touch it ad if the x - itercept will flatte out or ot Process for Graphig a Polyomial Determie the y - itercept Observe the leadig coefficiet to determie the behavior of the polyomial at the ed of the graph Plot a few more poits. This is left itetioally vague. The more poits that you plot the better the sketch. At the least you should plot at least oe at either ed of the graph ad at least oe poit betwee each pair of zeroes. We sketch the zeros

11 Begi by observig the power is 4+1+5=10 Sice this is eve ad the leadig coefficiet is eve, we kow its ed behavior We sketch the ed behavior We observe the multiplicity of each zero x=-3 has a multiplicity of 4 x=-2 has a multiplicity of 1 x=1 has a multiplicity of 5 11

12 x=-3 (multiplicity 4) meas the fuctio touches the x-axis flatly x=-2 (multiplicity 1) meas the fuctio crosses the x-axis x=1 (multiplicity 5) meas the fuctios crosses the x-axis flatly

13 We the rough i our sketch This is very rough but gives us a idea of the geeral shape We observe the graph matches our sketch 13