Analyzing Graphs of Polynomials

Given a graph of a polynomial function, we are able to observe several properties. For example, we may be able to determine any zeros or turning points the function may have. Moreover, we may be interested in determining the end behaviour of the function, or whether it is an odd or even function. In this article, we will go through the steps involved in analysing the graphs of polynomials.

Analyzing Graphs of Polynomials Analyzing Graphs of Polynomials

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Table of contents

    End Behaviour of Polynomial Functions

    Given a polynomial function, the end behaviour is what happens to the graph as x goes towards the boundaries of the domain. If we sketch the graph of a polynomial function, the end behaviour is what happens to the graph as we approach the "ends" of the real axis.

    End Behaviour of Polynomials, Graph Showing End Behaviour, GeogebraExample Showing End Behaviour of Polynomials, Jordan Madge- StudySmarter Originals

    Above is the graph of f(x)=x2-3x-4. Here we can see that as x gets larger and larger, the graph goes up. We say that as x tends to infinity, the function tends to infinity. Similarly, we say that as x approaches negative infinity, the function approaches positive infinity since as x gets smaller and smaller the graph also goes up.

    End Behaviour of Polynomial Functions, Graph Showing End Behaviour, GeogebraGraph showing End Behaviour of cubic polynomial, Jordan Madge- StudySmarter Originals

    Above is the graph of f(x)=x3-3x-2. Here we can see that as x approaches positive infinity, the function approaches positive infinity, however as x approaches negative infinity, the graph approaches negative infinity.

    A shorthand version of writing "x tends towards infinity" isx{"x":[[51,51,51,52,56,61,67,71,75,80,84,86,88,90,92,93,93,92,89,84,78,72,65,60,56,53,52,51,51,51,51],[119,119,118,116,114,111,109,108,107,106,106,105,104,103,102,102,101,100,99,99,98,97,97,97,97,96,96,96,96,96,96,96,96,97,99,101,103,105,108,110,112,114,115,117,118,120,121,123,124,126,127,128,128,129,129],[136,137,137,138,142,148,156,166,177,186,193,199,204,207,210,212,213,214,214,215,215,215,215,213,210,206,203,200,197,195,194,193,192,192,192,193,194,196,199,201,203,204,205,206,206,207,207,208,208,209,209,209,209,209,209,209,209,209,209,208,205,203,201,200,199,199,199,198,197],[236,236,236,236,237,238,240,242,244,247,249,251,254,256,258,259,261,262,263,265,266,267,268,269,271,272,274,275,277,279,281,284,287,289,292,294,296,297,299,300,300,301,301,301,301,301,301,301,300,299,298,297,296,295,294,293,293,292,291,291,289,288,286,283,280,278,277,275,274,273,272,271,271,270,269,268,267,266,264,262,260,257,255,254,253,251,250,249,248,248,247,246,246,245,245,244,244,243,242,241,241,240,240,239,239,239,239,238,238,238,237,237,237,236,236,236,236,236,236,236,237,237,238,238,239,239,239]],"y":[[74,74,74,74,74,74,74,74,74,74,74,76,78,82,87,93,100,107,114,121,127,131,134,135,136,137,137,137,136,136,136],[71,71,71,71,71,71,71,71,71,71,72,72,73,75,76,77,78,79,81,82,84,85,86,87,88,89,90,91,91,92,93,94,96,97,98,99,101,102,104,105,105,106,106,106,106,106,106,106,106,106,106,106,106,106,106],[77,77,77,77,77,77,75,73,71,69,68,68,67,67,67,67,67,67,67,67,67,66,66,66,65,64,63,62,61,60,60,60,60,60,59,59,59,59,59,59,59,59,60,60,60,61,61,61,61,62,62,62,62,62,62,62,63,63,65,67,70,74,76,78,79,79,79,79,79],[71,71,71,71,71,72,73,74,75,76,76,76,75,75,73,72,71,69,68,66,64,63,61,60,58,57,55,54,53,53,52,51,51,51,51,51,51,52,53,54,55,56,58,59,61,62,63,65,66,67,69,70,72,73,73,74,75,75,75,75,75,75,75,75,74,73,72,72,71,70,69,69,68,67,66,65,64,63,62,60,59,58,57,56,56,55,55,55,54,54,54,54,54,54,54,54,54,54,54,54,55,55,56,57,57,58,58,59,59,60,61,61,62,63,63,63,63,64,64,65,65,67,67,68,69,69,69]],"t":[[0,6,115,130,147,164,181,198,214,231,248,264,282,298,314,330,347,364,381,397,414,430,447,464,481,498,514,531,590,607,622],[1383,1523,1540,1556,1573,1589,1606,1623,1640,1656,1673,1690,1706,1722,1739,1756,1774,1789,1807,1823,1839,1857,1873,1889,1906,1922,1940,1957,1973,1990,2007,2023,2040,2057,2073,2089,2107,2123,2140,2156,2173,2189,2206,2223,2239,2256,2273,2289,2306,2323,2340,2355,2373,2389,2406],[2932,2938,3149,3164,3181,3198,3214,3230,3247,3264,3280,3297,3314,3330,3347,3364,3381,3397,3414,3430,3447,3499,3514,3531,3547,3573,3589,3606,3623,3640,3657,3674,3690,3707,3783,3798,3814,3831,3847,3864,3881,3897,3914,3931,3947,3964,3981,3998,4014,4031,4047,4073,4090,4114,4156,4173,4190,4206,4223,4240,4256,4273,4289,4307,4324,4340,4357,4373,4391],[4910,4910,5124,5139,5156,5173,5189,5207,5223,5240,5257,5273,5289,5307,5323,5339,5357,5374,5389,5406,5423,5440,5457,5473,5490,5506,5523,5540,5556,5573,5590,5605,5623,5639,5657,5674,5690,5707,5723,5741,5756,5774,5790,5806,5823,5840,5857,5873,5890,5907,5923,5940,5956,5974,5990,6006,6024,6040,6056,6073,6090,6107,6122,6140,6157,6174,6190,6206,6223,6239,6257,6273,6290,6307,6323,6340,6357,6373,6390,6407,6423,6440,6457,6473,6489,6505,6522,6540,6557,6574,6589,6607,6623,6641,6656,6673,6690,6707,6723,6740,6757,6774,6789,6806,6823,6840,6857,6874,6890,6906,6923,6940,6957,6973,6990,7007,7023,7040,7056,7074,7090,7107,7123,7139,7156,7174,7190]],"version":"2.0.0"}. So, in the above example, we could instead write: asx{"x":[[65,66,67,70,74,78,83,86,89,90,91,92,93,95,97,99,100,101,102,102,102,101,99,95,90,85,80,76,73,71,69,69,68,68,68,68,67,67,67,67],[120,119,118,117,115,113,112,111,110,110,109,108,106,105,105,104,104,104,104,104,104,104,105,106,107,107,108,109,110,111,114,117,120,124,126,128,129,129,130,130,130,130],[138,138,140,144,149,155,161,166,172,177,181,184,186,187,188,189,189,189,188,186,183,179,177,176,175,175,175,174,174,174,174,174,174,175,176,178,180,183,186,189,191,192,193,194,194,195,195,195,195,195,195,195,194,193,192,190,190,189,189,188,188,187,186,186,185,184,183,183,182,182,181],[207,207,207,207,207,207,207,207,208,209,210,211,212,214,215,217,219,220,221,222,223,224,225,226,227,227,228,228,229,229,229,230,230,231,231,232,233,234,236,238,240,243,245,248,250,253,254,256,257,257,258,259,259,260,261,261,261,261,261,261,261,260,260,259,258,257,256,254,252,250,248,247,245,244,243,241,240,239,239,238,237,236,235,234,233,232,231,230,229,229,228,228,227,226,226,226,225,225,224,224,224,224,223,223,223,222,221,220,219,217,216,214,214,213,212,212,211,211,211,210,210,209,209,208,208,207,207,207,207,207,207,207,207,207,207,207,207]],"y":[[63,63,63,63,63,63,63,63,64,65,67,69,73,77,82,88,92,96,99,101,103,106,108,112,115,117,119,121,121,122,122,122,122,121,121,121,120,120,120,120],[68,68,68,68,68,70,71,72,74,76,79,82,85,89,91,94,96,97,99,101,103,105,106,108,109,110,111,111,112,112,113,113,113,113,113,113,113,113,113,113,113,112],[83,83,83,83,83,83,83,83,83,83,83,83,83,83,83,83,83,82,81,80,77,75,73,72,71,71,71,71,70,70,70,70,70,71,71,72,73,75,77,79,80,81,82,83,83,84,84,84,84,84,85,85,86,87,89,90,91,91,92,92,93,93,94,94,95,95,96,96,96,96,96],[77,77,77,79,80,82,83,85,86,88,89,90,90,91,92,92,93,93,93,93,93,92,91,89,87,84,81,78,75,73,72,71,70,69,69,68,68,67,67,67,67,66,66,66,66,66,66,66,67,67,67,68,69,70,70,71,72,72,73,75,76,78,79,80,81,82,82,83,83,83,83,83,83,83,83,83,83,83,83,83,82,81,80,79,78,77,75,74,73,72,71,70,69,68,67,66,66,65,64,63,63,62,62,62,62,62,62,62,62,62,63,63,64,64,65,65,66,66,67,67,68,69,70,71,72,73,74,75,76,76,77,78,78,79,79,79,79]],"t":[[0,118,132,149,165,182,199,216,232,249,271,287,305,321,338,356,371,388,404,421,439,455,471,488,505,522,539,555,571,588,605,621,638,654,672,688,705,722,738,755],[1350,1499,1515,1532,1548,1565,1582,1598,1615,1632,1649,1665,1682,1698,1715,1732,1748,1771,1787,1805,1822,1837,1855,1872,1888,1905,1921,1937,1954,1971,1988,2004,2022,2039,2055,2072,2089,2105,2122,2138,2155,2171],[2716,2883,2898,2915,2932,2949,2965,2982,2999,3015,3031,3049,3065,3082,3098,3115,3183,3199,3215,3232,3249,3272,3289,3305,3323,3338,3355,3372,3388,3405,3422,3550,3565,3582,3598,3615,3632,3649,3665,3682,3698,3715,3732,3748,3772,3789,3806,3855,3871,3888,3905,3922,3938,3955,3972,3988,4005,4021,4038,4055,4073,4088,4105,4122,4139,4155,4171,4187,4205,4222,4238],[4759,4933,4949,4965,4983,4998,5015,5032,5049,5065,5081,5098,5115,5132,5148,5165,5182,5198,5216,5232,5249,5271,5288,5306,5322,5338,5355,5372,5388,5404,5423,5439,5456,5471,5488,5506,5522,5539,5554,5571,5588,5604,5623,5638,5656,5672,5688,5705,5723,5739,5754,5772,5788,5805,5821,5838,5855,5872,5889,5906,5922,5939,5955,5971,5988,6005,6021,6038,6055,6072,6089,6106,6122,6139,6155,6171,6189,6206,6222,6238,6255,6272,6289,6306,6323,6338,6355,6371,6388,6405,6422,6438,6455,6471,6488,6505,6522,6538,6555,6572,6588,6605,6622,6638,6655,6672,6689,6706,6722,6738,6755,6773,6788,6806,6821,6839,6855,6872,6889,6904,6922,6938,6955,6971,6988,7005,7022,7039,7055,7072,7088,7106,7122,7139,7155,7171,7188]],"version":"2.0.0"},fx{"x":[[111,111,111,110,110,108,107,106,105,104,103,101,99,97,95,93,91,90,89,87,87,86,86,86,86,86,86,86,86,86,86,86,86,86,86,85,85,85,84,84,84,84,84,83,82,80,78,76,74,73,73,72,72,72],[76,76,76,77,78,81,83,84,85,87,88,89,90,91,92,93,93,94,94],[145,145,144,144,143,142,140,139,137,136,136,135,135,135,135,135,135,135,135,136,136,136,137,138,139,141,142,143,144,145,145],[156,157,157,159,160,162,163,164,165,166,166,167,167,168,168,168,168,168,167,166,165,163,161,160,159,159,158,158,158,158,158,160,161,163,165,167,169,171,172,173,174,176,177,178,179,180,181,182,183,184,184,184,184,184,182,180,179,177,176,175,174,173,172,172,171,171,171,171,171,171,172,172,173,175,177,179,181,183,184,185],[197,197,198,199,200,201,203,205,207,208,209,210,210,211,211,211,211,211,210,208,206,203,201,199,198,197,197,196,196,195,194,194,193,193,192,192,192,192,192],[232,232,234,237,241,244,248,252,256,260,263,265,267,269,270,271,272,272,273,273,272,270,267,264,261,260,259,258,258,258,258,259,260,262,264,266,268,270,271,273,273,274,275,275,275,275,275,275,273,272,269,268,266,265,265,264,264,264],[314,314,314,314,314,314,314,314,314,314,314,315,316,317,318,319,320,321,322,323,325,326,328,329,331,333,334,335,337,338,339,340,341,342,342,343,344,345,345,346,347,348,349,349,350,352,353,354,355,356,358,359,360,362,363,365,366,367,368,369,370,371,372,372,372,372,372,372,372,372,372,372,372,372,372,371,371,371,370,370,369,368,367,366,365,364,363,362,361,360,359,358,357,354,351,348,345,343,341,340,340,339,338,338,337,336,335,335,334,333,332,331,330,329,328,327,325,324,323,322,322,321,320,319,318,318,317,316,315,314,313,312,311,311,310,310,310]],"y":[[57,57,57,57,57,57,57,57,57,57,57,57,58,60,61,63,66,69,73,78,84,89,94,98,102,106,110,113,116,119,121,123,126,128,130,133,134,136,137,139,139,140,140,140,140,140,139,137,136,134,132,131,130,130],[101,101,101,101,101,101,101,101,101,101,101,101,101,101,101,101,101,101,101],[55,55,55,56,58,61,64,67,70,73,77,80,84,88,92,95,98,100,102,104,106,107,108,109,110,111,112,112,112,112,112],[85,85,85,85,85,85,85,85,85,85,85,86,87,88,90,91,93,94,95,96,97,97,98,99,99,100,100,101,101,101,101,101,100,99,98,96,95,93,92,90,89,88,87,87,86,85,85,85,84,84,84,84,84,84,84,84,84,84,85,86,87,88,90,91,92,93,93,93,94,94,94,95,95,96,96,96,96,96,96,96],[43,43,43,43,43,44,47,52,56,60,65,68,70,72,74,76,78,80,82,85,89,93,96,99,101,102,103,104,104,105,105,106,107,107,108,109,109,109,109],[83,83,83,83,83,83,83,83,83,83,83,83,83,83,83,83,83,83,83,83,83,81,79,76,74,73,73,72,72,72,72,72,73,74,75,77,78,79,80,82,83,83,84,85,85,85,85,86,87,89,90,92,93,94,95,95,95,94],[62,62,63,64,66,68,69,71,73,75,77,78,80,81,82,83,84,85,86,86,87,87,87,87,87,86,86,84,83,82,81,79,77,76,74,72,71,70,69,68,67,66,65,64,63,62,62,61,61,61,61,61,61,61,61,61,61,61,61,61,62,62,63,63,64,65,66,67,68,69,70,71,73,75,76,77,78,78,78,79,79,80,80,81,81,81,82,82,82,82,82,82,81,79,76,73,71,69,68,67,67,66,66,65,64,64,63,62,62,61,59,59,58,57,57,56,56,55,55,55,55,55,55,55,55,56,56,57,58,60,62,64,65,66,67,67,67]],"t":[[0,150,165,182,198,215,232,249,265,282,299,316,332,349,366,382,399,415,432,450,465,482,499,515,532,550,565,591,606,624,640,657,674,690,707,724,740,756,773,791,807,824,840,857,873,890,907,924,940,957,973,991,1007,1023],[1502,1503,1633,1648,1665,1682,1698,1716,1732,1749,1765,1782,1799,1815,1833,1849,1865,1882,1899],[2501,2683,2699,2715,2732,2749,2766,2783,2799,2816,2833,2850,2865,2882,2899,2915,2933,2949,2965,2982,2999,3016,3032,3049,3066,3091,3107,3124,3140,3157,3174],[3568,3717,3732,3749,3765,3781,3798,3815,3832,3848,3865,3881,3899,3915,3932,3949,3965,3981,3998,4015,4031,4048,4065,4090,4108,4124,4140,4157,4173,4248,4265,4282,4298,4315,4332,4348,4365,4382,4399,4415,4432,4448,4465,4481,4498,4515,4531,4548,4565,4590,4607,4623,4699,4715,4731,4748,4765,4782,4798,4815,4832,4849,4865,4881,4898,4915,4931,4948,4964,4981,4998,5015,5032,5048,5066,5091,5106,5124,5141,5157],[5618,5750,5765,5781,5798,5815,5831,5848,5865,5881,5898,5915,5932,5949,5965,5981,5998,6014,6031,6048,6065,6092,6108,6124,6140,6158,6174,6190,6207,6224,6241,6258,6274,6290,6307,6325,6351,6359,6374],[6801,6957,6974,6991,7007,7024,7040,7057,7075,7090,7108,7124,7141,7158,7175,7191,7208,7224,7241,7291,7308,7324,7340,7358,7374,7391,7407,7424,7441,7457,7550,7565,7590,7608,7624,7641,7658,7674,7690,7707,7725,7741,7758,7773,7790,7808,7824,7841,7857,7874,7891,7907,7924,7941,7957,7974,7990,8008],[8499,8666,8681,8698,8714,8731,8748,8765,8781,8798,8814,8832,8849,8865,8880,8897,8914,8931,8948,8964,8981,8998,9015,9031,9048,9065,9091,9107,9124,9141,9158,9174,9191,9207,9224,9241,9258,9274,9290,9308,9324,9340,9358,9375,9391,9408,9424,9442,9458,9474,9492,9507,9524,9541,9558,9574,9591,9608,9624,9640,9657,9674,9690,9707,9725,9741,9757,9774,9791,9808,9824,9840,9857,9874,9891,9907,9924,9940,9957,9974,9991,10007,10024,10041,10057,10074,10090,10108,10124,10140,10157,10173,10190,10207,10224,10240,10258,10274,10291,10308,10324,10341,10358,10373,10390,10407,10424,10441,10457,10475,10491,10508,10525,10541,10558,10574,10591,10607,10623,10640,10657,10674,10690,10708,10725,10741,10758,10774,10791,10807,10823,10840,10857,10874,10890,10907,10925]],"version":"2.0.0"} and asx-{"x":[[58,58,58,59,61,64,68,70,73,75,76,78,79,80,81,82,83,84,85,85,85,85,85,85,83,81,79,77,75,74,72,70,67,65,63,61,60,59,58],[106,106,105,104,103,101,100,99,97,96,94,92,89,87,85,84,82,81,80,79,79,79,78,78,78,78,78,79,80,82,85,88,90,93,95,97,99,101,102,103,104,105,106,106,107,107,107,108],[115,115,116,118,121,127,133,138,144,149,155,160,164,168,170,172,173,174,174,174,174,174,173,170,167,164,161,159,157,156,155,155,155,156,158,161,165,168,172,175,177,179,180,180,181,181,181,181,181,180,179,178,177,176,176,176,175],[198,198,199,200,203,206,209,213,215,217,218,219,219,219,220,220,221],[245,245,245,245,245,245,245,245,245,246,247,249,250,253,255,257,259,261,262,264,265,267,268,270,272,273,274,275,276,277,278,279,280,282,284,286,288,290,292,294,296,297,298,299,300,301,302,303,303,304,304,304,304,304,304,304,304,304,303,302,302,301,299,298,296,293,290,287,284,282,280,279,277,274,272,269,267,265,264,262,261,259,259,258,257,257,256,255,255,254,254,253,253,252,251,251,250,249,248,247,246,246,245,245,245,244,244,244,244,244,244,244]],"y":[[72,72,72,72,72,72,72,72,72,72,72,72,73,75,77,79,81,83,85,87,90,92,96,100,104,107,110,112,114,115,116,117,118,118,118,118,118,118,118],[73,73,73,73,73,73,74,75,76,78,79,80,82,83,84,85,87,88,89,90,91,93,95,97,99,101,103,105,108,110,112,114,116,117,118,118,118,118,118,118,118,117,117,117,116,116,116,115],[87,87,87,87,87,87,87,87,87,87,87,87,87,87,87,87,87,87,87,87,86,85,83,81,78,76,75,74,73,73,73,73,73,73,73,75,77,79,82,84,86,87,88,89,89,90,90,91,91,92,93,93,94,94,95,95,95],[85,85,85,85,85,85,85,85,85,85,85,85,85,85,85,85,84],[71,71,71,72,74,75,76,77,78,78,79,79,79,79,79,79,79,78,77,76,74,73,71,69,67,65,63,62,61,60,58,57,56,56,55,54,54,54,54,54,54,54,55,55,56,57,58,59,60,61,62,63,64,65,67,69,71,73,75,76,77,78,79,80,81,82,82,82,82,82,82,81,80,77,74,71,69,66,63,61,59,57,56,55,54,54,53,53,53,52,52,52,52,52,52,52,53,54,55,57,58,60,61,62,63,63,64,64,64,64,64,65]],"t":[[0,2,117,132,149,165,183,199,216,232,248,266,282,299,316,332,349,366,383,399,415,432,448,466,483,499,516,532,549,566,582,601,616,632,649,666,682,699,716],[1549,1716,1732,1749,1765,1783,1799,1816,1832,1849,1865,1882,1899,1916,1932,1949,1965,1983,1999,2016,2032,2049,2066,2083,2099,2128,2145,2162,2179,2196,2212,2229,2245,2262,2280,2296,2312,2328,2345,2362,2379,2395,2411,2428,2445,2463,2479,2496],[3050,3062,3199,3215,3232,3249,3266,3283,3298,3315,3332,3349,3366,3383,3399,3416,3432,3449,3465,3482,3500,3516,3532,3548,3565,3583,3599,3629,3646,3662,3679,3694,3750,3765,3783,3799,3816,3833,3849,3865,3883,3900,3916,3933,3949,3966,3983,3999,4015,4032,4048,4066,4082,4100,4115,4132,4149],[4534,4662,4679,4695,4712,4728,4745,4763,4779,4794,4812,4829,4845,4862,4879,4896,4911],[5283,5462,5478,5495,5512,5528,5545,5563,5579,5594,5612,5629,5645,5662,5679,5695,5711,5728,5745,5762,5778,5796,5812,5828,5845,5862,5879,5895,5912,5928,5945,5961,5978,5995,6012,6029,6044,6062,6079,6095,6112,6128,6145,6162,6179,6195,6211,6229,6245,6262,6279,6295,6312,6328,6345,6362,6378,6395,6412,6428,6445,6463,6479,6495,6512,6528,6545,6562,6578,6596,6612,6628,6645,6662,6679,6696,6712,6728,6745,6762,6779,6795,6812,6828,6845,6861,6879,6895,6912,6928,6945,6962,6979,6995,7012,7028,7045,7062,7078,7095,7111,7129,7145,7163,7179,7195,7212,7228,7245,7262,7278,7332]],"version":"2.0.0"}, fx-{"x":[[98,98,98,98,97,96,95,94,93,93,92,91,90,90,89,88,86,85,84,83,83,83,84,86,89,91,93,94,96,96,96,96,96,96,94,92,79,76,74,73,72,72,72],[71,72,76,82,87,90,93,95,96,97],[152,151,150,148,145,143,141,140,138,137,136,135,133,132,131,130,130,130,130,131,132,134,136,137,139,141,141,142],[156,156,157,158,159,161,161,162,163,164,165,165,166,167,167,167,167,167,165,163,161,159,158,156,155,154,154,153,153,153,153,153,155,158,161,164,166,168,170,172,173,174,175,176,178,179,180,180,181,182,183,184,184,184,179,177,171,170,169,168,168,168,168,168,169,170,173,176,180,183,187,189,191,191,191,191],[201,201,202,204,208,213,218,224,228,232,234,235,237,238,239,239,239,239,239,238,237,235,233,232,231,230,230,229,226],[251,251,251,253,257,264,273,283,290,294,298,301,302,303,304,305,305,305,305,302,298,293,289,286,285,285,284,285,286,289,292,295,297,299,300,301,301,301,300,299,298,298,297,297,297,297],[336,336,338,342,348,353,358,362,365,368,371,372,373,373,372],[387,387,387,387,387,387,387,387,387,387,388,389,390,391,393,395,398,400,402,404,406,408,410,412,414,415,417,418,419,420,420,421,422,422,423,423,424,425,426,427,428,430,431,433,435,437,438,440,441,442,443,444,445,446,447,447,448,448,449,449,450,450,450,450,450,450,450,450,450,449,449,448,447,446,445,444,443,442,441,439,437,435,432,430,429,429,429,429,429,428,428,428,428,427,427,426,425,425,424,423,422,421,420,419,418,417,416,414,413,412,412,411,411,411,410,409,409,408,407,406,405,404,403,401,399,398,397,396,396,395,395,394,394,393,393,393,392,392,391,390,390,389,389,389,389,389,389]],"y":[[63,63,63,63,63,62,62,62,62,62,62,62,62,62,64,67,71,77,82,88,94,99,105,112,119,125,131,138,143,148,152,156,159,161,162,164,165,164,163,161,160,160,158],[111,111,111,111,111,111,111,111,111,111],[59,59,59,61,63,64,67,69,71,73,77,82,87,93,99,105,111,116,119,122,124,126,127,128,129,130,131,131],[79,79,79,79,79,79,79,79,78,79,79,80,82,83,85,88,90,93,96,100,102,105,106,108,109,110,110,111,111,111,111,110,107,104,101,98,95,93,90,89,87,86,84,83,82,81,80,80,80,80,80,80,80,80,82,83,88,90,91,93,94,95,96,97,98,99,100,100,100,100,100,100,100,100,100,99],[48,48,48,48,49,53,59,68,77,86,94,100,106,112,116,120,122,125,126,128,130,132,134,136,137,138,138,138,136],[95,95,95,95,95,95,95,95,95,95,95,95,95,95,95,95,95,95,95,93,90,87,84,83,82,82,82,82,83,84,86,87,88,89,90,91,93,96,100,103,106,108,109,110,110,110],[100,100,100,100,100,100,100,100,100,100,100,100,100,100,98],[88,88,88,89,90,92,93,95,95,96,97,97,98,98,99,100,100,100,101,101,101,100,99,98,96,95,94,92,90,89,87,86,85,83,82,81,79,78,77,76,75,74,74,74,74,74,74,74,74,74,75,76,77,77,78,79,80,81,83,84,85,87,89,90,92,93,94,96,98,100,101,102,103,103,104,105,105,105,106,106,106,105,104,103,102,102,101,101,101,101,101,100,100,99,98,96,95,94,92,91,89,87,85,83,82,81,79,78,76,75,74,73,73,72,72,71,71,71,70,70,70,70,70,71,72,73,73,74,75,75,76,76,77,78,78,78,79,80,80,82,82,83,84,84,84,84,84]],"t":[[0,3,117,132,149,166,182,199,216,232,249,266,283,299,316,333,349,366,382,399,416,432,449,466,483,499,516,547,564,580,597,614,631,647,665,683,746,764,782,797,814,830,848],[1267,1384,1399,1416,1433,1449,1466,1482,1499,1516],[1983,2117,2133,2149,2166,2182,2199,2216,2233,2249,2266,2289,2299,2316,2333,2349,2367,2382,2399,2416,2432,2449,2465,2483,2499,2515,2548,2569],[2950,3083,3099,3115,3132,3149,3165,3182,3199,3215,3232,3249,3266,3282,3298,3315,3332,3348,3366,3382,3399,3415,3432,3449,3466,3483,3498,3515,3547,3650,3666,3682,3700,3715,3732,3749,3766,3782,3798,3815,3832,3848,3866,3882,3899,3916,3932,3948,3965,3982,3999,4015,4047,4063,4198,4214,4264,4280,4296,4314,4330,4348,4364,4382,4396,4413,4431,4448,4463,4482,4497,4514,4531,4546,4563,4566],[4968,5083,5099,5115,5132,5148,5166,5183,5199,5216,5233,5249,5265,5282,5298,5315,5332,5348,5366,5382,5399,5415,5432,5449,5466,5481,5498,5514,5548],[6033,6036,6133,6149,6165,6182,6198,6215,6231,6248,6266,6282,6298,6315,6331,6348,6365,6382,6449,6466,6482,6499,6515,6548,6564,6581,6631,6646,6663,6682,6698,6714,6731,6747,6763,6781,6798,6814,6831,6848,6863,6880,6898,6914,6929,6949],[7349,7500,7515,7546,7564,7580,7597,7614,7630,7647,7664,7681,7698,7714,7731],[8254,8256,8350,8366,8381,8398,8414,8431,8449,8465,8482,8498,8515,8548,8565,8581,8597,8614,8629,8648,8664,8681,8697,8714,8731,8748,8764,8781,8797,8814,8831,8848,8864,8881,8897,8914,8930,8947,8964,8980,8997,9014,9031,9048,9065,9081,9098,9115,9131,9148,9165,9181,9198,9214,9231,9248,9265,9281,9298,9315,9331,9349,9365,9381,9398,9414,9431,9448,9466,9482,9498,9514,9546,9564,9581,9597,9614,9630,9647,9663,9681,9697,9714,9731,9747,9764,9780,9797,9814,9831,9847,9865,9880,9897,9914,9930,9947,9964,9981,9996,10013,10031,10047,10064,10080,10097,10114,10130,10147,10164,10181,10196,10214,10230,10248,10265,10280,10297,10314,10330,10347,10365,10381,10397,10414,10431,10447,10463,10481,10497,10514,10531,10547,10563,10581,10597,10614,10630,10647,10665,10680,10697,10714,10730,10747,10764,10781]],"version":"2.0.0"}

    What Determines the End Behaviour of a Polynomial Function?

    The degree of a polynomial is the highest power that a polynomial has. For example, the polynomial function is a degree 5 polynomial. The leading coefficient of a polynomial function is the term with the highest degree in the polynomial. So, for the polynomial, the leading coefficient is 7 and the degree is 5.

    As x gets really big or really small (asx{"x":[[65,66,67,70,74,78,83,86,89,90,91,92,93,95,97,99,100,101,102,102,102,101,99,95,90,85,80,76,73,71,69,69,68,68,68,68,67,67,67,67],[120,119,118,117,115,113,112,111,110,110,109,108,106,105,105,104,104,104,104,104,104,104,105,106,107,107,108,109,110,111,114,117,120,124,126,128,129,129,130,130,130,130],[138,138,140,144,149,155,161,166,172,177,181,184,186,187,188,189,189,189,188,186,183,179,177,176,175,175,175,174,174,174,174,174,174,175,176,178,180,183,186,189,191,192,193,194,194,195,195,195,195,195,195,195,194,193,192,190,190,189,189,188,188,187,186,186,185,184,183,183,182,182,181],[207,207,207,207,207,207,207,207,208,209,210,211,212,214,215,217,219,220,221,222,223,224,225,226,227,227,228,228,229,229,229,230,230,231,231,232,233,234,236,238,240,243,245,248,250,253,254,256,257,257,258,259,259,260,261,261,261,261,261,261,261,260,260,259,258,257,256,254,252,250,248,247,245,244,243,241,240,239,239,238,237,236,235,234,233,232,231,230,229,229,228,228,227,226,226,226,225,225,224,224,224,224,223,223,223,222,221,220,219,217,216,214,214,213,212,212,211,211,211,210,210,209,209,208,208,207,207,207,207,207,207,207,207,207,207,207,207]],"y":[[63,63,63,63,63,63,63,63,64,65,67,69,73,77,82,88,92,96,99,101,103,106,108,112,115,117,119,121,121,122,122,122,122,121,121,121,120,120,120,120],[68,68,68,68,68,70,71,72,74,76,79,82,85,89,91,94,96,97,99,101,103,105,106,108,109,110,111,111,112,112,113,113,113,113,113,113,113,113,113,113,113,112],[83,83,83,83,83,83,83,83,83,83,83,83,83,83,83,83,83,82,81,80,77,75,73,72,71,71,71,71,70,70,70,70,70,71,71,72,73,75,77,79,80,81,82,83,83,84,84,84,84,84,85,85,86,87,89,90,91,91,92,92,93,93,94,94,95,95,96,96,96,96,96],[77,77,77,79,80,82,83,85,86,88,89,90,90,91,92,92,93,93,93,93,93,92,91,89,87,84,81,78,75,73,72,71,70,69,69,68,68,67,67,67,67,66,66,66,66,66,66,66,67,67,67,68,69,70,70,71,72,72,73,75,76,78,79,80,81,82,82,83,83,83,83,83,83,83,83,83,83,83,83,83,82,81,80,79,78,77,75,74,73,72,71,70,69,68,67,66,66,65,64,63,63,62,62,62,62,62,62,62,62,62,63,63,64,64,65,65,66,66,67,67,68,69,70,71,72,73,74,75,76,76,77,78,78,79,79,79,79]],"t":[[0,118,132,149,165,182,199,216,232,249,271,287,305,321,338,356,371,388,404,421,439,455,471,488,505,522,539,555,571,588,605,621,638,654,672,688,705,722,738,755],[1350,1499,1515,1532,1548,1565,1582,1598,1615,1632,1649,1665,1682,1698,1715,1732,1748,1771,1787,1805,1822,1837,1855,1872,1888,1905,1921,1937,1954,1971,1988,2004,2022,2039,2055,2072,2089,2105,2122,2138,2155,2171],[2716,2883,2898,2915,2932,2949,2965,2982,2999,3015,3031,3049,3065,3082,3098,3115,3183,3199,3215,3232,3249,3272,3289,3305,3323,3338,3355,3372,3388,3405,3422,3550,3565,3582,3598,3615,3632,3649,3665,3682,3698,3715,3732,3748,3772,3789,3806,3855,3871,3888,3905,3922,3938,3955,3972,3988,4005,4021,4038,4055,4073,4088,4105,4122,4139,4155,4171,4187,4205,4222,4238],[4759,4933,4949,4965,4983,4998,5015,5032,5049,5065,5081,5098,5115,5132,5148,5165,5182,5198,5216,5232,5249,5271,5288,5306,5322,5338,5355,5372,5388,5404,5423,5439,5456,5471,5488,5506,5522,5539,5554,5571,5588,5604,5623,5638,5656,5672,5688,5705,5723,5739,5754,5772,5788,5805,5821,5838,5855,5872,5889,5906,5922,5939,5955,5971,5988,6005,6021,6038,6055,6072,6089,6106,6122,6139,6155,6171,6189,6206,6222,6238,6255,6272,6289,6306,6323,6338,6355,6371,6388,6405,6422,6438,6455,6471,6488,6505,6522,6538,6555,6572,6588,6605,6622,6638,6655,6672,6689,6706,6722,6738,6755,6773,6788,6806,6821,6839,6855,6872,6889,6904,6922,6938,6955,6971,6988,7005,7022,7039,7055,7072,7088,7106,7122,7139,7155,7171,7188]],"version":"2.0.0"} or x-{"x":[[58,58,58,59,61,64,68,70,73,75,76,78,79,80,81,82,83,84,85,85,85,85,85,85,83,81,79,77,75,74,72,70,67,65,63,61,60,59,58],[106,106,105,104,103,101,100,99,97,96,94,92,89,87,85,84,82,81,80,79,79,79,78,78,78,78,78,79,80,82,85,88,90,93,95,97,99,101,102,103,104,105,106,106,107,107,107,108],[115,115,116,118,121,127,133,138,144,149,155,160,164,168,170,172,173,174,174,174,174,174,173,170,167,164,161,159,157,156,155,155,155,156,158,161,165,168,172,175,177,179,180,180,181,181,181,181,181,180,179,178,177,176,176,176,175],[198,198,199,200,203,206,209,213,215,217,218,219,219,219,220,220,221],[245,245,245,245,245,245,245,245,245,246,247,249,250,253,255,257,259,261,262,264,265,267,268,270,272,273,274,275,276,277,278,279,280,282,284,286,288,290,292,294,296,297,298,299,300,301,302,303,303,304,304,304,304,304,304,304,304,304,303,302,302,301,299,298,296,293,290,287,284,282,280,279,277,274,272,269,267,265,264,262,261,259,259,258,257,257,256,255,255,254,254,253,253,252,251,251,250,249,248,247,246,246,245,245,245,244,244,244,244,244,244,244]],"y":[[72,72,72,72,72,72,72,72,72,72,72,72,73,75,77,79,81,83,85,87,90,92,96,100,104,107,110,112,114,115,116,117,118,118,118,118,118,118,118],[73,73,73,73,73,73,74,75,76,78,79,80,82,83,84,85,87,88,89,90,91,93,95,97,99,101,103,105,108,110,112,114,116,117,118,118,118,118,118,118,118,117,117,117,116,116,116,115],[87,87,87,87,87,87,87,87,87,87,87,87,87,87,87,87,87,87,87,87,86,85,83,81,78,76,75,74,73,73,73,73,73,73,73,75,77,79,82,84,86,87,88,89,89,90,90,91,91,92,93,93,94,94,95,95,95],[85,85,85,85,85,85,85,85,85,85,85,85,85,85,85,85,84],[71,71,71,72,74,75,76,77,78,78,79,79,79,79,79,79,79,78,77,76,74,73,71,69,67,65,63,62,61,60,58,57,56,56,55,54,54,54,54,54,54,54,55,55,56,57,58,59,60,61,62,63,64,65,67,69,71,73,75,76,77,78,79,80,81,82,82,82,82,82,82,81,80,77,74,71,69,66,63,61,59,57,56,55,54,54,53,53,53,52,52,52,52,52,52,52,53,54,55,57,58,60,61,62,63,63,64,64,64,64,64,65]],"t":[[0,2,117,132,149,165,183,199,216,232,248,266,282,299,316,332,349,366,383,399,415,432,448,466,483,499,516,532,549,566,582,601,616,632,649,666,682,699,716],[1549,1716,1732,1749,1765,1783,1799,1816,1832,1849,1865,1882,1899,1916,1932,1949,1965,1983,1999,2016,2032,2049,2066,2083,2099,2128,2145,2162,2179,2196,2212,2229,2245,2262,2280,2296,2312,2328,2345,2362,2379,2395,2411,2428,2445,2463,2479,2496],[3050,3062,3199,3215,3232,3249,3266,3283,3298,3315,3332,3349,3366,3383,3399,3416,3432,3449,3465,3482,3500,3516,3532,3548,3565,3583,3599,3629,3646,3662,3679,3694,3750,3765,3783,3799,3816,3833,3849,3865,3883,3900,3916,3933,3949,3966,3983,3999,4015,4032,4048,4066,4082,4100,4115,4132,4149],[4534,4662,4679,4695,4712,4728,4745,4763,4779,4794,4812,4829,4845,4862,4879,4896,4911],[5283,5462,5478,5495,5512,5528,5545,5563,5579,5594,5612,5629,5645,5662,5679,5695,5711,5728,5745,5762,5778,5796,5812,5828,5845,5862,5879,5895,5912,5928,5945,5961,5978,5995,6012,6029,6044,6062,6079,6095,6112,6128,6145,6162,6179,6195,6211,6229,6245,6262,6279,6295,6312,6328,6345,6362,6378,6395,6412,6428,6445,6463,6479,6495,6512,6528,6545,6562,6578,6596,6612,6628,6645,6662,6679,6696,6712,6728,6745,6762,6779,6795,6812,6828,6845,6861,6879,6895,6912,6928,6945,6962,6979,6995,7012,7028,7045,7062,7078,7095,7111,7129,7145,7163,7179,7195,7212,7228,7245,7262,7278,7332]],"version":"2.0.0"}), the leading coefficient becomes significant because that term will take over and grow significantly faster compared with the other terms. Therefore, to determine the end behaviour of a polynomial, we only need to look at the degree and leading coefficient to draw a conclusion. There are four possible scenarios.

    CaseDegreeLeading CoefficientEnd Behaviour Example
    1Even Positive As 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    End Behaviour of Polynomials, Even function, Geogebra

    2EvenNegativeAs 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    End Behaviour of Polynomials, Even function, Geogebra

    End Behaviour of Even Function, Jordan Madge- StudySmarter Originals.
    3OddPositiveAs 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    End Behaviour of Polynomials, Odd function, Geogebra

    End Behaviour of Odd Function, Jordan Madge- StudySmarter Originals.
    4OddNegativeAs 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    End Behaviour of Polynomials, Odd function, Geogebra

    End Behaviour of Odd Function, Jordan Madge- StudySmarter Originals.

    Determine the end behaviour of the polynomial 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    Solution:

    Here, the degree is 2 which is even and the leading coefficient is 1 which is positive. Therefore, we have case 1 and so as 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967,11984,12001,12018,12046,12068,12085,12102,12118,12135,12151,12168,12185,12202,12219,12235]],"version":"2.0.0"}.

    Determine the end behaviour of the polynomialfx=-x4+3x3+x-2{"x":[[156,157,157,156,155,154,152,150,147,143,140,137,134,131,129,127,125,125,125,125,127,129,131,132,132,132,131,130,128,126,125,123,122,120,119],[110,110,110,110,111,112,115,118,121,125,129,132,135,138,140,142],[202,200,197,195,193,190,188,185,182,179,177,175,173,172,172,172,174,177,180,183,187,190,192,194],[205,205,205,205,207,210,214,219,224,229,234,237,239,240,240,238,236,233,230,227,224,221,218,215,213,211],[252,250,247,244,242,240,238,237,236,236,236,236,238,240,244,248,252,257,262,264],[278,279,281,284,286,291,297,304,312,319,325,330,333,334,335,333,331,327,324,320,317,313,310,306,302,299,296],[385,384,384,384,386,387,390,392,396,399,403,406,409,410,411],[376,376,375,375,376,378,380,384,388,393,397,400,403,406,409],[464,461,459,457,456,455,455,456,457,458,460,462,464,466,470,474,480,485,491,496,500,503,504,505,505,506],[529,529,529,531,533,536,539,542,545,549,552,554,556,557,557,556,553,550,546,543,539,536,533,529,526,523,521,519,518,519],[577,577,576,576,575,573,571,569,566,563,561,559,556,555,554,553,554,557,561,565,569,573,576,579,581,582],[603,602,601,602,602,603,604,604,605,605,605,605,605,605,606,607,609,610,612,614,615,616],[618,618,618,619,619,620,619,618,617,616,615,615,615],[631,630,629,628,630,633,636,640,645,650,655,659,663,665,666,667,666],[647,646,645,645,645,645,645,646,646,645,644,644,644,645,647],[699,699,700,700,701,703,705,706,708,710,712,714,716,718,720,721,721,721,719,716,713,711,709,708,707,708,710,713,717,720,723,726,728,730,731,731,730,728,727,725,723,720,717,712,708,703,699,696,693,692,692,693,695],[745,747,748,750,751,753,754,756,757,758,760,760,760,759,757,754,750,747,744,742,740,739,739],[772,772,772,771,771,769,767,764,761,759,757,755,755,755,757,760,764,768,771,774,776,777],[789,789,791,792,794,796,798,800,801,801,801,801,799,797,796,794,794,794,794,795,796,797,797,797,796,795,793,791,789,787,786,785,784],[824,823,822,821,820,818,818,817,819,821,825,829,834,838,842,845,847,848,849],[834,833,833,833,834,835,836,836,836,836,836,837,838,840],[866,867,867,869,870,872,874,876,879,882,886,889,891,893,893,892,889,886,883,879,877,874,873,872,871,871,872,873],[903,904,905,905,904,902,900,898,896,894,893,893,893,895,898,902,906,910,914,917,921,922],[942,942,941,939,938,937,936,936,935,935,936,937,938,939,941,943,945,949,952,956,960,964,966,968,969,970,971,972,973,974],[991,990,990,990,990,991,992,994,997,1000,1005,1009,1014,1018,1022,1025,1027,1028,1028,1026,1023,1020,1016,1012,1010,1007,1005,1004,1002,1002,1004,1007,1011,1016,1021,1027,1032,1036,1040,1044,1047,1049,1051,1052,1052]],"y":[[198,196,194,190,188,186,184,182,180,178,175,173,171,171,172,175,180,186,193,202,210,218,227,234,240,247,253,259,264,267,269,270,271,271,270],[240,238,236,235,234,233,232,231,230,229,228,227,227,226,226,227],[183,182,180,180,180,182,185,188,192,196,201,208,215,223,232,239,246,251,254,255,256,256,255,253],[218,215,213,209,207,206,206,205,205,205,205,206,208,210,214,220,226,231,235,239,240,240,241,241,241,241],[201,200,198,198,198,201,204,207,211,215,220,225,231,236,241,244,246,247,246,245],[165,162,160,160,161,164,168,172,177,182,187,193,199,206,213,220,226,233,237,241,243,245,245,245,244,244,244],[181,180,179,180,182,183,185,186,188,189,190,190,191,191,192],[220,221,221,222,222,222,222,222,222,221,221,221,222,223,223],[197,197,197,197,197,198,199,200,201,202,202,202,202,202,202,201,200,199,197,196,195,194,194,194,193,193],[179,178,177,177,177,177,177,178,180,182,184,187,190,192,195,198,201,204,207,210,212,213,214,214,213,213,213,213,213,213],[180,178,175,174,174,175,176,178,180,183,185,188,193,197,203,209,213,216,217,217,216,214,211,208,205,204],[131,129,127,128,130,133,136,139,142,146,149,152,153,154,154,153,153,153,152,152,152,151],[137,138,140,143,149,154,159,163,165,168,170,171,172],[193,194,193,193,194,194,194,194,193,193,192,191,191,190,190,190,190],[172,172,171,172,176,181,187,193,199,203,207,209,211,211,211],[156,154,152,150,150,150,151,153,154,155,156,157,159,160,161,163,165,168,170,172,173,174,174,174,174,174,175,176,178,179,180,182,185,188,191,193,196,199,202,204,207,208,209,209,209,209,208,207,205,202,201,200,200],[179,177,175,175,176,177,179,180,182,183,184,186,188,190,193,195,198,199,200,201,201,200,199],[175,173,171,172,173,176,178,180,183,185,187,190,194,198,202,204,206,206,206,205,202,201],[118,117,115,115,115,116,117,118,119,121,122,123,124,125,127,128,129,130,131,132,134,135,137,138,139,140,141,141,141,141,141,140,140],[174,174,174,174,176,177,178,178,178,178,178,178,178,178,178,178,179,179,180],[159,159,164,170,176,182,188,191,193,194,195,195,195,194],[174,172,171,170,170,170,171,172,173,174,176,178,180,183,186,188,191,193,195,195,195,195,194,192,191,190,190,190],[176,173,172,171,172,174,176,178,181,184,187,190,192,193,194,194,193,193,192,191,190,190],[184,182,181,181,182,182,182,183,183,182,182,182,182,182,182,182,182,182,182,182,182,182,182,182,182,182,183,183,184,184],[154,153,151,150,149,149,148,148,148,147,147,148,149,150,152,155,158,162,165,169,173,176,180,184,187,190,192,194,195,196,196,196,196,196,196,197,198,199,199,199,199,200,200,200,201]],"t":[[0,11,16,19,26,31,45,48,59,64,74,81,91,98,108,114,124,132,145,148,159,165,174,181,191,198,208,214,225,232,245,248,258,265,271],[373,379,383,389,398,407,414,424,431,441,448,458,465,475,481,491],[712,720,727,730,736,740,748,757,764,774,781,791,798,808,815,825,831,842,848,859,865,875,882,892],[1011,1018,1024,1027,1040,1048,1058,1065,1075,1081,1092,1098,1108,1115,1126,1132,1144,1148,1160,1165,1177,1182,1193,1198,1210,1215],[1374,1384,1391,1394,1399,1408,1415,1423,1431,1441,1448,1458,1465,1475,1482,1492,1498,1509,1515,1523],[1703,1712,1719,1722,1727,1732,1740,1748,1759,1765,1776,1782,1792,1799,1809,1815,1826,1832,1842,1849,1859,1865,1876,1882,1893,1899,1909],[2070,2079,2087,2095,2101,2109,2115,2126,2132,2142,2149,2159,2165,2176,2182],[2336,2345,2352,2361,2365,2375,2382,2393,2399,2410,2415,2426,2432,2443,2449],[2679,2687,2692,2695,2700,2708,2715,2726,2732,2743,2749,2760,2766,2776,2782,2793,2799,2810,2816,2826,2832,2843,2849,2862,2866,2882],[4569,4579,4589,4608,4616,4626,4633,4643,4650,4660,4666,4677,4683,4694,4700,4710,4716,4727,4733,4744,4750,4761,4766,4777,4783,4794,4800,4811,4816,4828],[4959,4967,4973,4976,4983,4993,5000,5010,5016,5027,5033,5044,5050,5061,5067,5077,5083,5095,5100,5112,5117,5128,5133,5145,5150,5157],[5337,5352,5358,5383,5392,5400,5411,5417,5428,5433,5444,5450,5462,5467,5483,5494,5500,5511,5517,5528,5533,5545],[5675,5692,5700,5710,5717,5728,5733,5744,5750,5762,5767,5778,5784],[5957,5966,5973,5976,6000,6009,6017,6028,6033,6045,6050,6062,6067,6078,6083,6095,6100],[6209,6217,6223,6226,6233,6243,6250,6261,6267,6278,6284,6295,6300,6312,6317],[6475,6481,6486,6489,6495,6500,6510,6517,6525,6534,6544,6550,6562,6567,6578,6584,6595,6600,6612,6617,6628,6634,6646,6650,6662,6684,6695,6700,6712,6717,6729,6734,6745,6750,6762,6767,6779,6784,6796,6800,6812,6817,6829,6834,6846,6850,6862,6867,6879,6884,6896,6900,6912],[7051,7059,7066,7069,7075,7084,7092,7100,7110,7117,7128,7134,7145,7151,7162,7167,7179,7184,7196,7201,7213,7217,7230],[7372,7381,7388,7391,7398,7401,7410,7418,7429,7434,7446,7451,7463,7467,7479,7484,7496,7501,7513,7517,7529,7533],[7818,7827,7834,7837,7845,7851,7861,7867,7879,7884,7896,7901,7913,7918,7929,7934,7946,7951,7963,7968,7979,7984,7996,8001,8013,8018,8030,8034,8047,8051,8063,8068,8080],[8235,8244,8251,8254,8262,8267,8278,8284,8301,8312,8318,8330,8334,8347,8351,8363,8368,8380,8384],[8509,8518,8534,8543,8551,8562,8568,8580,8584,8596,8601,8614,8618,8630],[8747,8753,8759,8762,8770,8777,8784,8796,8801,8813,8818,8830,8835,8847,8851,8863,8868,8879,8885,8896,8901,8913,8918,8930,8934,8947,8962,8967],[9061,9070,9076,9078,9101,9110,9118,9130,9135,9147,9151,9163,9168,9180,9185,9197,9201,9213,9218,9230,9235,9242],[9352,9362,9371,9385,9394,9401,9413,9418,9435,9452,9463,9479,9485,9497,9502,9514,9518,9530,9535,9547,9551,9564,9568,9580,9585,9597,9601,9614,9618,9635],[9984,9994,10001,10004,10011,10028,10035,10047,10052,10064,10068,10081,10085,10098,10102,10114,10118,10131,10135,10143,10152,10163,10168,10181,10185,10197,10202,10214,10219,10231,10246,10252,10264,10268,10281,10285,10297,10302,10314,10318,10331,10335,10347,10352,10359]],"version":"2.0.0"}.

    Solution:

    Here, the degree is 4 which is even and the leading coefficient is -1 which is negative.

    Therefore, we have case 2 and so asx{"x":[[65,66,67,70,74,78,83,86,89,90,91,92,93,95,97,99,100,101,102,102,102,101,99,95,90,85,80,76,73,71,69,69,68,68,68,68,67,67,67,67],[120,119,118,117,115,113,112,111,110,110,109,108,106,105,105,104,104,104,104,104,104,104,105,106,107,107,108,109,110,111,114,117,120,124,126,128,129,129,130,130,130,130],[138,138,140,144,149,155,161,166,172,177,181,184,186,187,188,189,189,189,188,186,183,179,177,176,175,175,175,174,174,174,174,174,174,175,176,178,180,183,186,189,191,192,193,194,194,195,195,195,195,195,195,195,194,193,192,190,190,189,189,188,188,187,186,186,185,184,183,183,182,182,181],[207,207,207,207,207,207,207,207,208,209,210,211,212,214,215,217,219,220,221,222,223,224,225,226,227,227,228,228,229,229,229,230,230,231,231,232,233,234,236,238,240,243,245,248,250,253,254,256,257,257,258,259,259,260,261,261,261,261,261,261,261,260,260,259,258,257,256,254,252,250,248,247,245,244,243,241,240,239,239,238,237,236,235,234,233,232,231,230,229,229,228,228,227,226,226,226,225,225,224,224,224,224,223,223,223,222,221,220,219,217,216,214,214,213,212,212,211,211,211,210,210,209,209,208,208,207,207,207,207,207,207,207,207,207,207,207,207]],"y":[[63,63,63,63,63,63,63,63,64,65,67,69,73,77,82,88,92,96,99,101,103,106,108,112,115,117,119,121,121,122,122,122,122,121,121,121,120,120,120,120],[68,68,68,68,68,70,71,72,74,76,79,82,85,89,91,94,96,97,99,101,103,105,106,108,109,110,111,111,112,112,113,113,113,113,113,113,113,113,113,113,113,112],[83,83,83,83,83,83,83,83,83,83,83,83,83,83,83,83,83,82,81,80,77,75,73,72,71,71,71,71,70,70,70,70,70,71,71,72,73,75,77,79,80,81,82,83,83,84,84,84,84,84,85,85,86,87,89,90,91,91,92,92,93,93,94,94,95,95,96,96,96,96,96],[77,77,77,79,80,82,83,85,86,88,89,90,90,91,92,92,93,93,93,93,93,92,91,89,87,84,81,78,75,73,72,71,70,69,69,68,68,67,67,67,67,66,66,66,66,66,66,66,67,67,67,68,69,70,70,71,72,72,73,75,76,78,79,80,81,82,82,83,83,83,83,83,83,83,83,83,83,83,83,83,82,81,80,79,78,77,75,74,73,72,71,70,69,68,67,66,66,65,64,63,63,62,62,62,62,62,62,62,62,62,63,63,64,64,65,65,66,66,67,67,68,69,70,71,72,73,74,75,76,76,77,78,78,79,79,79,79]],"t":[[0,118,132,149,165,182,199,216,232,249,271,287,305,321,338,356,371,388,404,421,439,455,471,488,505,522,539,555,571,588,605,621,638,654,672,688,705,722,738,755],[1350,1499,1515,1532,1548,1565,1582,1598,1615,1632,1649,1665,1682,1698,1715,1732,1748,1771,1787,1805,1822,1837,1855,1872,1888,1905,1921,1937,1954,1971,1988,2004,2022,2039,2055,2072,2089,2105,2122,2138,2155,2171],[2716,2883,2898,2915,2932,2949,2965,2982,2999,3015,3031,3049,3065,3082,3098,3115,3183,3199,3215,3232,3249,3272,3289,3305,3323,3338,3355,3372,3388,3405,3422,3550,3565,3582,3598,3615,3632,3649,3665,3682,3698,3715,3732,3748,3772,3789,3806,3855,3871,3888,3905,3922,3938,3955,3972,3988,4005,4021,4038,4055,4073,4088,4105,4122,4139,4155,4171,4187,4205,4222,4238],[4759,4933,4949,4965,4983,4998,5015,5032,5049,5065,5081,5098,5115,5132,5148,5165,5182,5198,5216,5232,5249,5271,5288,5306,5322,5338,5355,5372,5388,5404,5423,5439,5456,5471,5488,5506,5522,5539,5554,5571,5588,5604,5623,5638,5656,5672,5688,5705,5723,5739,5754,5772,5788,5805,5821,5838,5855,5872,5889,5906,5922,5939,5955,5971,5988,6005,6021,6038,6055,6072,6089,6106,6122,6139,6155,6171,6189,6206,6222,6238,6255,6272,6289,6306,6323,6338,6355,6371,6388,6405,6422,6438,6455,6471,6488,6505,6522,6538,6555,6572,6588,6605,6622,6638,6655,6672,6689,6706,6722,6738,6755,6773,6788,6806,6821,6839,6855,6872,6889,6904,6922,6938,6955,6971,6988,7005,7022,7039,7055,7072,7088,7106,7122,7139,7155,7171,7188]],"version":"2.0.0"}, 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fx-{"x":[[78,78,78,78,77,76,74,72,71,69,68,67,66,65,65,64,63,62,60,59,57,56,55,55,55,55,55,55,56,57,58,59,60,61,62,63,63,63,63,63,63,63,63,61,57,54,50,48,47,46,46,45],[42,42,44,47,50,54,56,57,58,59,59,60],[131,130,130,129,128,128,127,127,126,124,123,121,120,119,118,118,117,117,116,116,116,115,115,115,115,115,115,116,118,120,123,125,128,130,131,133,134,135,136],[151,151,151,151,152,152,153,155,156,157,158,159,159,160,160,159,157,155,154,152,151,150,149,149,149,149,150,152,153,154,155,156,157,157,158,159,160,161,163,164,166,167,167,168,168,168,168,169,169,169,170,170,170,170,170,168,167,165,165,164,164,164,164,164,164,164,164,164,164,164,164,164,164,165,165,166,168,170,172,174,175,176,177,178,178],[181,181,182,183,186,189,192,194,197,199,201,202,203,204,205,206,207,207,207,207,207,207,206,205,203,202,200,199,198,197,196,196,196,196],[249,250,252,257,265,274,283,291,296,301,304,306,308,309,310,311,311,309,306,302,299,297,295,293,292,291,291,291,292,293,296,299,302,304,306,308,309,309,310,310,310,309,308,307,306,305,305,304,304,304],[334,334,334,336,340,343,347,350,352,353,355,356,356,357],[370,370,370,370,371,373,375,377,379,381,383,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,400,400,401,401,401,401,401,402,402,404,405,407,408,410,411,412,414,416,418,419,421,423,424,426,427,428,430,431,432,432,433,433,433,432,430,428,426,425,424,423,422,421,419,418,416,415,413,412,410,408,406,404,403,402,401,399,398,396,395,393,392,391,390,389,387,384,382,378,375,373,372,372,372,372,372,372,372,372,372]],"y":[[54,53,53,53,52,52,51,51,50,50,50,50,50,50,50,50,51,53,56,59,63,67,71,75,80,86,91,97,102,107,113,118,122,125,129,132,134,137,138,140,141,141,141,141,141,141,141,141,140,139,138,137],[94,94,94,94,94,94,94,94,94,94,94,94],[52,52,52,52,52,52,52,53,54,56,58,60,62,63,65,67,69,71,73,76,79,82,85,88,92,95,99,102,106,109,113,116,118,120,121,122,122,122,122],[81,81,81,81,81,81,81,82,83,85,87,88,90,92,94,95,97,99,100,101,102,102,103,103,103,103,102,101,100,98,96,95,93,92,91,90,88,87,86,85,85,84,84,84,84,84,84,84,84,84,84,84,84,84,84,85,86,87,88,88,89,89,89,89,90,90,91,92,93,94,94,94,94,95,95,95,96,96,96,96,96,96,96,96,96],[43,43,43,43,43,43,43,45,48,51,54,58,61,66,70,74,78,82,86,90,94,98,102,106,109,112,114,116,117,118,119,119,119,119],[86,86,86,86,86,86,86,86,86,86,86,86,86,86,86,86,86,85,83,82,80,78,77,76,76,75,75,75,75,76,77,79,81,82,84,85,86,86,87,88,89,90,92,94,95,96,96,96,96,96],[82,82,82,82,82,82,82,82,82,82,82,82,82,82],[75,75,76,77,78,80,82,84,85,86,87,88,89,89,90,90,90,90,90,90,90,90,90,89,87,84,81,78,76,75,75,74,74,74,73,72,71,70,70,69,69,69,69,69,69,69,69,69,69,69,70,71,71,72,74,76,78,81,83,86,88,90,91,92,93,93,93,93,93,93,92,90,89,87,85,84,82,80,79,78,77,75,74,72,71,69,68,68,67,67,67,67,67,67,67,67,68,69,69,69,69,70,70,70,70,70]],"t":[[0,250,268,286,301,317,334,351,367,385,401,418,434,451,467,484,501,518,534,551,567,586,601,619,635,652,668,684,701,718,736,752,768,785,801,818,834,851,868,884,901,919,936,951,968,984,1003,1018,1034,1051,1068,1084],[1585,1667,1685,1701,1718,1736,1752,1769,1784,1801,1819,1850],[2351,2518,2535,2552,2568,2586,2634,2652,2668,2686,2701,2718,2736,2752,2768,2785,2802,2818,2836,2852,2868,2886,2902,2918,2935,2953,2969,2985,3002,3018,3036,3053,3068,3085,3101,3119,3135,3152,3169],[3668,3867,3885,3902,3918,3934,3951,3969,3986,4002,4019,4035,4052,4068,4085,4102,4118,4135,4151,4168,4185,4202,4219,4236,4385,4402,4420,4435,4452,4468,4485,4502,4518,4535,4551,4568,4586,4602,4619,4636,4652,4669,4686,4702,4718,4750,4784,4801,4818,4835,4852,4868,4886,4967,4984,5001,5019,5035,5052,5069,5086,5101,5119,5136,5152,5168,5185,5202,5219,5236,5251,5269,5300,5319,5335,5353,5368,5385,5402,5418,5434,5452,5469,5484,5502],[6027,6117,6134,6152,6169,6185,6201,6219,6235,6252,6269,6285,6302,6318,6335,6352,6368,6385,6402,6418,6436,6452,6468,6486,6501,6518,6535,6552,6569,6586,6602,6619,6635,6652],[7180,7418,7436,7451,7468,7485,7501,7518,7534,7552,7569,7584,7601,7618,7635,7652,7750,7769,7785,7801,7819,7835,7851,7868,7886,7902,7919,8034,8051,8068,8085,8102,8118,8135,8152,8169,8185,8201,8219,8235,8252,8268,8284,8302,8318,8335,8352,8368,8433,8453],[9316,9466,9485,9502,9519,9535,9552,9568,9585,9602,9618,9635,9651,9669],[10239,10484,10501,10519,10536,10552,10568,10585,10602,10618,10636,10651,10669,10685,10701,10718,10735,10751,10768,10785,10801,10819,10836,10852,10868,10886,10902,10919,10935,10951,10968,10984,11002,11018,11035,11051,11069,11085,11102,11119,11135,11151,11168,11185,11201,11219,11236,11253,11268,11285,11301,11319,11336,11353,11368,11385,11402,11418,11436,11452,11468,11485,11502,11520,11536,11552,11570,11585,11601,11618,11635,11652,11670,11686,11703,11718,11736,11752,11769,11785,11802,11819,11836,11852,11868,11885,11902,11918,11936,11952,11969,11984,12001,12018,12035,12051,12068,12086,12101,12118,12136,12153,12169,12185,12202,12219]],"version":"2.0.0"}.

    Determine the end behaviour of the polynomialfx=2x3+7x2{"x":[[222,219,217,212,211,208,205,203,200,198,196,196,196,196,199,203,208,212,216,217,216,213,210,206,201,197,193,188,184],[175,175,176,178,179,181,185,190,195,202,208,213,217,221],[296,296,295,293,291,288,284,279,275,271,267,265,264,264,265,268,272,276,282,288,295,298],[307,307,308,311,313,319,325,332,338,343,348,351,352,350,345,339,333,328,324,320,317,315,313,313],[371,368,367,364,363,359,356,354,354,354,357,361,366,372,379,385,392,396],[416,417,417,422,425,431,438,445,451,456,458,458,457,454,450,446,441,437,434,431,429,428,427,426,427],[492,493,494,497,501,506,510,515,520,523,527,530,532],[505,504,503,504,506,509,513,516,520,525,529],[558,557,557,557,558,561,565,568,572,576,581,587,594,601,606,609,610,608,604,598,592,586,581,575,571,568,565,565,564,566,569,574,581,588,594,600,605,610,614,616,617],[644,645,645,646,648,651,656,662,668,672,676,678,677,674,669,663,658,652,648,644,640,636,633,631],[692,692,691,689,688,685,682,678,675,672,670,669,669,673,678,685,693,700,708,713,717,720,721],[709,709,709,711,714,717,721,724,727,728,729,728,726,724,722,721,721,722,723,724,725,725,725,722,718,714,711,708,707,706],[746,746,746,748,749,752,753,757,762,767,773,778,782,785,788],[771,770,770,770,770,770,771,772,773,775,776,778,779,779,779],[813,813,813,813,814,814,815,818,820,825,830,837,845,852,858,863,866,869,871,872,873,873,873,873,873,871,870,869,869,869,870,870,870,868,867],[820,822,823,830,832,839,843,851,857,864,868,873,876,879],[898,900,904,907,908,912,915,918,921,924,927,929,929,929,926,921,915,909,904,899,896,895,894,894,894,895,897],[942,943,944,942,940,937,934,932,930,930,930,933,937,942,947,954,960,965,968,971,971],[955,955,954,955,957,960,965,972,979,986,991,994,995,994,992,988,985,982,981,981,981,983,986,989,993,996,999,1002,1005,1007,1008]],"y":[[214,209,204,198,198,197,197,198,200,204,209,216,225,235,248,261,273,283,293,300,307,314,320,324,327,327,327,326,325],[284,280,277,274,273,273,272,270,268,266,264,263,262,262],[205,203,202,200,200,202,206,210,216,223,232,241,251,261,271,279,286,290,292,294,295,295],[251,248,244,241,241,240,240,241,243,245,248,251,256,261,267,273,279,282,283,284,284,284,283,282],[235,234,233,232,233,237,242,247,253,258,263,268,273,277,280,282,283,284],[196,194,193,192,194,198,203,210,217,225,232,241,249,258,266,273,279,284,287,288,289,288,286,284,282],[240,236,233,234,235,237,238,238,238,238,237,236,236],[257,256,254,254,255,257,258,259,260,260,259],[190,186,184,181,181,181,181,181,181,181,181,181,184,188,192,198,205,212,219,226,233,239,244,248,252,255,258,261,262,263,264,265,265,265,265,265,264,263,262,261,260],[213,212,210,209,210,213,217,220,224,227,229,232,235,238,242,245,248,249,250,250,249,249,249,250],[212,210,207,207,207,209,212,215,218,222,226,232,239,244,248,251,251,251,249,247,243,240,239],[152,149,146,146,146,147,148,148,149,150,151,153,155,157,158,158,159,160,161,163,165,167,168,170,172,172,173,173,172,171],[203,201,200,201,202,204,205,206,206,206,206,206,206,206,205],[186,186,189,191,193,197,204,207,210,215,217,221,224,227,228],[172,169,167,165,165,166,168,169,169,169,168,167,166,166,166,166,167,167,168,170,171,174,178,185,193,202,212,221,229,235,240,244,247,248,248],[209,206,204,202,203,207,208,209,209,208,207,207,207,207],[200,197,197,198,199,202,206,209,213,217,221,225,229,232,234,235,236,236,235,234,233,231,228,227,226,225,225],[205,202,200,200,202,204,207,210,213,217,221,224,228,230,232,233,234,234,232,230,229],[149,145,142,145,146,146,147,149,151,153,155,157,160,162,164,167,169,171,172,173,174,176,177,178,179,180,181,182,182,182,182]],"t":[[0,10,17,20,27,32,39,49,56,66,73,83,90,100,106,117,123,133,140,150,156,167,173,184,190,200,206,217,223],[320,327,333,336,341,349,356,366,373,383,389,400,406,417],[586,593,600,603,609,616,623,633,640,650,656,667,673,684,690,701,706,717,723,734,740,747],[862,870,876,880,886,890,899,906,917,923,934,940,950,957,967,973,984,990,1001,1007,1017,1023,1034,1040],[1182,1188,1191,1193,1198,1206,1216,1223,1234,1240,1251,1257,1267,1273,1284,1290,1302,1306],[1454,1461,1466,1469,1475,1483,1490,1501,1507,1517,1523,1534,1540,1551,1557,1568,1573,1585,1590,1602,1607,1618,1624,1635,1640],[1766,1774,1781,1784,1791,1800,1806,1817,1823,1834,1840,1851,1857],[1954,1962,1968,1977,1984,1990,2001,2007,2018,2023,2035],[2166,2173,2178,2180,2190,2199,2207,2218,2224,2234,2240,2252,2257,2268,2274,2285,2290,2302,2307,2318,2324,2335,2340,2352,2357,2368,2374,2385,2390,2402,2407,2419,2424,2435,2440,2452,2457,2469,2474,2485,2489],[2637,2645,2651,2653,2659,2666,2674,2684,2690,2701,2707,2718,2724,2735,2741,2752,2757,2769,2774,2785,2791,2802,2807,2819],[2954,2962,2969,2972,2979,2982,2990,3002,3007,3018,3024,3036,3041,3052,3057,3069,3074,3086,3091,3102,3107,3119,3123],[3283,3291,3297,3300,3308,3316,3324,3334,3341,3352,3357,3369,3374,3386,3391,3403,3407,3432,3441,3449,3458,3469,3474,3486,3491,3503,3508,3519,3524,3536],[3704,3713,3721,3723,3728,3736,3741,3752,3758,3769,3774,3786,3791,3803,3808],[3905,3919,3925,3931,3936,3936,3944,3952,3953,3961,3968,3974,3985,3991,4003],[4119,4126,4132,4135,4144,4151,4158,4170,4174,4187,4191,4203,4208,4220,4225,4237,4241,4253,4258,4270,4275,4286,4291,4303,4308,4320,4325,4337,4341,4353,4358,4370,4375,4387,4390],[4516,4523,4530,4532,4540,4544,4551,4558,4570,4575,4586,4591,4603,4608],[4729,4737,4744,4747,4754,4759,4769,4775,4787,4791,4803,4808,4820,4825,4836,4841,4853,4858,4870,4875,4886,4891,4903,4908,4920,4925,4936],[5028,5035,5043,5058,5069,5075,5087,5091,5104,5108,5120,5125,5137,5141,5154,5158,5170,5175,5187,5192,5199],[5358,5359,5365,5367,5375,5383,5391,5403,5408,5420,5425,5437,5441,5453,5458,5470,5475,5487,5492,5503,5508,5520,5525,5537,5541,5553,5558,5570,5575,5587,5590]],"version":"2.0.0"}.

    Solution:

    Here, the degree is 3 which is odd and the leading coefficient is 2 which is positive. Therefore, we have case 3 and so as 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fx-{"x":[[88,88,88,88,87,86,86,85,84,83,81,79,77,75,74,72,71,70,68,67,66,65,64,64,63,63,64,65,67,70,71,73,74,75,75,75,75,75,73,71,69,66,62,58,54,52,49,48,48,48,48],[52,53,56,61,67,72,76,79,80,81,81,82],[134,134,134,131,130,128,127,126,125,124,123,121,118,116,114,111,110,110,109,109,110,110,112,114,117,119,122,126,128,129],[140,140,141,142,143,144,146,148,151,153,156,157,158,159,160,160,159,156,152,149,146,144,143,143,145,148,153,157,161,163,165,166,166,166,166,167,167,168,168,169,169,170,170,171,171,172,172,171,169,165,162,160,159,158,157,157,156,156,156,157,159,164,170,175,180,183,185],[200,200,201,204,208,211,214,217,218,218,218,218,218,218,218,218,217,217,217,216,216,216,215,215],[254,255,258,266,277,290,301,310,317,323,326,328,328,329,329,328,324,319,312,306,302,299,298,298,298,299,301,305,310,313,316,318,319,320,320,320,320,320,320,320,320,320,320,320,319,318,317,317,316,316],[349,349,351,354,356,359,360,361,361],[379,379,379,379,379,379,379,379,380,382,384,387,390,393,397,400,403,405,408,410,412,415,416,418,418,419,420,421,423,425,427,429,432,434,437,439,441,444,445,447,448,449,450,451,451,451,449,445,439,432,426,420,415,412,410,408,405,403,402,400,398,397,395,394,393,391,390,389,387,387,386,386,386,385,384,382,380,378,377,377,377]],"y":[[39,39,39,38,38,37,36,36,35,35,34,34,34,34,34,34,34,34,34,35,37,41,46,51,57,63,68,74,79,84,89,94,100,105,111,117,122,126,128,129,130,131,131,131,130,127,125,122,119,116,114],[92,92,92,92,92,92,92,92,92,92,92,92],[35,35,35,34,35,36,37,38,39,40,42,47,52,57,64,71,78,83,88,93,97,100,104,107,110,113,114,115,115,115],[89,89,89,89,89,89,89,89,89,89,89,90,92,95,99,102,106,108,111,112,113,113,113,113,113,111,108,104,101,98,97,96,95,95,95,95,95,94,94,94,94,93,93,93,93,93,93,93,93,94,95,96,97,98,100,102,104,105,106,106,107,107,106,104,101,100,99],[26,26,26,26,27,30,36,45,55,63,70,77,83,88,93,96,99,100,101,102,103,104,104,104],[75,75,75,75,75,74,72,71,71,70,70,70,70,70,70,69,67,64,61,59,57,57,56,56,56,56,56,57,58,60,62,64,66,67,68,68,69,69,69,70,70,71,72,74,77,78,80,81,82,82],[69,69,69,69,69,69,69,69,69],[62,63,63,63,64,65,66,67,69,70,72,73,75,76,77,77,78,78,78,76,74,70,67,64,61,58,56,53,51,49,47,45,45,44,44,44,45,46,48,50,52,55,57,60,62,64,67,69,71,72,73,73,73,73,73,71,69,66,64,62,59,57,54,53,51,50,49,48,48,47,47,47,47,48,49,51,53,55,56,57,57]],"t":[[0,2,166,181,198,215,232,248,265,282,298,315,331,348,365,382,398,415,432,448,465,482,499,515,532,549,573,591,607,623,641,657,674,690,706,724,740,756,774,789,807,823,840,857,873,890,907,923,940,956,973],[1417,1498,1514,1531,1547,1574,1590,1606,1624,1640,1749,1765],[2283,2289,2407,2423,2440,2457,2473,2489,2506,2523,2539,2556,2574,2590,2606,2623,2640,2657,2674,2689,2706,2724,2740,2757,2774,2790,2806,2837,2847,2865],[3266,3384,3398,3415,3431,3449,3464,3480,3497,3514,3531,3547,3573,3590,3606,3623,3639,3656,3674,3690,3706,3723,3740,3848,3865,3881,3897,3914,3930,3947,3964,3981,3997,4014,4031,4047,4072,4089,4106,4123,4139,4156,4172,4190,4206,4223,4273,4289,4306,4323,4340,4356,4372,4389,4407,4423,4439,4456,4473,4490,4506,4523,4540,4556,4573,4591,4606],[5049,5148,5165,5182,5198,5215,5231,5248,5264,5281,5298,5315,5331,5348,5365,5381,5398,5414,5432,5448,5465,5481,5497,5515],[5999,6248,6264,6281,6298,6314,6331,6348,6364,6380,6397,6413,6430,6447,6506,6523,6540,6556,6573,6590,6606,6623,6639,6657,6673,6723,6739,6756,6773,6789,6806,6823,6840,6856,6873,6889,6906,6923,6939,6956,6973,6989,7006,7023,7040,7056,7073,7089,7107,7165],[7815,7931,7947,7964,7980,7997,8014,8031,8123],[8651,8790,8807,8823,8841,8858,8874,8890,8906,8924,8939,8957,8973,8990,9007,9024,9039,9057,9074,9090,9106,9125,9140,9158,9174,9189,9208,9225,9241,9257,9274,9290,9306,9323,9341,9357,9374,9390,9407,9424,9440,9456,9474,9491,9507,9523,9541,9556,9574,9590,9606,9624,9640,9656,9674,9690,9706,9723,9741,9756,9774,9791,9806,9823,9841,9857,9873,9890,9907,9924,9941,9957,9980,9997,10014,10030,10047,10074,10090,10106,10124]],"version":"2.0.0"}.

    Determine the end behaviour of the polynomial fx=-7x5+3x2-1{"x":[[217,214,210,203,197,193,188,184,182,181,180,180,182,185,189,192,194,195,194,191,188,185,182,179,175,171,169],[163,163,165,171,174,183,192,197,202,205],[301,301,301,299,298,294,290,284,279,273,267,262,257,254,253,252,254,257,261,266],[311,308,307,307,308,311,313,316,320,324,330,336,340,343,343,342,339,334,329,324,319,315,312,309,308,307],[355,354,354,353,352,350,347,344,342,340,339,340,342,345,349,353,358],[386,385,385,387,390,394,399,405,413,420,424,424,421,416,410,405,401,397,394,391,389,388],[449,450,453,459,462,468,473,477,481,485,489,494,500],[463,461,460,459,460,462,466,470,473,476,480,484,486],[523,523,525,527,531,534,538,543,548,554,559,565,570,575,576],[570,568,567,568,570,573,578,582,587,592,598,603,608,614,618,621,623,624,626,627,628,628,627,624,622,619,617,615,615,615,616,617,617],[602,601,599,598,600,604,607,610,614,617,620,624,628],[662,663,664,669,671,675,677,679,681,682,682,682,681,678,674,669,665,661,658,655,653,651,650],[698,697,696,694,692,689,687,685,684,685,687,690,694,697,701,705,707],[722,722,722,722,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,737,737,736,735,733,732,730,728,726,724,722,721,720,721,722,722,722,722,722,723,723,723,723,724,725,726,728,730,733,735,737,739,739,738,735,732,728,725,721,719,716,715],[738,739,741,742,743,746,750,755,759,763,767,770,772],[755,754,754,754,754,755,756,757,757,757,757,758,759,760,762],[798,800,801,805,810,814,817,819,819,819,817,814,809,805,801,798,796,796,798,801,806,810,814,818,821,824,826,826,826,824,821,818,815,811,807,803,800,797,796,796],[838,841,842,843,845,850,854,858,861,863,864,864,863,860,857,854,850,847,844,841,838,837,836,837,840,844],[877,879,880,881,880,879,876,874,871,869,869,869,871,874,877,881,886,890,892],[908,908,907,908,909,912,915,917,919,921,921,921,920,918,916,914,913,913,913,914,915,916,917,918,919,921],[937,938,939,940,942,945,949,954,959,964,968,970],[994,995,996,997,998,999,999,1001,1002,1004,1006,1007,1007,1007,1007,1007,1006,1006,1005]],"y":[[155,149,144,140,140,141,142,144,148,153,160,169,179,190,201,212,223,232,242,251,258,264,269,272,274,275,274],[219,215,212,211,211,211,210,210,210,210],[143,140,139,139,139,142,145,149,153,158,167,177,190,204,216,227,236,244,249,254],[199,195,193,191,191,190,190,189,189,189,189,191,194,197,202,207,213,219,224,228,231,233,234,235,235,235],[189,186,184,184,185,188,192,197,202,209,216,224,231,236,239,241,242],[141,139,137,137,140,144,150,157,165,175,186,197,211,224,236,247,255,262,266,267,266,265],[202,198,197,198,198,199,200,201,201,201,201,200,199],[226,226,226,227,228,229,230,231,231,231,231,230,230],[208,209,211,213,214,214,214,213,212,211,210,210,210,209,209],[154,151,150,151,151,153,154,155,156,156,156,156,156,157,158,160,163,167,172,180,188,197,205,212,220,227,235,243,248,252,256,256,255],[209,208,205,203,203,203,202,200,199,198,198,198,198],[190,189,189,189,190,193,194,198,202,207,212,218,223,229,233,236,238,239,239,238,237,235,233],[196,195,196,199,202,207,211,215,219,223,226,229,230,231,230,228,226],[121,120,119,118,119,120,120,120,120,120,121,121,121,121,121,121,121,121,121,121,120,121,121,121,121,121,122,123,124,125,125,126,126,126,126,127,128,129,131,133,135,137,138,139,139,139,140,140,142,144,147,150,152,155,156,157,157,157,156,154,153,153],[202,201,201,201,201,201,201,201,202,203,203,203,203],[178,179,181,183,189,198,206,211,216,220,222,223,224,222,221],[160,156,154,153,153,154,155,157,158,161,164,168,171,175,177,179,181,182,184,186,189,191,194,197,200,204,208,212,216,219,221,222,222,222,220,219,218,216,215,214],[184,182,182,182,182,184,187,189,192,195,198,202,205,209,213,215,217,218,218,217,216,214,213,211,209,207],[185,183,182,181,182,184,187,192,197,202,208,213,217,219,221,221,220,216,214],[136,134,132,131,131,132,132,132,132,133,134,136,139,142,146,149,151,154,155,156,156,156,155,154,153,153],[189,189,189,190,190,191,191,190,189,189,189,189],[153,150,149,150,154,158,164,172,181,191,199,206,211,214,217,219,219,218,217]],"t":[[0,9,16,18,27,37,42,54,59,71,76,87,92,104,109,121,126,137,142,154,159,171,176,187,192,204,208],[322,329,338,345,353,359,371,376,387,393],[548,556,561,563,567,576,586,593,604,609,620,626,637,643,654,659,671,676,687,693],[840,848,855,858,864,868,876,887,893,904,909,921,926,937,943,954,959,971,976,988,993,1004,1010,1021,1026,1033],[1165,1171,1176,1178,1193,1203,1209,1220,1226,1237,1243,1253,1259,1271,1276,1287,1293],[1660,1669,1676,1679,1685,1693,1703,1710,1720,1726,1738,1743,1754,1760,1771,1776,1788,1793,1805,1810,1822,1825],[1965,1972,1978,1980,1987,1993,2003,2010,2021,2026,2038,2043,2055],[2185,2192,2197,2199,2205,2210,2221,2227,2238,2243,2255,2260,2267],[2394,2402,2418,2426,2436,2443,2454,2460,2471,2477,2488,2493,2505,2510,2517],[2657,2665,2672,2675,2682,2687,2693,2705,2710,2722,2727,2738,2743,2755,2760,2772,2777,2789,2793,2805,2810,2822,2827,2838,2843,2855,2860,2872,2877,2889,2894,2905,2909],[3003,3012,3019,3022,3035,3043,3054,3060,3072,3077,3089,3094,3106],[3227,3233,3241,3244,3250,3255,3260,3270,3277,3288,3293,3305,3310,3322,3327,3339,3344,3355,3360,3372,3377,3389,3394],[3519,3530,3536,3544,3552,3560,3571,3577,3588,3594,3605,3610,3622,3627,3639,3644,3651],[3837,3844,3851,3853,3885,3894,3905,3910,3922,3939,3944,3961,3972,3977,3986,3994,4005,4011,4023,4039,4044,4077,4087,4106,4111,4123,4127,4140,4144,4156,4161,4173,4177,4211,4227,4239,4244,4256,4261,4273,4278,4289,4294,4311,4327,4339,4344,4356,4361,4373,4377,4390,4394,4406,4411,4423,4427,4439,4444,4456,4461,4468],[4641,4653,4660,4667,4669,4678,4689,4694,4706,4711,4723,4727,4740],[4860,4871,4876,4883,4888,4894,4906,4911,4923,4927,4940,4944,4956,4961,4968],[5071,5078,5084,5087,5094,5104,5111,5123,5128,5140,5144,5157,5161,5173,5178,5190,5194,5207,5211,5223,5228,5240,5244,5253,5261,5272,5278,5290,5294,5306,5311,5323,5328,5340,5344,5357,5361,5373,5378,5390],[5496,5502,5507,5509,5516,5521,5528,5540,5544,5557,5561,5574,5578,5590,5595,5607,5611,5624,5628,5640,5644,5657,5661,5674,5678,5690],[5748,5754,5763,5772,5789,5795,5806,5811,5823,5828,5840,5845,5856,5861,5873,5878,5890,5895,5902],[6033,6041,6047,6050,6058,6063,6072,6078,6090,6095,6107,6111,6124,6128,6140,6145,6157,6161,6174,6178,6190,6195,6207,6211,6224,6228],[6375,6383,6390,6393,6402,6403,6411,6423,6428,6441,6445,6452],[6578,6587,6594,6596,6605,6611,6622,6628,6641,6645,6657,6661,6674,6678,6691,6695,6707,6723,6728]],"version":"2.0.0"}

    Solution:

    Here, the degree is 5 which is odd and the leading coefficient is -7 which is negative. Therefore, we have case 4 and so as 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    Zeros of Polynomial Functions

    The zeros of a polynomial are the x values that make the polynomial equal to zero. Often, we denote a polynomial function using the notation p(x). Therefore, the zeros can be found by equating p(x) to zero and solving for x. The zeros are also the x-intercepts of the function.

    Find the zeros of the polynomial function px=x-1x-2x-3

    Solution:

    Equating p(x) to zero, we obtainx-1x-2x-3=0{"x":[[161,157,153,149,147,144,139,136,131,126,121,117,113,109,108,109,111,116,121,127,134,142,145],[153,152,151,151,153,155,159,163,168,173,179,185,190,194,195,195,193,190,187,183,179,175,171,167,164,161,159,159,160],[209,209,209,209,207,205,203,200,197,195,194,194,194,195,197,200,203,207,211,215,218],[242,243,244,247,250,254,258,261,263,264,265,267,268],[292,293,293,295,297,299,300,300,300,299,298,297,297,296,296],[304,303,303,305,307,311,316,321,328,334,342,352,359,365,371,374,377,378,378,376,374,370,365,359,354,349,344,341,339,338],[440,439,438,438,437,436,434,431,427,422,417,411,406,402,400,398,398,400,402,405,409,414,418,423],[448,447,447,447,450,451,455,460,465,471,475,478,480,479,477,475,472,468,464,460,456,452,450,447,446,446],[495,496,494,492,490,487,484,482,480,479,478,479,480,482,484,487,489,492,495,498],[526,526,525,526,527,529,531,532,534,535,537,539,541],[561,561,561,563,565,568,571,574,576,579,582,585,588,590,591,593,593,593,591,588,584,580,576,573,570,567,566,565,565,566,568,570,574,578,583,587,593,598,603,607,611,614,616],[611,609,608,609,612,615,619,624,631,639,647,652,654,655,655,655,653,652,650,644,642,639,637,636,635],[726,726,726,726,726,725,724,722,719,716,712,708,704,701,698,695,694,694,694,696,698,701,704,708,710],[734,734,734,735,736,738,741,744,748,752,755,758,761,761,761,758,755,751,747,744,741,738,736,734,733,732,733,734],[772,774,774,775,775,775,774,772,770,769,767,766,765,765,765,766,768,770,771,773,775,779,780],[800,800,801,802,804,804,806,808,811,813,816,819,822,824,827,830],[847,847,848,848,849,851,853,856,858,861,867,868,868,868,866,864,861,858,856,854,852,851,850,851,853,856,860,865,870,875,879,883,886,887,887,887,885,883,879,875,871,867,863,858,853,849,845,841,839],[873,874,875,879,881,884,887,891,895,899,905,909,913,916,918,920,922,923,923,923,921,919,917,914,911,909,907,904,901,899],[946,947,949,952,953,955,956,958,960,962,965,967,969,971,972,973,975,975],[950,950,951,953,955,958,962,965,969,973,976,980,983],[1043,1043,1043,1042,1042,1041,1039,1036,1032,1028,1024,1020,1015,1012,1009,1005,1003,1001,999,998,997,997,997,1000,1003,1008,1013,1019,1024,1030,1035,1041,1046,1050,1054,1057,1061,1063,1066,1068,1069,1069,1069,1069,1067,1066,1063,1061,1058,1056,1053,1050,1047,1044,1040,1035,1029,1026]],"y":[[122,122,122,122,124,125,131,134,141,149,159,170,183,197,209,221,231,240,247,254,260,265,267],[200,196,191,188,187,186,185,184,183,184,185,187,189,192,196,200,205,210,213,215,217,218,219,220,220,221,221,222,222],[178,176,174,173,173,176,179,184,189,194,199,205,211,216,222,225,227,229,229,228,227],[206,205,205,205,205,206,207,208,208,208,208,207,205],[170,171,172,177,183,191,197,203,209,213,218,224,229,233,236],[129,126,124,124,124,125,128,132,138,143,149,155,162,170,178,186,195,204,212,220,226,231,235,238,239,240,240,240,240,240],[128,126,125,124,124,126,128,130,133,137,142,150,160,171,184,197,207,217,228,237,245,252,257,261],[195,195,194,193,192,192,192,193,194,196,198,201,205,210,215,220,224,228,231,232,233,234,235,235,236,237],[191,191,192,193,196,199,202,207,212,219,225,230,235,237,239,240,240,239,238,235],[213,212,211,211,212,213,213,214,214,214,214,214,214],[183,181,178,177,175,174,172,170,169,168,168,169,172,175,179,183,187,191,196,200,204,208,211,214,216,218,219,220,221,222,223,224,225,225,224,223,222,221,220,220,220,220,220],[131,130,128,129,131,134,137,140,145,152,160,170,178,187,197,206,215,223,229,240,241,242,242,241,240],[128,126,124,121,120,120,121,123,127,131,137,145,155,166,177,188,197,205,213,218,223,227,229,231,231],[176,174,173,172,172,172,172,173,174,176,179,182,186,190,195,199,203,207,209,210,210,210,210,210,210,211,211,211],[177,175,174,173,174,175,176,177,179,181,184,187,191,196,202,207,211,213,215,215,215,213,212],[195,194,193,193,194,195,195,195,196,196,196,197,197,197,196,195],[167,164,163,160,160,160,160,160,160,161,163,165,167,169,172,175,177,179,182,183,184,185,185,186,187,188,189,190,192,195,199,203,208,213,217,221,224,227,229,231,232,233,233,232,230,228,225,222,219],[119,117,116,115,116,117,119,122,125,130,136,144,154,164,173,183,194,203,212,221,228,235,240,245,247,248,248,246,243,240],[177,176,175,174,174,174,174,174,175,175,175,176,177,177,178,178,179,180],[193,194,195,195,196,196,197,197,197,197,196,195,194],[142,139,136,135,136,136,138,139,140,142,143,145,148,151,156,160,165,170,176,181,186,192,196,200,204,207,210,212,212,212,211,208,206,204,202,199,197,194,191,188,185,181,177,172,168,163,160,157,154,153,152,151,150,149,149,149,148,148]],"t":[[0,9,13,21,29,30,38,43,54,60,70,77,91,93,105,110,120,126,137,143,154,160,167],[296,302,311,319,326,337,343,354,360,370,379,391,393,405,410,420,426,437,443,454,460,471,480,491,493,505,510,520,526],[647,653,661,670,678,687,693,704,710,720,726,737,743,754,760,771,776,788,793,804,809],[936,949,960,969,977,987,993,1004,1010,1022,1027,1038,1043],[1182,1199,1205,1210,1221,1227,1238,1243,1255,1260,1272,1277,1288,1294,1305],[1473,1481,1486,1489,1495,1503,1510,1521,1527,1538,1544,1555,1560,1572,1577,1589,1594,1605,1610,1622,1627,1639,1644,1655,1660,1672,1677,1689,1694,1705],[2083,2091,2098,2101,2108,2111,2119,2127,2138,2144,2155,2160,2172,2177,2189,2194,2206,2210,2222,2227,2236,2244,2255,2261],[2428,2435,2441,2443,2455,2460,2472,2477,2489,2494,2506,2511,2523,2527,2539,2544,2556,2561,2573,2577,2589,2594,2606,2611,2623,2627],[2778,2787,2803,2811,2821,2827,2839,2844,2856,2861,2872,2878,2889,2894,2906,2911,2923,2928,2939,2944],[3069,3077,3086,3094,3103,3111,3122,3127,3139,3144,3156,3161,3173],[3311,3320,3327,3330,3337,3344,3354,3361,3372,3378,3390,3394,3403,3411,3422,3428,3440,3444,3456,3461,3473,3478,3490,3494,3507,3511,3523,3528,3540,3544,3557,3561,3573,3578,3590,3594,3607,3611,3623,3628,3640,3645,3657],[3816,3824,3830,3839,3844,3855,3861,3873,3878,3890,3895,3907,3911,3923,3928,3940,3945,3957,3961,3989,3995,4003,4013,4023,4028],[4214,4222,4229,4231,4239,4244,4255,4261,4273,4278,4290,4295,4307,4311,4324,4328,4340,4345,4357,4361,4374,4378,4391,4395,4402],[4558,4566,4571,4574,4580,4588,4595,4606,4611,4623,4628,4640,4645,4657,4661,4674,4678,4691,4695,4707,4712,4724,4728,4741,4745,4758,4761,4774],[4877,4885,4892,4894,4906,4915,4915,4924,4931,4941,4945,4957,4961,4974,4978,4991,4995,5007,5012,5024,5028,5040,5043],[5164,5169,5178,5190,5198,5206,5212,5224,5228,5240,5245,5257,5262,5274,5278,5290],[5410,5416,5422,5425,5432,5439,5445,5457,5462,5474,5496,5503,5512,5523,5528,5540,5545,5557,5562,5573,5578,5591,5595,5612,5623,5628,5640,5645,5657,5662,5674,5678,5690,5695,5707,5712,5724,5728,5741,5745,5757,5762,5774,5778,5790,5795,5807,5812,5824],[5964,5973,5980,5982,5990,5995,6003,6012,6024,6029,6042,6045,6058,6062,6075,6079,6091,6095,6108,6112,6125,6129,6137,6145,6156,6162,6174,6179,6191,6196],[6384,6393,6400,6403,6412,6415,6422,6429,6441,6445,6458,6462,6474,6479,6491,6495,6508,6512],[6630,6662,6672,6679,6692,6696,6709,6712,6725,6729,6742,6746,6759],[6959,6968,6975,6978,6987,6987,6996,7010,7012,7026,7029,7042,7046,7059,7062,7075,7079,7092,7096,7109,7112,7125,7129,7142,7146,7158,7162,7175,7179,7191,7196,7208,7212,7224,7229,7241,7246,7258,7262,7275,7279,7291,7296,7308,7312,7325,7329,7342,7346,7358,7362,7375,7379,7391,7396,7408,7412,7420]],"version":"2.0.0"}.

    Solving for x, we get x=1,x=2,x=3{"x":[[80,78,76,75,75,76,78,80,84,88,94,102,110,117,122,124,124,122,119,116,112,108,103,98,93,88,82,78,75,74,75,76],[155,155,154,153,150,147,144,140,137,134,132,129,127,126,126,127,129,133,136,140,145,150,155,161,164],[196,195,194,194,195,198,202,206,210,215,220,225,230,233,235],[214,210,208,207,207,208,209,212,214,217,219,222,226,230,234,239],[275,275,274,276,277,278,280,282,284,287,289,290,290,290,289,288,288,289,290,291],[339,337,335,334,335,336,338,340,342,344,346,347,347,346,344,341,338,335,332,330,328,327],[378,376,375,375,376,378,380,383,386,390,395,400,404,406,407,407,405,402,399,396,392,389,386,383,381,380,379],[443,442,442,441,440,437,433,428,425,422,419,417,415,412,411,411,411,413,416,420,425,429,435,439,442,446,447],[484,483,483,483,484,484,485,485,486,487,488,490,493,495,498,501,502],[483,483,483,484,485,486,489,492,495,498,501,504,506],[528,526,525,523,523,522,521,520,519,519,520,522,525,529,533,537,541,546,550,554,556,558,558,557,555,552,548,543,539,535,531,529,528,528,529,531,533,537,542,547,551,556,559,562,564,565,566],[611,611,611,611,611,611,611,610,609,607,605,602,600,597,595,595,594],[652,652,652,652,653,653,654,655,657,659,661,664,666,670,672,675,676,676,675,673,669,666,662,659,656,653,650,649,648],[693,693,693,693,692,689,686,684,681,679,677,676,675,675,675,677,679,681,684,688,693,698],[743,743,742,742,742,743,744,746,748,751,753,756,758,760,762,763],[739,740,741,743,744,746,749,752,756,759,762,765,768,770],[810,810,810,810,811,812,814,817,820,824,827,829,831,832,832,831,828,825,821,818,814,811,808,806,805,804,804,805,808,811,815,819,824,828,833,836,838,840,841,841,840,838,834,830,825,821,817,812,808,804,800,795,791,786]],"y":[[186,185,184,182,181,180,178,177,176,176,176,176,178,182,186,191,196,202,205,208,211,213,216,218,220,221,222,222,220,219,219,219],[169,168,167,166,166,166,168,170,173,176,178,183,188,194,200,206,210,214,216,218,219,219,218,218,217],[178,178,178,179,179,180,182,183,185,185,185,185,185,184,184],[200,200,200,201,202,203,204,205,206,207,207,208,207,207,205,204],[150,149,147,149,152,156,160,166,171,177,183,189,196,202,207,212,215,215,215,214],[214,214,214,213,212,212,212,212,213,214,215,217,219,222,225,228,230,232,232,232,232,232],[165,164,162,161,161,161,161,161,162,163,165,167,169,173,176,181,185,189,193,196,198,200,200,200,200,200,200],[154,152,149,149,149,150,152,155,157,161,165,170,176,183,189,194,199,201,203,204,204,204,203,202,200,197,196],[169,167,166,165,165,166,167,168,169,170,171,171,171,172,172,172,172],[188,189,190,191,191,191,192,192,192,191,190,189,188],[158,157,156,154,153,152,151,148,145,142,140,138,136,135,135,135,135,138,141,144,148,152,155,159,163,168,172,176,180,184,187,189,190,191,192,192,193,195,198,200,202,204,204,204,204,204,204],[210,212,213,217,218,220,221,224,225,227,227,228,228,228,227,226,226],[163,160,159,157,155,154,153,153,152,153,153,155,157,159,163,166,171,175,179,184,187,189,191,192,193,193,193,193,194],[158,156,155,154,154,156,157,159,162,165,169,174,180,185,191,194,196,197,197,196,194,192],[160,159,158,157,158,158,159,160,160,161,161,161,161,161,161,161],[182,182,183,183,183,184,184,184,183,181,180,178,177,176],[126,125,123,122,121,121,121,121,122,123,124,126,128,131,133,136,138,140,142,144,146,147,148,149,150,151,152,153,155,156,158,160,163,166,169,173,177,180,184,187,191,195,198,201,204,206,208,210,211,213,215,217,218,219]],"t":[[0,8,13,15,25,33,45,50,61,67,78,83,95,100,112,117,128,133,145,150,162,167,178,183,195,200,212,217,229,233,250,257],[401,409,413,416,428,436,446,453,462,470,479,483,495,500,512,517,528,533,545,550,561,567,578,583,590],[722,730,736,739,747,751,761,767,778,784,795,800,812,817,829],[919,924,930,936,945,953,953,962,970,979,984,995,1000,1012,1017,1029],[1168,1176,1183,1192,1200,1210,1217,1228,1234,1245,1250,1262,1267,1279,1284,1296,1301,1326,1334,1346],[1523,1530,1534,1544,1560,1567,1578,1584,1595,1600,1612,1617,1629,1634,1646,1651,1662,1667,1679,1684,1696,1701],[1905,1914,1919,1922,1928,1934,1944,1951,1962,1967,1979,1984,1996,2001,2013,2018,2030,2034,2046,2051,2063,2067,2080,2084,2096,2101,2113],[2256,2265,2272,2281,2285,2295,2301,2313,2317,2329,2334,2346,2351,2363,2367,2379,2384,2396,2401,2412,2418,2429,2434,2446,2451,2463,2466],[2573,2583,2589,2592,2604,2612,2621,2629,2634,2646,2651,2663,2668,2680,2684,2697,2700],[2842,2850,2859,2865,2871,2878,2884,2893,2901,2913,2918,2930,2933],[3068,3075,3081,3084,3091,3096,3101,3113,3118,3130,3134,3146,3151,3163,3168,3180,3184,3196,3201,3213,3218,3230,3234,3247,3251,3263,3268,3280,3285,3297,3301,3313,3318,3330,3346,3351,3363,3368,3380,3385,3397,3401,3413,3418,3430,3435,3442],[3633,3642,3648,3651,3658,3662,3668,3680,3685,3697,3701,3714,3718,3731,3735,3747,3750],[3917,3926,3933,3936,3943,3951,3962,3968,3980,3985,3997,4001,4014,4018,4030,4035,4047,4051,4064,4068,4081,4085,4097,4101,4114,4118,4131,4135,4147],[4278,4286,4292,4297,4310,4318,4329,4335,4347,4351,4363,4368,4380,4385,4397,4402,4414,4418,4430,4435,4447,4451],[4594,4602,4608,4611,4635,4645,4652,4664,4668,4681,4685,4698,4702,4714,4718,4725],[4897,4905,4912,4915,4924,4929,4935,4947,4952,4965,4969,4981,4985,4997],[5203,5210,5216,5219,5227,5252,5263,5269,5281,5285,5298,5302,5315,5319,5331,5335,5348,5352,5365,5369,5382,5385,5398,5402,5414,5419,5435,5447,5452,5465,5469,5481,5485,5498,5502,5515,5519,5531,5535,5548,5552,5565,5569,5581,5586,5597,5602,5614,5619,5630,5635,5647,5652,5664]],"version":"2.0.0"}.

    Find the zeros of the polynomial functionpx=(x+2)(x+1).

    Solution:

    Equating p(x) to zero, we obtain(x+2)(x+1)=0.

    Therefore,x=-2,-1{"x":[[96,96,96,99,101,107,116,125,135,144,152,159,164,166,166,162,158,153,147,141,134,127,120,113,108,104,102,101,102],[192,190,189,187,184,180,175,171,167,163,160,158,157,158,160,163,168,173,180,187,194,201,207,214,216],[248,246,245,244,244,246,249,252,255,259,263,267,271,275,279],[248,246,246,246,247,248,250,253,256,259,263,266,270,274],[314,313,312,313,315,317,319,320,323,325,328,331,333],[352,350,350,349,349,350,352,353,355,358,362,367,372,378,383,387,388,389,388,386,382,378,374,369,365,361,357,355,354,354,355,356,358,361,364,368,372,377,381,384,387,389,390,392,395],[441,441,442,443,444,444,445,444,443,441,437,434,430,427,425,423,422,423],[472,471,472,473,475,479,483,487,491,494,498,501,505,508,510],[551,551,551,551,552,553,554,555,557,558,559,560,562,563,564,565,565,566,565,564,563]],"y":[[154,151,150,147,146,145,144,144,145,147,149,152,157,162,168,181,187,193,197,200,203,205,207,207,207,207,206,205,204],[141,138,137,136,138,142,146,150,154,159,163,168,174,180,187,193,197,200,202,202,201,198,196,193,192],[144,144,144,144,145,148,150,152,153,153,152,151,149,148,148],[176,178,179,180,181,181,182,182,182,180,179,177,176,175],[166,166,166,167,168,169,169,169,169,169,168,167,166],[130,127,126,124,123,122,121,120,119,118,118,118,119,121,124,127,131,135,138,143,148,152,157,162,166,171,175,179,181,183,183,184,184,185,185,186,188,189,190,191,192,193,193,193,194],[192,193,194,196,198,200,202,205,207,208,211,212,214,216,217,218,218,217],[156,156,156,157,159,160,161,162,162,162,162,161,160,158,157],[115,114,113,115,118,122,129,137,147,158,168,175,182,188,193,197,200,202,204,206,206]],"t":[[0,7,13,14,21,31,37,48,53,64,70,81,87,98,103,126,130,136,147,153,164,170,180,186,197,203,214,220,231],[374,379,385,391,395,403,414,420,431,437,448,453,465,470,482,487,498,503,515,520,532,537,548,553,561],[683,693,699,702,716,720,732,737,749,754,765,770,782,787,799],[929,935,942,948,953,964,970,982,987,999,1004,1015,1020,1032],[1182,1190,1203,1214,1220,1232,1237,1249,1254,1266,1270,1282,1286],[1392,1399,1405,1407,1420,1437,1454,1465,1470,1483,1487,1499,1504,1516,1520,1533,1537,1549,1554,1566,1571,1583,1587,1599,1604,1616,1620,1632,1637,1649,1654,1666,1671,1683,1687,1699,1704,1716,1721,1733,1737,1749,1754,1766,1771],[1947,1955,1961,1964,1971,1981,1987,1998,2004,2016,2021,2032,2037,2049,2054,2066,2071,2094],[2971,2984,2991,2998,3004,3016,3021,3033,3038,3050,3054,3067,3071,3083,3086],[3243,3251,3259,3271,3281,3288,3300,3304,3316,3321,3333,3339,3349,3354,3366,3371,3383,3388,3400,3404,3412]],"version":"2.0.0"}

    Locating zeros

    Suppose we have the polynomial functionp(x)=-x2+2x+1. If we work out p(2), we getp(2)=-(2)2+2(2)+1=1which is positive. If we work out p(3), we get p(3)=-(3)2+2(3)+1=-2 which is negative.

    The location principle states that for the polynomial functionpx{"x":[[243,241,239,239,238,238,238,239,240,241,242,243,243,244,244,245,245,245,244,243,242,240,238,237,235,234,233,233,234,236,239,242,246,250,253,257,261,265,269,273,276,277,277,275,272,268,264,260,255,252,248,244,241,239,237],[339,337,335,331,327,322,318,314,310,306,304,302,300,299,299,300,302,306,311,317,320],[361,362,363,369,371,377,383,388,392,394,395,392,388,384,379,374,369,363,359,355,352,351,351],[410,405,403,400,398,395,393,391,390,389,390,391,393,396,399,404,409,415,422,430,438,444],[447,449,451,457,466,476,485,494,501,507,510,510,509,507,505,504,502,499,495,491]],"y":[[216,217,219,221,227,236,245,255,263,272,283,294,305,316,323,329,333,334,335,334,332,328,322,316,307,298,288,276,264,252,240,231,223,216,211,208,205,204,204,206,209,213,219,225,232,238,244,248,252,255,257,259,261,262,262],[201,200,199,198,201,205,209,215,220,227,235,245,256,268,278,286,293,298,303,308,310],[249,246,245,244,245,247,249,251,253,255,259,264,269,274,279,283,286,288,290,291,292,293,294],[240,238,238,238,239,241,244,247,251,254,258,262,267,273,279,283,287,289,290,289,287,285],[187,188,189,194,201,209,218,226,236,247,259,273,286,299,311,321,329,336,343,348]],"t":[[0,7,16,21,28,35,45,52,60,68,78,85,95,102,112,118,130,135,147,152,163,168,180,185,195,202,212,218,229,235,244,252,262,268,279,285,293,302,312,319,329,335,346,352,363,369,379,385,396,402,412,418,429,435,446],[622,629,633,637,644,652,662,668,679,685,697,702,714,719,730,735,746,752,763,769,776],[931,939,945,947,952,962,968,979,985,996,1002,1013,1019,1029,1035,1046,1052,1063,1069,1080,1086,1096,1102],[1261,1267,1272,1274,1279,1285,1295,1302,1313,1319,1330,1335,1346,1352,1363,1369,1380,1386,1397,1402,1413,1419],[1579,1590,1595,1602,1612,1619,1630,1636,1646,1652,1664,1669,1680,1686,1697,1702,1714,1719,1730,1736]],"version":"2.0.0"}, if pa<0{"x":[[236,235,235,235,235,235,235,235,235,235,235,235,235,235,236,236,237,237,237,237,236,235,234,233,232,231,231,231,231,231,231,231,232,234,236,238,240,242,244,246,248,251,253,256,258,261,265,268,272,276,279,281,282,283,283,282,279,275,272,268,265,261,258,256,253,251,250,248],[364,353,348,343,338,334,329,323,319,314,309,306,303,301,301,301,301,301,301,302,304,307,311],[394,384,381,378,375,372,368,366,363,360,357,354,353,351,350,350,350,350,350,352,354,357,360,363,367,371,374,378,381,385,387,390,392,393,395,396,397,398,399,400,400,402,403,404,406,408,411,414,416,419],[436,438,443,448,453,459,464,469,473,476,479,481,482,482,482,482,479,476,474,471,468,465,462,459,457,455],[581,582,579,575,570,565,559,553,548,542,538,534,531,529,527,527,527,527,528,531,534,538,543,548,553,558,563,567,572,576,580,583,586,589],[669,667,665,662,659,655,651,647,642,639,635,632,629,627,625,625,624,623,623,623,623,623,623,625,628,630,633,637,640,644,648,653,657,662,666,671,676,680,684,687,691,693,696,698,699,700,700,700,700,700,700,700,699,697,696,694,692,690,688,686,683,680]],"y":[[156,168,174,183,194,207,220,235,252,268,283,298,311,323,332,338,343,347,349,350,350,348,342,335,325,313,301,288,272,257,242,227,215,203,192,184,176,169,165,161,158,157,156,156,156,157,159,162,166,170,174,179,185,190,195,201,205,210,213,216,219,221,222,223,224,225,226,227],[151,157,161,166,172,178,185,192,201,210,219,229,240,250,259,268,277,284,290,296,300,304,307],[238,219,219,219,219,219,221,224,229,234,239,245,251,257,264,269,273,278,280,282,283,283,283,282,278,274,269,263,257,250,244,238,234,230,228,227,227,227,230,234,240,246,252,261,269,277,283,290,294,298],[172,177,182,187,194,201,208,216,224,233,242,250,257,264,271,277,283,289,293,296,299,300,301,301,301,303],[175,175,177,181,185,189,193,198,202,206,210,214,217,219,221,222,223,224,225,225,226,227,228,228,230,230,232,233,233,233,234,234,234,235],[162,156,156,156,158,161,165,169,174,179,185,191,197,203,209,214,219,225,231,236,241,245,248,251,254,256,258,259,260,260,260,259,255,252,247,242,238,233,228,222,216,211,206,200,195,190,185,180,175,170,166,162,159,156,153,151,151,150,150,150,150,150]],"t":[[0,16,30,39,46,56,63,74,79,91,96,107,113,124,129,140,146,157,163,174,191,196,207,213,224,229,240,246,257,263,274,280,288,296,307,313,324,329,340,346,357,363,374,380,388,396,407,413,424,430,441,446,458,463,474,479,491,496,508,513,523,530,540,546,557,563,573,580],[793,822,830,840,846,857,863,874,880,891,896,907,913,924,930,941,946,958,963,975,980,991,996],[1152,1209,1222,1230,1241,1247,1258,1263,1274,1280,1291,1297,1308,1313,1324,1330,1341,1347,1355,1363,1374,1380,1391,1397,1408,1413,1424,1430,1441,1447,1457,1463,1474,1480,1491,1497,1513,1530,1542,1547,1558,1563,1574,1580,1591,1597,1608,1613,1625,1630],[1863,1900,1914,1923,1930,1941,1947,1958,1963,1975,1981,1992,1997,2008,2014,2025,2030,2042,2047,2058,2063,2075,2080,2091,2097,2108],[2304,2373,2381,2389,2397,2408,2414,2425,2430,2439,2447,2458,2464,2475,2480,2492,2497,2514,2525,2530,2542,2547,2559,2564,2575,2580,2592,2597,2609,2614,2625,2630,2642,2647],[2843,2892,2906,2914,2925,2931,2942,2947,2959,2964,2975,2980,2992,2997,3009,3014,3025,3031,3042,3047,3059,3064,3075,3080,3092,3097,3109,3114,3126,3131,3142,3147,3158,3164,3175,3180,3192,3197,3206,3214,3224,3231,3242,3248,3258,3264,3275,3281,3292,3297,3309,3314,3325,3331,3342,3347,3359,3364,3376,3381,3392,3397]],"version":"2.0.0"} and pb>0{"x":[[259,261,261,262,263,264,264,265,264,263,261,259,257,255,254,253,252,254,256,259,262,265,268,271,275,280,284,288,291,293,293,292,290,286,281,276,271,266,261,257,253,249,246,243],[355,352,351,346,344,341,336,330,324,319,314,312,309,308,307,308,310,313,317,322,328,334,337],[397,395,395,394,394,395,396,397,398,399,399,400,400,399,399,398,398,398,399,400,403,406,409,413,418,423,429,433,438,441,443,444,444,443,441,437,433,429,424,420,415,410,405,400,396,393,391,390,391],[448,448,449,450,453,458,464,470,477,485,492,499,503,505,506,506,504,501,499,496,493,490,487,483,479,475],[571,570,569,569,568,570,572,574,578,581,586,591,596,601,605,609,612,613,614,614,613,612,610,607,604,601,597,593,588,583,579,576,573,571,569,567,566],[673,672,672,671,670,668,666,663,660,656,652,649,644,641,638,637,637,638,641,645,650,657,663,669,677,683,690,696,702,707,712,716,719,721,723,723,723,721,718,714,709,703,697,690,682,671,660,648,638,635]],"y":[[243,261,266,279,286,290,291,292,291,288,284,278,270,260,249,237,223,210,199,189,181,174,167,163,160,158,158,159,162,165,170,176,183,191,199,206,212,217,219,221,221,221,221,220],[161,158,157,157,158,159,161,165,170,177,185,194,204,215,226,236,245,252,258,263,266,267,268],[169,170,171,177,180,184,191,198,205,211,217,222,227,230,231,231,229,226,222,218,213,209,206,202,200,200,201,204,207,211,215,220,226,231,237,242,246,249,252,253,253,252,250,246,243,240,239,238,237],[151,150,149,149,149,151,155,159,165,172,178,185,193,201,210,220,229,238,246,253,258,261,264,266,268,270],[172,172,172,171,171,172,173,174,175,176,177,179,182,185,188,190,193,195,196,197,197,199,200,202,205,208,212,216,221,225,229,233,236,238,240,241,242],[178,176,175,174,174,174,176,177,178,180,183,187,191,197,203,210,216,223,230,236,241,246,249,250,251,251,249,246,242,236,231,225,218,212,206,199,193,187,182,178,174,172,170,170,170,173,176,180,183,185]],"t":[[0,11,17,20,27,35,46,52,63,69,79,85,96,102,113,119,129,135,146,152,162,169,179,185,196,202,213,219,231,235,247,252,264,269,281,285,297,302,313,319,331,335,347,352],[519,531,537,539,544,552,563,569,579,585,596,602,613,619,629,635,646,652,663,669,679,686,693],[869,876,882,885,890,894,902,913,919,930,936,946,952,963,969,980,986,996,1002,1013,1019,1030,1036,1046,1052,1063,1069,1081,1086,1098,1102,1113,1119,1130,1136,1149,1152,1163,1169,1180,1186,1198,1202,1214,1219,1227,1236,1246,1252],[1744,1750,1759,1764,1769,1777,1786,1796,1802,1813,1819,1830,1836,1847,1853,1863,1869,1881,1886,1897,1903,1914,1919,1931,1936,1947],[2736,2742,2746,2748,2753,2769,2780,2786,2797,2803,2814,2820,2830,2836,2848,2853,2865,2870,2881,2886,2898,2903,2915,2920,2931,2936,2948,2953,2964,2970,2983,2986,2998,3003,3016,3020,3037],[3229,3235,3240,3242,3248,3263,3270,3281,3286,3298,3303,3314,3320,3331,3337,3348,3353,3365,3370,3382,3387,3398,3403,3415,3420,3432,3437,3448,3453,3465,3470,3482,3486,3499,3503,3515,3520,3532,3537,3549,3553,3565,3570,3582,3587,3598,3603,3615,3620,3627]],"version":"2.0.0"}, then there must be a zero between a and b.

    Locating Zeroes, Example, Geogebra Locating Zeroes Example, Jordan Madge- StudySmarter Originals

    Above is the graph of fx=x2+3x+2{"x":[[270,270,270,270,270,270,270,269,269,268,267,266,265,263,261,259,256,253,250,247,245,242,240,238,236,235,234,234,233,233,234,236,237,240,243,245,248,251,253,255,256,257,256,255,253,251,247,243,240,235,232,229,227,225,222,220,217,215,213,211,209,208,208,208],[222,222,223,225,228,234,240,245,250,254,256,258,260,262,264,266,268],[333,332,330,328,326,324,323,320,318,314,311,307,304,300,297,293,290,289,288,288,290,293,297,301,306,311,316,318],[340,340,340,340,342,345,349,354,359,363,367,369,369,369,367,364,360,356,352,349,346,344,343,342],[382,380,378,376,375,373,371,369,367,366,365,365,365,367,369,373,377,383,389,395,401,408,411],[411,410,409,409,412,417,421,427,434,440,446,452,456,460,463,463,463,461,458,454,451,448,445,442,440,439,439,439],[496,496,496,498,501,505,510,514,518,522,525,529,533,536],[505,503,502,502,504,506,510,513,517,521,525,528],[573,573,573,573,575,577,580,584,588,593,597,601,604,604,604,603,601,599,597,594,591,586,582,579,576,574,572,571,572],[627,625,623,621,619,616,613,610,608,606,605,604,604,604,605,608,612,618,624,630,637,643,648,654],[659,659,659,660,662,666,670,673,674,675,675,675,673,670,667,665,663,662,662,662,662,664,665,667,669,671,673,676],[690,689,689,690,692,695,700,705,711,716,721,725,728,730,731],[704,703,703,703,705,706,708,711,712,713,715,715,716,718,719,721,723],[751,752,753,755,756,758,761,763,764,765,765,765,764,762,759,757,754,752,751,751,752,753,755,757,760,764,768,771,775,779,781,783,784,784,784,783,780,777,774,769,765,761,757,753,751,748,746,744,743],[803,803,803,804,806,807,809,811,814,816,819,822,824,826,826,827,826,824,822,821,819,818,816,813,811,809,807,806,806,806,807],[842,843,843,843,842,841,839,837,835,832,830,828,826,824,823,822,822,824,826,830,834,838,843,847,850,853],[870,870,872,873,875,878,881,884,889,893,896,900,903,905,906,907],[882,882,883,884,885,886,888,889,890,891,892,893,894,895,895,895,895,895,895],[925,925,925,925,926,927,928,929,931,933,934,936,938,940,943,945,948,951,954,957,961,964,966,968,969,969,968,966,964,961,958,954,951,948,946,945,944,944,945,946,948,952,956,962,969,974,980,984,987,989,990,991,991,992]],"y":[[235,234,232,231,230,229,228,227,226,225,224,221,220,219,218,217,216,216,216,216,217,218,220,219,221,223,225,228,232,235,239,242,245,247,250,250,252,254,257,260,264,268,273,279,286,292,300,304,310,316,320,322,322,322,321,319,316,314,311,306,303,300,298,297],[269,268,265,264,264,264,263,261,260,259,259,259,259,258,259,260,260],[213,212,210,209,209,209,210,211,212,215,218,223,229,238,247,256,267,276,285,293,300,305,309,311,313,314,315,315],[264,263,261,260,259,258,257,257,257,258,260,262,265,268,272,277,281,285,288,291,292,293,293,294],[254,253,252,252,253,255,257,260,264,268,273,278,285,290,294,297,299,300,301,300,298,294,292],[209,209,207,206,207,210,213,217,223,229,236,244,251,258,265,272,280,288,295,301,305,308,309,309,309,309,308,307],[251,248,247,246,246,247,248,249,250,250,251,251,251,251],[272,273,273,274,274,275,275,275,274,273,272,272],[238,236,235,234,234,235,236,238,240,242,244,247,251,255,259,263,267,271,274,276,277,279,280,280,280,280,280,280,280],[240,239,238,238,239,241,244,246,249,251,255,259,263,267,271,274,277,279,280,280,280,278,275,272],[186,183,182,180,179,179,181,182,183,184,186,188,191,194,198,202,205,207,209,211,212,212,213,213,213,213,213,213],[256,255,254,254,255,256,256,256,255,255,254,254,254,254,254],[234,235,236,238,243,248,253,257,260,262,264,265,266,266,266,266,265],[216,215,213,211,211,211,212,213,214,215,216,218,220,222,225,228,230,232,233,234,234,234,234,234,234,236,237,239,242,244,246,249,253,256,260,263,266,268,269,270,270,269,268,266,264,262,259,257,256],[228,227,226,226,226,227,228,229,230,231,232,233,235,236,239,241,243,246,248,250,251,251,251,251,251,251,250,250,249,248,248],[229,228,227,226,226,226,226,227,227,228,229,231,234,236,239,242,245,248,250,252,252,253,253,252,251,250],[238,237,237,237,237,237,237,237,236,236,236,236,236,236,236,236],[226,225,224,224,225,227,228,231,234,237,240,243,245,247,248,249,250,251,252],[201,200,199,198,198,198,198,198,197,195,194,193,192,191,191,190,190,190,190,191,193,195,198,202,206,211,216,221,226,232,236,240,244,248,251,253,254,255,255,255,256,256,257,258,259,259,260,260,260,261,262,262,263,263]],"t":[[0,5,10,18,24,32,38,49,55,66,72,84,88,98,105,118,121,133,138,149,155,166,172,184,188,199,206,220,222,233,238,249,255,266,272,284,288,300,307,320,322,333,338,349,355,366,372,383,388,399,407,419,422,433,438,449,455,465,472,483,488,498,508,512],[670,676,688,697,705,719,722,733,738,748,755,765,772,782,788,799,804],[985,989,998,1005,1018,1022,1033,1039,1049,1055,1066,1072,1082,1089,1100,1105,1114,1122,1133,1139,1149,1155,1166,1172,1183,1189,1201,1205],[1375,1382,1393,1397,1405,1416,1422,1433,1439,1449,1455,1466,1472,1483,1489,1500,1505,1514,1524,1533,1539,1549,1555,1566],[1736,1739,1747,1755,1765,1772,1783,1789,1800,1805,1816,1822,1836,1839,1851,1855,1866,1872,1883,1890,1900,1906,1913],[2063,2069,2074,2081,2100,2106,2116,2122,2136,2139,2151,2156,2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If we look at f(-1.5), we see it is negative. If we look at f(0), we see it is positive. Clearly, there must be a zero between x=-1.5 and x=0 because the graph must cross the x-axis at some point in order to go from being negative to positive. This is the theory behind the location principle. It is really useful for graphs where it may be more difficult to locate the zeros using conventional methods for solving, such as quartics (order 4 polynomials), quintics (order 5 polynomials) or higher-order polynomials.

    Use the location principle to show that for the function px=x3+3x+1, there is a root between x=-1 and x=0.

    Solution:

    p-1=-1-3+1=-3,

    p(0)=0+0+1=1

    Graphing Polynomial Functions

    In this section, we will put some of what we have already discussed together to graph polynomial functions. To graph any polynomial function, there are four main steps:

    Step 1: Determine any zeros that the graph may have.

    Step 2: Draw up a table of values.

    Step 3: Determine the end behaviour of the polynomial function.

    Step 4: Use the above information to draw out the graph.

    Sketch the graph of the polynomial function f(x)=x3-1.

    Solution:

    Step 1: Determine any zeros.

    Equating the function to zero, we obtain x3-1=0. We can factorise this to get x-1x2+x+1=0{"x":[[21,4,5,19],[133,148,147,131],[30,54],[29,55],[73,93],[107,123,123,123],[172,156,156,173],[403,418,417,401],[182,207],[181,206],[220,220,227,232,233,227,220,218,218,233],[246,285],[266,264],[300,325],[299,324],[343,363],[377,392,393,391],[433,460],[434,460],[491,480,476,475,479,496,503,504,503,501,490],[351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014,351.64038561681014]],"y":[[143,121,39,8],[143,120,37,8],[50,115],[115,50],[83,84],[29,8,9,114],[144,126,35,8],[143,120,37,8],[51,116],[116,51],[24,15,13,16,25,40,59,66,65,65],[84,84],[59,108],[51,116],[116,51],[82,82],[31,9,9,115],[72,72],[94,93],[8,10,35,80,113,116,103,68,34,13,8],[65.7710072210961,65.7710072210961,65.7710072210961,65.7710072210961,65.7710072210961,65.7710072210961,65.7710072210961,65.7710072210961,65.7710072210961,65.7710072210961,65.7710072210961,65.7710072210961,65.7710072210961,65.7710072210961,65.7710072210961,65.7710072210961,65.7710072210961,65.7710072210961,65.7710072210961,65.7710072210961,65.7710072210961,66.3909461773933,67.63082408998773,68.25076304628493,68.87070200258214,69.49064095887934,70.73051887147376,71.35045782777097,71.97039678406817,72.59033574036538,73.21027469666258,74.450152609257,75.07009156555421,75.69003052185141,76.92990843444584,77.54984739074304,78.78972530333745,80.02960321593187,81.26948112852628,81.88942008482348,82.50935904112069,83.74923695371511,84.36917591001232,84.98911486630952,86.22899277890394,86.22899277890394,86.84893173520115,87.46887069149835,87.46887069149835,87.46887069149835,88.08880964779556,88.08880964779556,88.08880964779556]],"t":[[0,0,0,0],[0,0,0,0],[0,0],[0,0],[0,0],[0,0,0,0],[0,0,0,0],[0,0,0,0],[0,0],[0,0],[0,0,0,0,0,0,0,0,0,0],[0,0],[0,0],[0,0],[0,0],[0,0],[0,0,0,0],[0,0],[0,0],[0,0,0,0,0,0,0,0,0,0,0],[1645081703008,1645081703020,1645081703021,1645081703028,1645081703036,1645081703041,1645081703051,1645081703060,1645081703067,1645081703084,1645081703085,1645081703101,1645081703101,1645081703117,1645081703118,1645081703125,1645081703135,1645081703142,1645081703150,1645081703159,1645081703167,1645081703185,1645081703187,1645081703202,1645081703204,1645081703219,1645081703221,1645081703235,1645081703243,1645081703252,1645081703258,1645081703270,1645081703275,1645081703286,1645081703292,1645081703302,1645081703309,1645081703319,1645081703325,1645081703336,1645081703342,1645081703352,1645081703359,1645081703369,1645081703375,1645081703386,1645081703391,1645081703402,1645081703409,1645081703418,1645081703425,1645081703436,1645081703451]],"version":"2.0.0"} and we, therefore, get x=1 as the only real zero.

    Step 2: Draw up a table of values.

    x-10123
    f(x)-2-10726

    Above I have chosen some values of x and worked out the corresponding values of f(x).

    Step 3: Determine the end behaviour of the polynomial function.

    This polynomial has an odd degree and the leading polynomial is 1 which is positive. Therefore, we have as x{"x":[[65,66,67,70,74,78,83,86,89,90,91,92,93,95,97,99,100,101,102,102,102,101,99,95,90,85,80,76,73,71,69,69,68,68,68,68,67,67,67,67],[120,119,118,117,115,113,112,111,110,110,109,108,106,105,105,104,104,104,104,104,104,104,105,106,107,107,108,109,110,111,114,117,120,124,126,128,129,129,130,130,130,130],[138,138,140,144,149,155,161,166,172,177,181,184,186,187,188,189,189,189,188,186,183,179,177,176,175,175,175,174,174,174,174,174,174,175,176,178,180,183,186,189,191,192,193,194,194,195,195,195,195,195,195,195,194,193,192,190,190,189,189,188,188,187,186,186,185,184,183,183,182,182,181],[207,207,207,207,207,207,207,207,208,209,210,211,212,214,215,217,219,220,221,222,223,224,225,226,227,227,228,228,229,229,229,230,230,231,231,232,233,234,236,238,240,243,245,248,250,253,254,256,257,257,258,259,259,260,261,261,261,261,261,261,261,260,260,259,258,257,256,254,252,250,248,247,245,244,243,241,240,239,239,238,237,236,235,234,233,232,231,230,229,229,228,228,227,226,226,226,225,225,224,224,224,224,223,223,223,222,221,220,219,217,216,214,214,213,212,212,211,211,211,210,210,209,209,208,208,207,207,207,207,207,207,207,207,207,207,207,207]],"y":[[63,63,63,63,63,63,63,63,64,65,67,69,73,77,82,88,92,96,99,101,103,106,108,112,115,117,119,121,121,122,122,122,122,121,121,121,120,120,120,120],[68,68,68,68,68,70,71,72,74,76,79,82,85,89,91,94,96,97,99,101,103,105,106,108,109,110,111,111,112,112,113,113,113,113,113,113,113,113,113,113,113,112],[83,83,83,83,83,83,83,83,83,83,83,83,83,83,83,83,83,82,81,80,77,75,73,72,71,71,71,71,70,70,70,70,70,71,71,72,73,75,77,79,80,81,82,83,83,84,84,84,84,84,85,85,86,87,89,90,91,91,92,92,93,93,94,94,95,95,96,96,96,96,96],[77,77,77,79,80,82,83,85,86,88,89,90,90,91,92,92,93,93,93,93,93,92,91,89,87,84,81,78,75,73,72,71,70,69,69,68,68,67,67,67,67,66,66,66,66,66,66,66,67,67,67,68,69,70,70,71,72,72,73,75,76,78,79,80,81,82,82,83,83,83,83,83,83,83,83,83,83,83,83,83,82,81,80,79,78,77,75,74,73,72,71,70,69,68,67,66,66,65,64,63,63,62,62,62,62,62,62,62,62,62,63,63,64,64,65,65,66,66,67,67,68,69,70,71,72,73,74,75,76,76,77,78,78,79,79,79,79]],"t":[[0,118,132,149,165,182,199,216,232,249,271,287,305,321,338,356,371,388,404,421,439,455,471,488,505,522,539,555,571,588,605,621,638,654,672,688,705,722,738,755],[1350,1499,1515,1532,1548,1565,1582,1598,1615,1632,1649,1665,1682,1698,1715,1732,1748,1771,1787,1805,1822,1837,1855,1872,1888,1905,1921,1937,1954,1971,1988,2004,2022,2039,2055,2072,2089,2105,2122,2138,2155,2171],[2716,2883,2898,2915,2932,2949,2965,2982,2999,3015,3031,3049,3065,3082,3098,3115,3183,3199,3215,3232,3249,3272,3289,3305,3323,3338,3355,3372,3388,3405,3422,3550,3565,3582,3598,3615,3632,3649,3665,3682,3698,3715,3732,3748,3772,3789,3806,3855,3871,3888,3905,3922,3938,3955,3972,3988,4005,4021,4038,4055,4073,4088,4105,4122,4139,4155,4171,4187,4205,4222,4238],[4759,4933,4949,4965,4983,4998,5015,5032,5049,5065,5081,5098,5115,5132,5148,5165,5182,5198,5216,5232,5249,5271,5288,5306,5322,5338,5355,5372,5388,5404,5423,5439,5456,5471,5488,5506,5522,5539,5554,5571,5588,5604,5623,5638,5656,5672,5688,5705,5723,5739,5754,5772,5788,5805,5821,5838,5855,5872,5889,5906,5922,5939,5955,5971,5988,6005,6021,6038,6055,6072,6089,6106,6122,6139,6155,6171,6189,6206,6222,6238,6255,6272,6289,6306,6323,6338,6355,6371,6388,6405,6422,6438,6455,6471,6488,6505,6522,6538,6555,6572,6588,6605,6622,6638,6655,6672,6689,6706,6722,6738,6755,6773,6788,6806,6821,6839,6855,6872,6889,6904,6922,6938,6955,6971,6988,7005,7022,7039,7055,7072,7088,7106,7122,7139,7155,7171,7188]],"version":"2.0.0"}, 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    Step 4: Use the above information to draw out the graph.

    Graphing Polynomials, Example, Geogebra

    Graphing Polynomials Example, Jordan Madge- StudySmarter Originals

    Odd and Even Functions

    Once we have drawn the graph of our polynomial function, we may wish to determine whether it is an odd or even function. An even function occurs when we have symmetry about the y-axis. In other words, a function is even when f(x)=f(-x) for all values of x. An odd function occurs when -f(x)=f(-x).

    Odd and Even Funcitons, Example, Geogebra Odd and Even Functions Example, Jordan Madge- StudySmarter Originals

    Above is the graph of y=x4-3{"x":[[201,199,210,212,217,221,226,230,234,237,239,241,243,243,244,245,246,247,248,247,245,244,243,242,242,243,244,244,244,241,238,235,232,229,227,224,220,216,212,208,203,199,195,192,189,188,187,187],[283,282,282,283,285,289,294,299,306,312,317,322],[284,284,283,284,286,289,293,298,303,308,312,314],[342,342,343,344,346,350,354,359,365,370,375,381,385,389,391,391,391,390,387,383,378,374,369,364,360,356,353,350,349,351,353,357,361],[408,407,407,405,404,402,399,396,393,390,388,387,386,385,385,387,390,394,399,404,410,415,419,423,426],[431,430,429,429,429,428,428,427,427,426,425,424,424,424,424,425,427,429,432,434,437,440,442,444,445],[437,436,436,436,436,437,437,438,439,439,440,440,440,440],[457,457,456,456,457,460,463,466,471,476,481,485,489,492,494,495,496],[515,515,515,517,518,521,525,529,533,537,540,542,543,543,542,540,538,534,531,527,524,521,520,521,522,524,527,531,535,539,545,549,553,557,561,564,566,567,567,567,565,562,558,555,551,546,542,539,535,532,529,525,521]],"y":[[135,134,197,198,199,199,196,191,186,180,174,168,163,158,154,149,145,141,138,140,144,150,157,164,172,182,191,201,212,224,237,249,257,263,268,271,273,274,273,270,267,262,255,249,243,237,233,231],[178,177,176,176,176,176,176,176,175,175,174,173],[211,212,212,212,212,211,210,208,207,205,203,202],[182,179,176,175,173,172,171,171,171,172,173,175,177,180,183,188,193,198,204,208,211,213,214,215,215,215,214,213,212,212,212,212,211],[170,168,167,166,166,166,168,171,174,178,181,186,191,196,200,205,208,211,213,215,215,215,214,212,210],[130,128,126,127,129,131,133,136,139,142,145,148,150,152,153,154,154,154,154,154,153,153,153,152,152],[136,136,138,139,143,147,152,156,160,163,166,169,171,173],[183,182,182,181,181,181,181,181,181,180,179,179,179,179,179,179,179],[145,143,142,141,141,141,141,141,141,141,141,141,142,144,146,149,153,156,159,162,164,165,166,166,166,166,166,166,166,166,167,169,172,175,179,183,188,192,197,203,208,213,218,222,225,227,228,228,228,229,230,232,236]],"t":[[0,2,20,22,30,38,48,55,65,72,82,88,99,107,118,122,133,138,149,172,182,189,199,208,218,222,233,239,249,255,266,272,282,289,299,309,319,322,333,338,349,355,366,372,382,389,399,409],[602,614,620,628,630,639,650,655,666,672,683,689],[838,844,849,864,872,883,889,899,905,915,922,929],[1101,1107,1121,1122,1130,1139,1150,1156,1167,1172,1184,1189,1200,1206,1217,1222,1234,1239,1251,1256,1267,1272,1284,1289,1301,1306,1318,1322,1334,1351,1356,1368,1372],[1498,1503,1510,1512,1519,1523,1532,1539,1550,1556,1567,1572,1584,1589,1601,1606,1614,1623,1634,1639,1651,1656,1664,1673,1684],[1870,1877,1885,1906,1914,1923,1934,1939,1951,1956,1968,1973,1984,1989,2001,2006,2018,2023,2034,2039,2051,2056,2068,2073,2089],[2245,2256,2263,2270,2273,2283,2289,2301,2306,2318,2323,2334,2339,2351],[2535,2542,2550,2553,2561,2566,2573,2584,2589,2601,2606,2618,2623,2635,2640,2652,2656],[2835,2842,2850,2853,2859,2866,2873,2884,2889,2902,2906,2918,2923,2935,2940,2952,2956,2968,2973,2985,2990,3002,3006,3031,3040,3052,3056,3069,3073,3085,3090,3102,3106,3118,3123,3135,3140,3152,3156,3169,3173,3185,3190,3202,3206,3218,3223,3235,3240,3252,3256,3268,3273]],"version":"2.0.0"}. We can see that it is an even function because there is symmetry about the y-axis. Whenever we have symmetry about the y-axis, we have that f(x)=f(-x).

    Odd and Even Functions, Example,  Geogebra

    Odd and Even Functions Example, Jordan Madge- StudySmarter Originals

    Above is the graph of y=12x5{"x":[[137,135,134,135,137,138,139,140,141,142,143,143,143,143,143,143,143,143,143,144,145,147,149,152,154,157,160,162,165,167,168,170,173,176,179,183,186,188,189,191,192,193,194,194,195,196,197,198,199,200,201,201,199,197,192,187,182,176,171,167,163,159,156,153,151,149,149],[247,247,250,252,257,263,270,276,281],[258,256,255,254,254,256,259,264,269,274,279,282],[339,339,339,339,339,339,339,339,341,343,345,346,346],[319,319,320,323,326,333,341,347,351,355,359,363],[320,317,316,315,315,317,320,324,329,335,342,348,353,357,358,359,359,357,354,350,345,340,334,328,322,317,314,313,314,318,322,327,334,340,347,353,359,364,367,368,367],[403,400,399,398,398,400,402,405,408,411,415,418,421,424,425,425,423,421,418,415,412,409,406,404,401,400,399],[442,439,437,433,431,429,429,428,427,427,427,426,426,426,427,428,431,435,440,446,451,456,460,463,464,465],[521,519,517,516,515,514,511,508,504,500,495,492,488,485,483,482,480,479,479,478,478,479,479,479,480,480,480,481,481,482,483,484,485,486,487,488,489,490,492,493,495,497,498,500,501,501,500,499,497,494,492,490,487,485,481,479]],"y":[[148,148,148,148,147,147,147,147,146,146,146,147,148,151,154,158,165,172,180,190,199,207,215,220,223,225,225,225,224,222,219,215,209,202,194,186,179,171,166,160,156,154,153,155,160,166,175,184,194,206,217,231,247,262,277,291,300,306,309,309,307,302,297,290,282,276,273],[188,187,189,190,191,192,193,193,194],[215,215,215,216,217,218,218,217,216,215,214,213],[151,150,152,154,156,159,165,169,174,179,183,186,188],[220,219,218,217,216,214,212,210,209,208,207,206],[254,253,252,252,251,250,249,247,245,244,243,243,244,246,249,253,257,262,267,271,275,278,280,282,284,285,286,286,286,286,286,286,285,285,285,285,285,285,285,285,286],[187,185,185,184,183,183,182,181,181,181,182,184,187,190,193,197,201,204,208,211,213,215,216,216,216,216,215],[178,178,178,180,181,183,185,188,192,195,200,204,208,212,216,218,219,219,219,218,217,216,215,214,213,212],[120,118,116,116,115,115,116,117,118,120,121,123,124,125,126,127,128,129,131,133,135,137,139,140,140,141,142,142,143,143,142,141,140,140,140,139,139,139,140,140,141,142,144,146,148,150,152,154,156,157,159,159,159,160,161,162]],"t":[[0,11,19,33,47,57,69,74,80,90,96,107,113,124,130,141,146,158,163,174,180,191,196,208,213,225,230,241,246,257,263,274,280,291,297,307,313,324,330,341,347,357,363,374,380,391,397,407,413,424,430,441,447,459,463,475,480,492,497,509,513,525,530,542,547,559,562],[773,780,793,800,808,813,825,830,842],[968,975,982,985,992,997,1007,1013,1024,1030,1038,1046],[1194,1206,1213,1219,1225,1225,1233,1241,1247,1258,1264,1275,1279],[1411,1416,1425,1432,1434,1442,1450,1458,1463,1473,1480,1490],[1645,1650,1656,1658,1664,1674,1680,1691,1697,1708,1714,1725,1730,1742,1747,1759,1764,1776,1781,1792,1797,1809,1814,1826,1831,1843,1847,1859,1876,1881,1893,1897,1909,1914,1926,1931,1943,1947,1959,1964,1981],[2184,2192,2199,2202,2210,2214,2224,2231,2242,2247,2259,2264,2276,2281,2292,2297,2309,2314,2326,2331,2342,2347,2359,2364,2376,2381,2393],[2554,2560,2568,2571,2578,2584,2590,2598,2609,2614,2626,2631,2643,2647,2659,2664,2676,2681,2693,2698,2710,2714,2727,2731,2743,2748],[2978,2985,2993,2996,3003,3008,3014,3026,3031,3043,3048,3060,3064,3076,3081,3093,3098,3110,3114,3126,3131,3143,3148,3160,3164,3176,3193,3198,3210,3226,3231,3243,3248,3260,3265,3276,3281,3293,3298,3309,3314,3325,3331,3342,3348,3359,3364,3376,3381,3392,3398,3409,3414,3426,3431,3438]],"version":"2.0.0"}. We can see that it is an odd function because -f(x)=f(x) for all values of x.

    Turning Points of Polynomial Functions

    A turning point on a graph is a point where the graph goes from increasing to decreasing. In other words, the graph will change from going “upwards“ to “downwards” or vise versa. The local maxima are specifically when the graph changes from increasing to decreasing. The local minima are specifically when the graph changes from decreasing to increasing.

    The plural for maximum is maxima and the plural for the minimum is minima. The umbrella term for the two words is extrema.

    Turning Points, Example, Geogebra

    Turning Points Example, Jordan Madge- StudySmarter Originals

    On the above graph, we can see a turning point that occurs at (0,0). At this point, the graph goes from increasing to decreasing and so it is a local maximum.

    Analyzing Graphs of Polynomials - Key takeaways

    • The end behaviour of a polynomial function is what happens to the graph as x approaches positive or negative infinity.
    • We can determine the end behaviour of any polynomial by looking at the leading coefficient and degree of the polynomial.
    • A zero of a polynomial function is the point where it crosses the x-axis. It can be determined by equating the polynomial function to zero and solving for x.
    • We can use the location principal to determine values that zeros lie in between.
    • Even functions are functions that have symmetry about the x-axis and f(x)=f(-x)
    • Odd functions are functions where f(-x)= -f(x)
    • A turning point is a point when a function goes from increasing to decreasing or vice versa.
    • A local minimum occurs where a function goes from decreasing to increasing.
    • A local maximum occurs where a function goes from increasing to decreasing.
    Frequently Asked Questions about Analyzing Graphs of Polynomials

    How do you write an analysis of a graph?

    First determine the end behaviour of the graph. Then work out any zeroes. Maybe work out if it is an odd or even function and determine any turning points the graph has. 

    What are the characteristics of a polynomial graph?

    It may have turning points, it may be increasing or decreasing, it may have some zeros, as x and y intercepts. 

    What are the examples of  graphs of a polynomial equation?

    Quadratic graphs, cubic graphs, quartic graphs and so on. 

    How to sketch graphs of a polynomial equation?

    Determine any zeros, draw a table of values, determine the end behaviour, draw out the graph using the information gathered. 

    How to analyze a graph of a polynomial function?

    Determine turning points, intercepts, zeroes and end behaviour. 

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