{"id":1312,"date":"2015-11-12T18:35:30","date_gmt":"2015-11-12T18:35:30","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=1312"},"modified":"2017-03-30T17:36:49","modified_gmt":"2017-03-30T17:36:49","slug":"determine-end-behavior","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/determine-end-behavior\/","title":{"raw":"Determine end behavior","rendered":"Determine end behavior"},"content":{"raw":"As we have already learned, the behavior of a graph of a polynomial function<\/strong> of the form\r\n
[latex]f(x)={a}_{n}{x}^{n}+{a}_{n - 1}{x}^{n - 1}+...+{a}_{1}x+{a}_{0}[\/latex]<\/div>\r\nwill either ultimately rise or fall as x<\/em>\u00a0increases without bound and will either rise or fall as x\u00a0<\/em>decreases without bound. This is because for very large inputs, say 100 or 1,000, the leading term dominates the size of the output. The same is true for very small inputs, say \u2013100 or \u20131,000.\r\n\r\nRecall that we call this behavior the end behavior<\/em> of a function. As we pointed out when discussing quadratic equations, when the leading term of a polynomial function, [latex]{a}_{n}{x}^{n}[\/latex], is an even power function, as x<\/em>\u00a0increases or decreases without bound, [latex]f(x)[\/latex] increases without bound. When the leading term is an odd power function, as\u00a0x<\/em>\u00a0decreases without bound, [latex]f(x)[\/latex] also decreases without bound; as x<\/em>\u00a0increases without bound, [latex]f(x)[\/latex] also increases without bound. If the leading term is negative, it will change the direction of the end behavior. The table below\u00a0summarizes all four cases.\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Even Degree<\/th>\r\nOdd Degree<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
\"11\"<\/a><\/td>\r\n\"12\"<\/a><\/td>\r\n<\/tr>\r\n
\"13\"<\/a><\/td>\r\n\"14\"<\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\n
","rendered":"

As we have already learned, the behavior of a graph of a polynomial function<\/strong> of the form<\/p>\n

[latex]f(x)={a}_{n}{x}^{n}+{a}_{n - 1}{x}^{n - 1}+...+{a}_{1}x+{a}_{0}[\/latex]<\/div>\n

will either ultimately rise or fall as x<\/em>\u00a0increases without bound and will either rise or fall as x\u00a0<\/em>decreases without bound. This is because for very large inputs, say 100 or 1,000, the leading term dominates the size of the output. The same is true for very small inputs, say \u2013100 or \u20131,000.<\/p>\n

Recall that we call this behavior the end behavior<\/em> of a function. As we pointed out when discussing quadratic equations, when the leading term of a polynomial function, [latex]{a}_{n}{x}^{n}[\/latex], is an even power function, as x<\/em>\u00a0increases or decreases without bound, [latex]f(x)[\/latex] increases without bound. When the leading term is an odd power function, as\u00a0x<\/em>\u00a0decreases without bound, [latex]f(x)[\/latex] also decreases without bound; as x<\/em>\u00a0increases without bound, [latex]f(x)[\/latex] also increases without bound. If the leading term is negative, it will change the direction of the end behavior. The table below\u00a0summarizes all four cases.<\/p>\n\n\n\n\n\n\n
Even Degree<\/th>\nOdd Degree<\/th>\n<\/tr>\n<\/thead>\n
\"11\"<\/a><\/td>\n\"12\"<\/a><\/td>\n<\/tr>\n
\"13\"<\/a><\/td>\n\"14\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
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