{"id":1157,"date":"2023-09-12T17:53:42","date_gmt":"2023-09-12T17:53:42","guid":{"rendered":"http:\/\/www.learnermath.com\/?page_id=1157"},"modified":"2024-05-08T09:35:11","modified_gmt":"2024-05-08T09:35:11","slug":"graphing-polynomial-functions-examples","status":"publish","type":"page","link":"https:\/\/www.learnermath.com\/graphing-polynomial-functions-examples","title":{"rendered":"Graphing Polynomial Functions Examples"},"content":{"rendered":"\n
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\n On this Page<\/b>:<\/u> \n
\n  1. Turning Points<\/b><\/a>\n
\n\t  2. Leading Terms, Coefficients<\/a>\n\t
\n\t  3. Leading Coefficient Test<\/a>\n
\n  4. Multiplicity of a Zero<\/a>\n
\n  5. Graphing Steps & Example<\/a><\/font>\n <\/div>\n
\n

Here we look at how to approach graphing polynomial functions examples, where we want to try to make as accurate a sketch as we can of a polynomial graph on an appropriate axis.\n

\nSome examples of graphing quadratic graphs which are polynomials of degree 2 can be seen here<\/i><\/a>.\n

\nWhen a polynomial is of larger degree though, sometimes we need to do a bit more work for some more information on what the shape of the graph should be.\n


\nHere are two examples of what a polynomial graph can look like.\n
\nPolynomial graphs are smooth graphs and have no holes or breaks in them.<\/p>\n\n\n\n

\"Types<\/figure>\n\n\n\n


\n<\/a>\n
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Turning Points<\/span><\/h2>\n\n\n\nThe curves on polynomial graphs are called the ‘turning points’.\n

\nA polynomial of degree  n<\/i><\/font>,\n
\nhas at most   n<\/i> − 1<\/font>   turning points on its graph.\n

\nSo the graph of a polynomial such as   f(x) = x^3 + 5x^2 \\space {\\text{–}} \\space 2x + 1<\/span>, \n
\nwill have at most  2  turning points.\n



\n<\/a>\n
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Leading Terms and Leading Coefficients:<\/span><\/h3>\n\n\n\nWith learning how to deal with graphing polynomial functions examples it’s important to be clear about the leading term and leading coefficient is in a polynomial.\n

\nFor a polynomial,   P(x) = a_{n}x^n + a_{n \\space {\\text{–}} \\space 1}x^{n \\space {\\text{–}} \\space 1} + … + a_{1}x + a_0<\/span>.\n

\na_n<\/span>  is the leading coefficient,\n
\na_{n}x^n<\/span>   is the leading term.\n

\nSo with   g(x) = 2x^3 + x^2 \\space {\\text{–}} \\space 2x + 5<\/span>,\n

\nthe leading coefficient is  2<\/span>,  the leading term is   2x^3<\/span>.\n



\n<\/a>\n
\n\n\n\n

Leading Coefficient Test:<\/span><\/h3>\n\n\n\nSomething called the ‘leading coefficient test’ can help us establish the behaviour of the polynomial graph at each end.\n

\nAs stated, for a polynomial   P(x) = a_{n}x^n + a_{n \\space {\\text{–}} \\space 1}x^{n \\space {\\text{–}} \\space 1} + … + a_{1}x + a_0<\/span>,\n
\nthe leading term is   a_{n}<\/span>.\n


\nThere are  4  cases we can have with the leading term, and what it can tell us.\n

\n1)<\/font><\/b><\/font>    a_{n} > 0 \\space\\space , \\space\\space n<\/span>  even.   The graph will increase with no bound to the left and the right.\n

\n2)<\/b><\/font><\/font>    a_{n} > 0 \\space\\space , \\space\\space n<\/span>  odd.   The graph will decrease to the left, and increase on the right with no bound.\n

\n3)<\/b><\/font><\/font>    a_{n} < 0 \\space\\space , \\space\\space n<\/span>  even.   The graph will decrease with no bound to the left and the right.\n

\n4)<\/b><\/font><\/font>    a_{n} < 0 \\space\\space , \\space\\space n<\/span>  odd.   The graph will increase on the left, and decrease to the right with no bound.\n

\n

\n\n\n\n
\"Typical<\/figure>\n\n\n\n



\n<\/a>\n\n\n\n

Multiplicity of a Zero:<\/span><\/h3>\n\n\n\nThe multiplicity of a zero is the number of times it appears in the factored polynomial form.\n

\nFor   (x \\space {\\text{–}} \\space 5)^3(x + 1)^2 = 0<\/span>.\n
\nThe zeros are  5<\/span>  and  {\\text{-}}1<\/span>.\n

\nOf which,  5<\/span>  has multiplicity  3,  and  {\\text{-}}1<\/span>  has multiplicity  2.\n

\nSo:\n

\nWhen  (x \\space {\\text{–}} \\space t)^m<\/span>  is a factor,  t<\/i><\/font>  is a zero.\n

\nIf  m<\/i><\/font>  is even, the  x<\/span>-intercept  at  x = t<\/span>  will only touch the  x<\/span>-axis,  but not cross it.\n

\nIf  m<\/i><\/font>  is odd, the  x<\/span>-intercept  at  x = t<\/span>  will cross the  x<\/span>-axis.\n


\nAlso if  m<\/i><\/font>  is greater than  1,  the curve of the polynomial graph will flatten out at the zero.\n





\n<\/a>\n\n\n\n
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Graphing Polynomial Functions Examples
Steps<\/span><\/h2>\n\n\n\n
\nSo in light of what we’ve seen so far on this page, to make a sketch of the graph of a polynomial function there are some steps to take.\n

\nA)<\/b><\/font>  Find the zeros of the polynomial and their multiplicity.\n

\nB)<\/b><\/font>  Use the leading coefficient test to establish the graph end behaviour.\n

\nC)<\/b><\/font>  Find the y<\/span>-intercept,  (0,P(0))<\/span>.\n

\nD)<\/b><\/font>  Establish the maximum number of turning points there can be,  n \\space {\\text{–}} \\space 1<\/span>.\n

\nE)<\/b><\/font>  Plot some extra points for some more information of graph shape between the zeros.\n


\nPart E) will become more clear in an example.\n




\nExample    <\/u><\/font><\/font><\/font><\/b>\n


\n(1.1) <\/i><\/b><\/font><\/u><\/font><\/font>\n

\nSketch the graph of the polynomial   f(x) = x^4 \\space {\\text{–}} \\space 3x^3 \\space {\\text{–}} \\space 9x^2 + 23x \\space {\\text{–}} \\space 12<\/span>.\n

\nSolution<\/i>   <\/b><\/font><\/font><\/u><\/font>\n

\nOften we are required to find the factors and zeros, but as to concentrate on sketching a graph the polynomial here, we will state them.\n

\nf(x) = x^4 \\space {\\text{–}} \\space 3x^3 \\space {\\text{–}} \\space 9x^2 + 23x \\space {\\text{–}} \\space 12 \\space = \\space (x \\space {\\text{–}} \\space 1)^{\\tt{2}}(x \\space {\\text{–}} \\space 4)(x + 3)<\/span>\n

\n(x \\space {\\text{–}} \\space 1)^{\\tt{2}}(x \\space {\\text{–}} \\space 4)(x + 3) \\space = \\space 0<\/span>       =>       x = 1 \\space , \\space x = 4 \\space , \\space x = {\\text{-}}3<\/span>\n

\nOf the zeros,  1<\/span> has multiplicity 2,  while 4<\/span> and {\\text{-}}3<\/span> have multiplicity 1.\n
\nSo the graph will cross the x<\/span>-axis at  –3<\/b>  and  4<\/b>,  but only touch at  1<\/b>.\n

\nThe leading coefficient is  1<\/b>,  which is  > 0<\/b>,  and  n<\/span> = 4<\/b>.\n
\nSo the graph will increase with no bound past both the left and right ends.\n


\nFor the y<\/span>-intercept.   f(0) = {\\text{-}}12<\/span>   =>   (0,-12)<\/b>\n

\nAs  n = 4<\/span>,  there can be at most  3  turning points, if our graph has any more we have done something wrong somewhere.\n

\nWith what we know thus far, we can make a start at a sketch of the polynomial graph, on a suitable axis.\n

\n\n\n\n
\"Suitable<\/figure>\n\n\n\n

\nNow we look to establish some extra information about graph behaviour between the zeros or other point son the x<\/span>-axis.\n

\nBetween  1  and  4.   f(3) = {\\text{-}}24 \\space , \\space f(2) = {\\text{-}}10<\/span>\n

\nSo there is a turning point between  1  and  4.\n


\nWith  n \\space {\\text{–}} \\space 1 = 3<\/span>,  there can only be one more turning point.\n

\nBetween  -3  and  0.   f({\\text{-}}2) = {\\text{-}}54 \\space , \\space f({\\text{-}}\\frac{3}{2}) = {\\text{-}}51.56<\/span>\n

\nThere is a turning point between  -3  and  0.\n

\n\n\n\n
\"Fully<\/figure>\n\n\n\n

\nIn calculus we can work out more exact turning points and make more exact drawings of polynomial graphs when dealing with graphing polynomial functions examples.\n

\nBut for a rough sketch of the graph, what we have above gives us a good idea and is fairly accurate.<\/font>\n





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