\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](\"http:\/\/www.learnermath.com\/wp-content\/uploads\/2023\/09\/graphing-polynomial-functions-examples.png\")
<\/figure>\n\n\n\n
\n<\/a>\n
\n\n\n\nTurning 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
\n\n\n\nLeading 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\nLeading 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](\"http:\/\/www.learnermath.com\/wp-content\/uploads\/2023\/09\/end-graph-behaviour.png\")
<\/figure>\n\n\n\n
\n<\/a>\n\n\n\nMultiplicity 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
\n\n\n\nGraphing 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](\"http:\/\/www.learnermath.com\/wp-content\/uploads\/2023\/09\/graph-axis.png\")
<\/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](\"http:\/\/www.learnermath.com\/wp-content\/uploads\/2023\/09\/graph-sketch.png\")
<\/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
\n\n\n