![\"Graph](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201554\/CNX_Precalc_Figure_03_04_0072.jpg\")
<\/center>\nThe behavior of a graph at an x-intercept can be determined by examining the multiplicity of the zero.<\/span><\/strong><\/p>\n The [latex]x[\/latex]-intercept [latex]x=-3[\/latex] is the solution to the equation [latex]\\left(x+3\\right)=0[\/latex]. The graph passes directly through the [latex]x[\/latex]-intercept at [latex]x=-3[\/latex]. The factor is linear (has a degree of [latex]1[\/latex]), so the behavior near the intercept is like that of a line; it passes directly through the intercept. We call this a single zero because the zero corresponds to a single factor of the function.<\/p>\n The [latex]x[\/latex]-intercept [latex]x=2[\/latex] is the repeated solution to the equation [latex]{\\left(x - 2\\right)}^{2}=0[\/latex]. The graph touches the axis at the intercept and changes direction. The factor is quadratic (degree [latex]2[\/latex]), so the behavior near the intercept is like that of a quadratic\u2014it bounces off of the horizontal axis at the intercept.<\/p>\n [latex]{\\left(x - 2\\right)}^{2}=\\left(x - 2\\right)\\left(x - 2\\right)[\/latex]<\/p>\n The factor is repeated, that is, the factor [latex]\\left(x - 2\\right)[\/latex] appears twice. The number of times a given factor appears in the factored form of the equation of a polynomial is called the multiplicity<\/strong>. The zero associated with this factor, [latex]x=2[\/latex], has multiplicity [latex]2[\/latex] because the factor [latex]\\left(x - 2\\right)[\/latex] occurs twice.<\/p>\n The [latex]x[\/latex]-intercept [latex]x=-1[\/latex] is the repeated solution of factor [latex]{\\left(x+1\\right)}^{3}=0[\/latex]. The graph passes through the axis at the intercept but flattens out a bit first. This factor is cubic (degree [latex]3[\/latex]), so the behavior near the intercept is like that of a cubic with the same S-shape near the intercept as the function [latex]f\\left(x\\right)={x}^{3}[\/latex]. We call this a triple zero, or a zero with multiplicity [latex]3[\/latex].<\/p>\n For zeros<\/strong> with even multiplicities, the graphs touch<\/em> or are tangent to the [latex]x[\/latex]-axis at these [latex]x[\/latex]-values. For zeros with odd multiplicities, the graphs cross<\/em> or intersect the [latex]x[\/latex]-axis at these [latex]x[\/latex]-values. See the graphs below for examples of graphs of polynomial functions with multiplicity [latex]1, 2,[\/latex] and [latex]3[\/latex].<\/p>\n <\/p>\n For higher even powers, such as [latex]4, 6,[\/latex] and [latex]8[\/latex], the graph will still touch and bounce off of the [latex]x[\/latex]-axis, but for each increasing even power the graph will appear flatter as it approaches and leaves the [latex]x[\/latex]-axis.<\/p>\n For higher odd powers, such as [latex]5, 7,[\/latex] and [latex]9[\/latex], the graph will still cross through the [latex]x[\/latex]-axis, but for each increasing odd power, the graph will appear flatter as it approaches and leaves the [latex]x[\/latex]-axis.<\/p>\n If a polynomial contains a factor of the form [latex]{\\left(x-h\\right)}^{p}[\/latex], the behavior near the [latex]x[\/latex]-intercept [latex]h[\/latex] is determined by the power [latex]p[\/latex]. We say that [latex]x=h[\/latex] is a zero of multiplicity<\/strong> [latex]p[\/latex].<\/p>\n <\/p>\n The graph of a polynomial function will touch the [latex]x[\/latex]-axis at zeros with even multiplicities. The graph will cross the [latex]x[\/latex]-axis at zeros with odd multiplicities.<\/p>\n <\/p>\n The sum of the multiplicities is the degree of the polynomial function.<\/p>\n<\/section>\n How To: Given a graph of a polynomial function of degree [latex]n[\/latex], identify the zeros and their multiplicities.<\/strong><\/p>\n Use the graph of the function of degree [latex]6[\/latex] to identify the zeros of the function and their possible multiplicities.<\/p>\n <\/p>\n [reveal-answer q=\"583908\"]Show Solution[\/reveal-answer] The polynomial function is of degree [latex]n[\/latex] which is [latex]6[\/latex]. The sum of the multiplicities must be [latex]6[\/latex].<\/p>\n Starting from the left, the first zero occurs at [latex]x=-3[\/latex]. The graph touches the [latex]x[\/latex]-axis, so the multiplicity of the zero must be even. The zero of [latex]\u20133[\/latex] has multiplicity [latex]2[\/latex].<\/p>\n The next zero occurs at [latex]x=-1[\/latex]. The graph looks almost linear at this point. This is a single zero of multiplicity [latex]1[\/latex].<\/p>\n The last zero occurs at [latex]x=4[\/latex]. The graph crosses the [latex]x[\/latex]-axis, so the multiplicity of the zero must be odd. We know that the multiplicity is [latex]3[\/latex] and that the sum of the multiplicities must be [latex]6[\/latex].<\/p>\n [\/hidden-answer]<\/p>\n<\/section>\n [ohm2_question hide_question_numbers=1]13819[\/ohm2_question]<\/p>\n<\/section>\n A continuous function<\/strong> has no breaks in its graph: the graph can be drawn without lifting the pen from the paper. A smooth curve<\/strong> is a graph that has no sharp corners. The turning points<\/strong> of a smooth graph must always occur at rounded curves. The graphs of polynomial functions are both continuous and smooth.<\/p>\n The degree of a polynomial function helps us to determine the number of [latex]x[\/latex]-intercepts and the number of turning points. A polynomial function of [latex]n[\/latex]th degree is the product of [latex]n[\/latex] factors, so it will have at most [latex]n[\/latex] roots or zeros, or [latex]x[\/latex]-intercepts. The graph of the polynomial function of degree [latex]n[\/latex] must have at most [latex]n \u2013 1[\/latex] turning points. This means the graph has at most one fewer turning point than the degree of the polynomial or one fewer than the number of factors.<\/p>\n Why do we use the phrase \"at most<\/em> [latex]n[\/latex]\" when describing the number of real roots ([latex]x[\/latex]-intercepts) of the graph of an [latex]n^{\\text{th}}[\/latex] degree polynomial? Can it have fewer?<\/p>\n [reveal-answer q=\"232068\"]more[\/reveal-answer] Ex. Consider the graph of the polynomial function [latex]f(x)=x^2-x+1[\/latex]. The function is a [latex]2^{\\text{nd}}[\/latex] degree polynomial, so it must have at most<\/em> [latex]n[\/latex] roots and [latex]n-1[\/latex] turning points.<\/p>\n We know this function has non-real roots since the discriminant of the quadratic formula is negative. This means that this [latex]2^{\\text{nd}}[\/latex] polynomial has no real roots (apply the quadratic formula to prove this to yourself if needed). That is, it has no [latex]x[\/latex]-intercepts. But it does have two distinct complex roots.<\/p>\n Can you picture the graph of a quadratic function with one distinct real root? Two? But you can also see that there will never be more than two [latex]x[\/latex]-intercepts. Since a parabola (the graph of a [latex]2^{\\text{nd}}[\/latex] degree polynomial) has only one turning point, it can't cross the [latex]x[\/latex]-axis more than twice.<\/p>\n [\/hidden-answer]<\/p>\n<\/section>\n Graphs behave differently at various [latex]x[\/latex]-intercepts. Sometimes the graph will cross over the [latex]x[\/latex]-axis at an intercept. Other times the graph will touch the [latex]x[\/latex]-axis and bounce off.<\/p>\n Suppose, for example, we graph the function [latex]f\\left(x\\right)=\\left(x+3\\right){\\left(x - 2\\right)}^{2}{\\left(x+1\\right)}^{3}[\/latex].<\/p>\n Notice in the figure below that the behavior of the function at each of the [latex]x[\/latex]-intercepts is different.<\/p>\n The behavior of a graph at an x-intercept can be determined by examining the multiplicity of the zero.<\/span><\/strong><\/p>\n The [latex]x[\/latex]-intercept [latex]x=-3[\/latex] is the solution to the equation [latex]\\left(x+3\\right)=0[\/latex]. The graph passes directly through the [latex]x[\/latex]-intercept at [latex]x=-3[\/latex]. The factor is linear (has a degree of [latex]1[\/latex]), so the behavior near the intercept is like that of a line; it passes directly through the intercept. We call this a single zero because the zero corresponds to a single factor of the function.<\/p>\n The [latex]x[\/latex]-intercept [latex]x=2[\/latex] is the repeated solution to the equation [latex]{\\left(x - 2\\right)}^{2}=0[\/latex]. The graph touches the axis at the intercept and changes direction. The factor is quadratic (degree [latex]2[\/latex]), so the behavior near the intercept is like that of a quadratic\u2014it bounces off of the horizontal axis at the intercept.<\/p>\n [latex]{\\left(x - 2\\right)}^{2}=\\left(x - 2\\right)\\left(x - 2\\right)[\/latex]<\/p>\n The factor is repeated, that is, the factor [latex]\\left(x - 2\\right)[\/latex] appears twice. The number of times a given factor appears in the factored form of the equation of a polynomial is called the multiplicity<\/strong>. The zero associated with this factor, [latex]x=2[\/latex], has multiplicity [latex]2[\/latex] because the factor [latex]\\left(x - 2\\right)[\/latex] occurs twice.<\/p>\n The [latex]x[\/latex]-intercept [latex]x=-1[\/latex] is the repeated solution of factor [latex]{\\left(x+1\\right)}^{3}=0[\/latex]. The graph passes through the axis at the intercept but flattens out a bit first. This factor is cubic (degree [latex]3[\/latex]), so the behavior near the intercept is like that of a cubic with the same S-shape near the intercept as the function [latex]f\\left(x\\right)={x}^{3}[\/latex]. We call this a triple zero, or a zero with multiplicity [latex]3[\/latex].<\/p>\n For zeros<\/strong> with even multiplicities, the graphs touch<\/em> or are tangent to the [latex]x[\/latex]-axis at these [latex]x[\/latex]-values. For zeros with odd multiplicities, the graphs cross<\/em> or intersect the [latex]x[\/latex]-axis at these [latex]x[\/latex]-values. See the graphs below for examples of graphs of polynomial functions with multiplicity [latex]1, 2,[\/latex] and [latex]3[\/latex].<\/p>\n <\/p>\n For higher even powers, such as [latex]4, 6,[\/latex] and [latex]8[\/latex], the graph will still touch and bounce off of the [latex]x[\/latex]-axis, but for each increasing even power the graph will appear flatter as it approaches and leaves the [latex]x[\/latex]-axis.<\/p>\n For higher odd powers, such as [latex]5, 7,[\/latex] and [latex]9[\/latex], the graph will still cross through the [latex]x[\/latex]-axis, but for each increasing odd power, the graph will appear flatter as it approaches and leaves the [latex]x[\/latex]-axis.<\/p>\n If a polynomial contains a factor of the form [latex]{\\left(x-h\\right)}^{p}[\/latex], the behavior near the [latex]x[\/latex]-intercept [latex]h[\/latex] is determined by the power [latex]p[\/latex]. We say that [latex]x=h[\/latex] is a zero of multiplicity<\/strong> [latex]p[\/latex].<\/p>\n <\/p>\n The graph of a polynomial function will touch the [latex]x[\/latex]-axis at zeros with even multiplicities. The graph will cross the [latex]x[\/latex]-axis at zeros with odd multiplicities.<\/p>\n <\/p>\n The sum of the multiplicities is the degree of the polynomial function.<\/p>\n<\/section>\n How To: Given a graph of a polynomial function of degree [latex]n[\/latex], identify the zeros and their multiplicities.<\/strong><\/p>\n Use the graph of the function of degree [latex]6[\/latex] to identify the zeros of the function and their possible multiplicities.<\/p>\n <\/p>\n![\"Three](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201556\/CNX_Precalc_Figure_03_04_0082.jpg\")
<\/center>\ngraphical behavior of polynomials at [latex]x[\/latex]-intercepts<\/h3>\n
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![\"Three](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201558\/CNX_Precalc_Figure_03_04_0092.jpg\")
<\/center>\n
\n[hidden-answer a=\"583908\"]<\/p>\nTurning Points<\/h3>\n
turning points of polynomial functions<\/h3>\n\nA turning point<\/strong> of a graph is a point where the graph changes from increasing to decreasing or decreasing to increasing.<\/div>\n<\/section>\n
Determining the Number of Turning Points and Intercepts from the Degree of the Polynomial<\/h3>\n
determining the number of turning points and intercepts from the degree of the polynomial<\/h3>\n\nA polynomial of degree [latex]n[\/latex] will have, at most, [latex]n[\/latex] [latex]x[\/latex]-intercepts and [latex]n \u2013 1[\/latex] turning points.<\/section>\n
\n[hidden-answer a=\"232068\"]<\/p>\nGraphs of Polynomial Functions<\/h2>\n
Zeros and Multiplicity<\/h3>\n
<\/div>\n<\/div>\ngraphical behavior of polynomials at [latex]x[\/latex]-intercepts<\/h3>\n
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