Graphs of Polynomial Functions

Zeros and Multiplicity

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.

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].

Notice in the figure below that the behavior of the function at each of the [latex]x[/latex]-intercepts is different.

The behavior of a graph at an x-intercept can be determined by examining the multiplicity of the zero.

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.

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—it bounces off of the horizontal axis at the intercept.

[latex]{\left(x - 2\right)}^{2}=\left(x - 2\right)\left(x - 2\right)[/latex]

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. 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.

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].

For zeros with even multiplicities, the graphs touch or are tangent to the [latex]x[/latex]-axis at these [latex]x[/latex]-values. For zeros with odd multiplicities, the graphs cross 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].

 

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.

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.

graphical behavior of polynomials at [latex]x[/latex]-intercepts

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 [latex]p[/latex].

 

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.

 

The sum of the multiplicities is the degree of the polynomial function.

How To: Given a graph of a polynomial function of degree [latex]n[/latex], identify the zeros and their multiplicities.

1. If the graph crosses the [latex]x[/latex]-axis and appears almost linear at the intercept, it is a single zero.
2. If the graph touches the [latex]x[/latex]-axis and bounces off of the axis, it is a zero with even multiplicity.
3. If the graph crosses the [latex]x[/latex]-axis at a zero, it is a zero with odd multiplicity.
4. The sum of the multiplicities is the degree [latex]n[/latex].

Use the graph of the function of degree [latex]6[/latex] to identify the zeros of the function and their possible multiplicities.

 

Show Solution

The polynomial function is of degree [latex]n[/latex] which is [latex]6[/latex]. The sum of the multiplicities must be [latex]6[/latex].

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]–3[/latex] has multiplicity [latex]2[/latex].

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].

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].

Turning Points

A continuous function has no breaks in its graph: the graph can be drawn without lifting the pen from the paper. A smooth curve is a graph that has no sharp corners. The turning points of a smooth graph must always occur at rounded curves. The graphs of polynomial functions are both continuous and smooth.

turning points of polynomial functions

A turning point of a graph is a point where the graph changes from increasing to decreasing or decreasing to increasing.

Determining the Number of Turning Points and Intercepts from the Degree of the Polynomial

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 – 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.

determining the number of turning points and intercepts from the degree of the polynomial

A polynomial of degree [latex]n[/latex] will have, at most, [latex]n[/latex] [latex]x[/latex]-intercepts and [latex]n – 1[/latex] turning points.

Why do we use the phrase “at most [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?

more

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 [latex]n[/latex] roots and [latex]n-1[/latex] turning points.

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.

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.