Figure 1. (credit: Jason Bay, Flickr)

Suppose a certain species of bird thrives on a small island. Its population over the last few years is shown below.

The population can be estimated using the function [latex]P\left(t\right)=-0.3{t}^{3}+97t+800[/latex], where [latex]P\left(t\right)[/latex] represents the bird population on the island t years after 2009. We can use this model to estimate the maximum bird population and when it will occur. We can also use this model to predict when the bird population will disappear from the island. In this section, we will examine functions that we can use to estimate and predict these types of changes.

Identify power functions

 Identify end behavior of power functions

Figure 2 shows the graphs of [latex]f\left(x\right)={x}^{2},g\left(x\right)={x}^{4}[/latex] and [latex]\text{and}h\left(x\right)={x}^{6}[/latex], which are all power functions with even, whole-number powers. Notice that these graphs have similar shapes, very much like that of the quadratic function in the toolkit. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.

Figure 2. Even-power functions

To describe the behavior as numbers become larger and larger, we use the idea of infinity. We use the symbol [latex]\infty[/latex] for positive infinity and [latex]-\infty[/latex] for negative infinity. When we say that “x approaches infinity,” which can be symbolically written as [latex]x\to \infty [/latex], we are describing a behavior; we are saying that x is increasing without bound.

With the even-power function, as the input increases or decreases without bound, the output values become very large, positive numbers. Equivalently, we could describe this behavior by saying that as [latex]x[/latex] approaches positive or negative infinity, the [latex]f\left(x\right)[/latex] values increase without bound. In symbolic form, we could write

[latex]\text{as }x\to \pm \infty , f\left(x\right)\to \infty\\ [/latex]

Figure 3 shows the graphs of [latex]f\left(x\right)={x}^{3},g\left(x\right)={x}^{5},\text{and}h\left(x\right)={x}^{7}[/latex], which are all power functions with odd, whole-number powers. Notice that these graphs look similar to the cubic function in the toolkit. Again, as the power increases, the graphs flatten near the origin and become steeper away from the origin.

Figure 3. Odd-power function

These examples illustrate that functions of the form [latex]f\left(x\right)={x}^{n}[/latex] reveal symmetry of one kind or another. First, in Figure 2 we see that even functions of the form [latex]f\left(x\right)={x}^{n}\text{, }n\text{ even,}[/latex] are symmetric about the y-axis. In Figure 3 we see that odd functions of the form [latex]f\left(x\right)={x}^{n}\text{, }n\text{ odd,}[/latex] are symmetric about the origin.

For these odd power functions, as x approaches negative infinity, [latex]f\left(x\right)[/latex] decreases without bound. As x approaches positive infinity, [latex]f\left(x\right)[/latex] increases without bound. In symbolic form we write

[latex]\begin{align}&\text{as } x\to -\infty , f\left(x\right)\to -\infty \\[1mm] &\text{as } x\to \infty , f\left(x\right)\to \infty \\ \text{ } \end{align}[/latex]

The behavior of the graph of a function as the input values get very small ( [latex]x\to -\infty[/latex] ) and get very large ( [latex]x\to \infty[/latex] ) is referred to as the end behavior of the function. We can use words or symbols to describe end behavior.

The table below shows the end behavior of power functions in the form [latex]f\left(x\right)=k{x}^{n}[/latex] where [latex]n[/latex] is a non-negative integer depending on the power and the constant.

How To: Given a power function [latex]f\left(x\right)=k{x}^{n}[/latex] where n is a non-negative integer, identify the end behavior.

1. Determine whether the power is even or odd.
2. Determine whether the constant is positive or negative.
3. Use Figure 4 to identify the end behavior.

Example 2: Identifying the End Behavior of a Power Function

Describe the end behavior of the graph of [latex]f\left(x\right)={x}^{8}[/latex].

Show Solution

The coefficient is 1 (positive) and the exponent of the power function is 8 (an even number). As x approaches infinity, the output (value of [latex]f\left(x\right)[/latex] ) increases without bound. We write as [latex]x\to \infty , f\left(x\right)\to \infty [/latex]. As x approaches negative infinity, the output increases without bound. In symbolic form, as [latex]x\to -\infty , f\left(x\right)\to \infty\\ [/latex]. We can graphically represent the function as shown in Figure 5.

Figure 4

Example 3: Identifying the End Behavior of a Power Function.

Describe the end behavior of the graph of [latex]f\left(x\right)=-{x}^{9}[/latex].

Show Solution

The exponent of the power function is 9 (an odd number). Because the coefficient is –1 (negative), the graph is the reflection about the x-axis of the graph of [latex]f\left(x\right)={x}^{9}[/latex]. The graph shows that as x approaches infinity, the output decreases without bound. As x approaches negative infinity, the output increases without bound. In symbolic form, we would write

Figure 5. [latex]\begin{cases}\text{as } x\to -\infty , f\left(x\right)\to \infty \\ \text{as } x\to \infty , f\left(x\right)\to -\infty \end{cases}[/latex]

Analysis of the Solution

We can check our work by using the table feature on a graphing utility.

x	f(x)
–10	1,000,000,000
–5	1,953,125
0	0
5	–1,953,125
10	–1,000,000,000

We can see from the table above that, when we substitute very small values for x, the output is very large, and when we substitute very large values for x, the output is very small (meaning that it is a very large negative value).

Try It

Describe in words and symbols the end behavior of [latex]f\left(x\right)=-5{x}^{4}[/latex].

Show Solution

As x approaches positive or negative infinity, [latex]f\left(x\right)[/latex] decreases without bound: as [latex]x\to \pm \infty , f\left(x\right)\to -\infty\\ [/latex] because of the negative coefficient.

 Identify polynomial functions

An oil pipeline bursts in the Gulf of Mexico, causing an oil slick in a roughly circular shape. The slick is currently 24 miles in radius, but that radius is increasing by 8 miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radius r of the spill depends on the number of weeks w that have passed. This relationship is linear.

We can combine this with the formula for the area A of a circle.

Composing these functions gives a formula for the area in terms of weeks.

Multiplying gives the formula.

This formula is an example of a polynomial function. A polynomial function consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.

 Identify the degree and leading coefficient of polynomial functions

Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the power of the variable. Although the order of the terms in the polynomial function is not important for performing operations, we typically arrange the terms in descending order of power, or in general form. The degree of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form. The leading term is the term containing the highest power of the variable, or the term with the highest degree. The leading coefficient is the coefficient of the leading term.

A General Note: Terminology of Polynomial Functions

Figure 6

We often rearrange polynomials so that the powers are descending.

When a polynomial is written in this way, we say that it is in general form.

Identifying End Behavior of Polynomial Functions

Knowing the degree of a polynomial function is useful in helping us predict its end behavior. To determine its end behavior, look at the leading term of the polynomial function. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as x gets very large or very small, so its behavior will dominate the graph. For any polynomial, the end behavior of the polynomial will match the end behavior of the term of highest degree.

Polynomial Function	Leading Term	Graph of Polynomial Function
[latex]f\left(x\right)=5{x}^{4}+2{x}^{3}-x - 4[/latex]	[latex]5{x}^{4}[/latex]
[latex]f\left(x\right)=-2{x}^{6}-{x}^{5}+3{x}^{4}+{x}^{3}[/latex]	[latex]-2{x}^{6}[/latex]
[latex]f\left(x\right)=3{x}^{5}-4{x}^{4}+2{x}^{2}+1[/latex]	[latex]3{x}^{5}[/latex]
[latex]f\left(x\right)=-6{x}^{3}+7{x}^{2}+3x+1[/latex]	[latex]-6{x}^{3}[/latex]

Example 6: Identifying End Behavior and Degree of a Polynomial Function

Describe the end behavior and determine a possible degree of the polynomial function in Figure 7.

Figure 7

Show Solution

As the input values x get very large, the output values [latex]f\left(x\right)[/latex] increase without bound. As the input values x get very small, the output values [latex]f\left(x\right)[/latex] decrease without bound. We can describe the end behavior symbolically by writing

[latex]\begin{align}&\text{as } x\to -\infty , f\left(x\right)\to -\infty \\ &\text{as } x\to \infty , f\left(x\right)\to \infty \end{align}[/latex]

In words, we could say that as x values approach infinity, the function values approach infinity, and as x values approach negative infinity, the function values approach negative infinity.

We can tell this graph has the shape of an odd degree power function that has not been reflected, so the degree of the polynomial creating this graph must be odd and the leading coefficient must be positive.

Try It

Describe the end behavior, and determine a possible degree of the polynomial function in Figure 9.

Figure 9

Show Solution

As [latex]x\to \infty , f\left(x\right)\to -\infty ; as x\to -\infty , f\left(x\right)\to -\infty [/latex]. It has the shape of an even degree power function with a negative coefficient.

Example 7: Identifying End Behavior and Degree of a Polynomial Function

Given the function [latex]f\left(x\right)=-3{x}^{2}\left(x - 1\right)\left(x+4\right)[/latex], express the function as a polynomial in general form, and determine the leading term, degree, and end behavior of the function.

Show Solution

Obtain the general form by expanding the given expression for [latex]f\left(x\right)[/latex].

[latex]\begin{align} f\left(x\right)&=-3{x}^{2}\left(x - 1\right)\left(x+4\right)\\ &=-3{x}^{2}\left({x}^{2}+3x - 4\right)\\ &=-3{x}^{4}-9{x}^{3}+12{x}^{2}\end{align}[/latex]

The general form is [latex]f\left(x\right)=-3{x}^{4}-9{x}^{3}+12{x}^{2}[/latex]. The leading term is [latex]-3{x}^{4}[/latex]; therefore, the degree of the polynomial is 4. The degree is even (4) and the leading coefficient is negative (–3), so the end behavior is

[latex]\begin{align}&\text{as } x\to -\infty , f\left(x\right)\to -\infty \\ &\text{as } x\to \infty , f\left(x\right)\to -\infty \end{align}[/latex]

Try It

Given the function [latex]f\left(x\right)=0.2\left(x - 2\right)\left(x+1\right)\left(x - 5\right)[/latex], express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.

Show Solution

The general form is [latex]f(x)=0.2x^3-1.2x^2+0.6x-2[/latex]

The leading term is [latex]0.2{x}^{3}[/latex], so it is a degree 3 polynomial. As x approaches positive infinity, [latex]f\left(x\right)[/latex] increases without bound; as x approaches negative infinity, [latex]f\left(x\right)[/latex] decreases without bound.

Identifying Local Behavior of Polynomial Functions

In addition to the end behavior of polynomial functions, we are also interested in what happens in the “middle” of the function. In particular, we are interested in locations where graph behavior changes. A turning point is a point at which the function values change from increasing to decreasing or decreasing to increasing.

Figure 10

We are also interested in the intercepts. As with all functions, the y-intercept is the point at which the graph intersects the vertical axis. The point corresponds to the coordinate pair in which the input value is zero. Because a polynomial is a function, only one output value corresponds to each input value so there can be only one y-intercept, [latex]\left(0,{a}_{0}\right)[/latex]. The x-intercepts occur at the input values that correspond to an output value of zero. It is possible to have more than one x-intercept. 

A General Note: Intercepts and Turning Points of Polynomial Functions

A turning point of a graph is a point at which the graph changes direction from increasing to decreasing or decreasing to increasing. The y-intercept is the point at which the function has an input value of zero. The x-intercepts are the points at which the output value is zero.

How To: Given a polynomial function, determine the intercepts.

1. Determine the y-intercept by setting [latex]x=0[/latex] and finding the corresponding output value.
2. Determine the x-intercepts by solving for the input values that yield an output value of zero.

Example 8: Determining the Intercepts of a Polynomial Function

Given the polynomial function [latex]f\left(x\right)=\left(x - 2\right)\left(x+1\right)\left(x - 4\right)[/latex], written in factored form for your convenience, determine the y– and x-intercepts.

Show Solution

The y-intercept occurs when the input is zero so substitute 0 for x.

[latex]\begin{align}f\left(0\right)&=\left(0 - 2\right)\left(0+1\right)\left(0 - 4\right) \\ &=\left(-2\right)\left(1\right)\left(-4\right) \\ &=8 \end{align}[/latex]

The y-intercept is (0, 8).

The x-intercepts occur when the output is zero.

[latex]\left(x - 2\right)\left(x+1\right)\left(x - 4\right)=0[/latex]

[latex]\begin{align} &x - 2=0 && \text{or} && x+1=0 && \text{or} && x - 4=0 \\ &x=2 && \text{or} && x=-1 && \text{or} && x=4 \end{align}[/latex]

The x-intercepts are [latex]\left(2,0\right),\left(-1,0\right)[/latex], and [latex]\left(4,0\right)[/latex].

We can see these intercepts on the graph of the function shown in Figure 11.

Figure 11

Example 9: Determining the Intercepts of a Polynomial Function with Factoring

Given the polynomial function [latex]f\left(x\right)={x}^{4}-4{x}^{2}-45[/latex], determine the y– and x-intercepts.

Show Solution

The y-intercept occurs when the input is zero.

[latex]\begin{align} f\left(0\right)&={\left(0\right)}^{4}-4{\left(0\right)}^{2}-45 \\ &=-45 \end{align}[/latex]

The y-intercept is [latex]\left(0,-45\right)[/latex].

The x-intercepts occur when the output is zero. To determine when the output is zero, we will need to factor the polynomial.

[latex]\begin{align}f\left(x\right)&={x}^{4}-4{x}^{2}-45 \\ &=\left({x}^{2}-9\right)\left({x}^{2}+5\right) \\ &=\left(x - 3\right)\left(x+3\right)\left({x}^{2}+5\right)\\ \text{ } \end{align}[/latex]

[latex]\left(x - 3\right)\left(x+3\right)\left({x}^{2}+5\right)=0[/latex]

[latex]x^2+5[/latex] can’t be 0, so we only consider the first two factors.

[latex]\begin{align}x - 3=0 && \text{or} && x+3=0 \\ x=3 && \text{or} && x=-3 \end{align}[/latex]

The x-intercepts are [latex]\left(3,0\right)[/latex] and [latex]\left(-3,0\right)[/latex].

We can see these intercepts on the graph of the function shown in Figure 12. We can see that the function is even because [latex]f\left(x\right)=f\left(-x\right)[/latex].

Figure 12

Try It

Given the polynomial function [latex]f\left(x\right)=2{x}^{3}-6{x}^{2}-20x[/latex], determine the y– and x-intercepts.

Show Solution

y-intercept [latex]\left(0,0\right)[/latex]; x-intercepts [latex]\left(0,0\right),\left(-2,0\right)[/latex], and [latex]\left(5,0\right)[/latex]

Try It

Comparing Smooth and Continuous Graphs

The degree of a polynomial function helps us to determine the number of x-intercepts and the number of turning points. A polynomial function of nth degree is the product of n factors, so it will have at most n roots or zeros, or x-intercepts. The graph of the polynomial function of degree n must have at most n – 1 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.

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.

A General Note: Intercepts and Turning Points of Polynomials

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

Example 10: Determining the Number of Intercepts and Turning Points of a Polynomial

Without graphing the function, determine the local behavior of the function by finding the maximum number of x-intercepts and turning points for [latex]f\left(x\right)=-3{x}^{10}+4{x}^{7}-{x}^{4}+2{x}^{3}[/latex].

Show Solution

The polynomial has a degree of 10, so there are at most 10 [latex]x[/latex]-intercepts and at most 9 turning points.

 

Try It

Without graphing the function, determine the maximum number of x-intercepts and turning points for [latex]f\left(x\right)=108 - 13{x}^{9}-8{x}^{4}+14{x}^{12}+2{x}^{3}[/latex]

Show Solution

There are at most 12 x-intercepts and at most 11 turning points.

Example 11: Drawing Conclusions about a Polynomial Function from the Graph

What can we conclude about the polynomial represented by the graph shown in the graph in Figure 13 based on its intercepts and turning points?

Figure 13

Show Solution

Figure 14

The end behavior of the graph tells us this is the graph of an even-degree polynomial. 

The graph has 2 x-intercepts, suggesting a degree of 2 or greater, and 3 turning points, suggesting a degree of 4 or greater. Based on this, it would be reasonable to conclude that the degree is even and at least 4.

Try It

What can we conclude about the polynomial represented by Figure 15 based on its intercepts and turning points?

Figure 15

Show Solution

The end behavior indicates an odd-degree polynomial function; there are 3 x-intercepts and 2 turning points, so the degree is odd and at least 3. Because of the end behavior, we know that the lead coefficient must be negative.

Example 12: Drawing Conclusions about a Polynomial Function from the Factors

Given the function [latex]f\left(x\right)=-4x\left(x+3\right)\left(x - 4\right)[/latex], determine the local behavior.

Show Solution

The y-intercept is found by evaluating [latex]f\left(0\right)[/latex].

[latex]f\left(0\right)=-4\left(0\right)\left(0+3\right)\left(0 - 4\right)=0 [/latex]

The y-intercept is [latex]\left(0,0\right)[/latex].

The x-intercepts are found by determining the zeros of the function.

[latex]-4x\left(x+3\right)\left(x - 4\right)=0[/latex]

[latex]\begin{align}x=0 && \text{or} && x+3=0 && \text{or} && x - 4=0 \\ x=0 && \text{or} && x=-3 && \text{or} && x=4\end{align}[/latex]

The x-intercepts are [latex]\left(0,0\right),\left(-3,0\right)[/latex], and [latex]\left(4,0\right)[/latex].

The degree is 3 so the graph has at most 2 turning points.

Try It

Given the function [latex]f\left(x\right)=0.2\left(x - 2\right)\left(x+1\right)\left(x - 5\right)[/latex], determine the local behavior.

Show Solution

The x-intercepts are [latex]\left(2,0\right),\left(-1,0\right)[/latex], and [latex]\left(5,0\right)[/latex], the y-intercept is [latex]\left(0,\text{2}\right)[/latex], and the graph has at most 2 turning points.

Graphing Polynomials

The revenue in millions of dollars for a fictional cable company from 2006 through 2013 is shown in the table below.

The revenue can be modeled by the polynomial function

where R represents the revenue in millions of dollars and t represents the year, with t = 6 corresponding to 2006. Over which intervals is the revenue for the company increasing? Over which intervals is the revenue for the company decreasing? These questions, along with many others, can be answered by examining the graph of the polynomial function. We have already explored the local behavior of quadratics, a special case of polynomials. In this section we will explore the local behavior of polynomials in general.

Recognize characteristics of graphs of polynomial functions

Polynomial functions of degree 2 or more have graphs that do not have sharp corners; recall that these types of graphs are called smooth curves. Polynomial functions also display graphs that have no breaks. Curves with no breaks are called continuous. Figure 1 shows a graph that represents a polynomial function and a graph that represents a function that is not a polynomial.

Figure 1

Example 1: Recognizing Polynomial Functions

Which of the graphs in Figure 2 represents a polynomial function?

Figure 2

Show Solution

The graphs of f and h are graphs of polynomial functions. They are smooth and continuous.

The graphs of g and k are graphs of functions that are not polynomials. The graph of function g has a sharp corner. The graph of function k is not continuous.

 Use factoring to find zeros of polynomial functions

Find zeros of polynomial functions

Recall that if f is a polynomial function, the values of x for which [latex]f\left(x\right)=0[/latex] are called zeros of f. If the equation of the polynomial function can be factored, we can set each factor equal to zero and solve for the zeros.

We can use this method to find x-intercepts because at the x-intercepts we find the input values when the output value is zero. For general polynomials, this can be a challenging prospect. While quadratics can be solved using the relatively simple quadratic formula, the corresponding formulas for cubic and fourth-degree polynomials are not simple enough to remember, and formulas do not exist for general higher-degree polynomials. Consequently, we will limit ourselves to three cases in this section:

1. The polynomial can be factored using known methods: greatest common factor and trinomial factoring.
2. The polynomial is given in factored form.
3. Technology is used to determine the intercepts.

Example 2: Finding the x-Intercepts of a Polynomial Function by Factoring

Find the x-intercepts of [latex]f\left(x\right)={x}^{6}-3{x}^{4}+2{x}^{2}[/latex].

Show Solution

We can attempt to factor this polynomial to find solutions for [latex]f\left(x\right)=0[/latex].

[latex]\begin{align} &{x}^{6}-3{x}^{4}+2{x}^{2}=0 && \\ &{x}^{2}\left({x}^{4}-3{x}^{2}+2\right)=0 && \text{Factor out the greatest common factor}. \\ &{x}^{2}\left({x}^{2}-1\right)\left({x}^{2}-2\right)=0 && \text{Factor the trinomial}. \\ &{x}^{2}\left(x+1\right)\left(x-1\right)\left({x}^{2}-2\right)=0 && \text{Factor the difference of squares}. \end{align}[/latex]

Now set each factor equal to zero and solve.

[latex]\begin{align} & {x}^{2}=0 && x+1=0 && x-1=0 && {x}^{2}-2=0 \\ &x=0 && x=-1 && x=1 && x=\pm \sqrt{2} \end{align}[/latex]

Figure 3

This gives us five x-intercepts: [latex]\left(0,0\right),\left(1,0\right),\left(-1,0\right),\left(\sqrt{2},0\right)[/latex], and [latex]\left(-\sqrt{2},0\right)[/latex]. We can see that this is an even function.

Example 3: Finding the x-Intercepts of a Polynomial Function by Factoring

Find the x-intercepts of [latex]f\left(x\right)={x}^{3}-5{x}^{2}-x+5[/latex].

Show Solution

Find solutions for [latex]f\left(x\right)=0[/latex] by factoring.

[latex]\begin{align} &{x}^{3}-5{x}^{2}-x+5=0 \\ &{x}^{2}\left(x - 5\right)-1\left(x - 5\right)=0 && \text{Factor by grouping}. \\ &\left({x}^{2}-1\right)\left(x - 5\right)=0 && \text{Factor out the common factor}. \\ &\left(x+1\right)\left(x - 1\right)\left(x - 5\right)=0 && \text{Factor the difference of squares}. \end{align}[/latex]

Now we set each factor equal to 0.

[latex]\begin{align}&x+1=0 && x - 1=0 && x - 5=0 \\ &x=-1 && x=1 && x=5 \end{align}[/latex]

Figure 4

There are three x-intercepts: [latex]\left(-1,0\right),\left(1,0\right)[/latex], and [latex]\left(5,0\right)[/latex].

Example 4: Finding the y– and x-Intercepts of a Polynomial in Factored Form

Find the y– and x-intercepts of [latex]g\left(x\right)={\left(x - 2\right)}^{2}\left(2x+3\right)[/latex].

Show Solution

The y-intercept can be found by evaluating [latex]g\left(0\right)[/latex].

[latex]g\left(0\right)={\left(0 - 2\right)}^{2}\left(2\left(0\right)+3\right)=12[/latex]

So the y-intercept is [latex]\left(0,12\right)[/latex].

The x-intercepts can be found by solving [latex]g\left(x\right)=0[/latex].

[latex]{\left(x - 2\right)}^{2}\left(2x+3\right)=0[/latex]

[latex]\begin{align}&{\left(x - 2\right)}^{2}=0 && 2x+3=0 \\ &x=2 &&x=-\frac{3}{2} \end{align}[/latex]

So the x-intercepts are [latex]\left(2,0\right)[/latex] and [latex]\left(-\frac{3}{2},0\right)[/latex].

Analysis of the Solution

We can always check that our answers are reasonable by using a graphing calculator to graph the polynomial as shown in Figure 5.

Figure 5

Example 5: Finding the x-Intercepts of a Polynomial Function Using a Graph

Find the x-intercepts of [latex]h\left(x\right)={x}^{3}+4{x}^{2}+x - 6[/latex].

Show Solution

This polynomial is not in factored form, has no common factors, and does not appear to be factorable using techniques previously discussed. Fortunately, we can use technology to find the intercepts. Keep in mind that some values make graphing difficult by hand. In these cases, we can take advantage of graphing utilities.

Looking at the graph of this function, as shown in Figure 6, it appears that there are x-intercepts at [latex]x=-3,-2[/latex], and 1.

Figure 6

We can check whether these are correct by substituting these values for x and verifying that the function is equal to 0.

Since [latex]h\left(x\right)={x}^{3}+4{x}^{2}+x - 6[/latex], we have:

[latex]h\left(-3\right)={\left(-3\right)}^{3}+4{\left(-3\right)}^{2}+\left(-3\right)-6=-27+36 - 3-6=0[/latex]

[latex]h\left(-2\right)={\left(-2\right)}^{3}+4{\left(-2\right)}^{2}+\left(-2\right)-6=-8+16 - 2-6=0[/latex]

[latex]h\left(1\right)={\left(1\right)}^{3}+4{\left(1\right)}^{2}+\left(1\right)-6=1+4+1 - 6=0[/latex]

Each x-intercept corresponds to a zero of the polynomial function and each zero yields a factor, so we can now write the polynomial in factored form.

[latex]h\left(x\right)={x}^{3}+4{x}^{2}+x - 6=\left(x+3\right)\left(x+2\right)\left(x - 1\right)[/latex]

Identify zeros and their multiplicities

Graphs behave differently at various x-intercepts. Sometimes, the graph will cross over the horizontal axis at an intercept. Other times, the graph will touch the horizontal axis and bounce off.

Suppose, for example, we graph the function

Notice in Figure 7 that the behavior of the function at each of the x-intercepts is different.

Figure 7. Identifying the behavior of the graph at an x-intercept by examining the multiplicity of the zero.

The x-intercept [latex]x=-3[/latex] is the solution of equation [latex]x+3=0[/latex]. The graph passes directly through the x-intercept at [latex]x=-3[/latex]. The factor is linear (has a degree of 1), 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 x-intercept [latex]x=2[/latex] is the repeated solution of 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 2), so the behavior near the intercept is like that of a quadratic—it bounces off of the horizontal axis at the intercept.

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 2 because the factor [latex]\left(x - 2\right)[/latex] occurs twice.

The x-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 3), so the behavior near the intercept is like that of a cubic—with the same S-shape near the intercept as the toolkit function [latex]f\left(x\right)={x}^{3}[/latex]. We call this a triple zero, or a zero with multiplicity 3.

For zeros with even multiplicities, the graphs touch or are tangent to the x-axis. For zeros with odd multiplicities, the graphs cross or intersect the x-axis. See Figure 8 for examples of graphs of polynomial functions with multiplicity 1, 2, and 3.

Figure 8

For higher even powers, such as 4, 6, and 8, the graph will still touch and bounce off of the horizontal axis but, for each increasing even power, the graph will appear flatter as it approaches and leaves the x-axis.

For higher odd powers, such as 5, 7, and 9, the graph will still cross through the horizontal axis, but for each increasing odd power, the graph will appear flatter as it approaches and leaves the x-axis.

Example 6: Identifying Zeros and Their Multiplicities

Use the graph of the function of degree 6 to identify the zeros of the function and their possible multiplicities.

Figure 9

Show Solution

The polynomial function is of degree n. The sum of the multiplicities must be n.

Starting from the left, the first zero occurs at [latex]x=-3[/latex]. The graph touches the x-axis, so the multiplicity of the zero must be even. The zero of –3 has multiplicity 2.

The next zero occurs at [latex]x=-1[/latex]. The graph looks almost linear at this point. This is a single zero of multiplicity 1.

The last zero occurs at [latex]x=4[/latex]. The graph crosses the x-axis, so the multiplicity of the zero must be odd. We know that the multiplicity is likely 3 and that the sum of the multiplicities is likely 6.

Try It

Use the graph of the function of degree 5 to identify the zeros of the function and their multiplicities.

Figure 10

Show Solution

The graph has a zero of –5 with multiplicity 1, a zero of –1 with multiplicity 2, and a zero of 3 with even multiplicity.

 Determine end behavior

As we have already learned, the behavior of a graph of a polynomial function of the form

will either ultimately rise or fall as x increases without bound and will either rise or fall as x 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 –100 or –1,000.

Recall that we call this behavior the end behavior 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 increases or decreases without bound, [latex]f\left(x\right)[/latex] increases without bound. When the leading term is an odd power function, as x decreases without bound, [latex]f\left(x\right)[/latex] also decreases without bound; as x increases without bound, [latex]f\left(x\right)[/latex] also increases without bound. If the leading term is negative, it will change the direction of the end behavior. The table below summarizes all four cases.

Even Degree	Odd Degree

Understand the relationship between degree and turning points

In addition to the end behavior, recall that we can analyze a polynomial function’s local behavior. It may have a turning point where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising). Look at the graph of the polynomial function [latex]f\left(x\right)={x}^{4}-{x}^{3}-4{x}^{2}+4x[/latex] in Figure 11. The graph has three turning points.

Figure 11

This function f is a 4^th degree polynomial function and has 3 turning points. The maximum number of turning points of a polynomial function is always one less than the degree of the function.

Example 7: Finding the Maximum Number of Turning Points Using the Degree of a Polynomial Function

Find the maximum number of turning points of each polynomial function.

1. [latex]f\left(x\right)=-{x}^{3}+4{x}^{5}-3{x}^{2}++1[/latex]
2. [latex]f\left(x\right)=-{\left(x - 1\right)}^{2}\left(1+2{x}^{2}\right)[/latex]

Show Solution

1. [latex]f\left(x\right)=-x{}^{3}+4{x}^{5}-3{x}^{2}++1[/latex]

   First, rewrite the polynomial function in descending order: [latex]f\left(x\right)=4{x}^{5}-{x}^{3}-3{x}^{2}++1[/latex]

   Identify the degree of the polynomial function. This polynomial function is of degree 5.

   The maximum number of turning points is 5 – 1 = 4.

2. [latex]f\left(x\right)=-{\left(x - 1\right)}^{2}\left(1+2{x}^{2}\right)[/latex]

[latex]a_{n}=-\left(x^2\right)\left(2x^2\right)=-2x^4[/latex]

First, identify the leading term of the polynomial function if the function were expanded.

Then, identify the degree of the polynomial function. This polynomial function is of degree 4.

The maximum number of turning points is 4 – 1 = 3.

 Graph polynomial functions

We can use what we have learned about multiplicities, end behavior, and turning points to sketch graphs of polynomial functions. Let us put this all together and look at the steps required to graph polynomial functions.

Example 8: Sketching the Graph of a Polynomial Function

Sketch a graph of [latex]f\left(x\right)=-2{\left(x+3\right)}^{2}\left(x - 5\right)[/latex].

Show Solution

This graph has two x-intercepts. At x = –3, the factor is squared, indicating a multiplicity of 2. The graph will bounce at this x-intercept. At x = 5, the function has a multiplicity of one, indicating the graph will cross through the axis at this intercept.

The y-intercept is found by evaluating f(0).

[latex]\begin{align} f\left(0\right)&=-2{\left(0+3\right)}^{2}\left(0 - 5\right) \\ &=-2\cdot 9\cdot \left(-5\right) \\ &=90 \end{align}[/latex]

The y-intercept is (0, 90).

Figure 13

Additionally, we can see the leading term, if this polynomial were multiplied out, would be [latex]-2{x}^{3}[/latex],
so the end behavior is that of a vertically reflected cubic, with the outputs decreasing as the inputs approach infinity, and the outputs increasing as the inputs approach negative infinity.

To sketch this, we consider that:

• As [latex]x\to -\infty [/latex] the function [latex]f\left(x\right)\to \infty [/latex], so we know the graph starts in the second quadrant and is decreasing toward the x-axis.
• Since [latex]f\left(-x\right)=-2{\left(-x+3\right)}^{2}\left(-x - 5\right)[/latex]
  is not equal to f(x), the graph does not display symmetry.
• At (-3,0), the graph bounces off of the x-axis, so the function must start increasing.

  At (0, 90), the graph crosses the y-axis at the y-intercept.

Figure 14

Figure 15

Somewhere after this point, the graph must turn back down or start decreasing toward the horizontal axis because the graph passes through the next intercept at (5, 0). 

As [latex]x\to \infty [/latex] the function [latex]f\left(x\right)\to \mathrm{-\infty }[/latex], so we know the graph continues to decrease, and we can stop drawing the graph in the fourth quadrant.

Using technology, we can create the graph for the polynomial function, shown in Figure 16, and verify that the resulting graph looks like our sketch in Figure 15.

Figure 16

Writing Formulas for Polynomial Functions

Now that we know how to find zeros of polynomial functions, we can use them to write formulas based on graphs. Because a polynomial function written in factored form will have an x-intercept where each factor is equal to zero, we can form a function that will pass through a set of x-intercepts by introducing a corresponding set of factors.

A General Note: Factored Form of Polynomials

If a polynomial of lowest degree p has horizontal intercepts at [latex]x={x}_{1},{x}_{2},\dots ,{x}_{n}[/latex], then the polynomial can be written in the factored form: [latex]f\left(x\right)=a{\left(x-{x}_{1}\right)}^{{p}_{1}}{\left(x-{x}_{2}\right)}^{{p}_{2}}\cdots {\left(x-{x}_{n}\right)}^{{p}_{n}}[/latex] where the powers [latex]{p}_{i}[/latex] on each factor can be determined by the behavior of the graph at the corresponding intercept, and the stretch factor a can be determined given a value of the function other than the x-intercept.

How To: Given a graph of a polynomial function, write a formula for the function.

1. Identify the x-intercepts of the graph to find the factors of the polynomial.
2. Examine the behavior of the graph at the x-intercepts to determine the multiplicity of each factor.
3. Find the polynomial of least degree containing all the factors found in the previous step.
4. Use any other point on the graph (the y-intercept may be easiest) to determine the stretch factor.

Example 13: Writing a Formula for a Polynomial Function from the Graph

Write a formula for the polynomial function shown in Figure 19.

Figure 19

Show Solution

his graph has three x-intercepts: x = –3, 2, and 5. The y-intercept is located at (0, 2). At x = –3 and x = 5, the graph passes through the axis linearly, suggesting the corresponding factors of the polynomial will be linear. At x = 2, the graph bounces at the intercept, suggesting the corresponding factor of the polynomial will be second degree (quadratic). Together, this gives us

[latex]f\left(x\right)=a\left(x+3\right){\left(x - 2\right)}^{2}\left(x - 5\right)[/latex]

To determine the stretch factor, we utilize another point on the graph. We will use the y-intercept (0, –2), to solve for a.

[latex]\begin{align}f\left(0\right)&=a\left(0+3\right){\left(0 - 2\right)}^{2}\left(0 - 5\right) \\ -2&=a\left(0+3\right){\left(0 - 2\right)}^{2}\left(0 - 5\right) \\ -2&=-60a \\ a&=\frac{1}{30} \end{align}[/latex]

The graphed polynomial appears to represent the function [latex]f\left(x\right)=\frac{1}{30}\left(x+3\right){\left(x - 2\right)}^{2}\left(x - 5\right)[/latex].

Try It

Given the graph in Figure 20, write a formula for the function shown.

Figure 20

Show Solution

[latex]f\left(x\right)=-\frac{1}{8}{\left(x - 2\right)}^{3}{\left(x+1\right)}^{2}\left(x - 4\right)[/latex]

Key Concepts

• A power function is a variable base raised to a number power.
• The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end behavior.
• The end behavior pf a power function depends on whether the power is even or odd.
• A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power.
• The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient.
• The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function.
• A polynomial of degree n will have at most n x-intercepts and at most n – 1 turning points.
• Polynomial functions of degree 2 or more are smooth, continuous functions.
• To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero.
• Another way to find the x-intercepts of a polynomial function is to graph the function and identify the points at which the graph crosses the x-axis.
• The multiplicity of a zero determines how the graph behaves at the x-intercepts.
• The graph of a polynomial will cross the horizontal axis at a zero with odd multiplicity.
• The graph of a polynomial will touch the horizontal axis at a zero with even multiplicity.
• The end behavior of a polynomial function depends on the leading term.
• The graph of a polynomial function changes direction at its turning points.
• A polynomial function of degree n has at most n – 1 turning points.
• To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most n – 1 turning points.