Graphs of Polynomial Functions

Learning Outcomes

Recognize characteristics of graphs of polynomial functions.
Identify zeros of polynomials and their multiplicities.
Determine end behavior.
Understand the relationship between degree and turning points.
Graph polynomial functions.
Solve polynomial inequalities.
Use the Intermediate Value Theorem.
Write the formula for a polynomial function.

Revenue Example

A fictional cable company's revenue (in millions of dollars) from 2006 through 2013 is modeled by the polynomial function:

$R(t) = -0.037t^4 + 1.414t^3 - 19.777t^2 + 118.696t - 205.332$

where R represents the revenue in millions of dollars and t represents the year, with t = 6 corresponding to 2006.

The graph of the polynomial function can be examined to determine when the revenue for the company is increasing or decreasing.

Characteristics of Polynomial Functions

Polynomial functions of degree 2 or more have graphs that do not have sharp corners; these graphs are called smooth curves.
Polynomial functions also display graphs that have no breaks. Curves with no breaks are called continuous.

Recognizing Polynomial Functions

Polynomial functions have graphs that are smooth and continuous.

Domain of Polynomial Functions

All polynomial functions have as their domain all real numbers.
Any real number is a valid input for a polynomial function.

Finding Zeros of Polynomial Functions

The values of x for which $f(x) = 0$ are called zeros of f.
X-intercepts can be found by finding the input values when the output value is zero.
Quadratics can be solved using the relatively simple quadratic formula, but 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.
Three cases for finding zeros:
- The polynomial can be factored using known methods: greatest common factor and trinomial factoring.
- The polynomial is given in factored form.
- Technology is used to determine the intercepts.

How To Find The X-Intercepts

Set $f(x)=0$ .
If the polynomial function is not given in factored form:
- Factor out any common monomial factors.
- Factor any factorable binomials or trinomials.
Set each factor equal to zero and solve to find the $x-$ intercepts.

Identifying 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.
The x-intercept $x = -3$ is the solution of equation $x + 3 = 0$ . The graph passes directly through the x-intercept at $x = -3$ . 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 $x = 2$ is the repeated solution of the equation $(x - 2)^2 = 0$ . 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 $(x - 2)$ 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, $x = 2$ , has multiplicity 2 because the factor $(x - 2)$ occurs twice.
The intercept $x = -1$ is the repeated solution of factor $(x + 1)^3 = 0$ . The graph passes through the axis at the intercept, but fattens out a bit frst. 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 $f(x) = x^3$ . 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.
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 fatter 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 fatter as it approaches and leaves the x-axis.

Graphical Behavior of Polynomials at X-Intercepts

If a polynomial contains a factor of the form $(x - h)^p$ , the behavior near the x-intercept h is determined by the power p. We say that $x = h$ is a zero of multiplicity p.

The graph of a polynomial function will touch the x-axis at zeros with even multiplicities.
The graph will cross the x-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 N, Identify the Zeros and Their Multiplicities.

If the graph crosses the x-axis and appears almost linear at the intercept, it is a single zero.
If the graph touches the x-axis and bounces off of the axis, it is a zero with even multiplicity.
If the graph crosses the x-axis at a zero, it is a zero with odd multiplicity.
The sum of the multiplicities is n. This includes non-real zeros.

Determine End Behavior

The behavior of a graph of a polynomial function of the form

$f(x) = an x^n + a{n-1} x^{n-1} + … + a1 x + a0$

will either ultimately rise or fall as $x$ increases without bound and will either rise or fall as $x$ decreases without bound.

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.
This behavior is called the end behavior of a function.
When the leading term of a polynomial function, $a_n x^n$ , is an even power function, as $x$ increases or decreases without bound, $f(x)$ increases without bound.
When the leading term is an odd power function, as $x$ decreases without bound, $f(x)$ also decreases without bound. If the leading term is negative, it will change the direction of the end behavior.

Understand the Relationship Between Degree and Turning Points

A polynomial function’s local behavior can be analyzed such as a turning point where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising).
The maximum number of turning points of a polynomial function is always one less than the degree of the function.

Interpreting Turning Points

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

A polynomial of degree n will have at most n – 1 turning points.

Graph Polynomial Functions

Multiplicities, end behavior, and turning points can be used to sketch graphs of polynomial functions.

How To: Given a Polynomial Function, Sketch the Graph.

Find the intercepts.
Check for symmetry.
- If the function is an even function, its graph is symmetrical about the y-axis, that is, f(–x) = f(x).
- If a function is an odd function, its graph is symmetrical about the origin, that is, f(–x) = –f(x).
Use the multiplicities of the zeros to determine the behavior of the polynomial at the x-intercepts.
Determine the end behavior by examining the leading term.
Use the end behavior and the behavior at the intercepts to sketch a graph.
Ensure that the number of turning points does not exceed one less than the degree of the polynomial.
Optionally, use technology to check the graph.

Solving Polynomial Inequalities

One application of our ability to find intercepts and sketch a graph of polynomials is the ability to solve polynomial inequalities. It is a very common question to ask when a function will be positive and negative.
Polynomial inequalities can be solved by either utilizing the graph, or by using test values.

Use the Intermediate Value Theorem

In some situations, we may know two points on a graph but not the zeros. If those two points are on opposite sides of the x-axis, we can confirm that there is a zero between them.
Consider a polynomial function f whose graph is smooth and continuous.
The Intermediate Value Theorem states that for two numbers a and b in the domain of f, if a < b and $f(a) \neq f(b)$ , then the function f takes on every value between $f(a)$ and $f(b)$ .
If a point on the graph of a continuous function f at $x=a$ lies above the x-axis and another point at $x=b$ lies below the x-axis, there must exist a third point between $x=a$ and $x=b$ where the graph crosses the x-axis. Call this point $(c, f(c))$ . This means that we are assured there is a solution c where $f(c)=0$ .
When a polynomial function changes from a negative value to a positive value, the function must cross the x-axis.

Intermediate Value Theorem

Let f be a polynomial function.

The Intermediate Value Theorem states that if $f(a)$ and $f(b)$ have opposite signs, then there exists at least one value c between a and b for which $f(c)=0$ .

Writing Formulas for Polynomial Functions

Zeros of polynomial functions can be used to write formulas based on graphs.
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.

Factored Form of Polynomials

If a polynomial of lowest degree p has horizontal intercepts at $x=x1, x2, \dots, x_n$ , then the polynomial can be written in the factored form:

$f(x) = a(x - x1)^{p1} (x - x2)^{p2} \cdots (x - xn)^{pn}$

where the powers $p_i$ 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.

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

Using Local and Global Extrema

For general polynomials, finding turning points is not possible without more advanced techniques from calculus. Even then, finding where extrema occur can still be algebraically challenging.
Estimate the locations of turning points using technology to generate a graph.
Each turning point represents a local minimum or maximum.
A turning point that represents the highest or lowest point on the entire graph is a global maximum or a global minimum (also referred to as the absolute maximum and absolute minimum values of the function).

Local and Global Extrema

A local maximum or local minimum at x = a (sometimes called the relative maximum or minimum, respectively) is the output at the highest or lowest point on the graph in an open interval around x = a.

If a function has a local maximum at a, then $f(a) \ge f(x)$ for all x in an open interval around x = a.
If a function has a local minimum at a , then $f(a) \le f(x)$ for all x in an open interval around x = a.

A global maximum or global minimum is the output at the highest or lowest point of the function.

If a function has a global maximum at a, then $f(a) \ge f(x)$ for all x.
If a function has a global minimum at a, then $f(a) \le f(x)$ for all x.

Polynomial Functions and Global Extrema

Only polynomial functions of even degree have a global minimum or maximum.

Key Concepts

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.
Graphing a polynomial function helps to estimate local and global extremas.
The Intermediate Value Theorem tells us that if $f(a) \text{and} f(b)$ have opposite signs, then there exists at least one value c between a and b for which $f(c)=0$ .

Glossary

global maximum: highest turning point on a graph; $f(a)$ where $f(a) \ge f(x)$ for all x.
global minimum: lowest turning point on a graph; $f(a)$ where $f(a) \le f(x)$ for all x.
Intermediate Value Theorem: for two numbers a and b in the domain of f, if a < b and $f(a) \neq f(b)$ , then $f$ takes on every value between $f(a)$ and $f(b)$ ; specifically, when a polynomial function changes from a negative value to a positive value, the function must cross the x-axis
multiplicity: the number of times a given factor appears in the factored form of the equation of a polynomial; if a polynomial contains a factor of the form $(x-h)^p$ , $x=h$ is a zero of multiplicity p.