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5.2 Polynomial Functions

5.2 Polynomial Functions

This is part of 5.1 in the current text

In this section, you will learn about the properties, characteristics, and applications of polynomial functions. Upon completion you will be able to: • Recognize the degree, leading coefficient, and end behavior of a given polynomial function. • Memorize the graphs of parent polynomial functions (linear, quadratic, and cubic). • State the domain of a polynomial function, using interval notation. • Define what it means to be a root/zero of a function. • Identify the coefficients of a given quadratic function and he direction the corresponding graph opens. • Determine the vertex and properties of the graph of a given quadratic function (domain, range, and

minimum/maximum value). • Compute the roots/zeroes of a quadratic function by applying the quadratic formula and/or by factoring. • Sketch the graph of a given quadratic function using its properties. • Use properties of quadratic functions to solve business and social science problems.

Describing Polynomial Functions

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

Definition A polynomial function is a function of the form

f (x) = anxn+an−1xn−1+ . . .+a2x2+a1x+a0,

where a0,a1, . . . ,an are real numbers and n ≥ 1 is a natural number. �

In a previous chapter we discussed linear functions, f (x) = mx = b, and the equation of a horizontal line, y = b. both of these are examples of polynomial functions.

N the equation of a horizontal line, y = b, can be written in function notation as y = f (x) = b or simply as f (x) = b.

Properties of a Polynomial Function

Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the powers 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.

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Definition Suppose f (x) is a polynomial function, in general form.

f (x) = anxn+an−1xn−1+ . . .+a2x2+a1x+a0, with an � 0

we say • The natural number n (highest power of the variable which occurs) is called the degree of the polynomial f (x). – If f (x) = a0, and a0 � 0, we say f (x) has degree 0. – If f (x) = 0, we say f (x) has no degree.

• The term anxn is called the leading term of the polynomial f (x). • The real number an is called the leading coefficient of the polynomial f (x). • The real number a0 is called the constant term of the polynomial f (x).

Leading coefficient �

Degree �

f (x) = anxn���� ↑

Leading term

+ . . . + a2x2 +a1x+ a0���� ↑

Constant

�

For one type of polynomial, linear functions, the general form is f (x) = mx+b.

f (x) =

Leading coefficient ↓ mx�������������������������� ↑

Leading term

+ b���� ↑

Constant term

While the power of x is unwritten, x = x1, so linear functions are polynomials of degree 1. Any polynomial of degree 1 is called a first degree polynomial.

For another type of polynomial, a horizontal line, the general form is f (x) = b���� ↑

Constant term

.

If b � 0, the degree is 0 and is called a constant polynomial.

If b = 0, there is no degree and is called the zero polynomial.

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5.2 Polynomial Functions

� Example 1 Determine if each function is a polynomial function. If the function is a polynomial, state its degree, leading coefficient, and the constant term.

a. f (x) = 6x2+4x−3+8

b. g(y) = y5(y−3)(2y+7)

c. h(p) = 2p7−3p8−5p 23 +19.6

�

Knowing the degree and leading coefficient of a polynomial function is useful in helping us predict its end behavior. The end behavior of a function is a way to describe what is happening to the function values (the y-values) as the x-values approach the ’ends’ of the x-axis. That is, what happens to y as x becomes small without bound (written x→−∞) and, on the flip side, as x becomes large without bound (written x→∞).

To determine a polynomial’s end behavior, you only need to look at the leading term of the function. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as x→−∞ or as x→∞, so its behavior will dominate the graph. The leading coefficient and the degree found in the leading term both play a role in the end behavior of the polynomial. The general end behavior is summarized below.

End Behavior of Polynomial Functions f (x) = anxn+ . . . +a0, n is even.

Suppose f (x) = anxn+ . . . +a0, where an � 0 is a real number and n is an even natural number. The end behavior of the graph of f (x) matches one of the following:

• for an > 0, as x→−∞, f (x)→∞ and as x→∞, f (x)→∞ • for an < 0, as x→−∞, f (x)→−∞ and as x→∞, f (x)→−∞

Graphically:

�...� �···�

an > 0 an < 0

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End Behavior of Polynomial Functions f (x) = anxn+ . . . +a0, n is odd.

Suppose f (x) = anxn+ . . . +a0, where an � 0 is a real number and n is an odd natural number. The end behavior of the graph of f (x) matches one of the following:

• for an > 0, as x→−∞, f (x)→−∞ and as x→∞, f (x)→∞ • for an < 0, as x→−∞, f (x)→∞ and as x→∞, f (x)→−∞

Graphically:

� · · ·� � · · ·� an > 0 an < 0

N While the end behavior is determined solely by the leading term of a polynomial, the interior behavior of the function is dependent on all terms of the polynomial, and, thus, is different for each polynomial. Therefore, the interior behavior is represented by the . . . in each graphical representation of the end behavior.

� Example 2 Describe the end behavior of f (x) and g(x), both symbolically and with a quick sketch.

a. f (x) = 11x+9− x2+12x6

b. y(x) = 58−7x3+10x

�

� Example 3 Describe the end behavior symbolically and determine a possible degree of the polynomial function below.

x -9-9 -8-8 -7-7 -6-6 -5-5 -4-4 -3-3 -2-2 -1-1 11 22 33 44 55 66 77 88 99 1010 1111 1212

y

-6-6

-5-5

-4-4

-3-3

-2-2

-1-1

11

22

33

44

55

66

77

00

ff

�

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5.2 Polynomial Functions

Despite having different end behavior, all polynomial functions are continuous. While this concept is formally defined using Calculus, informally, graphs of continuous functions have no ‘breaks’ or ‘holes’ in them. Moreover, based on the properties of real numbers, for every real number input into a polynomial function, a single real number will be output. Thus, the domain of any polynomial function is (−∞,∞).

The domain of any polynomial function is (−∞,∞)(−∞,∞)(−∞,∞).

� Example 4 State the domain of each function. a. f (x) = −7x4+2x2+3

b. f (x) = 286x1001−1

�

We will see these parent functions, combinations of parent functions, their graphs, and their transformations throughout this chapter. It will be very helpful if we can recognize these parent functions and their features quickly by name, formula, and graph. The graphs, sample table values, and domain are included with each parent polynomial function showing in Table 5.1. We will add to our parent functions as we introduce additional functions in this chapter.

117 © TAMU

Name Function Graph Table Domain

Constant f (x) = c

where c is a constant -9-9 -8-8 -7-7 -6-6 -5-5 -4-4 -3-3 -2-2 -1-1 11 22 33 44 55 66 77 88 99

-6-6

-4-4

-2-2

22

44

66

xxx f (x)f (x)f (x) -2 3 0 3 2 3

(−∞,∞)

Linear (Identity)

1st degree polynomial f (x) = x -9-9 -8-8 -7-7 -6-6 -5-5 -4-4 -3-3 -2-2 -1-1 11 22 33 44 55 66 77 88 99

-6-6

-4-4

-2-2

22

44

66

xxx f (x)f (x)f (x) -2 -2 0 0 2 2

(−∞,∞)

Quadratic 2nd degree polynomial

f (x) = x2 -9-9 -8-8 -7-7 -6-6 -5-5 -4-4 -3-3 -2-2 -1-1 11 22 33 44 55 66 77 88 99

-6-6

-4-4

-2-2

22

44

66

xxx f (x)f (x)f (x) -2 4 0 0 2 4

(−∞,∞)

Cubic 3rd degree polynomial

f (x) = x3 -6-6 -5-5 -4-4 -3-3 -2-2 -1-1 11 22 33 44 55 66

-4-4

-3-3

-2-2

-1-1

11

22

33

44

00

xxx f (x)f (x)f (x) -1 -1

-0.5 -0.125 0 0

0.5 0.125 1 1

(−∞,∞)

Table 5.1: A table with 5 rows and 5 columns. The first row includes labels: Name, Function, Graph, Table, and Domain. The second row includes the corresponding properties for a constant parent function, while the third, fourth, and fifth rows include the corresponding properties for the linear, quadratic, and cubic parent functions, respectively.

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5.2 Polynomial Functions

Intercepts of a Polynomial Function

Characteristics of the graph such