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. Uponcompletion 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 theproduct of a number, called the coefficient of the term, and a variable raised to a non-negative integer (a naturalnumber) power.

DefinitionA 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. bothof 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 powersof the variable. Although the order of the terms in the polynomial function is not important for performingoperations, we typically arrange the terms in descending order of power, or in general form.

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DefinitionSuppose 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 polynomialf (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 ofdegree 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) = 2p7−3p8−5p23 +19.6

�

Knowing the degree and leading coefficient of a polynomial function is useful in helping us predict its endbehavior. The end behavior of a function is a way to describe what is happening to the function values (they-values) as the x-values approach the ’ends’ of the x-axis. That is, what happens to y as x becomes small withoutbound (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 thepower of the leading term is the highest, that term will grow significantly faster than the other terms as x→−∞ oras x→∞, so its behavior will dominate the graph. The leading coefficient and the degree found in the leading termboth 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 behaviorof 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 behaviorof 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 thefunction is dependent on all terms of the polynomial, and, thus, is different for each polynomial. Therefore, theinterior 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 functionbelow.

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 formallydefined 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 realnumber 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 transformationsthroughout this chapter. It will be very helpful if we can recognize these parent functions and their features quicklyby name, formula, and graph. The graphs, sample table values, and domain are included with each parentpolynomial function showing in Table 5.1. We will add to our parent functions as we introduce additional functionsin this chapter.

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Name Function Graph Table Domain

Constantf (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 30 32 3

(−∞,∞)

Linear(Identity)

1st degree polynomialf (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 -20 02 2

(−∞,∞)

Quadratic2nd 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 40 02 4

(−∞,∞)

Cubic3rd 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 -0.1250 0

0.5 0.1251 1

(−∞,∞)

Table 5.1: A table with 5 rows and 5 columns. The first row includes labels: Name, Function, Graph, Table, and Domain. Thesecond row includes the corresponding properties for a constant parent function, while the third, fourth, and fifth rows includethe 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 as vertical and horizontal intercepts are part of the interior behavior of apolynomial function.

DefinitionLike with all functions, the vertical intercept, or yyy-intercept, is where the graph crosses the y-axis, and occurswhen the input value, x, is zero, f (0). Since a polynomial is a function, there can only be one y-intercept, whichoccurs at the point (0,a0).

The yyy-intercept of a function, f (x), is the point (0,y), where the graph intersects the y-axis.

The xxx-intercept(s) of a function, f (x),are where the graph crosses the x-axis and occur at the x-value(s) thatcorrespond with an output value of zero, f (x) = 0. It is possible to have more than one x-intercept, (xi,0). �

DefinitionThe real zeros, or roots of a function, f (x) are

1. the x-value(s) when f (x) = 0.2. the solution(s) to the equation f (x) = 0.3. the x-coordinate(s) of the xxx -intercept(s) of f (x).4. the x-value(s) where graph of f (x) touches the x-axis.

�

N In this text, our discussion only concerns real zeros.

To find real zeros (x-intercept) of a function, we need to solve for when the output will be zero. For generalpolynomials, this can be a challenging prospect. While quadratics can be solved using the quadratic formula whichwe will discuss in a moment, the corresponding formulas for cubic and 4th degree polynomials are not simpleenough to remember, and formula do not exist for general higher-degree polynomials. Consequently, we will limitourselves to three cases when discussing zeros of polynomials:

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 approximate the real zeros.

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� Example 5 Determine the y-intercept and all real zeros of each given polynomial function.

a. f (x) = x(x−4)(x+2)3

b. g(x) = x2−5x+6

�

Describing Quadratic Functions

We will now explore quadratic functions, a type of polynomial function. Quadratics commonly arise from problemsinvolving revenue and profit, providing some interesting applications.

DefinitionA quadratic function is a function of the form

f (x) = ax2+bx+ cf (x) = ax2+bx+ cf (x) = ax2+bx+ c

where aaa, bbb, and ccc are real numbers aaa � 0. �

Since a quadratic function is a type of polynomial function, the domain of a quadratic function is (−∞,∞).

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5.2 Polynomial Functions

Properties of a Quadratic FunctionThe graph of a quadratic function, f (x) = ax2+bx+ c, is a U-shaped curve called a parabola. One importantfeature of the graph is that it has an extreme point, called the vertex. If the parabola opens up, a > 0, the vertexrepresents the lowest point on the graph, or the minimum of the quadratic function. If the parabola opens down,a < 0, the vertex represents the highest point on the graph, or the maximum. In either case, the vertex is a turningpoint on the graph. The graph is also symmetric with a vertical line drawn through the vertex, called the axis ofsymmetry. These features are illustrated in Figures 5.2.1 and 5.2.2.

vertex

Figure 5.2.1: A parabola opening up with the axisof symmetry drawn and vertex labeled on the graph.

vertex

Figure 5.2.2: A parabola opening drawn with theaxis of symmetry drawn and vertex labeled on thegraph.

Properties of Quadratic Functions

• A quadratic function is a polynomial function of degree two.

• The graph of a quadratic function is a parabola.

• The domain is (−∞,∞).

• The parent quadratic function is f (x) = x2.

• The general form of a quadratic function is f (x) = ax2+bx+ c where a, b, and c are real numbers anda � 0.

• The vertex form of a quadratic function is f (x) = a(x−h)2+ k where a � 0.

• The vertex is located a (h,k) =�−b2a, f�−b2a

��.

• The axis of symmetry is the vertical line x = h =−b2a

.

• The range is dependent on the value of the leading coefficient a.

– For a > 0, range :�f�−b2a

�,∞�

– For a < 0, range : −∞,�f�−b2a

�,

�

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� The vertex of a parabola will inform us of what the maximum or minimum value of the output of a quadraticfunction is k and where it occurs at x = h.

� Example 6 If f (x) = 3(x+6)2−11, determine the vertex, axis of symmetry, the maximum value, the minimumvalue, domain, and range of the function.

�

Intercepts of a Quadratic FunctionAs previously discussed, the y-intercept of a function, f (x) is found by evaluating f (0) and the real zeros are foundwhere the function is equal to zero. The number of x-intercepts of a quadratic function which correspond to the realzeros of the function can vary depending upon the quadratic function’s position with relation to the x-axis. (SeeFigures 5.2.3, 5.2.4, 5.2.5 for examples)

-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

Figure 5.2.3: A coordinate plane witha parabola. The parabola does not in-tersect the x- axis

-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

Figure 5.2.4: A coordinate plane witha parabola. The parabola intersects thex-axis exactly once.

-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

Figure 5.2.5: A coordinate plane witha parabola. The parabola intersects thex-axis exactly twice.

� Example 7 Find the y-intercept, real zeros, and x-intercept(s) of the quadratic function f (x) = 4x2−4x+3.

�

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5.2 Polynomial Functions

DefinitionIf a, b, and c are real numbers with a � 0 , then the solutions to ax2 + bx+ c = 0 are given by the quadraticformula:

x =−b±

√b2−4ac2a

�

We know having negative numbers underneath the square root produces values outside the set of real numbers.Given that

√b2−4ac is part of the quadratic function, we will need to pay special attention to the radicand b2−4ac.

It turns out the quantity b2−4ac plays a critical role in determining the nature of the real zeros of a quadraticfunction and is given a special name.

DefinitionIf a, b, and c are real numbers with a � 0, then the discriminant of the quadratic equation ax2+bx+ c = 0 is thequantity b2−4acb2−4acb2−4ac. �

The discriminant ‘discriminates’ between the kinds of real zeros we get from a quadratic function. These cases, andtheir relation to the discriminant, are summarized below.

Theorem 5.1 Let a, b, and c be real numbers with a � 0.• If b2−4ac < 0 the equation ax2+bx+ c = 0 has no real solutions and f (x) = ax2+bx+ c has no real zeros.

(See Figure 5.2.3)• If b2−4ac = 0 the equation ax2+bx+ c = 0 has exactly one real solutions and f (x) = ax2+bx+ c has one

real zero. (See Figure 5.2.4)• If b2−4ac > 0 the equation ax2+bx+ c = 0 has exactly two real solutions and f (x) = ax2+bx+ c has two

real zeros. (See Figure 5.2.5)

� Example 8 Use the quadratic formula to compute the real zeros of f (x) = 4x2−4x+3.

�

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N If the reader is not confident in their factoring skills, the quadratic formula can be used to solve any quadraticequation of the form ax2+bx+ c = 0.

! The quadratic formula is only applicable when solving the equation ax2+bx+ c = 0. It does not apply to anyother trinomial equation.

� Example 9 Graph the quadratic function with the following properties:• As x→−∞, f (x)→∞, and as x→∞, f (x)→∞• There is a minimum value of -5.• There are roots at x = 0 and x = 7.• The axis of symmetry is x = 3.5.• The graph intersects the y-axis at y = 0.

Indicate the vertex and x-intercepts on your graph, and give the coordinates of each.

�

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5.2 Polynomial Functions

In the chapter on Linear Equations we discussed how to find a linear revenue function of a company, if the itembeing sold had a fixed selling price, p,⇒ R(x) = px. However, we also learned that the selling price of an item couldbe determined by consumers in the form of a price-demand function, p(x) = mx+b. So, general if you are given alinear price-demand function, p(x) then the revenue function be given by R(x) = px = (mx+b)x = mx2+bx, whichis a quadratic function. Since profit is given by P(x) = R(x)−C(x), if revenue is a quadratic function and costs arelinear, profit will also be a quadratic function.

� Example 10 The cirque is coming to town for one night only. The university auditorium holds 2,500 people.With a ticket price of $50, the estimated attendance (based on previous performances at the university) will be 1,750people. When the price dropped to $45, the estimated attendance is 2,250. Assuming that attendance is linearlyrelated to ticket price,

a. What is the revenue function for the sale of tickets?b. How many tickets should be sold to maximize revenue? What is the maximum revenue?c. At what price should tickets be sold in order to maximize revenue?

�

Reflection:

• Can you describe the properties of a given polynomial without the use of technology?• Can you select the graphs of the parent polynomial functions from a group of graphs?• Can explain how to find the domain of a polynomial function algebraically?• Can you explain what a root/zero of a function is, in terms a person outside of a mathematics class could

understand?• Given f (x) = ax2+bx+ c, what can you say about the graph of f(x) without the use of technology?• Given a quadratic function, can you differentiate when to apply the quadratic formula and when to factor?• Can you graph a quadratic function without the use of technology?• Can you explain a business or social science situation where you would develop and use a quadratic function?

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