5.2 Polynomial Functions 5.2 Polynomialfunctions

5.2 Polynomial Functions 5.2 PolynomialFunctions

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.

DescribingPolynomialFunctions

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

n n 1 2 f(x)=a n x +a n 1 x − +...+a 2 x +a 1 x+a 0, −

wherea ,a ,...,a are real numbers andn 1 is a natural number. � 0 1 n ≥

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.

113 © TAMU Deﬁnition Supposef(x) is a polynomial function, in general form.

n n 1 2 f(x)=a n x +a n 1 x − +...+a 2 x +a 1 x+a 0, witha n �0 −

we say • The natural number n (highest power of the variable which occurs) is called the degree of the polynomial f(x). – Iff(x)=a 0, anda 0 � 0, we sayf(x) has degree 0. – Iff(x)= 0, we sayf(x) has no degree. n • The terma n x is called the leading term of the polynomialf(x). • The real numbera n is called the leading coeﬃcient of the polynomialf(x). • The real numbera 0 is called the constant term of the polynomialf(x).

Leading coeﬃcient Degree � � n 2 f(x)=a n x +...+a 2 x +a 1 x+a 0

Leading↑ term Constant↑ ��

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

Leading coeﬃcient f(x)= mx↓ +b

Leading↑ term Constant↑ term � �� While the power ofx is unwritten,x=x 1, 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 isf(x)=b .

Constant↑ term �� Ifb� 0, the degree is 0 and is called a constant polynomial.

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

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. 2 3 a.f(x)=6x +4x − +8

b.g(y)=y 5(y 3)(2y+ 7) −

7 8 2 c.h(p)=2p 3p 5p 3 + 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 thex-values approach the ’ends’ of thex-axis. That is, what happens toy asx becomes small without bound (writtenx ) and, on the flip side, asx becomes large without bound (writtenx ). → −∞ →∞ 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 asx 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.

n End Behavior of Polynomial Functionsf(x)=a n x +...+a 0,n is even.

n Suppose f(x)=a n x +...+a 0, where an �0 is a real number and n is an even natural number. The end behavior of the graph off(x) matches one of the following: • fora > 0, asx ,f(x) and asx ,f(x) n → −∞ →∞ →∞ →∞ • fora < 0, asx ,f(x) and asx ,f(x) n → −∞ → −∞ →∞ → −∞ Graphically:

... ··· � � � �

an >0a n <0

115 © TAMU n End Behavior of Polynomial Functionsf(x)=a n x +...+a 0,n is odd.

n Suppose f(x)=a n x +...+a 0, where an �0 is a real number and n is an odd natural number. The end behavior of the graph off(x) matches one of the following: • fora > 0, asx ,f(x) and asx ,f(x) n → −∞ → −∞ →∞ →∞ • fora < 0, asx ,f(x) and asx ,f(x) n → −∞ →∞ →∞ →−∞ Graphically:

� � � ··· ···�

an >0a n <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 off(x) andg(x), both symbolically and with a quick sketch. a.f(x)= 11x+9 x 2 + 12x6 −

b.y(x)=5 8 7x 3 + 10x −

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

y 77

11 x -9-9 -8-8 -7-7 -6-6 -5-5 -4-4 -3-3 -2-2 -1-100 11 22 33 44 55 66 77 88 99 1010 1111 1212 -1-1

-2-2

-3-3 because there are 4 humps, -4-4 degree must be at least 5 -5-5 -6-6 ff

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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)= 7x 4 +2x 2 +3 −

b.f(x)= 286x 1001 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

44 x f(x) f(x)=c 22 -2 3 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 ( , ) wherec is a constant -2-2 0 3 −∞ ∞ -4-4 2 3 -6-6

44 x f(x) Linear 22 -2 -2 (Identity) 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 ( , ) -2-2 0 0 −∞ ∞ 1st degree polynomial -4-4 2 2 -6-6

44 x f(x)

22 Quadratic 2 -2 4 nd 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 ( , ) 2 degree polynomial -2-2 0 0 −∞ ∞ -4-4 2 4 -6-6

x f(x) 44 33 -1 -1 22 Cubic 3 11 -0.5 -0.125 rd f(x)=x -6-6 -5-5 -4-4 -3-3 -2-2 -1-100 11 22 33 44 55 66 ( , ) 3 degree polynomial -1-1 0 0 −∞ ∞ -2-2 -3-3 0.5 0.125 -4-4 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.

Intercepts of a Polynomial Function

Characteristics of the graph such as vertical and horizontal intercepts are part of the interior behavior of a polynomial function.

Deﬁnition Like with all functions, the vertical intercept, or y-intercept, is where the graph crosses the y-axis, and occurs when the input value, x, is zero, f (0). Since a polynomial is a function, there can only be one y-intercept, which occurs at the point (0,a 0).

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

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

Deﬁnition The real zeros, or roots of a function,f(x) are 1. thex-value(s) whenf(x)= 0. 2. the solution(s) to the equationf(x)= 0. 3. thex-coordinate(s) of thex x -intercept(s) off(x). 4. thex-value(s) where graph off(x) touches thex-axis.

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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 general polynomials, this can be a challenging prospect. While quadratics can be solved using the quadratic formula which we will discuss in a moment, the corresponding formulas for cubic and 4th degree polynomials are not simple enough to remember, and formula do not exist for general higher-degree polynomials. Consequently, we will limit ourselves 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.

119 © TAMU � Example 5 Determine they-intercept and all real zeros of each given polynomial function. a.f(x)=x(x 4)(x+ 2) 3 −

b.g(x)=x 2 5x+6 −

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DescribingQuadraticFunctions

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

Deﬁnition A quadratic function is a function of the form

f(x)=ax 2 +bx+c

whereaa,bb, andcc are real numbersaa�0. �

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

Properties of a Quadratic Function The graph of a quadratic function,f(x)=ax 2 +bx+c, is a U-shaped curve called a parabola. One important feature of the graph is that it has an extreme point, called the vertex. If the parabola opens up,a> 0, the vertex represents 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 turning point on the graph. The graph is also symmetric with a vertical line drawn through the vertex, called the axis of symmetry. These features are illustrated in Figures 5.2.1 and 5.2.2.

vertex

Figure 5.2.2: A parabola opening drawn with the Figure 5.2.1: A parabola opening up with the axis axis of symmetry drawn and vertex labeled on the of symmetry drawn and vertex labeled on the graph. graph.

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 isf(x)=x 2. • The general form of a quadratic function is f(x)=ax 2 +bx+c where a, b, and c are real numbers and a�0. • The vertex form of a quadratic function isf(x)=a(x h) 2 +k wherea� 0. − b b • The vertex is located a (h,k)= − ,f − . 2a 2a � � �� b • The axis of symmetry is the vertical linex=h= − . 2a • The range is dependent on the value of the leading coeﬃcienta. b – Fora> 0, range : f − , 2a ∞ � � � � b – Fora< 0, range : , f − , −∞ 2a � � � �

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

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

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

66 66 44 44 44 22 22 22 -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 -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 -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 -2-2 -2-2 -2-2 -4-4 -4-4 -4-4 -6-6 -6-6 -6-6

Figure 5.2.3: A coordinate plane with Figure 5.2.4: A coordinate plane with Figure 5.2.5: A coordinate plane with a parabola. The parabola does not in- a parabola. The parabola intersects the a parabola. The parabola intersects the tersect thex- axis x-axis exactly once. x-axis exactly twice.

2 � Example 7 Find they-intercept, real zeros, andx-intercept(s) of the quadratic functionf(x)=4x 4x+ 3. −

Review how to factor quadratics with a leading cofficieint that is different from 1 for next class

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Deﬁnition If a, b, and c are real numbers with a� 0 , then the solutions to ax2 + bx + c = 0 are given by the quadratic formula:

b √b2 4ac x= − ± − 2a

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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 quantityb 2 4ac plays a critical role in determining the nature of the real zeros of a quadratic − function and is given a special name.

Deﬁnition If a, b, and c are real numbers with a� 0, then the discriminant of the quadratic equation ax2 + bx + c = 0 is the 2 quantitybb 4ac. � −

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

Theorem 5.1 Leta,b, andc be real numbers witha� 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)

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

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

2 ! The quadratic formula is only applicable when solving the equation ax + bx + c =0. It does not apply to any other trinomial equation.

� Example 9 Graph the quadratic function with the following properties: • Asx ,f(x) , and asx ,f(x) → −∞ →∞ →∞ →∞ • There is a minimum value of -5. • There are roots atx= 0 andx= 7. • The axis of symmetry isx=3.5. • The graph intersects they-axis aty= 0. Indicate the vertex andx-intercepts on your graph, and give the coordinates of each.

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In the chapter on Linear Equations we discussed how to find a linear revenue function of a company, if the item being sold had a fixed selling price, p, R (x)= px. However, we also learned that the selling price of an item could ⇒ be determined by consumers in the form of a price-demand function,p(x)=mx+b. So, general if you are given a linear price-demand function, p(x) then the revenue function be given by R(x)= px =( mx + b)x = mx2 + bx, which is a quadratic function. Since profit is given byP(x)=R(x) C(x), if revenue is a quadratic function and costs are − linear, 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 1,750 people. When the price dropped to $45, the estimated attendance is 2,250. Assuming that attendance is linearly related 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? • Givenf(x)=ax 2 +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?