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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 verte$ 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. • !&etch the graph of a given quadratic function using its properties. • Use properties of quadratic functions to solve business and social science problems. DescribingPolynomialFunctions ' 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 ' polynomial function is a function of the form n n ) * f(x =a n x +a n ) x − +...+a * x +a ) x+a +, − wherea ,a ,...,a are real numbers andn ) is a natural number. � + ) 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 ) * f(x =a n x +a n ) x − +...+a * x +a ) x+a +, witha n �+ − 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 +, anda + � 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 + is called the constant term of the polynomialf(x). Leading coefficient Degree � � n * f(x =a n x +...+a * x +a ) x+a + Leading↑ term Constant↑ ���� ���� � /or 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 ), so linear����������� functions����������� are polynomials of degree 1. Any polynomial of degree ) is called a first degree polynomial. /or another type of polynomial, a horizontal line, the general form isf(x =b . Constant↑ term ���� Ifb� 0, the degree is + and is called a constant polynomial. Ifb= 0, there is no degree and is called the zero polynomial. © TAMU 114 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. * 3 a.f(x =1x +2x − +4 b.g(y =y 5(y 3)(2y+ 6 − 6 4 * c.h(p =*p 3p 5p 3 + )7.1 − − � 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 ). → −∞ →∞ -o 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 +,n is even. n Suppose f(x =a n x +...+a +, where an �+ 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 >+a n <+ 115 © TAMU n End Behavior of Polynomial Functionsf(x =a n x +...+a +,n is odd. n Suppose f(x =a n x +...+a +, where an �+ 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 >+a n <+ 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 =))x+7 x * +)*x 1 − b.y(x =5 4 6x 3 + )+x − � � Example 3 Describe the end behavior symbolically and determine a possible degree of the polynomial function below. y 77 66 55 44 33 22 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 � © TAMU 116 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 = 6x 2 +*x * +3 − b.f(x = *41x )++) ) − � 0e 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. 0e will add to our parent functions as we introduce additional functions in this chapter. 117 © TAMU Name Function Graph Table Domain 66 44 x f(x f(x =c 22 (* 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 + 3 −∞ ∞ -4-4 * 3 -6-6 66 44 x f(x Linear 22 (* (* (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 + + −∞ ∞ )st degree polynomial -4-4 * * -6-6 66 44 x f(x 22 Quadratic * (* 2 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 ( , * degree polynomial -2-2 + + −∞ ∞ -4-4 * 2 -6-6 x f(x 44 33 () () 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 + + −∞ ∞ -2-2 -3-3 0.5 0.125 -4-4 ) ) Table 5.1: ' table with 5 rows and 5 columns. The first row includes labels: >ame, Function, Graph, -able, and Domain. The second row includes the corresponding properties for a constant parent function, while the third, fourth, and #fth rows include the corresponding properties for the linear, quadratic, and cubic parent functions, respectively. © TAMU 118 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 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 (+,a + . Theyy(intercept of a function,f(x), is the point (0,y), where the graph intersects they-axis.
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