Mathematics
Evaluating and Graphing Polynomials
Evaluating and graphing polynomials involves finding the value of a polynomial expression for a given input and plotting the resulting points on a coordinate plane. The degree of a polynomial determines the number of possible roots or x-intercepts, which can be found using the Rational Root Theorem or synthetic division. The end behavior of a polynomial can also be determined by examining its leading coefficient and degree.
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7 Key excerpts on "Evaluating and Graphing Polynomials"
eBook - PDFCollege Algebra
Building Concepts and Connections 2E
- Revathi Narasimhan(Author)
- 2019(Publication Date)
- XYZ Textbooks(Publisher)
Example 1 Objectives ■ Define a polynomial function. ■ Determine end behavior. ■ Find x -intercepts and zeros by factoring. ■ Sketch a complete graph of a polynomial function. ■ Relate zeros, x -intercepts, and factors of a polynomial. ■ Solve applied problems using polynomials. Polynomial Functions and Their Graphs VIDEO EXAMPLES SECTION 3.2 246 Chapter 3 Polynomial and Rational Functions The rest of this section will be devoted to graphs of polynomial functions. These graphs have no breaks or holes, and no sharp corners. Figure 1 illustrates graphs of several functions, and indicates which are the graphs of polynomial functions. Polynomials of the Form x n and Their Transformations The graphs of polynomial functions can be quite varied. The simplest polynomial functions are those with just one term, x n . Consider the functions f ( x ) = x 3 and g ( x ) = x 4 . Table 1 gives a few of their function values. It is useful to examine the trend in the function value as the value of x increases to positive infinity ( x → ∞ ) or when the value of x decreases to negative infinity ( x → − ∞ ). This is known as end behavior and is summarized in Table 2. The graphs and their end behaviors are shown in Figure 2. We now summarize properties of the general function f ( x ) = x n , n a positive integer.
eBook - PDFIntermediate Algebra
A Guided Approach
- Rosemary Karr, Marilyn Massey, R. Gustafson, , Rosemary Karr, Marilyn Massey, R. Gustafson(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
We then examine several polynomial functions and their graphs. Classify a polynomial as a monomial, binomial, trinomial, or none of these. Algebraic terms are expressions that contain constants and/or variables. Some examples are 17, 9 x , 15 y 2 , and 2 24 x 4 y 5 The numerical coefficient of 17 is 17. The numerical coefficients of the remaining terms are 9, 15 , and 2 24 , respectively. A polynomial is a term or the sum of two or more algebraic terms whose variables have whole-number exponents. 1 Classify a polynomial as a monomial, binomial, trinomial, or none of these. Determine the degree of a polynomial. Evaluate a polynomial or a polynomial function. Graph a polynomial function. algebraic term polynomial monomial binomial trinomial degree of a polynomial descending order ascending order polynomial function Write each expression using exponents. 1. 3 aabb 2. 2 5 xxxy 3. 4 pp 1 7 qq 4. aaa 2 bb A polynomial in one variable (say, x ) is a term or the sum of two or more terms of the form ax n , where a is a real number and n is a whole number. The following expressions are polynomials in x . Note that 2 17 5 2 17 x 0 . 3 x 2 1 2 x , 3 2 x 5 2 7 3 x 4 2 8 3 x 3 , and 19 x 20 1 3 x 14 1 4.5 x 11 2 17 POLYNOMIAL IN ONE VARIABLE Section 5.1 Vocabulary Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 292 CHAPTER 5 Polynomials, Polynomial Functions, and Equations The following expressions are not polynomials.
- K. F. Riley, M. P. Hobson(Authors)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
In order to make accurate enough sketches for these purposes, it is important to be able to recognise the main features that will be possessed by the plots of given or deduced functions f i ( x ), without having to make detailed calculations for many values of x . Even if a precise (rather than approximate) value of x is required, and it is to be found using, say, numerical methods, preliminary sketches are always advisable, so that an appropriate numerical method may be selected, or a good starting point can be chosen for methods that depend upon successive approximation. Much of what is needed for drawing adequate sketches could be discussed at this point – and we will be sketching some graphs later in this chapter. However, as indicated in the introduction to Chapter 1, we will defer a general discussion of graph-sketching 52 53 2.1 Polynomials and polynomial equations until the end of Chapter 3 , so that the benefits that differential calculus has to offer can be included. 2.1 Polynomials and polynomial equations • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Firstly we consider the simplest type of equation, a polynomial equation , in which a polynomial expression in x , denoted by f ( x ), is set equal to zero and thereby forms an equation which is satisfied by particular values of x , called the roots of the equation: f ( x ) = a n x n + a n − 1 x n − 1 + · · · + a 1 x + a 0 = 0 . (2.1) Here n is an integer > 0, called the degree of both the polynomial and the equation, and the known coefficients a 0 , a 1 , . . . , a n are real quantities with a n = 0. Equations such as ( 2.1 ) arise frequently in physical problems, the coefficients a i being determined by the physical properties of the system under study. What is needed is to find some or all of the roots of ( 2.1 ), i.e.
eBook - PDF- James Stewart, Lothar Redlin, Saleem Watson, , James Stewart, Lothar Redlin, Saleem Watson(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
With the help of computer graphics, designers can see how good the “mathematical car” looks before they build the real one. Moreover, the mathematical car can be viewed from any perspective; it can be moved, rotated, or seen from the inside. These manipulations of the car on the com-puter monitor translate mathematically into solving large systems of linear equations. Mathematics in the Modern World Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. SECTION 3.2 ■ Polynomial Functions and Their Graphs 297 We plot the points in the table and connect them by a smooth curve to complete the graph, as shown in Figure 8. Test point → Test point → Test point → Test point → FIGURE 8 P 1 x 2 x 3 2 x 2 3 x x P x x c 2 10 1 0 1 2 7 8 0 0 1 4 2 6 3 0 4 20 y 0 x 1 5 Now Try Exercise 31 ■ EXAMPLE 6 ■ Finding Zeros and Graphing a Polynomial Function Let P 1 x 2 2 x 4 x 3 3 x 2 . (a) Find the zeros of P . (b) Sketch a graph of P . SOLUTION (a) To find the zeros, we factor completely. P 1 x 2 2 x 4 x 3 3 x 2 x 2 1 2 x 2 x 3 2 Factor x 2 x 2 1 2 x 3 21 x 1 2 Factor quadratic Thus the zeros are x 0, x 3 2 , and x 1. (b) The x -intercepts are x 0, x 3 2 , and x 1. The y -intercept is P 1 0 2 0 . We make a table of values of P 1 x 2 , making sure that we choose test points between (and to the right and left of) successive zeros. Since P is of even degree and its leading coefficient is negative, it has the fol-lowing end behavior.
eBook - PDF- Doug French(Author)
- 2004(Publication Date)
- Continuum(Publisher)
Whilst this may seem simpler as it is a more routine task, it starts from something that is unfamiliar, namely the equation, which can set up an immediate barrier because it looks strange and new and seems to have appeared for no apparent reason. Co-ordinates and their graphical representation should already be familiar and therefore provide a more reassuring start to a new idea. Once the idea of equations has been established as a way of describing straight lines and curves it is perfectly legitimate to explore the graphical representations of new functions as they arise, often using graphical calculators or graph-plotting software as a powerful tool, but in the beginning the major priority is to establish the meaning of equations and to develop the skills involved in moving between equation, table of values and graph. 82 Teaching and Learning Algebra Figure 6.1 displays three examples with the different starting points: equation, table of values or graph. In the early stages students will move between equation and graph via the table of values, but once that skill has been established the next major task is to establish direct links between them through recognition of the gradient and the intercept on the y axis. At this point, a graph plotter, either a graphical calculator or on a computer, becomes a valuable tool when combined with appropriate written tasks. A graph plotter enables the student to obtain an accurate plot of a graph and is therefore helpful in identifying the common properties of a set of related graphs. Although the availability of graph plotters may diminish the need to plot accurate graphs by hand it does not reduce the importance of being able to make quick sketches of graphs by hand. Indeed experiences with graph plotters should be designed to enhance the skills of graph sketching and the related task of recognizing the sort of equation that fits a given graph.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
The method transforms the problem into a sequence of eigenvalue problems involving tridiagonal matrices with only simple eigenvalues: such eigenvalues can easily be approximated by means of the QR algorithm. Graphs A polynomial function in one real variable can be represented by a graph. • The graph of the zero polynomial ________________________ WORLD TECHNOLOGIES ________________________ f ( x ) = 0 is the x -axis. • The graph of a degree 0 polynomial f ( x ) = a 0 , where a 0 ≠ 0, is a horizontal line with y -intercept a 0 • The graph of a degree 1 polynomial (or linear function) f ( x ) = a 0 + a 1 x , where a 1 ≠ 0, is an oblique line with y -intercept a 0 and slope a 1 . • The graph of a degree 2 polynomial f ( x ) = a 0 + a 1 x + a 2 x 2 , where a 2 ≠ 0 is a parabola. • The graph of a degree 3 polynomial f ( x ) = a 0 + a 1 x + a 2 x 2 , + a 3 x 3 , where a 3 ≠ 0 is a cubic curve. • The graph of any polynomial with degree 2 or greater f ( x ) = a 0 + a 1 x + a 2 x 2 + ... + a n x n , where a n ≠ 0 and n ≥ 2 is a continuous non-linear curve. The graph of a non-constant (univariate) polynomial always tends to infinity when the variable increases indefinitely (in absolute value). Polynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end behavior. The illustrations below show graphs of polynomials. ________________________ WORLD TECHNOLOGIES ________________________ Polynomial of degree 2: f ( x ) = x 2 - x - 2 = ( x +1)( x -2) Polynomial of degree 3: f ( x ) = x 3 /4 + 3 x 2 /4 - 3 x /2 - 2 = 1/4 ( x +4)( x +1)( x -2) ________________________ WORLD TECHNOLOGIES ________________________ Polynomial of degree 4: f ( x ) = 1/14 ( x +4)( x +1)( x -1)( x -3) + 0.5 Polynomials and calculus One important aspect of calculus is the project of analyzing complicated functions by means of approximating them with polynomials.
eBook - PDFPre-Calculus All-in-One For Dummies
Book + Chapter Quizzes Online
- Mary Jane Sterling(Author)
- 2023(Publication Date)
- For Dummies(Publisher)
The calculator will not only identify the zeros, but also show you the maximum and minimum values of the graph so that you can draw the best possible representation. Graphing when all the roots are real numbers When graphing a polynomial function, you want to use the following steps: 1. Plot the zeros (x-intercepts) on the coordinate plane and the y-intercept. 2. Determine which way the ends of the graph point. 3. Figure out what happens between the zeros by picking any value to the left and right of each intercept and plugging it into the function. 4. Plot the graph. Q. Plot the graph of the polynomial f x x x x x 2 9 21 88 48 4 3 2 . This is an example that appears earlier. The zeros are x 3 , x 1 2 , and x 4 . A. Use the following steps. 1. Plot the zeros (x-intercepts) on the coordinate plane. Mark the zeros that you found previously: x 3 , x 1 2 , and x 4 . Now plot the y-intercept of the polynomial. The y-intercept is always the constant term of the polynomial — in this case, y 48. If no constant term is written, the y-intercept is 0. 2. Determine which way the ends of the graph point. You can use a handy test called the leading coefficient test, which helps you figure out how the polynomial begins and ends. The degree and leading coefficient of a polyno- mial always explain the end behavior of its graph (see the section, “Understanding Degrees and Roots,” for more on finding degree): CHAPTER 5 Graphing Polynomial Functions 119 • If the degree of the polynomial is even and the leading coefficient is positive, both ends of the graph point up. • If the degree is even and the leading coefficient is negative, both ends of the graph point down. • If the degree is odd and the leading coefficient is positive, the left side of the graph points down and the right side points up. • If the degree is odd and the leading coefficient is negative, the left side of the graph points up and the right side points down.
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