Mathematics
Analyzing Graphs of Polynomials
Analyzing graphs of polynomials involves identifying the degree, leading coefficient, and end behavior of the polynomial function. It also involves finding the x-intercepts, y-intercepts, and any vertical or horizontal asymptotes. By analyzing these features, we can determine the behavior of the function as x approaches positive or negative infinity and sketch an accurate graph of the function.
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eBook - PDFCalculus
Late Transcendentals
- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
Thus, the goal of curve sketching is no longer the graph itself, but rather the information it reveals about the function. Polynomials are among the simplest functions to graph and analyze. Their significant features are symmetry, intercepts, relative extrema, inflection points, and the behavior as x → +∞ and as x →−∞. Figure 3.2.11 shows the graphs of four polynomials in x. The graphs in Figure 3.2.11 have properties that are common to all polynomials: • The natural domain of a polynomial is (−∞, +∞). • Polynomials are continuous everywhere. • Polynomials are differentiable everywhere, so their graphs have no corners or vertical tangent lines. For each of the graphs in Figure 3.2.11, count the number of x-intercepts, rela- tive extrema, and inflection points, and confirm that your count is consistent with the degree of the polynomial. • The graph of a nonconstant polynomial eventually increases or decreases without bound as x → +∞ and as x →−∞. This is because the limit of a nonconstant polyno- mial as x → +∞ or as x →−∞ is ±∞, depending on the sign of the term of highest degree and whether the polynomial has even or odd degree [see Formulas (13) and (14) of Section 1.3 and the related discussion]. • The graph of a polynomial of degree n (> 2) has at most n x-intercepts, at most n − 1 relative extrema, and at most n − 2 inflection points. This is because the x- intercepts, relative extrema, and inflection points of a polynomial p(x) are among the real solutions of the equations p(x) = 0, p (x) = 0, and p (x) = 0, and the polynomials in these equations have degree n, n − 1, and n − 2, respectively. Thus, for example, the graph of a quadratic polynomial has at most two x-intercepts, one relative extremum, and no inflection points; and the graph of a cubic polynomial has at most three x-intercepts, two relative extrema, and one inflection point.
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.
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.
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