CONDENSED LESSON Polynomial Degree and 7.1 Finite Differences In this lesson you will ● learn the terminology associated with polynomials ● use the finite differences method to determine the degree of a polynomial ● find a polynomial function that models a set of data A polynomial in one variable is any expression that can be written in the form n nϪ1 1 an x anϪ1x · · · a 1x a 0 where x is a variable, the exponents are nonnegative integers, the coefficients are n nϪ1 real numbers, and an 0. A function in the form f (x) anx anϪ1x · · · 1 a 1 x a 0 is a polynomial function. The degree of a polynomial or polynomial function is the power of the term with the greatest exponent. If the degrees of the terms of a polynomial decrease from left to right, the polynomial is in general form. The polynomials below are in general form. 1st degree 2nd degree 3rd degree 4th degree 3x 7 Ϫx 2 2x 1.8 9x 3 4x 2 x 11 Ϫ5x 4 A polynomial with one term, such as Ϫ5x 4, is called a monomial. A polynomial with two terms, such as 3x 7, is called a binomial. A polynomial with three terms, such as Ϫx 2 2x 1.8, is called a trinomial. Polynomials with more than three terms, such as 9x 3 4x 2 x 11, are usually just called polynomials. For linear functions, when the x-values are evenly spaced, the differences in the corresponding y-values are constant. This is not true for polynomial functions of higher degree. However, for 2nd-degree polynomials, the differences of the differences, called the second differences and abbreviated D2, are constant. For 3rd-degree polynomials, the differences of the second differences, called the third differences and abbreviated D3, are constant. This is illustrated in the tables on page 379 of your book. If you have a set of data with equally spaced x-values, you can find the lowest possible degree of a polynomial function that fits the data (if there is a polynomial function that fits the data) by analyzing the differences in y-values. This technique, called the finite differences method, is illustrated in the example in your book. Read that example carefully. Notice that the finite differences method determines only the degree of the polynomial. To find the exact equation for the polynomial function, you need to find the coefficients by solving a system of equations or using some other method. In the example, the D2 values are equal. When you use experimental data, you may have to settle for differences that are nearly equal. Investigation: Free Fall If you have a motion sensor, collect the (time, height) data as described in Step 1 in your book. If not, use these sample data. (The values in the last two columns are calculated in Step 2.) (continued) Discovering Advanced Algebra Condensed Lessons CHAPTER 7 93 ©2010 Key Curriculum Press DDAA2CL_010_07.inddAA2CL_010_07.indd 9933 11/13/09/13/09 22:42:55:42:55 PPMM Lesson 7.1 • Polynomial Degree and Finite Differences (continued) Complete Steps 2–6 in your book. The results Time (s) Height (m) given are based on the sample data. x y D1 D2 Step 2 The first and second differences, D1 and 0.00 2.000 D , are shown in the table at right. Ϫ0.012 2 0.05 1.988 Ϫ0.025 For these data, we can stop with the second Ϫ0.037 0.10 1.951 Ϫ0.024 differences because they are nearly constant. Ϫ0.061 0.15 1.890 Ϫ0.025 Ϫ0.086 0.20 1.804 Ϫ0.024 Ϫ0.110 0.25 1.694 Ϫ0.025 Ϫ0.135 0.30 1.559 Ϫ0.024 Ϫ0.159 0.35 1.400 Ϫ0.025 Ϫ0.184 0.40 1.216 Ϫ0.024 Ϫ0.208 0.45 1.008 Step 3 The three plots are shown below. (time, height ) (time 2, d 1) (time 3, d 2 ) Step 4 The graph of (time, height ) appears parabolic, suggesting that the correct model may be a 2nd-degree polynomial function. The graph of (time 2, d 1 ) shows that the first differences are not constant because they decrease in a linear fashion. The graph of (time 3, d 2 ) shows that the second differences are nearly constant, so the correct model should be a 2nd-degree polynomial function. Step 5 A 2nd-degree polynomial in the form y ax 2 bx c fits the data. Step 6 To write the system, choose three data points. For each point, write an equation by substituting the time and height values for x and y in the equation y ax 2 bx c. The following system is based on the values (0, 2), (0.2, 1.804), and (0.4, 1.216). c 2.000 ͭ 0.04 a 0.2b c 1.804 0.16a 0.4b c 1.216 One way to solve this system is by writing the matrix equation 0 0 1 a 2.000 0.04 0.2 1 ϭ 1.804 ΄ ΅ ΄ b ΅ ΄ ΅ 0.16 0.4 1 c 1.216 and solving using an inverse matrix. The solution is a 4.9, b 0, and c 2, so an equation that fits the data is y 4.9x 2 2. Read the remainder of the lesson in your book. 94 CHAPTER 7 Discovering Advanced Algebra Condensed Lessons ©2010 Kendall Hunt Publishing DDAA2CL_010_07.inddAA2CL_010_07.indd 9944 11/13/09/13/09 22:42:56:42:56 PPMM CONDENSED LESSON 7.2 Equivalent Quadratic Forms In this lesson you will ● learn about the vertex form and factored form of a quadratic equation and the information each form reveals about the graph ● use the zero-product property to find the roots of a factored equation ● write a quadratic model for a data set in vertex, general, and factored form A 2nd-degree polynomial function is called a quadratic function. In Lesson 7.1, you learned that the general form of a quadratic function is y ax 2 bx c. In this lesson you will explore other forms of a quadratic function. You know that every quadratic function is a transformation of y x 2. When ____y k ΂____x h΃2 ΂____x h΃2 a quadratic function is written in the form b a or y b a k, you can tell that the vertex of the parabola is (h, k) and that the horizontal and vertical scale factors are a and b. Conversely, if you know the vertex of a parabola and you know (or can find) the scale factors, you can write its equation in one of these forms. ΂____x h΃2 The quadratic function y b a k can be rewritten in the form b b __ 2 __ y a 2 ( x h) k. The coefficient a 2 combines the two scale factors into one vertical scale factor. In the vertex form of a quadratic equation, y a(x h)2 k, this single scale factor is simply denoted a. From this form, you can identify the vertex, (h, k), and the vertical scale factor, a. If you know the vertex of a parabola and the vertical scale factor, you can write an equation in vertex form. Work through Example A carefully. The zero-product property states that for all real numbers a and b, if ab 0, then a 0, or b 0, or a 0 and b 0. For example, if 3x (x 7) 0, then 3x 0 or x 7 0. Therefore, x 0 or x 7. The solutions to an equation in the form f (x) 0 are called the roots of the equation, so 0 and 7 are the roots of 3x (x 7) 0. The x-intercepts of a function are also called the zeros of the function (because the corresponding y-values are 0). The function y 1.4(x 5.6)(x 3.1), given in Example B in your book, is said to be in factored form because it is written as the product of factors. The zeros of the function are the solutions of the equation Ϫ1.4(x 5.6)(x 3.1) 0. Example B shows how you can use the zero-product property to find the zeros of the function. ΂ ΃΂ ΃ In general, the factored form of a quadratic function is y a x r1 x r2 . From this form, you can identify the x-intercepts (or zeros), r1 and r2, and the vertical scale factor, a. Conversely, if you know the x-intercepts of a parabola and know (or can find) the vertical scale factor, then you can write the equation in factored form. Read Example C carefully. Investigation: Rolling Along Read the Procedure Note and Steps 1–3 in your book. Make sure you can visualize how the experiment works. Use these sample data to complete Steps 4–8, and then compare your results to those below. (These data have been adjusted for the position of the starting line as described in Step 3.) (continued) Discovering Advanced Algebra Condensed Lessons CHAPTER 7 95 ©2010 Kendall Hunt Publishing DDAA2CL_010_07.inddAA2CL_010_07.indd 9955 11/13/09/13/09 22:42:56:42:56 PPMM Lesson 7.2 • Equivalent Quadratic Forms (continued) Time (s) Distance from Time (s) Distance from Time (s) Distance from x line (m), y x line (m), y x line (m), y 0.2 Ϫ0.357 2.2 3.570 4.2 3.309 0.4 Ϫ0.355 2.4 3.841 4.4 2.938 0.6 Ϫ0.357 2.6 4.048 4.6 2.510 0.8 Ϫ0.184 2.8 4.188 4.8 2.028 1.0 0.546 3.0 4.256 5.0 1.493 1.2 1.220 3.2 4.257 5.2 0.897 1.4 1.821 3.4 4.193 5.4 0.261 1.6 2.357 3.6 4.062 5.6 Ϫ0.399 1.8 2.825 3.8 3.871 5.8 Ϫ0.426 2.0 3.231 4.0 3.619 6.0 Ϫ0.419 Step 4 At right is a graph of the data.