March 20, S4.1q Polynomial Functions and Models

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1 MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial Division; The Remainder and Factor Theorems 4.4 Theorems about Zeros of Polynomial Functions 4.5 Rational Functions 4.6 Polynomial and Rational Inequalities 4.1 Polynomial Functions and Models Determine the behavior of the graph of a polynomial function using the leading term test. Factor polynomial functions and find the zeros and their multiplicities. Use a graphing calculator to graph a polynomial function and find its real number zeros its relative maximum and minimum values, and its domain and range. Solve applied problems using polynomial models; fit linear, quadratic, power, cubic, and quartic polynomial functions to data. The following PowerPoint presentation was developed by Dr. Moore. It covers material in sections as indicated below. Average of Students taking test Fall 2012 Spring 2013 Test 1 Test 2 Test 1 Test 2 D D D D D D D D Spring 2013 Test 2 D10 D11 D14 D Total Absent Section 4.1, Slides ppt Section 4.2, Slides ppt Section 4.3, Slides ppt Section 4.4, Slides ppt Oct 23 7:55 AM Technology for this section Click the globe to the left and visit SAS Curriculum Pathways for interactive programs on Polynomial Functions. User: able7oxygen Quick Launch: 1022 (Worksheet: Polynomial Patterns), 1441 (Exploring Graphs of Polynomial Functions) Polynomial Function A polynomial functionp is given by You may use the "Polynomial Roots" program to graph polynomial functions and find the real roots (zeros). where the coefficients a n, a n 1,, a 1, a 0 are real numbers and the exponents are whole numbers. Quadratic Function Mathematica Interactive Figures are available through Tools for Success, Activities and Projects in CourseCompass. You may access these through CourseCompass or from the Important Links webpage. You must Login to MML to use this link. 4.1 Polynomials and the Leading Term Test; 4.1 Zeros of Polynomial Functions 4.1 Volume of a Box; 4.1 Modeling Data: Regression You may use the "Interest Calculator" program to find the Interest and Total Amount of an investment with Simple Interest or Compound Interest. 1

2 Cubic Function Examples of Nonpolynomial Functions Examples of Polynomial Functions Polynomial Functions The graph of a polynomial function is continuous and smooth. The domain of a polynomial function is the set of all real numbers The Leading Term Test Graphs Example Using the leading term test, match each of the following functions with one of the graphs A D, which follow. a) b) c) d) Solution a) b) c) d) 2

3 Finding Real Zeros on a Calculator Finding Zeros of Factored Polynomial Functions If c is a real zero of a function (that is, f (c) = 0), then (c, 0) is an x intercept of the graph of the function. Find the zeros of f (x) = 0.2x 3 1.5x 2 0.3x + 2. Approximate the zeros to three decimal places. Solution Use a graphing calculator to create a graph. Look for points where the graph crosses the x axis. We use the ZERO feature to find them. Example: Find the zeros of Finding Zeros of Factored Polynomial Functions continued Solution: To solve the equation f(x) = 0, we use the principle of zero products, solving x 1 = 0 and x + 2 = 0. The zeros are approximately 1.164, 1,142, and The zeros of f(x) are 1 and 2. See graph on right. Even and Odd Multiplicity 309/2. Determine the leading term, the leading coefficient, and the degree of the polynomial. Then classify the polynomial function as constant, linear, quadratic, cubic, or quartic. f(x) = 15x x 4 7x 3 If (x c) k, k 1, is a factor of a polynomial function P(x) and (x c) k + 1 is not a factor and: k is odd, then the graph crosses the x axis at (c, 0); k is even, then the graph is tangent to the x axis at (c, 0). Example Find the zeros of f (x) = x 4 + 8x Solution We factor as follows: f (x) = x 4 + 8x 2 33 = (x )(x 2 3). Solve the equation f(x) = 0 to determine the zeros. We use the principle of zero products. 309/5. Determine the leading term, the leading coefficient, and the degree of the polynomial. Then classify the polynomial function as constant, linear, quadratic, cubic, or quartic. f(x) = 305x

4 309/8. Determine the leading term, the leading coefficient, and the degree of the polynomial. Then classify the polynomial function as constant, linear, quadratic, cubic, or quartic. f(x) = 2 x 2 309/12. Select one of the following four sketches to describe the end behavior of the graph of the function. f(x) = (1/4)x 4 + (1/2)x 3 6x 2 + x 5 309/10. Determine the leading term, the leading coefficient, and the degree of the polynomial. Then classify the polynomial function as constant, linear, quadratic, cubic, or quartic. f(x) = 12 + x 309/14. Select one of the following four sketches to describe the end behavior of the graph of the function. f(x) = (2/5)x 5 2x 4 + x 3 (1/2)x /19. Use the leading term test to match the function with one of the graphs (a) (d), which follow. f(x) = x 6 + 2x 5 7x 2 309/16. Select one of the following four sketches to describe the end behavior of the graph of the function. f(x) = x 3 + x 5 0.5x 6 309/18. Select one of the following four sketches to describe the end behavior of the graph of the function. f(x) = 2x + x 3 5x 5 310/20. Use the leading term test to match the function with one of the graphs (a) (d), which follow. f(x) = 2x 4 x

5 310/21. Use the leading term test to match the function with one of the graphs (a) (d), which follow. f(x) = x 5 + (1/10)x 3 310/23. Use substitution to determine whether 4, 5, and 2 are zeros of f(x) = x 3 9x x /22. Use the leading term test to match the function with one of the graphs (a) (d), which follow. f(x) = x 3 + x 2 2x + 4 f(5) = (5) 3 9(5) (5) + 24 = 125 9(25) = 6 Thus, x = 5 is NOT a zero of f(x). f( 2) = ( 2) 3 9( 2) ( 2) + 24 = 8 9(4) = 48 Thus, x = 2 is NOT a zero of f(x). 310/24. Use substitution to determine whether 2, 3, and 1 are zeros of f(x) = 2x 3 3x 2 + x /28. Find the zeros of the polynomial function and state the multiplicity of each f(x) = (x + 5) 3(x 4)(x + 1) 2 310/25. Use substitution to determine whether 2, 3, and 1 are zeros of f(x) = x 4 6x 3 + 8x 2 + 6x 9 roots: x = multiplicity: k = x axis: cross tangent 310/26. Use substitution to determine whether 1, 2, and 3 are zeros of f(x) = x 4 x 3 3x 2 + 5x 2 310/34. Find the zeros of the polynomial function and state the multiplicity of each f(x) = x 2(x + 3) 2 (x 4)(x + 1) 4 roots: x = multiplicity: k = x axis: cross tangent 5

6 310/38. Find the zeros of the polynomial function and state the multiplicity of each f(x) = x 4 10x roots: x = multiplicity: k = x axis: cross tangent 310/44. Using a graphing calculator, find the real zeros of the function f(x) = x 3 + 3x 2 9x 13 You may wish to use this program to check your work /42. Find the zeros of the polynomial function and state the multiplicity of each f(x) = 3x 3 + x 2 48x 16 roots: x = multiplicity: k = 310/46. Using a graphing calculator, find the real zeros of the function f(x) = x 4 2x You may wish to use this program to check your work. x axis: cross tangent 310/48. Using a graphing calculator, find the real zeros of the function f(x) = 3x 3 x 2 14x 10 You may wish to use this program to check your work /50. Using a graphing calculator, find the real zeros of the function f(x) = x 6 10x x 3 4x 2 5 You may wish to use this program to check your work. 6

7 310/52. Using a graphing calculator, estimate the real zeros, the relative maxima and minima, and the range of the function h(x) = ( 1/2)x 4 + 3x 3 5x 2 + 3x /55. Using a graphing calculator, estimate the real zeros, the relative maxima and minima, and the range of the function f(x) = x x + x 5 310/54. Using a graphing calculator, estimate the real zeros, the relative maxima and minima, and the range of the function h(x) = 2x 3 x /57. Determine whether true or false: If P(x) = (x 3) 4(x + 1) 3, then the graph of the polynomial function y = P(x) crosses the x axis at (3, 0). 310/56. Using a graphing calculator, estimate the real zeros, the relative maxima and minima, and the range of the function f(x) = 2x 4 5.6x /58. Determine whether true or false: If P(x) = (x + 2) 2(x 1/4) 5, then the graph of the polynomial function y = P(x) crosses the x axis at (1/4, 0). 7

8 311/59. Determine whether true or false: If P(x) = (x 2) 3(x + 5) 6, then the graph of the polynomial function y = P(x) crosses the x axis at ( 5, 0). roots: x = multiplicity: k = x axis: cross tangent 311/64. Threshold Weight. In a study performed by Alvin Shemesh, it was found that the threshold weight W, defined as the weight above which the risk of death rises dramatically, is given by W(h) = (h/12.3) 3, where W is in pounds and h is a persons height, in inches. Find the threshold weight of a person who is 5 ft 7 in. tall. 311/60. Determine whether true or false: If P(x) = (x + 4) 2(x 1) 2, then the graph of the polynomial function y = P(x) crosses the x axis at (1/4, 0). roots: x = 4 1 multiplicity: k = 2 2 even even x axis: cross tangent tangent tangent W(h) = (h / 12.3)3 Substitute 5 ft 7 in. = 67 in. into the formula for h = 67. Thus, W(67) = (67 / 12.3)3 = lbs. The threshold weight for a person 5 ft 7 in. is about lbs. 311/65. Projectile Motion. A stone thrown downward with an initial velocity of 34.3 m/sec will travel a distance of s meters, where s(t) = 4.9t t and t is in seconds. If a stone is thrown downward at 34.3 m/sec from a height of 294 m, how long will it take the stone to hit the ground? 311/??. Windmill Power. Under certain conditions, the power P, in watts per hour, generated by a windmill with winds blowing v miles per hour is given by P(v) = 0.015v 3. (a) Find the power generated by 15 mph winds. (b) How fast must the wind blow in order to generate 120 watts of power in 1 hr? 8

9 312/70. Interest Compounded Annually. When P dollars is invested at interest rate i, compounded annually, for t years, the investment grows to A dollars, where A = P(1 + i) t. When Sara enters the 11th grade, her grandparents deposit $10,000 in a college savings account. Find the interest rate i if the $10,000 grows to $11, in 2 years. For instructions of using TI to model, see the tutorial at 313/72. Determine which, if any, of the following functions might be used as a model for the data. a) Linear, f(x) = mx + b b) Quadratic, f(x) = ax 2 + bx + c, a > 0 c) Quadratic, f(x) = ax 2 + bx + c, a < 0 d) Polynomial, not linear or quadratic 313/73. Determine which, if any, of the following functions might be used as a model for the data. a) Linear, f(x) = mx + b b) Quadratic, f(x) = ax 2 + bx + c, a > 0 c) Quadratic, f(x) = ax 2 + bx + c, a < 0 d) Polynomial, not linear or quadratic 313/74. Determine which, if any, of the following functions might be used as a model for the data. a) Linear, f(x) = mx + b b) Quadratic, f(x) = ax 2 + bx + c, a > 0 c) Quadratic, f(x) = ax 2 + bx + c, a < 0 d) Polynomial, not linear or quadratic 313/76. Determine which, if any, of the following functions might be used as a model for the data. a) Linear, f(x) = mx + b b) Quadratic, f(x) = ax 2 + bx + c, a > 0 c) Quadratic, f(x) = ax 2 + bx + c, a < 0 d) Polynomial, not linear or quadratic 313/??. Unemployed. The table below shows the number of unemployed in the United States from 1996 through See TI tutorial at a) Use a graphing calculator to fit cubic and quartic functions to the data. Let x represent the number of years since b) Use the functions found in part (a) to estimate the number of unemployed in Compare the estimates and determine which model gives the more realistic estimate. 313/77. Foreign Adoptions. The number of foreign adoptions in the United States has declined in recent years, as shown in the table below. See TI tutorial at a) Use a graphing calculator to fit quadratic, cubic, and quartic functions to the data. Let x represent the number of years since Using R 2 values determine which function is the best fit. b) Use the functions found in part (a) to estimate the number of U.S. foreign adoptions in

10 313/78. U.S. Farm Acreage. As the number of farms has decreased in the United States, the average size of the remaining farms has grown larger, as shown in the table below. See TI tutorial at a) Use a graphing calculator to fit quadratic, cubic, and quartic functions to the data. Let x represent the number of years since Using R 2 values determine which function is the best fit. b) Using the functions found in part (a) estimate the average acreage in 1955, in 1998, and in /80. Dog Years. A dog s life span is typically much shorter than that of a human. Age equivalents for dogs and humans are listed in the table below. See TI tutorial at a) Use a graphing calculator to fit linear and cubic functions to the data. Which function has the better fit? [ Larger R 2 ] b) Using the function from part (a), estimate the equivalent human age for dogs that are 5, 10, and 15 years old. Year x Acreage See TI tutorial at 10