1.1 Investigating Polynomial Functions

Transcription

1 1.1 Investigating Polynomial Functions Overview: Objective: Terms: Materials: Procedures: These activities allow participants to investigate behavior of polynomial functions using the power of the graphing calculator. Precalculus TEKS (c.1) The student defines functions, describes characteristics of functions, and translates among verbal, numerical, graphical, and symbolic representations of functions, including polynomial, rational, radical, exponential, logarithmic, trigonometric, and piece-wise defined functions. Polynomial function, degree, root, zero, long run behavior, end behavior, short run behavior, power function Graphing calculators In these activities, participants will analyze functions to determine their long run and short run behavior. This sort of analysis is very helpful to students preparing for a calculus course. Activity 1: What is the Long Run? What is the Short Run? In this activity, we will use a graphing utility to examine the graph of polynomial functions of differing degrees. We will describe the end behavior and the short run behavior of these polynomial functions based on their degrees. Work with the whole group to get a complete picture of the polynomial function. Start with graphing using a standard graphing window. What can we do to get a more complete picture of the polynomial function? One option is to guess possible maximum and minimum values for y. Another option is to create a table of the function to determine the maximum and minimum possible values. TEXTEAMS Part 2: Precalculus TEXTEAMS Institute DRAFT 1

2 Seeing this table might suggest this new window and graph. The minimum value needs to be decreased. The table that allows one to enter the independent variable will help to determine what that minimum might be. This table might indicate the third choice for a window. How do you know the graph does not turn back down? Larger and smaller values of x may be tested or the window may be increased to see the graph does not turn back down. How would you describe the long run behavior? Long run behavior is often called end behavior. It describes what the graph of the function does as x grows very large or very small. Answers will vary, but TEXTEAMS Part 2: Precalculus TEXTEAMS Institute DRAFT 2

3 participants should be guided to understand that this polynomial function s long run behavior is most effected by the power function. P( x) = 8x 6 As x increases, this term increases much faster than any of the other terms of this polynomial function. f ( x) = 8x 6-24x 5-13x x 3-84x x + 52 Graphing these two functions in the original window emphasizes the differences in the short run behavior of these two functions, but increasing the window shows how alike the graphs are in the long run. The values of these functions for very large values of x may be examined using a table or computing values on the home screen. Note that the table rounds the values, but the home screen does not. Describe how these two functions are different. This question should lead to a discussion of the short run behavior of polynomial functions. Short run behavior includes how many roots the polynomial has and how many turning points (relative maximum and minimum points) the function has. A relative maximum point is the greatest value of the function in an open interval about the x-value of the point. A relative minimum point is the least value of the function in an open interval about the x value of the point. To illustrate this, one may look at one of the minimum points of the graph or by looking at a table around this x-value. TEXTEAMS Part 2: Precalculus TEXTEAMS Institute DRAFT 3

4 The roots or zeros of the polynomial function are those values of x for which f ( x) = 0. They are the x-coordinates of the x-intercepts of the polynomial functions. Zeroes will be examined more thoroughly in the next activity. The short run behavior of this polynomial function is that it has four roots and three relative maximum or minimum points. Divide the participants into groups of four and assign each group one of the sets of problems. Tell participants that they may need more than one viewing window. After the groups have illustrated the graphs and described the short run behavior and the long run behavior of the polynomial functions of their given degree, have each group report their conclusions to the whole group. Group 1: x x 20-70,000 y 70, ,000 y 400,000 6 roots 5 maximum or minimum points 0 x 3.5 Long run behavior like P( x) = x y 12 3 roots 5 maximum or minimum points Long Run behavior like P( x) = x 6 TEXTEAMS Part 2: Precalculus TEXTEAMS Institute DRAFT 4

5 3. -8 x x 5-50 y y 40 no roots 4 roots 3 maximum or minimum points 3 maximum or minimum points Long run behavior like P( x) = x 6 Long Run behavior like P( x) = -x 6 Group 2: x x y y roots 4 maximum or minimum points -65 x 2.5 Long run behavior like P( x) = x 5 10 y 26 1 root 2 maximum of minimum points Long Run behavior like P( x) = x x x 8-10 y y 5000 TEXTEAMS Part 2: Precalculus TEXTEAMS Institute DRAFT 5

6 5 roots 4 maximum or minimum points -2 x 1 Long run behavior like P( x) = x 5-48 y 24 1 root 2 minimum or maximum points Long Run behavior like P( x) = -3x 5 Group 3: x x y y roots 4 root 3 maximum or minimum points 3 minimum or maximum points Long run behavior like P( x) = x 4 Long Run behavior like P( x) = -2x 4 TEXTEAMS Part 2: Precalculus TEXTEAMS Institute DRAFT 6

7 x x 10-20,000 y 20, y roots 3 distinct roots (one double root) 1 maximum or minimum points 3 minimum or maximum points Long run behavior like P( x) = x 4 Long Run behavior like P( x) = x 4 Group 4: x x y y root 2 distinct roots (one double root) 2 maximum or minimum points 2 minimum or maximum points Long run behavior like P( x) = 5x 3 Long Run behavior like P( x) = -x x x y y 10 3 roots 3 distinct roots 2 maximum or minimum points 2 minimum or maximum points Long run behavior like P( x) = 9x 3 Long Run behavior like P( x) = -8x 3 The following questions may be asked as participate describe their conclusions: What is the relationship between the degree of the polynomial function and the number of roots? the number of relative maximum and Precalculus TEXTEAMS Institute 7

8 minimum points? The number of real roots is less than or equal to the degree of the polynomial functions. If the number of real roots is less than the degree, the number of real roots is the degree minus an even number. This is because complex roots always travel in conjugate pairs. The number of relative maximum and minimum points is 1 less than the degree of 1 less than the degree of minus an even number. What does the graph do when you have a double root? How did you know you had a double root? The graph bounces on the x-axis at the root- the graph is tangent to the x-axis at the root when you have a double root. You know you have a double root when a factor appears twice. How can you tell the difference between the graph of a polynomial function of even degree and the graph of a polynomial function of odd degree? The end behavior of an even degree polynomial function is that as x Æ, p( x) Æ. The end behavior of an old degree polynomial function is that as x Æ, p( x) Æ, and as x Æ -, p( x) Æ -. Why does this difference exist? Answers will vary. Why must a polynomial function of an even degree have an odd number of relative maximum and minimum points? Show two examples and count the number of turns. Are there any situations that are not covered by your set of polynomial functions? Answers will vary. What is the effect of the negative leading coefficient? The long run behavior is the opposite of a polynomial of the same degree and a positive leading coefficient. Can you tell by the graph of a polynomial which degree it might be? You can tell if whether its degree is odd or even. Activity 2: Zeroes of Polynomial Functions Before this session begins, share the program that simulates synthetic division with the participants. What methods can you use to locate the zeroes of the polynomial function? Graphing the polynomial function may help locate the roots. Precalculus TEXTEAMS Institute 8

9 One approach often used is the Rational Root Theorem. The possible rational roots are many, but based on the graph it does not make sense to test all of these roots! The possible roots may be tested by using the table (with the ask feature) The roots are 6, - 1 2, 5, and 5. 3 How do you know these are all of the roots? A fourth degree polynomial function can have at most four roots. Another approach is to use a program that models synthetic division. When this program is used, testing the root 5, it generates the factors: ( x - 5) ( 6x x 2-47x - 30). Continued testing determines the other roots. The advantage of using this program is that is will generate the quotient and the remainder. For example to determine the quotient and the remainder when ( 6x 4 - x x x +150) is divided by ( x - 8) use 8 in the synthetic division process. The quotient is 6x x x with remainder The value of the polynomial at 8 is the remainder Assign one of the remaining problems to each of the table groups. Ask them to report their findings to the whole group. 2. The graph of a polynomial function with a double zero will be tangent to the x-axis at that x value. Example: F( x) = ( x - 9) ( x + 3) 2 ( x - 4) Precalculus TEXTEAMS Institute 9

10 3. A polynomial function with a triple root will pass through that x-value (change from positive to negative or negative to positive value), but the curve will change its shape. Example: F( x) = ( x + 6) ( x - 2) 3 ( x + 4) 4. The polynomial function cannot be a third degree polynomial function because the long run behavior requires it to approach negative infinity for both positive and negative values of x. It must be at least a fourth degree polynomial function with a negative leading coefficient. If there are 3 rational roots, there is another root. Call it a. One possibility is P( x) = -( x - 2) ( x - 4) ( x + 3) ( x - a)) If the y intercept is - 96, then P( 0) = = -( 0-2) ( 0-4) ( 0 + 3) ( 0 - a) a = -4 P( x) = -( x - 2) ( x - 4) ( x + 3) ( x + 4) 5. The roots appear to be - 3, -1, 1, 4. Because the graph of the polynomial function opens down the graph must have a leading coefficient of 1. P( x) = -( x -1)( x - 4) ( x + 3) ( x +1) Answers to Reflect and Apply: 1. E 2. B 3. D 4. C 5. A 6. F Focus on the TEKS: What TEKS (Algebra II/ Precalculus) were introduced, reinforced, or extended in the activities? Justify your answer. These activities reinforce Precalculus TEKS (c.1.b, D, and E). The questions target analysis of end behavior and roots of the functions. How do these activities support subsequent courses? Analyzing the long run behavior of functions gets students ready to deal with limits at infinity in calculus. Finding the roots of complicated functions is Precalculus TEXTEAMS Institute 10

11 also very important in calculus. The roots of a function are often used as limits of integration or tell students which function is greater when comparing two functions. These skills are fundamental in calculus. Can you take this activity to the classroom? If so, how would you adapt it? These activities could be adapted to the classroom as a summary activity done in small groups. Activity 2 makes use of synthetic division through the use of a calculator program. This would need to be discussed with students prior to starting the activity. Summary: These activities have shown how the study of polynomial functions may be greatly enhanced and simplified with the use of a graphing utility. Precalculus TEXTEAMS Institute 11

12 I. Polynomial Functions 1.1 Investigating Polynomial Functions: Activity 1 Activity 1: What s the Long Run? What s the Short Run? I. Whole group discussion A polynomial function is a sum of power functions that have a nonnegative integer powers. Graph this polynomial function and describe its long run behavior. Long run behavior is often called end behavior. It describes what the graph of the function does as x grows very large or very small. f ( x) = 8x 6-24x 5-13x x 3-84x x + 52 II. Group activities Each group will be assigned one of these sets of polynomial functions. Graph the functions on the graphing calculator. Choose an appropriate window and sketch the graphs on large grid paper. Two graphs might be necessary in order to illustrate both the long run and short run behavior of the graph. Describe the long run and short run behavior of the functions of the given degree. The short run behavior discussion should include the number of roots and the number of maximum and minimum points a polynomial function of the given degree may have. Group 1: 1. P( x) = x 6 + 4x 5-95x 4-206x x x P( x) = x 6-6x 5-134x x x x P( x) = x 6-5x P( x) = -2x 6 + 6x 5-5x x 3-8x 2-24x +12 Group 2: 1. P( x) = x 5-17x x P( x) = 8x 5-44x x 3-73x x P( x) = x 5-7x 3 - x 2 + 6x 4. P( x) = -3x 5 + 2x 4-8x 2-12 Precalculus TEXTEAMS Institute 12

13 I. Polynomial Functions 1.1 Investigating Polynomial Functions: Activity 1 Group 3: 1. P( x) = x 4-8x 2 + 9x P( x) = -2x x x 2-29x P( x) = x 4-15x 3-68x 2-147x P( x) = x 4-3x 3-13x x - 36 Group 4: 1. P( x) = 5x 3-40x x P( x) = -1x x 2-135x P( x) = 9x x 2-12x P( x) = -8x 3 + 9x + 1 Precalculus TEXTEAMS Institute 13

14 I. Polynomial Functions 1.1 Investigating Polynomial Functions: Activity 2 Activity 2: Zeroes of Polynomial Functions 1. Determine the zeroes of the polynomial function. P x ( ) = 6x 4 - x 3-192x x How can you tell from the graph that the polynomial function has a double root? To test your hypothesis, create a polynomial function that has a double root. 3. Create a polynomial function that has a triple root. How can you tell from looking at a graph that a root is a triple root? 4. P( x) is a polynomial function such that the following are true: a. P( x) has zeroes x = 2, x - 4, and x = -3. b. As x Æ ±, P( x) Æ - c. The y-intercept of the function is 96. Find a possible formula for P( x). 5. Create a polynomial function that could have the following graph: The graphing window is -5 x y 100 Precalculus TEXTEAMS Institute 14

15 I. Polynomial Functions 1.1 Investigating Polynomial Functions: Reflect and Apply Reflect and Apply Without using a graphing utility, match each function with its graph. Explain your choice. 1. f ( x) = x 5-3x 4-2x 3 + 6x 2 + x f ( x) = -4x 2 + 6x f ( x) = x 3-5x 2 + 4x f ( x) = -2x 3-4x f ( x) = 3x 4-5x 2 + 5x f ( x) = -x 4 + x 3 + 4x 2 + 2x - 2 A B C B D E F Precalculus TEXTEAMS Institute 15

16 I. Polynomial Functions 1.1 Investigating Polynomial Functions: Student Activity Student Activity: Pill Fill Overview: Objective: Students will use their knowledge of polynomial functions to solve a problem. Algebra II TEKS Precalculus TEKS (c.1) The student defines functions, describes characteristics of functions, and translates among verbal, numerical, graphical, and symbolic representations of functions, including polynomial, rational, radical, exponential, logarithmic, trigonometric, and piecewise-defined functions. (c.3) The student uses functions and their properties to model and solve reallife problems. Terms: Materials: Procedures: Optimize, minimize, maximize Graphing calculators Activity: In this activity, students will write and solve a polynomial equation that fits a given real life situation. It may be necessary to provide students with the formulas for the volume of a sphere and the volume of a cylinder. V sphere = 4 3 pr3 V cylinder = pr 2 h Answers: 1. The independent variable is the radius of the capsule. 2. The dependent variable is the volume of the capsule. 3. The volume of the capsule can be broken into two parts: the volume of the cylinder and the volume of the two hemispheres or one sphere. V = 4 3 pr3 + pr 2 h h + 2r = 12 h = 12-2r V = 4 3 pr3 + p( 12-2r)r 2 V = 4 3 pr3 +12pr 2-2pr 3 V = pr3 +12pr 2 4. Graph the two functions f ( x) = 250 and V( x) = px3 + 12px 2. The volume occurs when the radius is approximately 2.8 cm. Precalculus TEXTEAMS Institute 16

17 I. Polynomial Functions 1.1 Investigating Polynomial Functions: Student Activity 5. In most real world situations, the values would not be negative. Precalculus TEXTEAMS Institute 17

18 I. Polynomial Functions 1.1 Investigating Polynomial Functions: Student Activity Student Activity: Pill Fill The Get-Well-Quick Pharmaceutical Company makes time-release capsules for various medications. This pharmaceutical company always uses capsules that are in the shape of a cylinder with hemispheres on each end. 12mm After extensive research and consumer testing, the Get-Well-Quick Company has determined that for ease of swallowing, the optimal length of its capsules should be 12 mm. With a new strain of a species of bacteria reported in the U.S., the Get- Well-Quick Company is hurriedly attempting to get a new product on the market. Get-Well-Quick s research lab has proven that their new capsule will be effective in combating the bacteria if the capsule can hold 250 mm 3 of the antibiotic medication. Write a polynomial function to model the volume of the capsule that the Get-Well-Quick Company can use to find the optimal radius for the capsule. 1. What is the independent variable? 2. What is the dependent variable? 3. Write a polynomial function to model the volume of the Get-Well- Pharmaceutical Company s capsule. (Remember that their capsules are 12 mm in length.) Precalculus TEXTEAMS Institute 18

19 I. Polynomial Functions 1.1 Investigating Polynomial Functions: Student Activity 4. If the capsule must hold 250 mm 3 of the antibiotic, what is the optimal radius for the capsule? Solve your polynomial equation graphically. 5. Why is it sometimes necessary to restrict the domain in a real-world situation? Precalculus TEXTEAMS Institute 19