For Review Purposes Only. Two Types of Definitions Closed Form and Recursive Developing Habits of Mind. Table H Output.

Transcription

1 79_CH_p-9.pdf Page 7. Two Types of Definitions Closed Form and Recursive The table below is from Exercise of Lesson.. Table H Developing You can describe a function that agrees with this table in more than one way. For example, If the input n is zero, the output is. To get the next output, add to the previous output. g(n) = n Loo for patterns. You will learn later on that Table H itself is a function. The set of possible inputs is,,,, and nothing else. When you thin of the table as a function, you can write H() =, but H() is not defined. In this lesson, you are looing for something different a way to express some regularity in the table. Finding and describing a pattern can allow you to extend the domain from,,,, to a larger set of numbers. Both bulleted descriptions above do this. When you match a table with a polynomial or another simple rule, you are uncovering a hidden relationship in the numbers. This is something that mathematicians really prize.. When the input is 9, what is the output of each of the two functions described above. Can you use each function to find the output when the input is Explain. You learned that the domain of a function is the set of allowable inputs. Chapter Functions

2 79_CH_p-9.pdf Page Facts and Notation A function definition such as f() = and any output is more than the previous output is a recursive definition. Recursive definitions are useful for expressing patterns in the outputs of a function. The notation below is a useful way to write a recursive definition. if n = if n. so Notice that f is a recursive function and g is a closed-form function. Do the two definitions give the same function os e f(n) = c f(n ) nl y A function definition such as g(n) = n is a closed-form definition. A closed-form definition lets you find any output g(n) for any input n by direct calculation. Developing rp Use a model. The closed-form definition below matches Table H from Exercise of the Getting Started lesson. g(n) = n Pu You can build a computer or calculator model for function g in your function-modeling language. Then you can experiment with the model. You can do the following: Evaluate the function. w Graph the function. How you build a model depends on your computer or calculator. See the TI-Nspire Handboo, p. 99. Mae a table of the function in a spreadsheet window. ev ie It is a good habit to build models lie these for the functions you use, especially when you want to get a feel for how the functions behave. Fo rr The magnifying glass points out giant sequoia rings that formed in the years A.D.,, and 9. Tree growth is a recursive process that a tree s rings record.. Two Types of Definitions Closed Form and Recursive 9

3 79_CH_p-9.pdf Page 9. Build a model for g in your function-modeling language.. Mae a table of your model for inputs between and.. Graph your model in your graphing environment. A recursive definition lets you calculate any output in terms of previous outputs. The simplest ind of recursive definition for a function f tells you how to compute f(n) for an integer n in terms of f(n ). You need a place to start. The definition below tells you to start at (when n = ) and to add to get from one output to the next. f (n) = c if n = f (n ) if n. f () =, because n = f () = f ( ) = f () = = f () = f ( ) = f () = = You can use function-modeling language to model a recursive definition. Your technology may even provide a template lie the one below. For help, see the TI-Nspire Handboo, p. 99. define f (n) = c j, j j, j You just fill in the boxes., if n = define f (n) = c f (n ), if n.. Build a model for f and experiment with it. What numbers will f accept as inputs Explain. How does g compare to f Difference Tables A difference table can help you see patterns that lead to recursive definitions. Use Table H from Exercise of Lesson.. Table H Read the second line as the current output is the previous output plus five. This recursively defined function fits all the entries in Table H. Recursive definitions tell you how the outputs are related. Closed-form definitions tell you how inputs are related to outputs. Each tells you something interesting. In some cases, you can convert one to the other. Later in this investigation you will see how. Chapter Functions

4 79_CH_p-9.pdf Page To mae a difference table, add a third column mared with the D symbol. Write the difference between one output and the next in the third column. Î The differences are exactly what you need to write a recursively defined function that matches Table H. In this case, all the differences are the same number,. The following recursive definition fits Table H. f (n) = c if n = f (n ) if n. In some tables the differences are not constant, as in the table below., n Table D, d(n) 7 9 In Lesson., you will see that you can still use the D column to find a recursive function that matches the table. Here is a recursive definition for d. d(n) = c if n = d(n ) n if n. 7. Show that the function d fits Table D. The D symbol is the capital Gree letter delta. It represents change or difference. What function matches the D column. Two Types of Definitions Closed Form and Recursive

5 79_CH_p-9.pdf Page Chec Your Understanding For Exercises, use Table B from Lesson.., n Table B, B(n). Mae a difference table for Table B.. Decide whether each recursive definition fits Table B. a. b(n) = c if n = b(n ) if n. b. b(n) = c if n = b(n ) (n ) if n. c. b(n) = c if n = b(n ) n if n. d. b(n) = c if n = b(n ) n if n.. Decide whether each closed-form definition fits Table B. a. b(n) = n b. b(n) = n n c. To find each output, tae the input and multiply by one more than the input. d. b(n) = n Chapter Functions

6 79_CH_p-9.pdf Page For Exercises, copy and complete each difference table Î 7. Use the recursive definition below.. f (n) = c if n = n f (n ) if n. a. Find the values of f() through f() for this function. b. What preprogrammed function on your calculator agrees with f. The table at the right is an incomplete inputoutput table for a function. You can use each rule to complete the table. Mae a completed table for each rule. a. To get each output, tae the previous output and add four. b. To get each output, tae the previous output and multiply by three. c. n A ( n ) d. To get each output, tae the input, multiply by four, and then add two. 9. Write About It Consider the tables in Lesson.. Find three tables that are related. Describe how they are related. You may find it helpful to mae difference tables. Remember... Remember... a = for any nonzero number a, so =.. Two Types of Definitions Closed Form and Recursive

7 79_CH_p-9.pdf Page On Your Own The triangular numbers are numbers determined by the pattern shown below. The number of dots in a triangular pattern with n dots on a side is the nth triangular number. Here is a table for the triangular numbers.. Mae a difference table for the triangular numbers.. a. Write a recursive function definition that fits the table of triangular numbers. b. Tae It Further Find a closed-form definition for a function that generates the triangular numbers.. a. Copy and complete the difference table below., x, ax b Chapter Functions b a b a b a b a b a b b. Find a formula for f (x ) f (x) when f (x) = ax b. Side Length Number of Dots This pattern results when you arrange the counting numbers in a spiral and color the triangular numbers. You can define f(x) = ax b in your computer algebra system (CAS) and as for f(x ) f(x). Mae sure you have not assigned any values to a, b, or x.

8 79_CH_p-9.pdf Page. a. Copy and complete the difference table below., x, ax bx c c a b c a b c 9a b c a b c a b c b. Find a formula for f (x ) f (x) when f (x) = ax bx c.. Standardized Test Prep Find the output of the function g(n) below for the input n =. g(n) = c if n = g(n ) n if n. A. B. C. D. Maintain Your Sills. In each table below, the input-output pairs represent points on the graph of a linear function. Find the slope of each graph. a. b. c. 7 d. Describe how you can find the slope of a linear function when you have a table for the function in which the inputs are consecutive integers Loo for relationships. What is the connection between the two parts of this exercise Go nline Remember... Remember... Slope is the ratio of the change in the y-coordinates to the change in the x-coordinates.. Two Types of Definitions Closed Form and Recursive

9 79_CH_p-9.pdf Page. Constant Differences In this lesson, you will explore a specific type of difference table. The inputs are consecutive integers and all the differences between the outputs are constant. Here is Table H from Lesson.. Below is a difference table for Table H. Minds in Action In Lesson., you saw that the recursively defined function below fits the table. f (n) = c if n = f (n ) if n. You can use a recursive definition that has constant differences to find a closed form that also fits the table. Leslie explains how she finds a closed-form definition that fits the table. Leslie Say I m looing for f() and all I now is the recursive definition. f () = f () So if I want to now f (), I just need to now what f () was. But f () depends on f (). Oh, I have to use the recursive definition again. f () = f () And I can combine those two: f () = f (). Two steps, two fives. Every step I tae is another five. So if I go all the way bac, that s four steps. f () = f () Table H Chapter Functions

10 79_CH_p-9.pdf Page The definition tells me f() =, so f() is plus fives. What s great about this is there isn t anything special about finding f(). If I want to find f(7), I add plus 7 fives. If I want to find f(n), I add plus n fives. f(n) = n I thin I d rather write that as f(n) = n. Developing Visualize. The two properties below apply to any difference table, whether or not it has constant differences. Up-and-over property. An output is the sum of two numbers above it: the output directly above and the difference above and to the right. Hocey stic property. An output is the sum of all the differences above it to the right and the single output at the top of the table. The properties are easier to see than to describe. Here is an example of the up-and-over property for Table Q from Lesson Î The output for an input of is the sum of the output for and the difference next to the output for. Here is an example of the hocey stic property for Table Q Î (9 9 9) The output for an input of is the sum of all the differences in the column leading up to Q() and the output for. Leslie used the numbers highlighted in this difference table to find f() = f(). When you highlight all the numbers you add up, it loos lie a hocey stic. Otherwise, this property has nothing to do with hocey!. Constant Differences 7

11 79_CH_p-9.pdf Page 7 The hocey stic property leads to Theorem.. Theorem. You can match an input-output table with constant differences with a linear function. The slope of the graph of the function is the constant difference in the table.. Show that the up-and-over property is a result of the way you construct difference tables.. Show that the hocey stic property is a result of the way you construct difference tables.. Use the hocey stic property to prove Theorem.. Theorem. is the converse of Theorem.. Theorem. If f(x) = ax b is a linear function, its differences are constant.. Prove Theorem.. You can use the hocey stic property to quicly find a closed-form definition for a function that fits a table with constant differences. Here is Table H again with a hocey stic illustrated.. Describe how you can find the number using the hocey stic property.. How can you use the hocey stic property to find a closed form that agrees with the table In this chapter, assume that the inputs in the table are,,,, c unless stated otherwise. Detect the ey characteristics. What is the value of the constant difference The output is the number at the tip of the hocey stic plus the sum of the numbers on the handle. Chapter Functions

12 79_CH_p-9.pdf Page Chec Your Understanding For Exercises, use Table G from this investigation s Getting Started lesson.. Mae a difference table for Table G.. Write a recursive definition for a function that fits Table G.. Write a closed-form definition for a function that fits Table G.. Use the functions from Exercises and. What does each function give as output for each input below a. b... What s Wrong Here Leslie built a difference table for the input-output table at the right. Leslie says, All the differences are. So, this table has constant differences. Then the rule for the table is f(n) = n. Hmm, that doesn t seem right. That rule says f() is, but it s not. Find the flaw in Leslie s logic.. The table at the right has constant differences. Copy and complete the table. 7. Suppose the table from Exercise continues with constant differences. Find the values of p(), p(), and p().. Standardized Test Prep Suppose you have a table with constant differences. The input gives the output. The input gives the output. What is the output for the input 7 A. B. 7 C. D. n n Table G p(n) 7 f(n) Î Loo for relationships. Model your recursive and closed-form rules in your function modeling language. Do they agree for all inputs. Constant Differences 9

13 79_CH_p-9.pdf Page 9 9. Use the function table for F(n). a. Build a difference table for F(n). b. How are the differences related to the outputs c. Predict the value of F() by extending the pattern in the table. On Your Own For Exercises, use Table M from Lesson... Mae a difference table for Table M.. Write a recursive definition for a function that fits Table M.. Write a closed-form definition for a function that fits Table M.. Model your recursive and closed-form definitions from Exercises and in your function-modeling language. Do they agree for all inputs. Use the function table for D(n). a. Build a difference table for D(n). b. How are the differences related to the outputs c. Predict the value of D() by extending the pattern. n Table M n F(n) 9 7 D(n) Can you model this function in your function-modeling language Experiment. In Exercise, what does each model give as an output when you input 7 What does each model give as an output when you input 7 Go nline You can use a eyboard or voicerecognition software. If the words you input are the same, then the outputs will be the same. Chapter Functions

14 79_CH_p-9.pdf Page. Write About It Describe how to find a closed-form definition for a function that fits a table with constant differences. Include an example.. Which of the Tables A V in Lesson. have constant differences 7. A table with integer inputs has constant differences. When the input is, the output is 9. When the input is, the output is. Calculate the constant difference.. Tae It Further Suppose you reverse the inputs and outputs of Table M. You get the inputoutput table at the right. a. Find a function that fits the table. b. Using your function, copy and complete the difference table below that has integer inputs from to. c. How is the constant difference in the table from part (b) related to the constant difference in the original Table M Maintain Your Sills 9. The table at the right has constant differences. Copy and complete the table.. Find the slope between each pair of points. a. A(, ) and B(, ) b. C(7, ) and D(, ) c. E(, ) and F(, ) Inverse of Table M 9 7 Î Organize what you now. It may help to mae the table, even if you cannot fill in very much of it at first.. Constant Differences

15 79_CH_p-9.pdf Page. Tables and Slope In the last lesson, you saw that you can fit a linear function to a table with constant differences. In this lesson, you will learn the relationship between difference tables and slope. Example Problem An input-output table has constant differences. When the input is, the output is 9. When the input is 7, the output is. Find the constant difference. Solution Suppose the constant difference is. You can mae this difference table. 7 9 Î Now that you have labeled the difference column, you can write other outputs as expressions in terms of. For example, you can use the up-and-over property of difference tables to find the output when the input is Î Assume that the inputs are consecutive integers. Since the table has a constant difference, you can label every entry in the D column. Chapter Functions

16 79_CH_p-9.pdf Page You can calculate the other outputs in the same way. When the input is, the output is 9. The completed table at the right shows the outputs written in terms of. Now you now two expressions for the output when the input is 7. The up-and-over property gives 9, but the output is supposed to be. These must be equal. 9 = Solve for to find the constant difference. 9 = = =. Can you use the hocey stic property, instead, to find the constant difference Explain.. Find a linear function that fits the table. Developing Consider more than one strategy. There is another way to find the constant difference. Theorem. says that you can match a table with constant differences with a linear function. The graph of the function is a line with slope equal to the constant difference in the table. You can calculate the slope of that line by finding the slope between any two points on the line. Any input-output pair from the table gives the coordinates of a point on that line. change in output Therefore, you can calculate the constant difference as change in input Î. Tables and Slope

17 79_CH_p-9.pdf Page. A table has constant differences. When the input is, the output is. When the input is, the output is also. Find the constant difference. Chec Your Understanding. An input-output table has constant differences. When the input is, the output is. When the input is 7, the output is. a. Find the constant difference. b. Find the output when the input is. c. Find the linear function that fits the table.. A line passes through the points (, ) and (7, ). a. Find the slope of the line. b. Find an equation for the line.. Write About It Is there a line that contains the points (, ), (, ), and (, ) Explain. y (, ) (, ) x O (, ) Chapter Functions

18 79_CH_p-9.pdf Page. Some entries are missing from the table below. Can a linear function generate the table Explain.. A linear function generated the table below left. Find the values of a and b. 7 a b. Find a linear function that agrees with the table above right. 7. An input-output table has constant difference. When the input is, the output is. a. Find the output when the input is 7. b. Find the output when the input is. c. Find a linear function that fits the input-output table.. Use differences to prove that you cannot find a linear function that matches Table K from Lesson.. Table K 7. Tables and Slope

19 79_CH_p-9.pdf Page 9. Copy and complete the difference table from Table K. Complete the last column by finding the differences of the numbers in the D column. On Your Own. A line contains the points (, ) and (, ). a. Find the slope of the line. b. Find the value of a such that the point (, a) lies on the line. c. Find the value of b such that the point (7, b) lies on the line.. Write About It Does the table below have constant differences Explain., x, y. A line connects the points (, 7) and (, 7). a. Draw the line on a coordinate plane. b. Find the slope of the line. c. Write an equation for the line.. A line connects the points (7, ) and (7, ). a. Draw the line on a coordinate plane. b. Explain why the slope of this line is undefined. c. Write an equation for the line.. Mae a difference table for Table O from Lesson.. Explain why Table O cannot come from a linear function. x K(x) 7 7 Table O The symbol D means D of the D. Find the differences of the difference column. Loo for patterns. What is the value of c such that (, c) lies on the line Go nline Chapter Functions

20 79_CH_p-9.pdf Page. Use differences to find a linear rule that agrees with Table P from Lesson.. Table P 9 7. Tae It Further Here is a difference table for Table Q from Lesson.. a. Describe the relationship between Tables O, P, and Q. b. Describe the relationship between the difference tables that come from Tables O, P, and Q. 7. The table below comes from the functionr(w) = w. You can find the entries in the D, D, and D difference columns by finding the differences of the numbers in the respective previous columns. Copy and complete the table. w 7 R(w) n Q(n) 9 97 The Sudbury Neutrino Observatory is enclosed in this sphere with a 9-m radius. You can use the cubic function V = pr to find the volume of the sphere Each difference column will have one entry fewer than the one before it.. Tables and Slope 7

21 79_CH_p-9.pdf Page 7. Derman is maing a table for the function below. d(x) = x x x x x d(x) a. Copy and complete Derman s table for him. b. Derman says, What Do I have a linear function How can you help Derman figure this out 9. Standardized Test Prep Ramon accidentally shredded his physics homewor. He nows the relationship he was graphing is linear. He is able to reconstruct two table values. For the input of, the output is 7. For the input of, the output is. What is the slope of the relationship Ramon was graphing A. 9 7 B. 7 C. 7 D. 9 Maintain Your Sills. Mae a difference table for each function. Include inputs through. a. a(x) = x b. b(x) = x c. c(x) = x d. d(x) = x e. Find an integer such that the function e(x) = x has the number in its difference column. f. Tae It Further Find all integers such that the function e(x) = x has the number in its difference column.. Mae a difference table for each function. Use inputs,,,, and. a. f(x) = x b. g(x) = f (x ) f (x) c. h(x) = x x d. (x) = h(x ) h(x) e. r(x) = x f. s(x) = r(x ) r(x) Organize what you now. How can you use your CAS for this exercise Chapter Functions