Chapter One. Quadratics 20 HOURS. Introduction

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1 Chapter One Quadratics 0 HOURS Introduction Students will be introduced to the properties of arithmetic and power sequences. They will use common differences between successive terms in a sequence to identify and distinguish between these types of sequences. They will use regression and other techniques to identify a rule used to generate a given sequence. Particular emphasis is placed on examining sequences generated by quadratic functions, as these are the simplest examples of power sequences. This chapter extends students previous experience with quadratics introduced in Mathematical Modeling, Book 1. Students will use the technique of completing the square to change the general form y ax bx c into the transformational form 1 k (y q) (x p). They will use the transformational form of the quadratic function to determine the vertex, the equation of the line of symmetry, and the minimum or maximum point of the parabola described by the quadratic function. Students will also derive and apply the quadratic formula and use the discriminant to determine whether the corresponding quadratic equation obtained by setting y 0 has two different real roots, two real and equal roots (which are also known as a double real root), or no real roots, which are also known as two complex roots. The chapter has four sections: 1. Section 1.1 introduces students to finite and infinite number sequences. Students will solve problems by identifying and extending patterns in number sequences. They will learn to associate an arithmetic sequence with a first-level common difference, a quadratic sequence with a second-level common difference, and a higherdegree power sequence with third- and higher-level common differences. Students will use a variety of techniques to create the rule used to generate a given arithmetic or quadratic sequence. Students in the enhanced course will determine the value of the coefficient a of the rule generating a quadratic sequence by dividing the secondlevel difference by : that is, 1 a. They also use and the solutions to two linear equations to determine the rule generating a quadratic sequence: that is, an bn c.. Section 1. extends students understanding of non-linear relationships and functions, especially quadratic relationships and functions. Students will use regression analysis to determine the equation of a curve of best fit given a number sequence or tabular data. They will also model real-world quadratic relationships to determine maximum or minimum values. 3. Section 1.3 explores the connection among the transformational form of a quadratic function, the standard form, and the general form of a quadratic function. Students will explore when the transformational form of the quadratic function is most Chapter 1 Quadratics 1

2 effective, when the general form or standard form of the quadratic function is most effective, and how to convert from one form to the other. 4. Section 1.4 explores the roots of the quadratic equation 0 ax bx c. Students will complete the square to find the roots and to derive the quadratic b b ac formula: x 4 a Students will use the quadratic formula to solve problems involving quadratic relationships. They will use the discriminant b 4ac to determine the nature of the roots of a quadratic equation; that is, whether the quadratic equation has two different real roots, two real and equal roots (double root), or no real roots (that is, two complex roots). Students use the roots to locate the x-intercepts of the parabola for a given quadratic function. Connections This chapter builds on and extends students knowledge of non-linear functions, in particular, quadratic functions. The maxima/minima problems involving quadratic functions are developed further in Chapter, Rate of Change. Problems involving combinations and permutations analyzed in this chapter are examined from another perspective in Chapter 5, Probability. Chapter 1 Quadratics

3 1.1 Number Patterns Suggested instruction time: 4 hours Purpose of the Section Students examine properties of number sequences. They solve problems involving sequences by identifying and extending patterns. They classify arithmetic and power sequences by forming a sequence of differences between successive terms in a sequence or table. Students use the first-level common difference to create a linear function used to generate a given arithmetic sequence. Students also examine and find quadratic functions that generate sequences. Students in the enhanced course use the second-level common difference and systems of linear functions to create a quadratic function used to generate a given quadratic sequence. CURRICULUM OUTCOMES (SCOs) demonstrate an understanding of patterns that are arithmetic, power, and geometric, and relate them to corresponding functions C4 analyze tables and graphs to distinguish between linear, quadratic, and exponential relationships C9 sketch graphs from descriptions, tables, and collected data C3 determine the equation of a quadratic function using finite differences C10Z RELATED ACTIVITIES solve problems by identifying and extending number sequences; use geometric and number patterns to create arithmetic sequences; explore graphs of arithmetic, quadratic, and higher-degree power sequences use common differences to distinguish between arithmetic and power sequences use D n, the n th -level common difference, to determine the degree of a power sequence explore the relationship between, the second-level common difference, and the coefficient a in the function an bn c used to generate a quadratic sequence use the second-level difference and the solution to two linear equations to determine the rule used to generate a quadratic sequence STUDENT BOOK p. p. 11 p. 11 p. 11 p Number Patterns 3

4 ASSUMED PRIOR KNOWLEDGE definitions and distinguishing characteristics of linear and quadratic functions using symbols to represent linear and quadratic functions calculation of slope of a line using technology to graph linear and non-linear functions and to find the equation of the curve of best fit NEW TERMS AND CONCEPTS terms of a sequence finite and infinite sequences common difference between successive terms in a sequence arithmetic sequence sequence of differences coefficient degree of a function quadratic function, quadratic sequence power sequence PAGE Investigation 1 Number Patterns That Grow, Part 1 [Suggested time: 60 min] [Text page ] sequence an ordered arrangement of numbers, symbols, or pictures in which each item or term follows another according to a rule term each item in a sequence. The symbol t 1 represents the first term in the sequence, t represents the second term, and so on. infinite sequence a sequence that continues indefinitely and can be written as {t 1, t, t 3, }. An ellipsis (three dots) indicates the continuation of an infinite sequence. finite sequence a sequence that eventually terminates and can be written as {t 1, t, t 3,, } where the represents the last or n th term Purpose Students will explore the properties of arithmetic sequences and discover that the differences between successive terms form another sequence of constanumbers called the common difference. They will use this constanumber and the value of the first term to develop a rule to describe any term of the arithmetic sequence. The general expression, t 1 (n 1)d, will be developed in Question 6. Management Suggestions and Materials Students need coordinate grid paper or graphing calculators to create the graphs. Have students copy the definitions and terminology associated with sequences into their notebooks, accompanied by illustrative examples. Procedure Steps A to C Students can draw adjoining squares on scrap paper or grid paper to represent the joining of multiple sections of fencing. Ensure that students use the notation {4, 7, 10, 13, 16, 19} to describe the first six terms of the sequence. You might ask them to identify the second term and have them use the notation t to represent this term. Steps D and E Students should observe that, although each section of fencing contains four metal rods, the total number of rods in the fence increases by only three when another section is added because one rod acts as a common side for two adjacent sections. 4 Chapter 1 Quadratics

5 Some students will likely use a recursive pattern to solve the problem by repeatedly adding 3 to create a sequence with 63 terms. They should understand that the extension of a recursive pattern is a practical way to solve a problem involving a sequence only when there is a limited number of terms. Non-recursive patterns are more useful when the number of terms is great. A non-recursive pattern can be established by noting that, because the 10 th term equals 31, the rule for generating the sequence might be 3 times the value of the term number (10) plus 1. Consequently, a 63-section fence will contain rods. Other terms can be used to confirm this hypothesis. Students might develop a non-recursive pattern through the following reasoning: Because the first section of the fence contains 4 metal rods and because the remaining 6 sections require metal rods, the total number of rods is Similar reasoning can be used at the end of Question 6 to develop the rule for an arithmetic sequence t 1 (n 1)d, where t 1 is the initial term and d is the constanumber (common difference) added to each term. Students should also recognize that the expression t 1 (n 1)d can be expanded and rewritten as t 1 nd d (t 1 d) nd, where t 1 is the initial term and d is the constant number (common difference) added to each term. Management Tip Ensure that you identify the various methods used by students to determine the number of rods needed to construct 63 sections. Ask students to explain their methods. Investigation Questions QUESTION 1 Page 3 Students may draw the graph on graph paper or use graphing technology. Ensure that they choose a suitable scale. The graph shown below has been extended to include ten terms. Encourage students to explain why the graph does not intersect the origin. Students should explain why the graph of the sequence lies on a straight line. They might want to connect the points on the graph; however, the domain of this graph is only the positive integers. Ensure that students recognize that the slope of the line equals the constant difference because the difference between each pair of successive term numbers is constant and the difference between each pair of term numbers is 1. Students who use graphing technology might find the equation of the curve of best fit using LinReg to confirm that the slope of the line is (a) The discrete points are arranged on a straight line. Think about... Questions 1 to 4 Each sequence is generated by the simple arithmetic of adding the same number to each successive term. Remind students that adding a negative integer is equivalent to subtracting a positive number. Therefore, an arithmetic sequence can be created by adding any real number. Later in this Investigation, students will examine sequences created by adding any real number. 1.1 Number Patterns 5

6 (b) The domain is the set of natural numbers and the range is natural-number multiples of 3 plus 1. In seotation: D {x x N}; R {y y 3x 1, x N}. (c) The slope of the line is 3. QUESTION Page 3 Students should note that each additional section of the fence in part (a) increases the total number of rods by four, not five (the number of rods in a single section), because adjacent sections share a common rod. For the same reason, each additional section of the fence in part (b) increases the total number of rods by five, not six. Some students will likely solve the problems using a recursive pattern by extending each sequence to 63 terms. However, some students might develop a nonrecursive pattern by reasoning that the remaining 6 sections of the fence in part (a) contain rods. Therefore, the total number of rods needed to build the fence is A similar process can be used to calculate the number of rods needed to build the fence in part (b): 6 (6 5) (a) 53 rods (b) 316 rods QUESTION 3 Page 3 Notebook Entry Have students record their definitions of an arithmetic sequence in their notebooks along with some illustrative examples that show adding positive and negative numbers. Note To avoid confusion, have students distinguish between a sequence of differences and the common difference. Subtraction of successive terms in an arithmetic sequence results in a sequence of differences in which each term is a constanumber called the common difference; for example, {3, 3, 3,...}. Students should note that any additional section of drywall increases the total number of studs by n 1, where n is the number of sides in the polygon, because adjacent sections share one common side/stud. Students might use different methods to determine the number of studs needed to construct a 100-section drywall. Ensure that the students present and discuss these different methods. 3. (a) example: For octagonal sections, the sequence is {8, 15,, 9, 36, 43}. (b) example: For octagonal sections, the total number of studs is QUESTION 4 Page 3 Answer 4. example: {10, 19, 8, 37, 46, 55} ; ; Each term is 9 more than the one before. QUESTION 5 Page 3 Students should note that each sequence of differences has a common or constant term equivalent to the constanumber added to generate each arithmetic sequence. Thus, another definition of an arithmetic sequence is one in which each term in the sequence of differences is the same number. In Investigation, this sequence will be called the sequence of first-level differences, and be represented by the notation D 1. 6 Chapter 1 Quadratics

7 5. (a) Question 1: {3, 3, 3, 3, 3} Question.(a): {4, 4, 4, 4, 4} (b): {5, 5, 5, 5, 5} will vary for Questions 3 and 4. (b) Each term in the sequences in (a) is a constant or the same number. (c) example: The sequence of first differences in an arithmetic sequence is a constanumber. sequence of differences a sequence created from another sequence by subtracting the value of each term in the original sequence from the next term in that sequence. For example, the sequence of differences for {1, 3, 5, 7, 9, } is {3 1, 5 3, 7 5, 9 7, } or {,,, }. QUESTION 6 Page 3 The purpose of this question is to develop a general rule for generating an arithmetic sequence given the initial term and the common difference. Students should note that each additional section of the bridge railing increases the total number of rods by two rather than three because adjacent sections share a common side. Note that the number of twos being added is one less than the number of terms: for example, the fourth term contains three twos. These observations can be expressed in words as the n th term is 3 plus n 1 twos and symbolically as 3 (n 1). This can be simplified to n 1. Students should again note that although n, the term number, could be any real number, in practice the domain is restricted to the natural numbers or positive integers. Encourage students to represent the domain and range using seotation. Challenge Yourself An arithmetic sequence can be represented as 1 d, where d is a constanumber. Therefore, the difference between successive terms in an arithmetic sequence must be the constant d because 1 d. 6. (a) example: Each successive term is found by adding to the previous term. (b) t 6 3 5, t 7 3 6, t 8 3 7, t 9 3 8, t (c) 3 (n 1) or n 1 (d) t 00 3 (00 1) 401 (e) The domain is the set of all natural numbers. The range is the set of all odd natural numbers greater than or equal to 3. D {x x N}, R {y y x 1, x N} (f) t 1 (n 1)d Check Your Understanding [Completion and discussion: 50 min] QUESTION 7 Page 4 Each additional hexagonal section increases the total number of rods needed by five. Encourage students to use words to express the relationship between the term number and the number of metal rods in the railing. For example, The total number of metal rods n can be found by adding (n 1) fives to the initial number, six. This can be represented symbolically as 6 (n 1) 5 or 5n Number Patterns 7

8 7. (a) {6, 11, 16, 1, 6, 31, 36, 41, 46, 51} (b) 6 (n 1) 5 or 5n 1 (c) t 00 6 (00 1) QUESTION 8 Page 4 Challenge Yourself Students should develop a formal proof such as the following. A sequence of differences can be represented as the difference between the successive terms 1 and : 1 [d(n 1) t 1 ] [dn t 1 ] [dn d t 1 ] [dn t 1 ] d, where d is the common difference. Students should note that the slope d represents the common difference in an arithmetic sequence generated by a linear function. Students can identify each arithmetic sequence by finding either a sequence of differences or the constanumber being repeatedly added to each term of the sequence. 8. (a) arithmetic sequence; common difference 4; 4 (n 1) ( 4) or 4n (b) arithmetic sequence; common difference 3; 3 (n 1) 3 or 3n (c) arithmetic sequence; common difference 5; (n 1) 5 or 5n 3 (d) not an arithmetic sequence; no constant difference (e) arithmetic sequence; common difference ; 6.5 (n 1) or n 4.5 (f) not an arithmetic sequence; no constant difference (g) arithmetic sequence; common difference 1 4 ; 1 (n 1) 1 or n 1 4 or n 1 4 (h) arithmetic sequence; common difference 5; 4 (n 1) 5 or 5n 9 QUESTION 9 Page 5 The purpose of this question is to ensure that students understand that the common difference can be any real number, not just a positive integer. Did You Know? Students could investigate and report on figurate numbers after Investigation. A large amount of useful information on figurate numbers can be found through the Internet: for example, swarthmore.edu/dr.math/ Management Tip Provide Blackline Master (Dot Paper) to assist students in creating the dot patterns. 9. (a) example: {3,, 7, 1,...} (b) example: {0, 3.5, 7, 10.5,...} (c) example: 1, 1 4, 1, 5 4,... (d) example: {100, 50, 00, 350,...} QUESTION 10 Page 5 Students can note that each term in the square dot pattern in part (a) is generated by squaring the term number, and that each successive term in the triangle dot pattern in part (b) is generated by adding the term number to the previous number of dots. Consequently, neither sequence is an arithmetic sequence because the difference between successive terms is not a constant but increases as the number of terms increases. You might want students to simplify the expression for the dot pattern in part (c) from 3 (n 1) to n 1. 8 Chapter 1 Quadratics

9 10. (a) {1, 4, 9, 16}; not an arithmetic sequence (b) {1, 3, 6, 10}; not an arithmetic sequence (c) {3, 5, 7, 9}; arithmetic sequence; common difference is ; 3 (n 1) n 1 QUESTION 11 Page 5 Encourage students to explain their procedures for finding and to simplify the expression. They should be able to find the common difference, and use it as the slope of the function. Answer 11. example: The common difference is 4 in the sequence {3, 7, 11, 15, 19, 3,...}. Therefore, 3 (n 1) 4 4n 1. QUESTION 1 Page 5 Students should make the connection here between the common difference for the tems of the sequence and the slope of the graph of the generating function for the sequence. Both measure the constant rate of change in the terms with respect to the term number. 1. (a) The graph points lie on a straight line; that is, the rate of change in the terms is constant, which indicates a common difference between terms of the sequence. (b) slope 3 (c) common difference 3 (d) The common difference is equal to the slope. QUESTIONS 13 AND 14 Page 6 Assist students in understanding why each expression generates an arithmetic sequence by writing each expression in words; for example, 3n can be interpreted as adding successive multiples of 3 to an initial number. Students can use graphing technology to construct each graph. They can calculate the vertical distance between successive points on the graph. They will note that each difference is 3. Because each horizontal distance between successive points is 1, the slope must be 3. The graphs of the three functions are parallel lines because the three slopes are equal and the common differences are equal. That is, the rate of change is the same for all terms in each of the three sequences. slope r ise 3 run 1 3 Technology Students might want to use a graphing calculator to check their answers by using the TRACE feature to check values. Management Tip You might want to review the calculation of slope and its relationship to the linear equation y ax b. Slope a r ise run Challenge Yourself This open-ended item has many possible solutions. Encourage students to use words to describe the n th term in each sequence. This will help them in determining the corresponding linear function that can be used to generate each arithmetic sequence. examples: (a) {0, 3, 6, 9, 1} can be generated by the linear function 3n 3 or 3(n 1). (b) There are five common differences between 1 and 31. Therefore, the common difference is (31 1) 6. The 5 sequence {1, 7, 13, 19, 5, 31} can be generated by the linear function 1 (n 1) 6 6n 5. (c) Start with a negative number for t 1 and simply add a constant: { 4,, 0,, 4}. This sequence is generated by (n 1) 4 (n 3). (d) 1, 3 4, 1, 5 4, 3,... has a common difference of 1. The sequence can be 4 generated by 1 4 (n 1) 1 1 (n 1). 4 (e) There are ten common differences between the first term, which is my age, (say) 17, and the eleventh term, 100. Therefore, the common difference is (100 17) The sequence {17, 5.3, 33.6,..., 100} can be generated by the linear function 17 (n 1) n Number Patterns 9

10 Assessment Ask students to write a journal entry describing a procedure to determine whether or not a sequence is an arithmetic sequence. They should also explain why the slope in a linear function is equivalent to the common difference. Technology You might want to demonstrate how to use the recursive function on a graphing calculator to generate an arithmetic sequence. The calculator can be set in sequence mode by pressing MODE and then selecting Seq. To define an arithmetic sequence such as u(n) 3 5(n 1), the Y key is pressed and the function is entered by pressing the appropriate numerical keys and using the key labelled X,T,θ,n to enter the variable n. If the first term of the sequence is 3, u(nmin) is set to equal 3. Entering u(1,10) by pressing nd [u] (just above the 7 key) displays a few of the first ten terms of the arithmetic sequence. 13. (a) In each sequence, successive terms differ by 3, indicating an arithmetic sequence. (b) Each graph is a straight line because each function is a linear function. (c) The lines are parallel because they have the same slope. (d) They are equal QUESTION 15 Page 6 For the sequences that remain arithmetic, students should determine the rule for generating the new sequence and the common difference. Students should realize that adding and subtracting a constanumber leaves the common difference unchanged. Answer 15. Sequences (a), (b), (c), and (d) will be arithmetic sequences, but (e) will not. QUESTION 16 Page (a) An arithmetic sequence can be represented as nd t 1. Adding or subtracting a constanumber a in each term yields the functions nd t 1 a and nd t 1 a. Because t 1 a and t 1 a are constant numbers, the functions nd t 1 a and nd t 1 a will generate arithmetic sequences with a common difference of d. (b) Multiplying or dividing each term of an arithmetic sequence by a constant number a yields the functions a(nd) at 1 and n d t1. These can be a a expressed as (ad)n at 1 and d a n t 1. Because ad, at a 1, d a and t 1 are a constanumbers, each function will generate an arithmetic sequence with a common difference of ad and d, respectively. a Investigation Number Patterns That Grow, Part [Suggested time: 60 min] [Text page 7] The remaining terms can be displayed by pressing the key repeatedly. To find a particular term, such as the twentieth term, we can enter u(0) and the solution 98 is displayed. Purpose Students will explore the common differences in power sequences and discover that a quadratic sequence of the form y ax bx c produces a sequence of common or constant differences at the second level, while other power sequences produce a sequence of common differences at higher levels. Students will also explore the relationship between the coefficient a and the second-level common difference in a quadratic sequence generated by the function y ax bx c. Students also compare graphs of quadratic and cubic sequences to note similarities and differences. 10 Chapter 1 Quadratics

11 Management Suggestions and Materials Students will need chairs or desks, graphing technology, and graph paper. Procedure Steps A and B The diagrams in the Student Book show two different ways that Alice and Beatrice can buy lunch from Outlets 1 and. Alice has two choices for buying lunch: Outlet 1 or Outlet. In the first diagram, Alice buys lunch from Outlet 1 and therefore Beatrice s only choice is to buy lunch from Outlet. In the second diagram, Alice buys lunch from Outlet and therefore Beatrice must buy lunch from Outlet 1. Since this exhausts Alice s choices, the two diagrams show all the possible ways for two food outlets to be chosen. Students might use a similar systematic method for three, four, and five food outlets. You may want to suggest that students try using some problem-solving strategies such as the following: acting out the problem; looking for patterns; or starting with simple cases and then working toward more difficult cases. For example, you could have students act out this problem by arranging to 10 chairs or desks to represent the 10 fast-food outlets. Students can number squares on square paper to represent outlets and use the letters A and B to represent Alice and Beatrice. Encourage students to use a systematic method, like the one described above for two outlets, to identify all possible lunch patronage choices. Students may realize that the total number of possible lunch choices is twice what it would be if Alice and Beatrice did not care which one of them went to any given food outlet for example, if they intended to share their different types of lunches. For example, there are six ways (shown in bold in the 4 outlets, 1 choices diagram at right) that they can buy different types of lunch from among four outlets, if they don t distinguish between which of them is doing the buying. If they do distinguish, there are 1 ways. The sequence begins {, 6, 1, 0}. Students mighote the following types of patterns evident in the sequence: Each number is an even number. The differences between successive terms are 4, 6, 8,..., which form an arithmetic sequence. 1.1 Number Patterns 11

12 Think about... Step C Students should recognize that the sequence of the number of lunch choices is not an arithmetic sequence because the terms of the sequence of first-level differences D 1 are not constant. Steps C and D Introduce the notation D 1 and to represent the sequence of first- and second-level differences between successive terms in a sequence. Students will recognize that the sequence has constant terms of. Step E Students might solve the problem by extending the pattern evident in D 1 to ten food outlets. Some students might suggest an alternative method of solution, which is presented in Question 17. For example, Alice can choose any one of the ten fast-food outlets, leaving Beatrice to choose from among the remaining nine outlets. Therefore, there are ways that they can choose different types of lunch. Management Tip You might want to have students reconsider this problem after they begin the study of permutations and combinations presented in Chapter 5, Probability. If no students make this suggestion, you might want to present this reasoning now and have students determine its validity. For example, this pattern becomes evident when the data are placed in a table. Outlets Ways to buy lunches Adjacent terms in the first row have a product in the second row; for example, This pattern can be stated as the rule n(n 1). Do not give students the rule now; they are asked to formulate it themselves as part (a) of Question 17. Investigation Questions QUESTION 17 Page 8 You might want to have two students roleplay this situation in order to determine that there are ways for them to buy two different types of lunch from among ten food outlets. You might also have them use a systematic list or a table to record each purchasing choice. 17. (a) n(n 1) (b) Chapter 1 Quadratics

13 QUESTION 18 Page 8 You might want students to act out the problem starting with, 3, 4, and 5 teams. Students should recognize the similarity between the relationship connecting the number of teams in a round-robin tournament and the number of games and the relationship connecting the number of different ways that two people can eat lunch at different food outlets and the number of outlets. The value of each term in the game sequence should be one-half of the value in the food-outlet sequence. As in the food-outlet problem, each of n teams plays n 1 other teams for a total of n(n 1) games. However, Team A playing Team B is the same as Team B playing Team A. Therefore the total number of games can be expressed as n(n 1). A table such as the following shows the relationship visually: Team A Team B Team C Team D Team E Team A Team B Team C Team D Team E There are five teams and games. 18. (a) {1, 3, 6, 10, 15} (b) Each number in the game sequence is one-half the number in the food-outlet sequence. (c) 10 9 = 45 games (d) = n(n 1) QUESTION 19 Page 8 Students should note that the number of games would be triple the number in a single-game round-robin tournament. Consequently, the function generating the sequence can be expressed as = 3n(n 1). Answer 19. The function would change from = n(n 1) to t n = 3n(n 1). 1.1 Number Patterns 13

14 QUESTION 0 Page 9 Students can draw each graph manually, but it is more practical to use graphing technology. Question 17 Question 18 Question 19 For example, for Question 17, once the function Y X(X 1) has been entered using the Y key on the TI-83 graphing calculator, students can use the WINDOW settings in the margin for each graph. They will observe that the graph of the function over the domain of positive integers is a reflection image (in the line x 0.5) of the graph over the domain of zero and the negative integers. You might mention that the second graph is a parabola, whose properties will be investigated later in the chapter. 0. (a) example: each graph is part of a parabola; each graph rises upward buot in a straight line (b) Each graph is not a straight line. (c) parabolas with line of symmetry at x 0.5 (d) Each graph in part (a) is the right-hand portion of each corresponding graph in part (c). QUESTION 1 Page 9 Refer the students to the definition of a quadratic function. You might mention that the constant term c can also be considered a coefficient because it can be expressed as cx 0 (for x 0). quadratic function a function that can be represented by y ax bx c, where a and b are coefficients, a 0, and c is a constanumber. Because the greatest exponent in the function is, the function is said to have a degree of. Suggest that students expand each function to demonstrate more clearly that each is a quadratic. Students might graph both functions such as y x(x 1) and y x x on the same coordinate system to verify that the functions are equivalent. Also ensure that they identify the value of each coefficient and the constant term c. 1. (a) example: The function in Question 17, = n(n l) = n n, is a quadratic function of the form y ax bx c, where a 0. The function in Question 18, = n(n 1) 1 n 1 n, is a quadratic function of the form y ax bx c, where a 0. The function in Question 19, = 3n(n 1) 3 n 3 n, is a quadratic function of the form y ax bx c, where a Chapter 1 Quadratics

15 (b) For n n, a 1. For 1 n 1 n, a 1. For 3 n 3 n, a 3. (c) In each case, the second-level common difference is twice the coefficient a. Check Your Understanding coefficient the constant part of a term in an expression; for example, in the expression x 3x 9, is the coefficient in the term x [Completion and discussion: 60 min] QUESTION Page 9 Encourage students to compare the sequences in a variety of ways. Students in the enhanced course must also discuss the coefficinet of a and the value.. examples: The graph of a quadratic sequence is a parabola but the graph of an arithmetic sequence is a line. The terms of a quadratic sequence do not increase by a constant but the terms of an arithmetic sequence do. The terms in the sequence of second-level differences are constant for a quadratic sequence, but the terms in the sequence of first-level differences D 1 are constant for an arithmetic sequence. The coefficient a in the function generating a quadratic sequence is twice the value of each term in the sequence of second-level differences, but the coefficient a or slope in the function generating an arithmetic sequence equals the value of each term in the sequence of first-level differences D 1. QUESTION 3 Page 9 The purpose of this activity is to help students develop the relationship between the second-level common difference of a quadratic sequence and the coefficient a in the quadratic function generating the sequence. You might want to assign each quadratic to a different group of students to hasten the calculation process. Students in the enhanced course can then compare their results to determine that each term of the sequence of the second-level differences is a constant a. You might suggest to students who use decimals to represent the first- and second-level differences for the function 1 3 n 3n that they convert the decimals to fractions, to better see the relationship between the second-level difference and the coefficient a. 3. (a) and (b) n n {1, 6, 13,, 33, 46, 61, 78, 97, 118} D 1 {5, 7, 9, 11, 13, 15, 17, 19, 1} {,,,,,,, } n n 1 {3, 11, 3, 39, 59, 83, 111, 143, 179, 19} D 1 {8, 1, 16, 0, 4, 8, 3, 36, 40} {4, 4, 4, 4, 4, 4, 4, 4} quadratic sequence a sequence whose terms are generated by a quadratic function Notebook Entry Have students record in their notebooks the relationship between the coefficient a in a quadratic function and the common difference, along with an illustrative example. 1.1 Number Patterns 15

16 Technology You might assist students in constructing sequences of differences using the LIST editor. For example, to construct sequences of differences for the sequence generated by the function 1 3 n 3n, the following steps can be used. Generate 10 terms of the sequence: Enter the LIST editor and move the cursor to L 1 and press ENTER. Press nd List and display the LIST OPS menu. Press 5 to select 5:seq(. Enter the following data using the comma and ALPHA keys: seq(n /3 3N,N,1,10). Press ENTER. Generate 10 terms of the sequence of first-level differences: Move the cursor to L and press ENTER. Press nd List and display the LIST OPS menu. Press 7 to select 7: List(. Enter the instructions L 1 ) and press ENTER. Generate 10 terms of the sequence of second-level differences: Move the cursor to L 3 and press ENTER. Press nd List and display the LIST OPS menu. Press 7 to select 7: List(. Enter the instructions L ) and press ENTER. The following data will be displayed under lists L 1, L, and L n 3n 1 0,, 1, 5, 7 0, ,, 13 6, 54, D 1 4, 1 4, 16, ,,, 8, 6, n n D 1 3, 3, 3, 3, 3, 3, 3, 3 {, 1, 30, 56, 90, 13, 18, 40, 306, 380} { 10, 18, 6, 34, 4, 50, 58, 66, 74} { 8, 8, 8, 8, 8, 8, 8, 8} 0.n 1 { 1., 1.8,.8, 4., 6, 8., 10.8, 13.8, 17., 1} D 1 { 0.6, 1, 1.4, 1.8,.,.6, 3, 3.4, 3.8} { 0.4, 0.4, 0.4, 0.4, 0.4, 0.4, 0.4, 0.4} (c) Each term in, the common difference, equals a. (d) a 5 3 ; therefore, each term in, the common difference, equals QUESTION 4 Page 9 You might want to have students display their sequences and sequences of firstand second-level differences on poster board. Answer 4. will vary. QUESTION 5 Page 9 Students should provide a symbolic proof such as the following. Answer 5. Any three consecutive terms of any quadratic can be expressed, for all n, as: an bn c 1 a(n 1) b(n 1) c a(n ) b(n ) c 3 a(n 3) b(n 3) c and so on. Students could also use a spreadsheet to generate ten terms and each difference sequence D 1 and. 16 Chapter 1 Quadratics

17 D 1 and can be determined after expanding and collecting like terms: D 1 an bn c 1 an an a bn b c an a b an 4an 4a bn b c an 3a b a 3 an 6an 9a bn 3b c an 5a b a Each term in the sequence of second-level differences equals a. The next six questions introduce students to higher-order power sequences, starting with sequences generated by cubic functions and continuing with sequences generated by quartic functions, and so on. The patterns that students have already encountered in this section will continue; for example, a sequence generated by a cubic function has a common third-level sequence of differences D 3. QUESTION 6 Page 10 Students can act out the lunch-buying arrangements, draw diagrams, or make organized lists. They might also build on the reasoning used to find the number of ways two people can buy different types of lunches from among n food outlets: The number of ways that two people can choose two different outlets is n(n 1). This leaves Person C with n outlets to choose from. Therefore, the number of ways that three people can choose to patronize different food outlets is n(n 1)(n ). Students should realize that, because the variable n appears three times in the expression, the product should contain the cube, n 3. Therefore, the degree of the function is 3, and the function is referred to as a cubic function. Students will need graphing technology to create the graph of the cubic function in part (d). Have them compare its shape to the graph of the quadratic created in Question 0. Students with a TI-83 graphing calculator might graph both Y X(X 1) and Y X(X 1)(X ) using first different and then the same WINDOW settings, to get an idea of how each graph looks and how they compare. The following WINDOW settings produce these graphs: Have students discuss why the quadratic function has only y-values greater than or equal to 0.5 (this can be verified using the TRACE function) and why the cubic function takes all positive and negative y-values. They should also explain why each function intersects the origin. They should also see that for most positive y-values, the cubic function grows more rapidly than the quadratic function. For parts (e) and (f), because the cubic function was developed from the different lunch choices of three people among n food outlets, the first term number should 1.1 Number Patterns 17

18 Think about... Question 6(a) Students can use expanding and collecting like terms to determine the equivalence of the two functions, once they have created the function n(n 1)(n ) in part (a). n(n 1)(n ) (n n)(n ) n 3 n n n n 3 3n n The greatest exponent in the function is 3; therefore, the degree is 3. be 3 rather than 1. Students can substitute values of n from 3 to 1 in the function n(n 1)(n ) or n 3 3n n. 6. (a) n(n 1)(n ) (b) The degree is 3. (c) (d) example: The graph of a cubic function includes all negative and positive y-values; the graph of a quadratic function does not. (e) and (f) Term number t(n) n(n 1)(n ) D D (g) D 3 is a constanumber, 6. QUESTION 7 Page 10 Management Tip You might want students to compare the graphs of a variety of cubic and quadratic functions for real values of x in order for them to develop some generalizations about the shape of each graph. The cubic function investigated in Question 6 has a coefficient a 1 and the third-level common difference is 6. This activity allows students to explore the relationship between a in a cubic function and the common difference D 3. You might want to assign each cubic function to a different group of students to hasten the calculation process. Students can then compare their results to determine that each term of the sequence of third-level differences is a constant 6a. Encourage students to use a graphing calculator or spreadsheet to generate ten terms and all sequences of differences D 1,, and D 3. For example, the calculations for the sequence 1 6 n3 n 1 are shown below. Term number t(n) 1 6 n3 n D D (a) and (b) n 3 n 1 {4, 19, 58, 133, 56, 439, 694, 1033, 1468, 011} D 1 {15, 39, 75, 13, 183, 55, 339, 435, 543} {4, 36, 48, 60, 7, 84, 96, 108} D 3 {1, 1, 1, 1, 1, 1, 1} 18 Chapter 1 Quadratics

19 3n 3 n {4, 6, 84, 196, 380, 654, 1036, 1544, 196, 3010} D 1 {, 58, 11, 184, 74, 38, 508, 65, 814} {36, 54, 7, 90, 108, 16, 144, 16} D 3 {18, 18, 18, 18, 18, 18, 18} 4n 3 n 1 {, 9, 104, 51, 494, 857, 1364, 039, 906, 3989} D 1 { 7, 75, 147, 43, 363, 507, 675, 867, 1083} { 48, 7, 96, 10, 144, 168, 19, 16} { 4, 4, 4, 4, 4, 4, 4} D n3 n 1 See spreadsheet in previous discussion. (c) Each sequence has constant terms. (d) Each term in the sequence D 3 has a value of 6a. QUESTION 8 Page 10 Students can arrange the terms of the sequence an 3 bn cn d, D 1,, and D 3 in columns. They can substitute values of n from 1 to n. Answer 8. QUESTION 9 Page 10 Students might label the function in Question 9(d) as a quintic function. The relationship between the common difference and the degree of a function generating a quartic and quintic sequence becomes apparent when a table is used to summarize students investigations of the sequence of differences for arithmetic, quadratic, and cubic sequences. Sequence of Differences Sequence Function Degree with Common Term Arithmetic an b 1 D 1 Quadratic an bn c Cubic an 3 bn cn d 3 D 3 Students can surmise that a quartic with degree 4 will have a common difference D 4, and a quintic with degree 5 will have a common difference D 5. Students can generate each sequence by substituting values of n. The LIST editor on the TI-83 or a spreadsheet can be used to hasten the calculation process. Management Tip You might ask students to list words that have the prefix quart to help them recognize the relationship between the exponent 4 and the term quartic. quarterly quartet 1.1 Number Patterns 19

20 Challenge Yourself Students can draw an analogy to Question 6 where they developed a function to find the number of ways that three people could choose different types of lunches from n food outlets. There is no significant difference between choosing different seats and choosing different restaurants. If three people can choose in n(n 1)(n ) ways, then the fourth person can choose from n 3 remaining outlets, or seats in this case. Therefore, four people in n seats can be developed into the quartic function n(n 1)(n )(n 3) or n 4 6n 3 11n 6n. Term number t(n) n D D D Term number t(n) n 5 4n 3 5n D D D D Students should note that the growth of the sequence increases as the degree of the generating function increases. 9. (a) Quart refers to four parts, and the quartic function has degree 4. (b) Each term in D 4 is constant. (c) The function has degree 5; therefore, each term in D 5 will be constant. QUESTION 30 Page 11 Notebook Entry Have students record the definition of a power sequence in their notebooks, along with an example of their own. A power sequence is a sequence generated by a function with greatest exponent or power (degree) n greater than 1. Based on the results of Investigation, students can hypothesize that a power sequence generated by a function with degree n has the n th -level sequence of differences D n as the sequence of common differences. That is, D n is the lowest-level sequence of differences that consists of a constant number. 30. (a) will vary. (b) If the degree of the function generating a power sequence is n, then the sequence of differences D n will be the sequence of common differences. QUESTION 31 Page 11 To simplify the calculation of the sequences of differences, you might ask students to create only quadratic, cubic, or quartic sequences. Answer 31. will vary. 0 Chapter 1 Quadratics

21 QUESTION 3 Page 11 Students can determine that the new sequence is quadratic by finding the sequence of differences D 1 and. Calculating technology such as a TI-83 or a spreadsheet can hasten the calculation process. For example, Excel was used to generate the arithmetic sequences 3n 4 and n 1. The product of each term and the sequences of differences D 1 and were then calculated as shown. Term number t(n) 3n t(n) n t(n) ( 3n 4)(n 1) D (a) quadratic (b) Each term is the product of a multiple of n; therefore, a square of n will result. Functions with an exponent of are quadratic. QUESTION 33 Page 11 Students in the enhanced course should be expected to develop a formal proof. Answer 33. example: If an b and cn d for any real values a, b, c, and d, where a 0 and c 0, then the product is (an b)(cn d) acn and bcn bd (ac)n (ad bc)n (bd). Each expression in the parentheses is a real number; therefore, the function must generate a quadratic sequence. A Creating Quadratic Functions [Suggested time: 60 min] [Text page 11] Purpose Students use the common difference in the sequence of second-level differences and equation solving to determine the quadratic function used to generate a given quadratic sequence. Note that this method of finding the equation of the quadratic function always works when the elements in the domain increase in equal steps, of 1 or otherwise. For example, if the domain were {1, 3, 5, 7,...} instead of {1,, 3, 4,...}, the sequence {t 1, t 3, t 5, t 7,...} {1, 6, 15, 8,...}, D 1 {5, 9, 13,...}, and {4, 4,...}, the sequence of common differences. Challenge Yourself This open-ended item has many possible solutions. Encourage students to explain their reasoning. examples: (a) {3 1, 3, 3 3, 3 4,...} or {3, 1, 7, 48,...}; {6, 6, 6,...} (b) If each term in D 3 is negative, then a must be negative: { , 0.5 3, , ,...} or { 0.5, 4, 13.5, 3,...}; D 3 { 3, 3, 3,...} (c) { 1 1 4, 1 4, 1 3 4, 1 4 4,...} or { 1, 16, 81, 56,...}; D 4 { 4, 4, 4, 4,...} (d) If the second term is 1, then 4 4a 1 4 and a 1. The 16 quadratic sequence can be , 1 6 1, 1 6 3, or ,... 1, , 9, 1,... 16, resulting in 1 8, 1 8, 1 8, 1 8,.... (e) { 1,, 3, 4,...} or {, 8, 3,...}; { 4, 4, 4,...} Management Tip You might have students who are not in the enhanced course omit Focus A because it centres on outcome C4Z, determine a quadratic equation using finite differences. However, they might try to solve the problems in Questions 34, 38, 39, and 41 as well as the Challenge Yourself on page 13 by using patterns. They can also determine that each problem involves a quadratic sequence, without using common differences and equation solving to determine the actual quadratic. 1.1 Number Patterns 1

22 Think about... The Sequences of Differences (a) Students can refer to the results of Investigations 1 and to conclude that the sequence of triangular numbers is not an arithmetic sequence because each term in the sequence of first-level differences D 1 is not a constant. (b) Because each term in the sequence of secondlevel differences is a constant 1, the sequence must be generated by a quadratic function. (c) The coefficient a is Management Tip Students can develop another approach to determining the quadratic function by determining the sequence of first-level differences and second-level differences for the general quadratic function an bn c as shown. n an bn c D 1 1 a(1) b(1) c a b c a() b() c 4a b c 3 a(3) b(3) c 9a 3b c 4 a(4) b(4) c 16a 4b c In solving the triangular number problem, the sequence of first-level differences, D 1, is {, 3, 4, 5, 6,...}. Therefore 3a b corresponds to and 5a b corresponds to 3. This results in the two linear equations: 3a b 3 5a b The solution of the two equations yields a 1, b 1 and c 0. Thus, the function 1 n 1 n n(n 1) will generate the triangular number sequence for positive integers n. 3a b 5a b 7a b a a Challenge Yourself Each term in the sequence of second differences,, is 3. (a) b 0 and c 0 Because a 3, 3 n (b) b 0 and c 0 3 n bn and, therefore, quadratic functions such as the following would satisfy the conditions: 3 n 4n 3 n n 3 n 1 n Students should note that each of these functions can be factored. Some students might want to use technology to calculate the terms of each sequence. Term number t(n) 3 n 4n D Term number t(n) 3 n n D Term number t(n) 3 n 1 n D (c) b 0 and c 0 3 n c and, therefore, quadratic functions such as the following would satisfy the conditions: 3 n 7 3 n 1 3 n 3 (d) b 0 and c 0 3 n bn c and, therefore, quadratic functions such as the following would satisfy the conditions: 3 n n 4 3 n 1 n 1 3 n 9.8n 0.6 Focus Questions QUESTIONS 34 AND 35 Page 1 You might want to present a brief review of the methods that can be used to solve a system of two linear equations. This method of finding quadratic functions always works when the domain increases in equal steps, of 1 or otherwise. 34. (a) Equation ➀: 1 1 b c; 1 b c Equation ➁: 3 b c; 1 b c (b) c 0; b n 1 n Chapter 1 Quadratics

23 QUESTIONS 36 AND 37 Page n 1 n by factoring out the common term 1 n 1 n(n 1) n(n 1) 37. t Check Your Understanding [Completion and discussion: 5 min] QUESTION 38 Page 1 Have students explain why the relationship exists only for non-collinear points described in the Student Book. (If three points lie in a straight line, then one line segment connects all three.) You might have them use diagrams to find the next few terms of the sequence and encourage them to find and extend a pattern to create a sequence with six terms. For example, if students recognize that the terms in D 1 are the counting numbers, they can extend the pattern to find other terms in the sequence. Students should note that each additional point increases the number of line segments by n 1. For example, the seventh point will increase the number of line segments by six. Therefore, seven non-collinear points can be connected by line segments. This pattern is evident when the data are displayed in a table. Number of points Number of lines Think about... Choosing Terms t 1 and t You might have different groups of students create the quadratic by using different terms, including those that are not successive. Each group can present its results to verify that any pair of terms can be used to develop the corresponding quadratic function. For example, the following equations are obtained using the successive terms t and t 3 and the non-successive terms t 3 and t () b() c 6 1 (3) b(3) c 1 b c 1.5 3b c 1.5 3b 1 b b 1 b 0.5 or 1 c (3) b(3) c 1 1 (6) b(6) c 1.5 3b c 3 6b c 1.5 3b b 0.5 or 1 c 0 t n(n 1) 38. (a) {0, 1, 3, 6, 10, 15} (b) n(n 1) (c) t lines 1 n 1 n QUESTION 39 Page 13 Remind students that the number of diagonals rather than the number of sides is found and that the term numbers in the sequence are 3, 4, 5,... rather than 1,, 3,.... You might have students use diagrams to find the next few terms of the sequence, then encourage them to find and extend a pattern to create a sequence 1.1 Number Patterns 3

24 Management Tip Students should be cautioned that the term number n or the number of sides of a polygon begins with 3 rather than 1. Therefore, the first equation is with several terms. For example, if students recognize that the terms in D 1 are the counting numbers from onward, they can extend the pattern to find other terms in the sequence. 0 1 (3) b(3) c, not 0 1 (1) b(1) c. The pattern is evident when the data are displayed in a table. Number of sides Number of diagonals t n(n 3) for n 3 Students should be expected to use the second-level common difference to determine the coefficient a, and substitution and equation solving to determine the coefficients b and c. The coefficient a is 1 because the second-level common difference is 1. Any two terms can now be used to generate a pair of equations that can be solved for b and c. Substitute two values of n, say 3 and 4, into the equation for : 0 1 (3) b(3) c 1 (4) b(4) c These two equations involving b and c can then be solved. The coefficients b 3 or 1.5 and c 0 result in the equation n(n 3). Students should note that this quadratic has a domain of counting numbers from 3 onward and can only be applied to finding the number of diagonals for n t diagonals t n n(n 3) for n 3 QUESTION 40 Page 13 Students could solve this problem by solving second-level differences and seeing that a 4.9. Students could also set up a system of three equations and three unknowns. Students can use the second-level difference of 9.8 and equation solving to determine the quadratic function generating the sequence b c Equation ➀ for n 1 s: (1) b(1) c b c Equation ➁ for n s: () b c b Subtract Equation ➀ from Equation ➁ b 0 and c 1000 The quadratic function is n. The rock strikes the ground when 0; therefore, n and n 14.9 seconds. 40. (a) is a constant 9.8 (b) n (c) about 14.9 s 4 Chapter 1 Quadratics

25 QUESTIONS 41 AND 4 Pages These questions are similar to the questions posed in the rest of this Check Your Understanding. They could be assigned for homework with the solutions being presented the next day. 41. (a) Second-level differences are the same. (b) n 6n (c) about 3.30 s 4. (a) Second-level differences are the same. (b) n 10n 80 (c) 55 m Challenge Yourself This is a classic problem found in puzzle or recreational math books over the decades. Most hobby and toy stores sell a puzzle in which two different-coloured pegs are moved from hole to hole in a wooden board. Provide large grid paper for students to locate their coins. Suggest that they simplify the problem to one pair of coins, then two pairs of coins, rather than trying to solve the puzzle with five pairs of coins. (a) It takes 3 moves to exchange one pair of coins. The movement of the coins can be recorded as PDP. Management Tip Students will need coins and squared paper to play the game and solve the problem outlined in the Challenge Yourself on page 13; graphing technology is recommended. It takes 8 moves to exchange two pairs of coins. The movement of the coins can be recorded as PDDPPDDP. It takes 15 moves to exchange three pairs of coins, which can be represented as PDDPPPDDDPPPDDP. These representations can be compared to help identify a pattern for four and five pairs of coins. PDP PDDPPDDP PDDPPPDDDPPPDDP PDDPPPDDDDPPPPDDDDPPPDDP PDDPPPDDDDPPPPPDDDDDPPPPPDDDDPPPDDP 3 moves 8 moves 15 moves 4 moves 35 moves (b) The second-level common difference in the sequence of moves is, which indicates that the sequence is a quadratic with coefficient a 1. Substitution and equation solving results in the quadratic m p p p(p ), where m is the minimum number of moves and p is the number of pairs of coins. (c) Ten pairs of coins can be exchanged in moves. Encourage students to look for patterns and to develop a way to record the number of moves. 1.1 Number Patterns 5

26 Chapter Project Fencing Gardens This is the introduction of the Chapter Project, which continues in Sections 1., 1.3, and 1.4. Management Tip Ensure that students keep accurate and legible records of each part of the Chapter Project because each continuation builds on the results of the previous sections. Students not in the enhanced course will need guidance, and need to determine only that the relationship is quadratic. Students in the enhanced course should be expected to develop the function from using and equation solving. For All Students Students not in the enhanced course can create a sequence or table by calculating the area of the two garden plots when the length of a cut varies from 1 to 10 m. Students will have to realize that the two lengths, x and 100 x, are the perimeters of the two squares. Each perimeter must be divided by 4 to find the dimensions of each square and these dimensions must be squared to find the area. For example, when the 100-m length of fencing is cut into lengths of 1 m and 99 m, the two square garden plots have dimensions of 0.5 m and 4.75 m respectively. The areas of the squares are m and m, for a total area of m. (a) Suggest that students who are not in the enhanced course use a spreadsheet or the LIST or TABLE editor to enter data and simplify calculations. The table below shows the first 10 terms of the sequence. Perimeter of one square Area of both squares These data can be represented as a sequence: {61.65, 600.5, , 577, , 554.5, , 533, 5.65, 51.5} (b) The sequences of first-level and second-level differences are as follows: D 1 { 1.15, , 11.65, , 11.15, , 10.65, , 10.15} {0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5} The common difference 0.5 indicates that the function is a quadratic function and that a must be Students should also note that when a problem involves area, the underlying sequence is very likely to be quadratic because the dimensions are squared. Students in the enhanced course should create the quadratic generating the sequence from 0.5 and a 0.15 and equation solving. 6 Chapter 1 Quadratics

27 Equation ➀: (1) b(1) c; 61.5 b c Equation ➁: () b() c; 600 b c Solving Equation ➀: c 61.5 b Substituting c in Equation ➁ yields: 600 b 61.5 b b 1.5 c 61.5 ( 1.5) 65 The quadratic function generating the sequence is y 0.15x 1.5x 65. Students in the Enhanced Course (c) Students will need guidance in developing a table or sequence to determine that the relationship is quadratic. Again, have them find the total area of two circles when the fencing is cut at x 1 m, m, 3 m and so forth. For example, when the fencing is cut into two lengths of 1 m and 99 m, the circles have circumferences of 1 m and 99 m respectively. The radius of each circle can be found as follows: C C πr and, therefore, r π. 1 r 1 π and r 99 π The total area of both circles is π(0.159) π(15.756) m. A spreadsheet or calculator can also be used to generate a table or sequence as shown. Circumference of one circle Total area The table can be extended to include first- and second-level differences as shown. Note Students will likely obtain slightly differing values for the total area of the two circles because of rounding differences and the use of π. Minimize this difference by using the π key on their calculators. Total area D Because is a constant 0.318, there is a quadratic relationship between the length of the cut x and the total area of both circles. Students in the enhanced course should create the quadratic generating the sequence from and a and equation solving. Equation ➀: (1) b(1) c; b c Equation ➁: () b() c; b c Solving Equation ➀: c b Substituting c in Equation ➁ yields: b b b c ( ) The quadratic function generating the sequence is y 0.159x x Number Patterns 7

28 1. Non-linear Relationships and Functions Suggested instruction time: 3 hours Purpose of the Section Students will determine the equation of the curve of best fit for collected and given non-linear real-world data, especially data involving quadratic relationships. Students will be encouraged to make informed decisions about the type of regression to apply, based on the shape of the graph, the nature of the physical phenomenon being modeled, and the properties of non-linear sequences investigated in Section 1.1. Students can use the technical knowledge acquired in Section 1. to develop further understanding of quadratic functions presented in the final two sections. Students will also determine a quadratic function by modeling the problem under investigation. They will also learn that an equation of the curve of best fit is often restricted to a limited domain. CURRICULUM OUTCOMES (SCOs) model real-world phenomena using quadratic functions C1 sketch graphs from descriptions, tables, and collected data C3 analyze scatter plots, and determine and apply the equations for curves of best fit, using appropriate technology F1 describe and translate between graphical, tabular, written, and symbolic representations of quadratic relationships C8 analyze tables and graphs to distinguish between linear, quadratic, and exponential relationships C9 describe and interpret domains and ranges using seotation A7 RELATED ACTIVITIES students use a CBR or CBL system to collect and analyze non-linear data involving moving objects; they use the shape of a graph, the nature of a real-world phenomenon, and common differences to determine the type of curve of best fit compare stopping distances of vehicles on various surfaces model garden areas with fixed perimeters use technology to find maxima and minima STUDENT BOOK p. 15 p. 17 p. 0 p. 0 8 Chapter 1 Quadratics

29 ASSUMED PRIOR KNOWLEDGE using technology to graph linear and non-linear functions and to find the equation of the curve of best fit using symbols to represent or model real-world relationships using common differences to classify non-linear data NEW TERMS AND CONCEPTS properties of quadratic functions properties of accelerating and decelerating objects PAGE Suggested Introduction You might have a class discussion to help students distinguish between velocity and acceleration. Investigation 3 Analyzing Non-linear Data [Suggested time: 60 min] [Text page 15] Purpose Students will collect and analyze non-linear data and use regression analyses to find curves of best fit. Management Suggestions and Materials Small groups work best for this Investigation. Several groups could perform the Investigation and compare data, or one group could demonstrate for the class. Blackline Master 1..1 contains typical data of the sort that would be collected by the measuring device, in case it is more convenient to use the data without conducting the experiment. A Calculator-Based Ranger is the recommended form of measuring device. Students will need: toy car or ball board or plastic eavestrough; eavestrough is preferable because it will prevent the car or ball from falling off the side of the ramp a measuring device such as a Calculator-Based Ranger calculator graphing calculator Blackline Master 1..1, Ramp Data Blackline Master 1.., Ranger-to-Calculator Instructions Management Tip If you have access to a Texas Instruments CBL (Computer- Based Laboratory) System and the book Real-World Math with the CBL System by Chris Brueningsen et al., follow the instructions for the activity What Goes Up (pp ) as an alternative to the experiment suggested in the Student Book. If you do not have access to a CBL or CBR, provide copies of sample ramp data on Blackline Master Information on the CBR can be found on the Texas Instruments Web site at docs/cbr.htm Likewise, information on the CBL is located at calc/docs/cbl.htm 1. Non-linear Relationships and Functions 9

30 Procedure Think about... Step B Have students describe the shape of each graph and justify each shape. You might ask probing questions such as the following: Why is each graph non-linear? [example: The distance does not increase at a uniform rate.] Which graph has a maximum y-value? Which graph has a minimum y-value? [example: The graph in (a) will have a minimum value because the car or ball reaches a minimum distance from the Ranger when it stops rolling and returns to and possibly passes the starting point. The graph in (b) will have a maximum value because the car or ball reaches a maximum distance from the starting point when it stops rolling and returns to and possibly passes the starting point.] In Section 1.1, we constructed graphs of linear (arithmetic), quadratic, cubic, and other power relationships. Which of those graphs is more likely to represent the relationship between time and distance of an object rolling up and down a ramp? [example: likely quadratic or cubic because each had maximum and minimum values] Steps A and B You might ask a pair of students to conduct the experiment and have other students transfer the resulting graph and data to their graphing calculators. Instructions for transferring this information are on Blackline Master 1... Use a large ball with a reasonably hard surface to reflect the sound waves emerging from the Ranger. If a toy car is used, you might have to tape an index card or aluminum foil to it in order to increase the reflectivity of the surface. Students will likely have to experiment with the angle of the ramp and take several samples. Ensure that students do not assist the car or ball, and have them understand that the data only become meaningful after the car or ball is released. They might also have to try various smoothing settings in order to obtain a useful graph. Investigation Questions QUESTIONS 1 AN Page 15 Depending on the angle of the ramp, students should obtain a parabola similar to the following graph, which represents a sample distance time graph of a toy car rolling up and down a plastic eavestrough. The displayed x- and y-values indicate that the car was located 0.84 m or 8.4 cm from the Ranger at the beginning of the experiment. Students should note that y represents the distance from the Ranger, not the distance from the starting point. Therefore, the graph should show that the distance slowly decreases and then increases until it reaches and even surpasses the starting distance from the Ranger. From their work in Section 1.1, students can see that the distance does not increase at a uniform rate, which indicates a non-linear relationship. Students should realize that the shape of the graph looks similar to the quadratic and cubic functions examined in Section 1.1. Have them compare the graph of the data to the graph they sketched in Think about... Step B, part (a), for similarities and differences. A table similar to the following is obtained when TRACE is used to determine time and distance data. Time (s) Distance (m) from the Ranger By examining sequences of differences, students should see a second-level common difference, indicating a quadratic relationship between time and distance. To create a curve of best fit, students can enter the data into the LIST editor of a graphing calculator or into a spreadsheet. Because a cubic function has a shape 30 Chapter 1 Quadratics

31 similar to a quadratic function over a limited domain, students should be encouraged to perform both quadratic and cubic regression. Note Because of measurement errors, will likely be close to a common value. However, students can use the general shape of the curve to deduce that the relationship is likely a quadratic one. As can be seen by comparing the two trendlines for these experimental data, the coefficient of determination R for the cubic function can be greater than the R for the quadratic function. If this happens, help students understand that a quadratic function makes more sense over the entire domain of positive numbers. For example, as the time rolling up and down the ramp increases, the distance travelled by the ball or toy car must increase. Students can graph the quadratic and cubic functions for longer travel times and compare the graphs. Management Tip Students might find that cubic regression provides a better fit because of the limited domain. The discussion here shows the results of both a cubic and quadratic regression. Students can note that both graphs are good approximations of the data, but as the time exceeds.5 s, the height of the quadratic graph continues to increase while the height of the cubic graph begins to decrease. The quadratic graph represents a situation in which the distance between the rolling object and the Ranger 1. Non-linear Relationships and Functions 31

32 Management Tip Perhaps the most important lesson that students can learn from this example is that regression analyses musot be a substitute for thinking critically about the phenomenon under investigation. gradually decreases and then increases as it rolls back, exceeding the original distance if the ramp is long enough. The cubic graph shows a gradual decrease in distance as the rolling object approaches the Ranger and a gradual increase as it rolls away from the Ranger. However, after the ball rolls for more than.5 s, the ball or toy car appears to reverse direction and travel past the Ranger, as indicated by the negative values of y. Therefore, a cubic model is an unrealistic description of the behaviour of a ball or car rolling up and down a ramp. 1. will vary.. to (a), (b), and (c) will vary, but the graph and equation of the curve of best fit should represent a quadratic relationship with a minimum y-value. For Students in the Enhanced Course Ask students to read and discuss the article Teaching Statistics with Data of Historic Significance: Galileo s Gravity and Motion Experiments (by David A. Dickey and J. Tim Arnold, North Carolina State University, Journal of Statistics Education v.3, n.1, 1995), available on the Web site jse/v3n1/datasets.dickey.html. The authors use experimental data on rolling balls down an inclined ramp that were gathered by Galileo during his studies of falling bodies and projectiles. They demonstrate the importance of giving thought to parsimony, independent and dependent variables, and the importance of understanding the scientific nature of the experiment when selecting regression models. The opportunities for class discussion are especially rich in this understandable and real experiment, particularly when coupled with graphical analysis. QUESTION 3 Page 16 You might want students to replicate the experiment using a ramp with a steeper angle to verify their predictions. Answer 3. If the ramp is steeper and the same force is used to propel the ball or toy car, the rolled object will come to a complete stop in less time than it did in the initial experiment. Therefore, the minimum y-value or vertex of the graph will be to the left of the vertex of the original graph. It will also likely show greater y-values beyond the minimum value than the original graph, because a steep ramp will accelerate the toy car or ball and it should travel a greater distance down the ramp in a given amount of time. QUESTION 4 Page 16 Students can create new distance data by subtracting each distance from the starting point. For example, each distance in the third row in the following table was found by subtracting each distance in the second row from 0.84 m. Time (s) Distance from measuring device (m) Distance from start (m) Chapter 1 Quadratics

33 Students can compare the graphs by graphing both curves of best fit on the same coordinate system. Students can see that the curve drawn by the Ranger has a minimum value because each y-value represents the distance from the Ranger. The other curve has a maximum value because each y-value represents the distance from the starting point. You might have them note that the graph with a minimum value has a coefficient a that is positive, and the graph with a maximum value has a coefficient a that is negative. 4. to (a), (b), and (c) will vary, but the graph should be a quadratic with a maximum rather than a minimum y-value. Check Your Understanding [Completion and discussion: 45 min] Think about... The Giant Drop Ride Data In the first second the gondola drops 4.9 m, and in the next second it drops m 4.90 m 14.7 m, indicating an increasing rather than a uniform growth rate. The relationship between time and drop height of a falling body is non-linear. Time (s) Drop distance (m) Because the data in the table will be graphed, have students enter the data into the LIST editor of a graphing calculator or into a spreadsheet. Then they can use the calculator or spreadsheet to determine each sequence of differences. Time (s) Drop distance (m) D Students can see that the relationship must be quadratic because the sequence of second-level differences has a constant term. The constant difference 9.8 is the acceleration of gravity and the coefficient a must be = Non-linear Relationships and Functions 33

34 Management Tip Students can use the following Web sites to gather the vital statistics of several types of theme park rides. The second site contains a video clip of the Giant Drop ride. com/thrillrides/superman/ index.html drop.html The following Web site suggests a simple experiment with water that can be used to illustrate the properties of a free-fall ride. learner.org/exhibits/ parkphysics/freefall.html QUESTION 5 Page 16 Students will realize from the shape of the scatter plot that a non-linear relationship must exist between time and drop distance. Students can use the results of the calculated common differences and the shape of the graph to conclude that a quadratic regression should be performed. Because of the procedures used by spreadsheets to construct a trendline or curve of best fit, very low values of the coefficients b and c might be displayed as shown. Both b and c are negligible; therefore, the equation of the curve of best fit is y 4.9x. 5. (a) Management Tip Some students mighote that after 5 s, the gondola appears to have traveled 1.5 m, although it was raised to a height of only 119 m. Explain that air friction and other types of friction will slow the gondola slightly and it will have an acceleration very close to buot equivalent to that of gravity. Students might also note that the gondola has to slow down just before it hits the ground, to avoid harming the passengers. When the brakes are applied, the graph will reflect this, becoming less steep. This would decrease the last drop-distance value in the table. (Some gondolas actually drop below ground level in a hollowed-out space before slowing their descent, which increases the thrill of the ride.) (b) quadratic regression (c) y 4.9x QUESTION 6 Page 16 Students can use the completed table to note that doubling the time does not double the drop distance. They can compare the distance at 1 s (4.9 m) and s (19.6 m) to conclude that the change in drop distance increases by the square of the change in time. In this case, the drop distance in s is 4 times the drop distance in 1 s. Drop distances in the table are rounded to the nearest centimetre. 6. (a) Time (s) Drop distance (m) (b) See the previous discussion. 34 Chapter 1 Quadratics

35 QUESTION 7 Page 16 Students should realize that a graph of drop distance versus time is a representation of the average velocity of the falling gondola. As shown by the graph, the average velocity is not uniform. When average velocity versus time is graphed, however, the graph is a straight line representing the uniform acceleration of the gondola under the influence of gravity. The velocity calculations and graphing can be completed quickly if data have been entered into a graphing calculator or spreadsheet. 7. (a), (b), (c), and (d) Management Tip Spreadsheets will often calculate non-existent values of b in performing linear regression, as shown by the value of in the graph on the right. Therefore, the equation of the curve of best fit is y 4.9x. QUESTION 8 Page 17 Students should use graphing or sequences of differences to conclude that quadratic regression should be performed to determine the equation of the curve of best fit, as shown in the margin. The very small values of b and c can be ignored and the function can be represented as y 3.17x. Students in the enhanced course should use common differences and equation solving to determine that the coefficient a 3.17 and that b and c are both equal to or close to 0. Students who have studied Mathematical Modeling, Book, could also use three-by-three matrices to find the quadratic function. Students can use the information to create a matrix equation. Using y ax bx c, substitute x 1, x, and x 3 to give the following equations: a b c 3.17; 4a b c 1.70; 9a 3b c a b = c 8.57 Graphing calculators can be used to find the inverse of , which is Non-linear Relationships and Functions 35