MODELING WITH FUNCTIONS

Transcription

1 MATH HIGH SCHOOL MODELING WITH FUNCTIONS

2 Copright 05 b Pearson Education, Inc. or its affiliates. All Rights Reserved. Printed in the United States of America. This publication is protected b copright, and permission should be obtained from the publisher prior to an prohibited reproduction, storage in a retrieval sstem, or transmission in an form or b an means, electronic, mechanical, photocoping, recording, or otherwise. For information regarding permissions, request forms and the appropriate contacts within the Pearson Education Global Rights & Permissions department, please visit This work is solel for the use of instructors and administrators for the purpose of teaching courses and assessing student learning. Unauthorized dissemination, publication or sale of the work, in whole or in part (including posting on the internet) will destro the integrit of the work and is strictl prohibited. PEARSON and ALWAYS LEARNING are exclusive trademarks in the U.S. and/or other countries owned b Pearson Education, Inc. or its affiliates. Copright 05 Pearson Education, Inc.

3 CONTENTS LESSON : WETLAND PLANTS... 5 LESSON : MODELING RELATIONSHIPS... 7 LESSON : DIFFERENT WAYS OF GROWING... LESSON 4: COMPARING GROWTH... LESSON 5: DOMAIN AND RANGE... 7 LESSON 6: COMPARING GRAPHS OF FUNCTIONS... LESSON 7: THREE KINDS OF GROWTH...5 LESSON 8: LINEAR AND EXPONENTIAL DECAY...9 LESSON 9: RECURSIVE RULES FOR SEQUENCES... LESSON 0: SEQUENCES...5 LESSON : PUTTING IT TOGETHER... 7 LESSON : MODELING RELATIONSHIPS...40 LESSON : DIFFERENT WAYS OF GROWING... 4 LESSON 4: COMPARING GROWTH...4 LESSON 5: DOMAIN AND RANGE...45 LESSON 6: COMPARING GRAPHS OF FUNCTIONS...46 LESSON 7: THREE KINDS OF GROWTH...49 Copright 05 Pearson Education, Inc.

4 CONTENTS LESSON 8: LINEAR AND EXPONENTIAL DECAY... 5 LESSON 9: RECURSIVE RULES FOR SEQUENCES... 5 LESSON 0: SEQUENCES...55 LESSON : PUTTING IT TOGETHER...56 Copright 05 Pearson Education, Inc. 4

5 LESSON : WETLAND PLANTS. Write what ou alread know about functions. Share our summar with a classmate. Did ou write the same things?. Write our wonderings about modeling with functions. Share our wonderings with a classmate.. Write a goal stating what ou plan to accomplish in this unit. 4. Based on our previous work in math, write three things that ou will do during this unit to increase our success. For example, consider was ou will participate in classroom discussions, our stud habits, how ou will organize our time, what ou will do when ou have a question, and so on. Copright 05 Pearson Education, Inc. 5

6

7 LESSON : MODELING RELATIONSHIPS. For each graph determine whether the graph is a function or not a function. Graph A: Graph B: f(x) f(x) x x Graph C: f(x) x Copright 05 Pearson Education, Inc. 7

8 LESSON : MODELING RELATIONSHIPS. For each table determine whether the table is a function or not a function. Table A: x Table B: x The circumference of a circle C depends on its diameter d according to the formula C = π d. Write the diameter formula as a function of circumference, where d is diameter and C is circumference. C d π x ( ) = 4. Determine whether this graph represents a function. Explain how ou know. f(x) x Copright 05 Pearson Education, Inc. 8

9 LESSON : MODELING RELATIONSHIPS 5. The formula for the area of a circle is A(r) = π r. Calculate A(4). 6. The formula for the area of a circle is A(r) = π r. Write the function r(a) that expresses r as a function of A. 7. Write a function that describes this situation. Double the input and subtract 7. Use function notation. 8. The circumference of a circle C depends on its diameter d according to the formula C = π d. Which of these statements is true about the relationship between d and C? There ma be more than one correct relationship. A The relationship between d and C is a function. B As C increases, d decreases. C This graph represents the relationship between C and d. f(x) x D This table represents the relationship. d 4 C π π π 4π Challenge Problem 9. Look at this function. U(x) = A(x) A(x ), where A(x) = x Compute U(0), U(), U(), U(), U(4). What tpe of function is it? Copright 05 Pearson Education, Inc. 9

10

11 LESSON : DIFFERENT WAYS OF GROWING. What tpe of function is f(x)? f(x) = x A Linear B Quadratic C Exponential D Cubic. Which of these graphs represents a linear function? A B C D. Compare two quantities where the input is time and it grows in this wa. Begin with 50 and add 0 for each unit of time. Make a table using time (x) values between 0 and Compare two quantities where the input is time and it grows in this wa. Begin with 50 and add 0% for each unit of time. Make a table using time (x) values between 0 and 6. Round our values to the nearest hundredth. Copright 05 Pearson Education, Inc.

12 LESSON : DIFFERENT WAYS OF GROWING 5. Compare our tables for the two functions. Function A: Begin with 50 and add 0 for each unit of time. Function B: Begin with 50 and add 0% for each unit of time. Which one reaches 00 first? Support our conclusion mathematicall. 6. Compare two quantities where the input is time and it grows in this wa. Begin with and add for each unit of time. Draw a graph using time (x) values between 0 and Compare two quantities where the input is time and it grows in this wa. Begin with and multipl b for each unit of time. Draw a graph using time (x) values between 0 and Compare the graphs of the two functions. 0 5 f(x): Begin with and add for each unit of time. g(x): Begin with and multipl b for each unit of time x When are the functions equal? Support our conclusion b referencing points on the graph. Challenge Problem 9. Write formulas (equations) for the two growth functions. f(x): Begin with and add for each unit of time. g(x): Begin with and multipl b for each unit of time Copright 05 Pearson Education, Inc.

13 LESSON 4: COMPARING GROWTH. Determine if each scenario describes linear or exponential growth. a. Sarah has $650 in her pigg bank. She adds $50 each month. b. David has $400 in his bank. He earns 0% interest each ear. c. Cell division takes place as shown in the diagram. Each cell divides into a new cell ever hour. d. Miki collects souvenirs related to Michael Jackson. She allows herself to bu a new one ever months. She alread has 7 items.. Write an equation for a function that describes this relationship. Use function notation. Sarah has $650 in her pigg bank. She adds $50 each month.. Look at this graph = x What is the growth factor of the function = x between the values and? A B C 6 D x Copright 05 Pearson Education, Inc.

14 LESSON 4: COMPARING GROWTH 4. Look at this graph = x 4 What is the constant difference of the function A 4 4 B C D 4 = x 4? 4 5 x 5. Look at these two graphs = x = 4 x 4 5 x 4 5 x Examine the growth of each function from 0 to. Explain wh over time the growth of an exponential function will alwas exceed the growth of a linear function. Copright 05 Pearson Education, Inc. 4

15 LESSON 4: COMPARING GROWTH 6. Look at this graph x 5 0 What is the constant difference of this function? A B C.5 D.5 7. Draw the graph of = x. 8. Draw the graph of = 5 x. Challenge Problem 9. a. Draw the graphs of = 4x and = 4x in the same coordinate plane. b. What are the points of intersection for the graphs? c. Can ou prove the points of intersection algebraicall? Copright 05 Pearson Education, Inc. 5

16

17 LESSON 5: DOMAIN AND RANGE. Rosa throws a basketball. This graph shows the height of the ball as a function of the distance. Coordinate values are rounded to the nearest tenth. h (.5, 4.5) (0, ) (5., 0) d What is the domain of this function? A 0 d 4.5 B 0 d 5. C h 4.5 D h 5.. Rosa throws a basketball. This graph shows the height of the ball as a function of the distance. Coordinate values are rounded to the nearest tenth. h (.5, 4.5) (0, ) (5., 0) d What is the range of this function? A 0 d 4.5 B 0 d 5. C 0 h 4.5 D 0 h 5. Copright 05 Pearson Education, Inc. 7

18 LESSON 5: DOMAIN AND RANGE. Rosa throws a basketball. This graph shows the height of the ball as a function of the distance. h (.5, 4.5) (0, ) (5., 0) d The formula for this function is h = d + 0d +. What is the meaning of the + in the context of this basketball situation? A The height when the ball was released B The maximum height that the ball reaches C The minimum height that the ball falls D The distance from the hoop to where the ball was released 4. Jason starts recording cell division when there are alread cells. His results are shown in this table. t (hours) N(t) N(t) Explain wh in this context the domain can be extended to include t values greater than 4 but not less than Jason starts recording cell division when there are alread cells. His results are shown in this table. t (hours) N(t) N(t) Describe some of the laws of exponents ou can observe in this table. Copright 05 Pearson Education, Inc. 8

19 LESSON 5: DOMAIN AND RANGE 6. A growth process is described b the function f(x) = 8.5 x. What is the value for x = 0? A 0 B.7 C 8 D 7 7. A growth process is described b the function f(x) = 8.5 x. a. What is the value for x = 4? b. What is the value for x =? 8. A growth process is described b the function f(x) = 8.5 x. What is the domain of this function? 9. A growth process is described b the function f(x) = 8.5 x. What is the range of this function? Challenge Problem 0. The number of people living in Laos between 000 and 005 can be described b this formula with growth factor g. f(x) = 5.0 g x where x is time in ears and f(x) is measured in millions of people In 00 there were 5.4 million people, and b 00 the population grew to 5.0 million. Tr to predict the population of Laos in 05. Copright 05 Pearson Education, Inc. 9

20

21 LESSON 6: COMPARING GRAPHS OF FUNCTIONS. Which of these graphs represents a function? There ma be more than one correct graph. A 0 5 x = x 5 0 B 0 5 = log x x 5 0 C 0 5 x = x 5 0 (continues) Copright 05 Pearson Education, Inc.

22 LESSON 6: COMPARING GRAPHS OF FUNCTIONS (continued) D 0 5 x = x 5 0. Where does the function defined in Table A intersect the -axis? Table A x A = 0 B = C =.5 D It does not cross the -axis.. Where does the function defined in Table B intersect the -axis? Table B x A = 0 B = C =.5 D It does not cross the -axis. Copright 05 Pearson Education, Inc.

23 LESSON 6: COMPARING GRAPHS OF FUNCTIONS 4. Look at the functions defined in Tables A and B. Table A x Table B x a. Does the function in Table A increase b a constant amount over each unit interval? If so, state the constant amount. b. Does the function in Table B increase b a constant amount over each unit interval? If so, state the constant amount. 5. Here are two functions in the domain x, where x is an integer. f(x) = x + g(x) = x Fill in the table to compare the functions. x 0 f(x) g(x) 6. Here are two functions in the domain x, where x is all real numbers. f(x) = x + g(x) = x Use a graph tool to graph both functions in one coordinate sstem. a. Where in this domain is f(x) > g(x)? b. What is the range of each function for the domain of all real numbers? Copright 05 Pearson Education, Inc.

24 LESSON 6: COMPARING GRAPHS OF FUNCTIONS 7. Here are two functions in the domain 5 x 5, where x is all real numbers. f(x) = x + 4 g(x) = x Use a graph tool to graph both functions in one coordinate sstem. a. Where in this domain is f(x) > g(x)? b. What is the range for each function for the domain of all real numbers? c. Where do the functions intersect? 8. a. Find the domain and range of =. b. Explain wh x = is not a function. Challenge Problem 9. a. Explain wh functions like f(x) = x and g( x)= and range. x all have the same domain b. Explain wh functions like f(x) = x, g( x)= x, and h(x) = x all have the same domain and range. Copright 05 Pearson Education, Inc. 4

25 LESSON 7: THREE KINDS OF GROWTH. Match each situation with the tpe of function that models it. Tpe of Functions Linear Function Cubic Function Exponential Function Quadratic Function Situations The area of a square with a side length of x The speed of a car moving at constant speed, with respect to time The volume of a sphere with a radius of r The number of insects in a population that doubles each da. N(t) = 5, x This function is a function. A linear B quadratic C cubic D exponential. N(t) = 5, x a. What is the domain of this function? b. What is the range of this function? 4. x 4 0 f(x) This table represents a function. A linear C cubic B quadratic D exponential 5. x 4 0 f(x) a. What is the domain of this function? b. What is the range of this function? Copright 05 Pearson Education, Inc. 5

26 LESSON 7: THREE KINDS OF GROWTH 6. There were,00 members in a soccer club in 008, and 5 members leave each ear. This situation can be represented b a A linear C cubic function. B quadratic D exponential 7. There were,00 members in a soccer club in 008, and 5 members leave each ear. a. What is the domain of this function? b. What is the range of this function? 8. The change in the volume of the cube in relationship to the side length. This situation can be represented b a A linear C cubic function. B quadratic D exponential 9. This diagram shows the distance traveled b a ball ever of a second. 0 a. This situation can most likel be represented b a function A linear B quadratic C cubic D exponential b. Explain wh Copright 05 Pearson Education, Inc. 6

27 LESSON 7: THREE KINDS OF GROWTH Challenge Problem 0. This diagram shows the distance traveled b a ball ever of a second. 0 a. Sketch a graph of the function. 0 4 b. Compute the distance of the ball after sec Copright 05 Pearson Education, Inc. 7

28

29 LESSON 8: LINEAR AND EXPONENTIAL DECAY. Which of these statements are true about decreasing linear functions? There ma be more than one true statement. A The slope is a negative number. B The -intercept is positive. C The graph slopes downward from left to right. D The x-intercept is positive. E The slope is a fractional number less than.. Which of these statements are true about decreasing exponential functions in the form = a x? There ma be more than one true statement. A The growth factor, a, is greater than, and the exponent, x, is positive. B The -intercept is positive. C The graph slopes downward from left to right. D The growth factor, a, is less than, and the exponent, x, is positive. E The graph goes through (0, 0).. The half-life of Uranium-8 is an incredible 4.5 billion ears. The age of rock formations can be calculated using this fact. Look at this graph. Amount of Uranium-8 Remaining (%) ,500 Half-Life of Uranium-8 9,000,500 8,000,500 Time (millions of ears) Onl 70% of Uranium-8 remains in a certain piece of rock. Approximatel how old is the piece of rock? x Copright 05 Pearson Education, Inc. 9

30 LESSON 8: LINEAR AND EXPONENTIAL DECAY 4. The half-life of Uranium-8 is an incredible 4.5 billion ears. The age of rock formations can be calculated using this fact. Look at this graph. Amount of Uranium-8 Remaining (%) ,500 Half-Life of Uranium-8 9,000,500 8,000,500 Time (millions of ears) Explain what ou know about the formula for the function that represents this graph. x 5. Graph these functions using a graph tool. = x = x = x 6 = x 6 Explain the similarities and differences between our graphs. Include the terms domain, range, increase, and decrease in our explanation. 6. Graph these functions using a graph tool. = x = x Explain the similarities and differences between our graphs. Include the terms domain, range, increase, and decrease in our explanation. Copright 05 Pearson Education, Inc. 0

31 LESSON 8: LINEAR AND EXPONENTIAL DECAY 7. A bathtub has 5 gallons of water in it. It is empting at a constant rate of gallons per minute. Which equation represents this situation? A = t + 5 B = 5t C = t + 5 D = 5t 8. The population of a small town can be modeled b the equation P() = 4,65(0.97), where P is the number of people living in the small town and is the number of ears after 00. What kind of function represents this situation? A Increasing linear function B Decreasing linear function C Increasing exponential function D Decreasing exponential function 9. The population of a small town can be modeled b the equation P() = 4,65(0.97), where P is the number of people living in the small town and is the number of ears after 00. B what percent does the population of the small town decrease each ear according to this model? A % per ear B % per ear C 97% per ear D 4.65% per ear 0. The population of a small town can be modeled b the equation P() = 4,65(0.97), where P is the number of people living in the small town and is the number of ears after 00. What is the population in 00? Challenge Problem. A certain material decas at a rate of 50% per ear. If ou start with 0 grams of the material, how much will be left after 4 ears? Copright 05 Pearson Education, Inc.

32

33 LESSON 9: RECURSIVE RULES FOR SEQUENCES. Look at the following sequence., 7,, 5 Write the next three terms in this sequence.. Look at the following sequence., 7,, 5 What is the 0th term (n = 0) of this sequence? A B 5 C 9 D 49. Look at the following sequence., 7,, 5 What is the recursive rule for this sequence written in function notation? A f() = 0, f(n) = f(n ) + for n > B f() =, f(n) = f(n ) + 4 for n > C f() =, f(n) = f(n ) + 5 for n > D f() = 4, f(n) = f(n ) + for n > 4. Look at the following sequence. 7, 4,,, 9 Describe in words the recursive rule for this sequence. 5. Look at the following sequence. 7, 4,,, 9 Fill in the missing terms. How man are there? 6. Look at the following sequence. 7, 4,,, 9 Write the recursive rule for this sequence using function notation. Copright 05 Pearson Education, Inc.

34 LESSON 9: RECURSIVE RULES FOR SEQUENCES 7. Look at the following sequence. 40, 0, 0, 5, Describe in words the recursive rule for this sequence. A Start with 40, subtract 0 and multipl b, subtract 0 and multipl b, subtract 5 and multipl b. B Start with 0, add 40, and then multipl b. C Start with 40, then divide each term b to get the next term. D Start with 40, then multipl the last term b to get the next term. 8. Look at the following sequence. 40, 0, 0, 5, a. Write the first 0 terms of this sequence. b. Plot these values in the coordinate plane. 9. Look at the following sequence. 40, 0, 0, 5, Write the recursive rule for this sequence using function notation. 0. Look at the following sequence.,,, 4, 7,, Find the first 0 terms of this sequence. Challenge Problem. a. Look at the following sequence. 40, 0, 0, 5, Suppose f(n) gives the term value for term number n. What is the smallest n such that < f(n) <? b. Compare the sequence,,, 4, 7,, with the sequence n, where n is the term number. For what values of n is f(n) > n? Copright 05 Pearson Education, Inc. 4

35 LESSON 0: SEQUENCES. Tell if each of these sequences is an arithmetic sequence or a geometric sequence., 4, 7, 0,,,, 9, 7, 8, 8,, 6, 0, 6, 6, 5, 4, 0, 9, 6,, 7, 7, 7 9,. An arithmetic sequence starts with the number and increases b. Find the first five terms of this sequence.. An arithmetic sequence starts with the number and increases b. Write a recursive formula for finding the nth term. 4. A geometric sequence starts with 8, and the factor is. Find the first five terms of this sequence. 5. A geometric sequence starts with 8, and the factor is. Write a recursive function using function notation for finding the nth term. 6. A geometric sequence starts with 8, and the factor is. Find the first five terms of this sequence. 7. An arithmetic sequence starts with 7 and has a constant difference of. Find the first five terms of this sequence. 8. A geometric sequence has a first term (n = 0) of 6 and a constant (growth) factor of 4. What is the third term (n = ) of this sequence? A 4 B 6 C 48 D 96 Copright 05 Pearson Education, Inc. 5

36 LESSON 0: SEQUENCES 9. Which graph best matches this sequence? A, 5, 0, 7, 6 B Value Value 4 5 Term 4 5 Term C D Value Value 4 5 Term 4 5 Term Challenge Problem 0. A sequence of numbers is described as follows. Add the two previous numbers, and then multipl b. a. Write a recursive rule for this sequence. b. Determine the next three numbers in this sequence.,, 6,, Copright 05 Pearson Education, Inc. 6

37 LESSON : PUTTING IT TOGETHER. Read our Self Check and think about our work in this unit. Write down three things ou have learned during the unit. Share our answer with a classmate. Does our classmate understand what ou wrote?. Make a chart that summarizes each tpe of function ou have learned about. In our chart, include the general form of the function and a picture of what the graph of that functions looks like. Include arithmetic and geometric sequences in our chart. Set up our chart as follows. Tpe of Function General Form of the Equation General Form of the Graph. Review the notes ou took during the lessons about functions. Add an additional ideas ou have about functions including sequences to our notes. 4. Complete an exercises from this unit ou have not finished. Copright 05 Pearson Education, Inc. 7

38

39 MATH HIGH SCHOOL MODELING WITH FUNCTIONS FOR

40 LESSON : MODELING RELATIONSHIPS F.IF.. Graph A: Not a function Graph B: Not a function Graph C: Function F.IF.. Table A: Function C F.IF.. d( C) = π Table B: Not a function F.IF. 4. No, this graph is not a function. There is more than one -value for an x-value. Note: This graph shows the equation x = 4. F.IF. 5. A(r) = π r A(4) = π 4 A(4) = π6 A(4) = 6π F.IF. 6. ra ( )= A for A > 0 π F.IF. 7. f(x) = x 7 F.IF. 8. A The relationship between d and C is a function. D This table represents the relationship. d 4 C π π π 4π Challenge Problem F.IF. 9. U(0) = U() = U() = U() = 5 U(4) = 7 It is a linear function. Copright 05 Pearson Education, Inc. 40

41 LESSON : DIFFERENT WAYS OF GROWING F.IF.. C Exponential F.LE..b. A F.LE. F.LE.. x F.LE. F.LE. 4. x F.LE. 5. Function A reaches 00 at t = 5, while Function B is onl 80.5 when t = 5. Function B shows multiplicative growth, so it is an exponential function. Function A shows additive growth, so it is a linear function. Over time, an exponential function eventuall exceeds a linear function, but in this case, Function A reaches 00 quicker than Function B. F.IF.7.a 6. f(x) x Copright 05 Pearson Education, Inc. 4

42 LESSON : DIFFERENT WAYS OF GROWING F.IF.7.e 7. f(x) x F.LE. 8. The graphs show that the functions are equal somewhere between x =.5 and x = 6. The point of intersection is (.5, 6.05). Challenge Problem F.LE. 9. f(x) = x + g(x) = x Copright 05 Pearson Education, Inc. 4

43 LESSON 4: COMPARING GROWTH F.LE.. a. Sarah has $650 in her pigg bank. She adds $50 each month. Linear b. David has $400 in his bank. He earns 0% interest each ear. Exponential c. Cell division takes place as shown in the diagram. Each cell divides into a new cell ever hour. d. Miki collects souvenirs related to Michael Jackson. She allows herself to bu a new one ever months. She alread has 7 items. Exponential Linear F.LE.. f(x) = x, where x is time in months. F.LE..a. C 6 F.LE..a 4. A 4 F.LE. 5. In the exponential function = x, as x increases from 0 to, increases from to 9 or at a rate of change of 4. In the linear function = x, as x increases from 0 to, 4 increases from 0 to 6 4, or at a rate of change of 4. An exponential function will also exceed a linear function in growth over time because the linear function increases at a constant rate while the exponential function is influenced b the number taken to a power. F.LE..a 6. D.5 F.IF x Copright 05 Pearson Education, Inc. 4

44 LESSON 4: COMPARING GROWTH F.IF (5, 60) 00 (4, 80) x Challenge Problem F.IF.7.a 9. a. 5 = 4x 0 = 4x 5 x b. Points of intersection: (0, 0) and (, 4) c. This solution uses algebra ou might not have learned et. The point of intersection is where the two expressions are equal. 4x = 4x 4x 4x = 0 4x(x ) = 0 Thus: 4x = 0 or x = 0 x = 0 or x = If x = 0, then = 4x or = 4 0, = 0. If x =, then = 4x or = 4, = 4. The points of intersection are (0, 0) and (, 4). Copright 05 Pearson Education, Inc. 44

45 LESSON 5: DOMAIN AND RANGE F.IF.5. B 0 d 5. F.IF.5. C 0 h 4.5 F.F.4 N.Q.. A The height when the ball was released F.IF < N(t) < for t values less than 5; ou cannot have a fraction of a cell. F.IF.9 5. Here are some examples. ( ) = 8 0 = F.IF. 6. C 8 F.IF. 7. a. f(4) = = 9.5 b. f ( ) = 8 5. = 8 = 8 = F.IF.5 8. Domain: all real numbers F.IF.5 9. Range: = f(x) > 0 Challenge Problem F.IF Approximate g b using Then f(x) = 5.0 (.0) x f(5) = 5.0 (.0) 5 f(5) 7.8 The population of Laos would be around 7.8 million in 05. Copright 05 Pearson Education, Inc. 45

46 LESSON 6: COMPARING GRAPHS OF FUNCTIONS F.IF.. A 0 5 x = x 5 0 B 0 5 = log x x 5 0 F.IF.. B = F.IF.. B = F.IF. 4. a. In Table A, increases b a constant amount of over each unit interval of x (add ). b. In Table B, increases b a constant factor of over each unit interval of x (multipl b ). F.IF. 5. x f(x) g(x) Copright 05 Pearson Education, Inc. 46

47 LESSON 6: COMPARING GRAPHS OF FUNCTION F.IF f(x) a. In the domain x, f(x) is greater than g(x) for all values x > 0, and g(x) is greater than f(x) for all values x < b. The range of f(x) is all real numbers. The range of g(x) is all real numbers > 0. g(x) x 4 5 F.IF a. In the domain 5 x 5, f(x) is alwas greater than g(x). b. The range of f(x) is all real numbers. The range of g(x) is all real numbers < 0. c. The functions never intersect. 4 f(x) = x + 4 g(x) = x x F.IF. 8. a. The domain of the function = includes all real numbers, but the range is just ; is equal to for ever value of x included in the domain. Challenge Problem b. x = is not a function because its domain is just x =, and that single x-value generates a range that includes all real -values. Copright 05 Pearson Education, Inc. 47

48 LESSON 6: COMPARING GRAPHS OF FUNCTION F.IF. 9. a. The domain of the functions is all real numbers. The range is the same for both and cannot be less than 0, since the exponential term ields a smaller and smaller fraction as x becomes more and more negative. b. Linear functions like these have the same domain and range of all real numbers since each line continues infinitel in both the positive and negative -directions. Copright 05 Pearson Education, Inc. 48

49 LESSON 7: THREE KINDS OF GROWTH F.LE..b. Tpe of Functions Linear Function Cubic Function Exponential Function Quadratic Function Situations The area of a square with a side length of x The speed of a car moving at constant speed, with respect to time The volume of a sphere with a radius of r The number of insects in a population that doubles each da F.IF.. D exponential F.IF.. a. The domain is all real numbers. b. The range is > 0. F.IF. 4. B quadratic F.IF. 5. a. The domain is all real numbers. b. The range is 0. Note: The formula is f(x) = x. F.IF. 6. A linear F.IF. 7. a. Domain: 0 x 8. The domain is usuall not negative when ou are dealing with time. b. Range: 0,00 with being a whole number The range cannot be negative because the context is about people. Note: The formula is N(t) =,00 5t, where t is the number of ears after 008. F.IF. 8. C cubic Copright 05 Pearson Education, Inc. 49

50 LESSON 7: THREE KINDS OF GROWTH F.IF. 9. a. B quadratic b. When ou look at the numbers, ou see squares. Challenge Problem F.IF.7.a F.LE. 0. a. 75 Distance x Time b. = x, where x = of a second 0 sec will be 40 units. = 40 =,600 units The distance after sec is,600 units. Copright 05 Pearson Education, Inc. 50

51 LESSON 8: LINEAR AND EXPONENTIAL DECAY F.IF.4. A The slope is a negative number. C The graph slopes downward from left to right. F.IF.4. C The graph slopes downward from left to right. D The growth factor, a, is less than, and the exponent, x, is positive. F.IF.4 N.Q.. From the graph, the rock looks to be about,000 million, or billion, ears old. F.LE..c 4. The formula is a function that represents exponential deca. The exponent is negative and is multiplied b 00 (the -intercept). F.IF.7 5. All the functions are linear. The functions with 5 the negative constants, = x and = x 4 = x = x 6, are decreasing linear functions, while = x and = x 6 are increasing linear functions. The domain of all the = x 6 = x 6 functions is all real numbers, and the range of all the functions is all real numbers. = x and x = x are proportional relationships because the go through the point (0, 0). = x 6 and = x 6 both intersect the -axis at (0, 6). 4 5 F.IF.7 6. = x x = x Both functions are exponential. The function = x has a growth factor greater than and x is increasing. The function = has a growth factor between 0 and and is decreasing. The domain of both functions is all real numbers, and the range of both functions is all real numbers greater than 0. The both intersect the -axis at (0, ). Copright 05 Pearson Education, Inc. 5

52 LESSON 8: LINEAR AND EXPONENTIAL DECAY F.BF. 7. A = t + 5 F.LE.5 8. D Decreasing exponential function F.LE.5 A.SSE..c 9. B % per ear F.IF. 0. 4,65 Challenge Problem F.LE g or 7.5 g Copright 05 Pearson Education, Inc. 5

53 LESSON 9: RECURSIVE RULES FOR SEQUENCES F.LE.. 9,, 7 F.LE.. C 9 F.IF. F.IF. F.LE. F.IF. F.IF. F.LE.. B f() =, f(n) = f(n ) + 4 for n > 4. You start with 7 and ou add to the last term to get the next term. F.LE. 5. There are nine missing terms: 8, 5,,, 4, 7, 0,, 6 F.IF. F.IF. F.LE. 6. f() = 7, f(n) = f(n ) + for n > F.LE. 7. D Start with 40, then multipl the last term b to get the next term. F.IF. F.LE. 8. a. 40, 0, 0, 5,.5,.5, 0.65, 0.5, 0.565, b x Copright 05 Pearson Education, Inc. 5

54 LESSON 9: RECURSIVE RULES FOR SEQUENCES F.IF. F.IF. F.LE. 9. f() = 40, f( n) = f( n ) for n> F.IF. F.LE. 0.,,, 4, 7,, 4, 44, 8, 49 Challenge Problem F.IF. F.IF. F.LE.. a. n = 7; f(7) = 0.65 b. f(n) > n for n > 9 n n f(n) Copright 05 Pearson Education, Inc. 54

55 LESSON 0: SEQUENCES F.LE.., 4, 7, 0,, Arithmetic sequence,, 9, 7, 8, Geometric sequence 8,, 6, 0, 6, Arithmetic sequence 6, 5, 4, 0, 9, Arithmetic sequence 7 6,, 7,,, 7 9 Geometric sequence F.LE.., 5, 47, 59, 7, F.IF.. f() =, f(n) = f(n ) + for n > F.LE. 4. 8, 64,, 6, 8, F.IF. 5. f() = 8, f( n) = f( n ) for n> F.LE. 6. 8, 6,, 64, 8, F.LE. 7. 7, 69, 66, 6, 60,. F.LE. 8. D 96 F.LE. 9. C Value 4 5 Term Challenge Problem F.IF. 0. a. f() =, f() =, f(n) = [f(n ) + f(n )] for n > b. 8; 06;,6 Copright 05 Pearson Education, Inc. 55

56 LESSON : PUTTING IT TOGETHER F.IF.4 F.LE. F.LE.. Tpe of Function General Form of the Equation General Form of the Graph Linear f(x) = mx + b x Exponential f(x) = ab x x Quadratic f(x) = ax + bx + c x (continues) Copright 05 Pearson Education, Inc. 56

57 LESSON : PUTTING IT TOGETHER (continued) F.IF.4 F.LE. F.LE.. Tpe of Function General Form of the Equation General Form of the Graph Arithmetic Sequence Recursive Form: an = a n + d where d is the constant difference and a is the term x Geometric Sequence Recursive Form: B n = cr (n ) where r is the constant ratio and c is the initial value x Copright 05 Pearson Education, Inc. 57