Sequences and Functions

Transcription

1 Chapter 1 TABLE OF CONTENTS 1-1 Arithmetic Sequences 1- Geometric Sequences LAB Fibonacci Sequence 1-3 Other Sequences Mid-Chapter Quiz Focus on Problem Solving 1-4 Functions 1-5 Linear Functions 1-6 Eponential Functions 1-7 Quadratic Functions LAB Eplore Cubic Functions 1-8 Inverse Variation Problem Solving on Location Math-Ables Technology Lab Study Guide and Review Chapter 1 Test Performance Assessment Standardized Test Prep Sequences and Functions Growth Rates of E. coli Bacteria Doubling Time Conditions (min) Optimum temperature (30 C) and 0 growth medium Low temperature (below 30 C) 40 Low nutrient growth medium 60 Low temperature and low 10 nutrient growth medium Bacteriologist Bacteriologists study the growth and characteristics of microorganisms. They generally work in the fields of medicine and public health. Bacteria colonies grow very quickly. The rate at which bacteria multiply depends upon temperature, nutrient supply, and other factors. The table shows growth rates of an E. coli bacteria colony under different conditions. Chapter Opener Online go.hrw.com KEYWORD: MP4 Ch1 588 Chapter 1

2 Choose the best term from the list to complete each sentence. 1. An equation whose solutions fall on a line on a coordinate plane is called a(n)?.. When the equation of a line is written in the form y m b, m represents the? and b represents the?. 3. To write an equation of the line that passes through (1, 3) and has slope, you might use the? of the equation of a line. linear equation point-slope form slope -intercept y-intercept Complete these eercises to review skills you will need for this chapter. Number Patterns Find the net three numbers in the pattern , 3, 4,... 5., 3, 6, 11, 18, , 8, 5, , 1, 1,... Evaluate Epressions Evaluate each epression for the given values of the variables. 8. a (b 1)c for a 6, b 3, c 4 9. a b c for a, b 4, c 10. (ab) c for a 3, b, c 11. (a b) c for a 1, b 4, c 10 Graph Linear Equations Use the slope and the y-intercept to graph each line. 1. y y y y y y 5 Simplify Ratios Write each ratio in simplest form Sequences and Functions 589

3 1-1 Arithmetic Sequences Learn to find terms in an arithmetic sequence. Vocabulary sequence term arithmetic sequence common difference Joaquín received 5000 bonus miles for joining a frequent-flier program. Each time he flies to visit his grandparents, he earns 150 miles. The number of miles Joaquín has in his account is 650 after 1 trip, 7500 after trips, 8750 after 3 trips, and so on. After 1 trip After trips After 3 trips After 4 trips ,000 Difference Difference Difference 10, A sequence is a list of numbers or objects, called terms, in a certain order. In an arithmetic sequence, the difference between one term and the net is always the same. This difference is called the common difference. The common difference is added to each term to get the net term. EXAMPLE 1 You cannot tell if a sequence is arithmetic by looking at a finite number of terms, because the net term might not fit the pattern. Identifying Arithmetic Sequences Determine if each sequence could be arithmetic. If so, give the common difference. A 8, 13, 18, 3, 8, , Subtract each term from the term before it. The sequence could be arithmetic with a common difference of 5. B 1,, 4, 8, 16, , Chapter 1 Sequences and Functions Subtract each term from the term before it. The sequence is not arithmetic.

4 Determine if each sequence could be arithmetic. If so, give the common difference. C 100, 93, 86, 79, 7, , Subtract each term from the term before it. The sequence could be arithmetic with a common difference of 7. D 1, 3,, 5,3, 7,4, , Subtract each term from the term before it. The sequence could be arithmetic with a common difference of 1. E 5, 1, 3, 7, 11, , Subtract each term from the term before it. The sequence could be arithmetic with a common difference of 4. Subscripts are used to show the positions of terms in the sequence. The first term is a 1, the second is a, and so on. Suppose you wanted to know the 100th term of the arithmetic sequence 5, 7, 9, 11, 13,.... If you do not want to find the first 99 terms, you could look for a pattern in the terms of the sequence. Term Name a 1 a a 3 a 4 a 5 a 6 Term Pattern 5 0() 5 1() 5 () 5 3() 5 4() 5 5() The common difference d is. For the nd term, one is added to a 1. For the 3rd term, two s are added to a 1. The pattern shows that for each term, the number of s added is one less than the term number, or (n 1). The 100th term is the first term, 5, plus 99 times the common difference,. a () FINDING THE nth TERM OF AN ARITHMETIC SEQUENCE The nth term a n of an arithmetic sequence with common difference d is a n a 1 (n 1)d. 1-1 Arithmetic Sequences 591

5 EXAMPLE Finding a Given Term of an Arithmetic Sequence Find the given term in each arithmetic sequence. A 15th term: 5, 7, 9, 11,... B 3rd term: 5, 1, 17, 13,... a n a 1 (n 1)d a n a 1 (n 1)d a 15 5 (15 1) a 3 5 (3 1)( 4) a a 3 63 C 1th term: 9, 5, 1, 3,... D 0th term: a 1 3, d 15 a n a 1 (n 1)d a n a 1 (n 1)d a 1 9 (1 1)4 a 0 3 (0 1)15 a 1 35 a 0 88 You can use the formula for the nth term of an arithmetic sequence to solve for other variables. EXAMPLE 3 Travel Application Joaquín received 5000 bonus miles for signing up for an airline s frequent-flier program. He earns 150 miles each time he purchases a round-trip ticket to visit his grandparents. How many trips does he have to make to collect 5,000 frequent-flier miles? Identify the arithmetic sequence: 650, 7500, 8750,... a Let a frequent flier miles after first trip. d 150 a n 5,000 Let n represent the trip number in which Joaquín will have earned a total of 5,000 miles. Use the formula for arithmetic sequences. a n a 1 (n 1)d Solve for n. 5, (n 1)150 Distributive Property 5, n 150 Combine like terms. 5, n Subtract 5000 from both sides. 0, n Divide both sides by n After 16 trips, Joaquín will have collected 5,000 frequent-flier miles. Think and Discuss 1. Eplain how to determine if a sequence might be an arithmetic sequence.. Compare your answers for the 10th term of the arithmetic sequence 5, 7, 9, 11, 13,... by finding all of the first 10 terms and by using the formula. 59 Chapter 1 Sequences and Functions

6 1-1 Eercises FOR EXTRA PRACTICE see page 666 Homework Help Online go.hrw.com Keyword: MT4 1-1 See Eample 1 See Eample See Eample 3 GUIDED PRACTICE Determine if each sequence could be arithmetic. If so, give the common difference. 1. 4, 6, 8, 10, 1, , 1, 11, 9, 8, , 1 3, 4 9, 5 9, 3, , 9, 85, 78, 71, , 1 4, 1 8, 1 1, 16 3, , 6, 3, 0, 3,... Find the given term in each arithmetic sequence th term: 5, 7, 9, 11, th term:, 6, 10, 14, st term: 4, 8, 1, 16, th term: a 1 11, d Postage for a first-class letter costs $0.37 for the first ounce and $0.3 for each additional ounce. If a letter costs $1.5 to mail, how many ounces is it? See Eample 1 INDEPENDENT PRACTICE Determine if each sequence could be arithmetic. If so, give the common difference. 1. 1, 1, 1 1,, 1, ,, 1, 0, 1, , 3 8, 7 8, 1 1 8, 1 5 8, , 9, 5, 75, 98, , 1 1 5, 1 3 5,, 1, , 0.4, 0.7, 1, 1.3,... 5 See Eample See Eample 3 Find the given term in each arithmetic sequence th term: 5, 3, 1, 1, rd term: 0.1, 0.15, 0., th term: a 1 1, d 1. 18th term: a , d 3.5. Mariano received a bonus of $50 for working the day after Thanksgiving, plus his regular wage of $9.45 an hour. If his total wages for the day were $135.05, how many hours did he work? PRACTICE AND PROBLEM SOLVING Write the net three terms of each arithmetic sequence , 14, 17, 0, , 8,, 4, , 90, 79, 68, , 5 8, 3 4, 7, , 18, 30, 4, , 0.4, 0.3, 0.,... 8 Write the first five terms of each arithmetic sequence. 9. a 1 1, d a 1 3, d a 1 0, d a 1 100, d a 1 3, d a 1 6, d Arithmetic Sequences 593

7 35. The 5th term of an arithmetic sequence is 134. The common difference is 14. What are the first four terms of the arithmetic sequence? 36. The 1st term of an arithmetic sequence is 9. The common difference is 11. What position in the sequence is the term 163? 37. Julia s watch loses 5 minutes each day. At noon on Sunday, her watch read 11:55. Write the first four terms of an arithmetic sequence modeling the situation. (Assume a 1 11:55.) 38. RECREATION The rates for a mini grandpri course are shown in the flyer. a. What are the first 5 terms of the arithmetic sequence that represents the fees for the course? b. What would the rate be for 9 laps around the course? c. If the cost of a license plus n laps is $11, find n. 39. BUSINESS A law firm charges an administrative fee of $75, plus a $5.50 fee for each half hour of consultation. a. What are the first 4 terms of an arithmetic sequence that represents the rates of the law firm? b. How long was a consultation if the total bill came to $390? 40. WRITE ABOUT IT Eplain how to find the common difference of an arithmetic sequence. What can you say about the terms of a sequence if the common difference is positive? if the common difference is negative? 41. WRITE A PROBLEM Write an arithmetic sequence problem using a 7 15 and d CHALLENGE The 1st term of an arithmetic sequence is 4, and the common difference is 5. Find two consecutive terms of the sequence that have a sum of 103. What positions are the terms in the sequence? Solve each inequality. (Lesson 10-4) p p p 18 p c 11 14c d d 49. TEST PREP A right triangle has vertices at (0, 0), (4, 0), and (4, 10). What is the slope of the hypotenuse? (Lesson 5-5) A.5 B 0.4 C D Chapter 1 Sequences and Functions

8 1- Geometric Sequences Learn to find terms in a geometric sequence. Vocabulary geometric sequence common ratio Joey mows his family s yard every week. His mother offers him a choice of $10 per week, or 1 the first week, the second week, 4 the third week, and so on. Week 1 Week Week 3Week Ratio Ratio Ratio The weekly amounts Joey would get paid in this plan form a geometric sequence. In a geometric sequence, the ratio of one term to the net is always the same. This ratio is called the common ratio. The common ratio is multipied by each term to get the net term. EXAMPLE 1 Identifying Geometric Sequences Determine if each sequence could be geometric. If so, give the common ratio. A 96, 48, 4, 1, 6, , Divide each term by the term before it. The sequence could be geometric with a common ratio of 1. B 5, 5, 5, 5, 5, , Divide each term by the term before it. The sequence could be geometric with a common ratio of 1. C 5, 7, 9, 11, , Divide each term by the term before it. The sequence is not geometric. 1- Geometric Sequences 595

9 Determine if each sequence could be geometric. If so, give the common ratio. D 4, 6, 9, 13.5, 0.5, , Divide each term by the term before it. The sequence could be geometric with a common ratio of 1.5. Suppose you wanted to find the 15th term of the geometric sequence, 6, 18, 54, 16,.... If you do not want to find the first 14 terms, you could look for a pattern in the terms of the sequence. Term Name a 1 a a 3 a 4 a 5 a 6 Term Pattern (3) 0 (3) 1 (3) (3) 3 (3) 4 (3) 5 The common ratio r is 3. For the nd term, a 1 is multiplied by 3 once. For the 3rd term, a 1 is multiplied by 3 twice. The pattern shows that for each term, the number of times 3 is multiplied is one less than the term number, or (n 1). The 15th term is the first term,, times the common ratio, 3, raised to the 14th power. a 15 (3) 14 (4,78,969) 9,565,938 FINDING THE nth TERM OF A GEOMETRIC SEQUENCE The nth term a n of a geometric sequence with common ratio r is a n a 1 r n 1. EXAMPLE Finding a Given Term of a Geometric Sequence Find the given term in each geometric sequence. A 1th term: 6, 18, 54, 16,... B 57th term: 1, 1, 1, 1, 1,... r r a 1 6(3) 11 1,06,88 a 57 1( 1) 56 1 C 10th term: 5, 5, 5 4, 5 8, 5,... D 0th term: 65, 500, 400, 30, r 1 r a a 0 65(0.8) Chapter 1 Sequences and Functions

10 EXAMPLE 3 Money Application For mowing his family s yard every week, Joey has two options for payment: (1) $10 per week or () 1 the first week, the second week, 4 the third week, and so on, where he makes twice as much each week as he made the week before. If Joey will mow the yard for 15 weeks, which option should he choose? If Joey chooses $10 per week, he will get a total of 15($10) $150. If Joey chooses the second option, his payment for just the 15th week will be more than the total of all the payments in option 1. a 15 ($0.01)() 14 ($0.01)(16,384) $ Option 1 gives Joey more money in the beginning, but option gives him a larger total amount. Think and Discuss 1. Compare arithmetic sequences with geometric sequences.. Describe how you find the common ratio in a geometric sequence. 1- Eercises FOR EXTRA PRACTICE see page 754 Homework Help Online go.hrw.com Keyword: MP4 1- See Eample 1 GUIDED PRACTICE Determine if each sequence could be geometric. If so, give the common ratio. 1. 4,, 0,, 4,...., 6, 18, 54, 16, , 3, 3, 3, 3, , 1.5,.5, 3.375, , 3 6 8, 3 4, 3,... 6., 4, 8, 16,... See Eample Find the given term in each geometric sequence. 7. 1th term: 3, 6, 1, 4, 48, st term: 1 3, 1 3, 1 3, 1 3, 1 3, nd term: a 1 6,144, r th term: 1, 4, 16, 64, 56,... See Eample Heather makes $6.50 per hour. Every three months, she is eligible for a % raise. How much will she make after years if she gets a raise every time she is eligible? 1- Geometric Sequences 597

11 See Eample 1 See Eample INDEPENDENT PRACTICE Determine if each sequence could be geometric. If so, give the common ratio , 8, 4,, 1, , 1 8, 1 4, 1, , 6, 9, 1, , 384, 19, 96, , 3, 9, 7, 81, ,, 3, 9,... Find the given term in each geometric sequence th term: 1, 1,, 4, th term: a , r th term: a 1 1, r th term: 3, 6, 1, 4, nd term: 3, 6 1, 1 8 9,, th term: 1, 1.5,.5, 3.375,... See Eample 3 4. A tank contains 54,000 gallons of water. One-third of the water remaining in the tank is removed each day. How much water is left in the tank on the 15th day? PRACTICE AND PROBLEM SOLVING Find the net three terms of each geometric sequence. 5. a 1 4, common ratio 1 6. a 1 4, common ratio 1 7. a 1 8 1, common ratio 3 8. a 1 3, common ratio.5 Find the first five terms of each geometric sequence. 9. a 1 1, r a 1 5, r a 1 100, r a 1 64, r a 1 10, r a 1 64, r Find the 1st term of the geometric sequence with 6th term and common ratio. 36. Find the 3rd term of the geometric sequence with 7th term 56 and common ratio Find the 1st term of the geometric sequence with 5th term and common ratio Find the 1st term of a geometric sequence with 4th term 8 and common ratio. 39. Find the 5th term of a geometric sequence with 3rd term 8 and 4th term Find the 3rd term of a geometric sequence with 4th term 5400 and 6th term Find the 1st term of a geometric sequence with 3rd term 7 and 5th term Chapter 1 Sequences and Functions

12 4. ECONOMICS A car that was originally valued at $16,000 depreciates at 15% per year. This means that after each year, the car is worth 85% of its worth the previous year. What is the value of the car after 6 years? Round to the nearest dollar. 43. LIFE SCIENCE Under controlled conditions, a culture of bacteria doubles in size every days. How many cells of the bacteria are in the culture after weeks if there were originally 3 cells? 44. PHYSICAL SCIENCE A rubber ball is dropped from a height of 56 ft. After each bounce the height of the ball is recorded. Height of Bouncing Ball Number of Bounces Height (ft) a. Could the heights in the table form a geometric sequence? If so, what is the common ratio? b. Estimate the height of the ball after the 8th bounce. Round your answer to the nearest foot. 45. WRITE ABOUT IT Compare a geometric sequence with a 1 and r 3 with a geometric sequence with a 1 3 and r. 46. WHAT S THE ERROR? A student is asked to find the net three terms of the geometric sequence with a 1 10 and common ratio 5. His answer is, 5,. 5 What error has the student made, and what is the correct answer? 47. CHALLENGE The 5th term in a geometric sequence is 768. The 10th term is 786,43. Find the 7th term. Find the appropriate conversion factor. (Lesson 7-3) 48. meters to millimeters 49. quarts to gallons 50. gallons to pints 51. grams to centigrams 5. kilograms to grams 53. yards to inches 54. TEST PREP On a blueprint, a window is.5 inches wide. If the actual window is 85 inches wide, what scale factor was used to create the blueprint? (Lesson 7-5) 1 A 8 B 1 17 C 1 1 D Geometric Sequences 599

13 Fibonacci Sequence 1A Use with Lesson 1-3 Lab Resources Online go.hrw.com KEYWORD: MP4 Lab1A WHAT YOU NEED: Square tiles Activity 1 Use square tiles to model the following numbers: Place the first stack of tiles on top of the second stack of tiles. What do you notice? The first two stacks added together are equal in height to the third stack. 3 Place the second stack of tiles on top of the third stack of tiles. What do you notice? The second stack and the third stack added together are equal in height to the fourth stack. This sequence is called the Fibonacci sequence. By adding two successive numbers you get the net number in the sequence. The sequence will go on forever. Think and Discuss 1. If there were a term before the 1 in the sequence, what would it be? Eplain your answer.. Could the numbers 144, 33, 377 be part of the Fibonacci sequence? Eplain. Try This 1. Use your square tiles to find the net two numbers in the sequence. What are they?. The 18th and 19th terms of the Fibonacci sequence are 584 and What is the 0th term? 600 Chapter 1 Sequences and Functions

14 1-3 Other Sequences Learn to find patterns in sequences. Vocabulary first differences second differences Fibonacci sequence The first five triangular numbers are shown below To continue the sequence, you can draw the triangles, or you can look for a pattern. If you subtract every term from the one after it, the first differences create a new sequence. If you do not see a pattern, you can repeat the process and find the second differences. Term Triangular Number First differences Second differences First and second differences can help you find terms in some sequences. EXAMPLE 1 Using First and Second Differences Use first and second differences to find the net three terms in each sequence. A 1, 9, 4, 46, 75, 111, 154,... Sequence st Differences nd Differences The net three terms are 04, 61, 35. B 5, 5, 7, 13, 5, 45, 75,... Sequence st Differences nd Differences The net three terms are 117, 173, Other Sequences 601

15 By looking at the sequence 1,, 3, 4, 5,..., you would probably assume that the net term is 6. In fact, the net term could be any number. If no rule is given, you should use the simplest recognizable pattern in the given terms. EXAMPLE Finding a Rule, Given Terms of a Sequence Give the net three terms in each sequence using the simplest rule you can find. A 1, 1, 1 3, 1 4, 1 5,... One possible rule is to add 1 to the denominator of the previous term. This could be written as the algebraic rule a n 1 n. The net three terms are 1 6, 1 7, 1 8. B 1, 1,,, 3, 3,... Each positive term is followed by its opposite, and the net term is 1 more than the previous positive term. The net three terms are 4, 4, 5. C, 3, 5, 7, 11, 13, 17,... The rule for the sequence could be the prime numbers from least to greatest. The net three terms are 19, 3, 9. D 1, 4, 9, 16, 5, 36,... The rule for the sequence could be perfect squares. This could be written as the algebraic rule a n n. The net three terms are 49, 64, 81. Sometimes an algebraic rule is used to define a sequence. EXAMPLE 3 Finding Terms of a Sequence, Given a Rule n Find the first five terms of the sequence defined by a n n. 1 a a a a a The first five terms are 1, 3, 3 4, 4 5, Chapter 1 Sequences and Functions

16 A famous sequence called the Fibonacci sequence is defined by the following rule: Add the two previous terms to find the net term. 1, 1,, 3, 5, 8, 13, 1, EXAMPLE 4 Using the Fibonacci Sequence Suppose a, b, c, and d are four consecutive numbers in the Fibonacci sequence. Complete the following table and guess the pattern. a, b, c, d bc ad 1, 1,, 3 1() 1(3) 3 3, 5, 8, 13 5(8) 40 3(13) 39 13, 1, 34, 55 1(34) (55) , 89, 144, 33 89(144) 1,816 55(33) 1,815 The product of the two middle terms is either one more or one less than the product of the two outer terms. Think and Discuss 1. Find the first and second differences for the sequence of pentagonal numbers: 1, 5, 1,, 35, 51, 70, Eercises FOR EXTRA PRACTICE see page 754 Homework Help Online go.hrw.com Keyword: MP4 1-3 See Eample 1 See Eample GUIDED PRACTICE Use first and second differences to find the net three terms in each sequence. 1. 1, 7,, 46, 79, 11, 17,.... 5, 10, 30, 65, 115, 180, , 1, 15, 4, 4, 7, 117, , 8, 19, 48, 104, 196, 333,... Give the net three terms in each sequence using the simplest rule you can find. 5. 1, 3, 3 4, 4 5, 5 6, 6, , 6, 7, 8, 9, 10, 11, , 5, 6, 4, 5, 6, 4, , 8, 7, 64, 15, Other Sequences 603

17 See Eample 3 See Eample 4 Find the first five terms of each sequence defined by the given rule. 4n 9. a n n 10. a n (n )(n 3) 11. a n n 1 n 1. Suppose a, b, and c are three consecutive numbers in the Fibonacci sequence. Complete the following table and guess the pattern. a, b, c ac b 1, 1, 3, 5, 8 13, 1, 34 55, 89, 144 See Eample 1 See Eample See Eample 3 See Eample 4 INDEPENDENT PRACTICE Use first and second differences to find the net three terms in each sequence ,, 34, 47, 61, 76, , 11,, 1, 31, 55, , 6, 9, 35, 45, 60, 81, , 0.0, 0.08, 0.4, 0.55,... Give the net three terms in each sequence using the simplest rule you can find ,,, 3, 3, 3, 4, 4, 4, 4, 5, , 1, 4, 1, 5, 9, , 1.01, 1.001, , , 1 4, 1 9, 1 16, 1 1, 5 3,... 6 Find the first five terms of each sequence defined by the given rule. 1. a n n 1 3n. a n 1 n n(n 1) n 3. a n n 1 4. Suppose a, b, c, d, and e are five consecutive numbers in the Fibonacci sequence. Complete the following table and guess the pattern. a, b, c, d, e ae bd c 1, 1,, 3, 5 3, 5, 8, 13, 1 13, 1, 34, 55, 89 55, 89, 144, 33, 377 PRACTICE AND PROBLEM SOLVING The first 14 terms of the Fibonacci sequence are 1, 1,, 3, 5, 8, 13, 1, 34, 55, 89, 144, 33, Where in this part of the sequence are the even numbers? Where do you think the net four even numbers will occur? 6. Where in this part of the sequence are the multiples of 3? Where do you think the net four multiples of 3 will occur? 604 Chapter 1 Sequences and Functions

18 A 0.8 Hz Music Pitch is the frequency of a musical note, measured in units called hertz (Hz). The lower the frequency of a pitch, the lower it sounds, and the higher the frequency of a pitch, the higher it sounds. A pitch is named by its octave. A 4 is in the 4th octave on the piano keyboard and is often called middle A. 55 HZ 110 HZ A 1 A 7. What kind of sequence is represented by the frequencies of A 1, A, A 3, A 4,...? Write a rule to calculate these frequencies. 165 HZ E 8. What is the frequency of the note A 5, which is one octave higher than A 4? C # 3 0 HZ A 3 When a string of an instrument is played, its vibrations create many different frequencies at the same time. These varying frequencies are called harmonics. 75 HZ? HZ E 3 Frequencies of Harmonics on A HZ A 4 Harmonic Fundamental (1st) nd 3rd 4th 5th Note A 1 A E A 3 C # 3 9. What kind of sequence is represented by the frequencies of different harmonics? Write a rule to calculate these frequencies. 30. What is the frequency of the note E 3 if it is the 6th harmonic on A 1? 31. CHALLENGE In music an important interval is a fifth. As you progress around the circle of fifths, the pitch frequencies are approimately as shown (rounded to the nearest tenth). What type of sequence do the frequencies form in clockwise order from C? Write the rule for the sequence. If the rule holds all the way around the circle, what would the frequency of the note F be? B? Hz E? Hz A? Hz F? Hz D? Hz KEYWORD: MP4 Pitch C 65.4 Hz F# 745. Hz G 98.1 Hz B Hz D 147. Hz E 331. Hz? HZ A 5 A 6 A 7 A 8 Find the -intercept and y-intercept of each line. (Lesson 11-3) y y y y TEST PREP If y varies directly with and y 5 when 15, find the equation of direct variation. (Lesson 11-5) A y 3 5 B y 5 C y 15 D y Other Sequences 605

19 Chapter 1 Mid-Chapter Quiz LESSON 1-1 (pp ) Determine if each sequence could be arithmetic. If so, give the common difference , 11, 13, 16,.... 7, 4, 1, 18, ,, 33, 44, , 60, 103, 177,... Find the given term in each arithmetic sequence. 5. 8th term: 5, 8, 11, 14, th term: 7, 6.9, 6.8, th term: 9, 9 1 4, 9 1, th term: 8, 15,, 11, Frank deposited $5 in an account the first week. Each week, he deposits $5 more than the previous week. In which week will he deposit $100? LESSON 1- (pp ) Mid-Chapter Quiz Determine if each sequence could be geometric. If so, give the common ratio , 5, 5, 15, , 5, 1, 19, , 7, 9, 3, , 18, 5.4, 1.6,... Find the given term in each geometric sequence th term: 1, 36, 108, th term: 36, 1, 4, th term: 3, 3, 6, th term: 1000, 100, 10, The purchase price of a machine at a factory was $500,000. Each year, the value of the machine depreciates by 5%. To the nearest dollar, what is the value of the machine after 6 years? LESSON 1-3 (pp ) Find the first five terms of each sequence, given its rule. 19. a n 3n 5 0. a n n 1 1. a n ( 1) n 3n. a n (n 1) 1 Use first and second differences to find the net three terms in each sequence. 3. 9, 9, 11, 15, 1, , 10, 1, 36, 55, , 11, 13, 1, 8, , 4, 11,, 38, 60,... Give the net three terms in each sequence using the simplest rule you can find. 7. 1, 3 4, 5 6, , 8, 7, 64, Chapter 1 Sequences and Functions

20 Solve Eliminate answer choices When answering a multiple-choice question, you may be able to eliminate some of the choices. If the question is a word problem, check whether any answers do not make sense in the problem. Eample: Gabrielle has a savings account with $15 in it. Each week, she deposits $5 in the account. How much will she have in 1 weeks? A $65 B $185 C $14 D $190 The following sequence represents the weekly balance in dollars: 15, 130, 135, 140, 145, The amount will be greater than $15, so it cannot be A. It will also be a multiple of 5, so it cannot be C. Read each question and decide whether you can eliminate any answer choices before choosing an answer. Eplain your reasoning. 1 An art gallery has 400 paintings. Each year, the curator acquires 15 new paintings. How many paintings will the gallery have in 7 years? A 450 C 505 B 6000 D 95 There are 360 deer in a forest. The population increases each year by 10% over the previous year. How many deer will there be after 9 years? A 849 C 34 B 450 D Donna is in a book club. She has read 4 books so far, and she thinks she can read 3 books a week during the summer. How many weeks will it take for her to read a total of 60 books? A 0 weeks C 3 weeks B 1 weeks D 60 weeks Oliver has $30.00 in a savings account that earns 6% interest each year. How much will he have in 1 years? A $30.00 C $ B $ D $46.81 Focus on Problem Solving 607

21 1-4 Functions Input Domain Learn to represent functions with tables, graphs, or equations. Vocabulary function input output domain range function notation A function is a rule that relates two quantities so that each input value corresponds to eactly one output value. The domain is the set of all possible input values, and the range is the set of all possible output values. Function One input gives one output. 3 Function Range Output Not a Function One input gives more than one output. 16 y = y = Eample: The output is times the input. Eample: The outputs are the square roots of the input. Functions can be represented in many ways, including tables, graphs, and equations. If the domain of a function has infinitely many values, it is impossible to represent them all in a table, but a table can be used to show some of the values and to help in creating a graph. EXAMPLE 1 Finding Different Representations of a Function Make a table and a graph of y + 1. Make a table of inputs and outputs. Use the table to make a graph. 1 y y ( ) ( 1) 1 0 (0) (1) 1 O () Chapter 1 Sequences and Functions To determine if a relationship is a function, verify that each input has eactly one output.

22 EXAMPLE Identifying Functions Determine if each relationship represents a function. A y B y O Each input has only one output y. The relationship is a function. The input 0 has two outputs, y 3 and y 3. Other -values also have more than one y-value. The relationship is not a function. C y Make an input-output table and use it to graph y. y y ( ) ( 1) 1 0 (0) 0 1 (1) 1 O () 4 Each input has only one output y. The relationship is a function. f() is read f of. f(1) is read f of 1. You can describe a function using function notation. In function notation, the output value of the function f that corresponds to the input value is written as f(). The epression f() means the rule of f applied to the value of, not f multiplied by. y f() The output y is the rule of f applied to. f(1) 1 1 f(1) means evaluate f() for 1. EXAMPLE 3 Evaluating Functions For each function, find f(0), f(), and f( 1). A y 1 f() 1 Write in function notation. f(0) (0) 1 1 f() () 1 3 f( 1) ( 1) Functions 609

23 For each function, find f(0), f(), and f( 1). y B Read the graph to find y for each. O 4 6 f() y f(0) 4 f() f( 1) 5 C y Read the table to find y for each. f() y f(0) f() 8 f( 1) Think and Discuss 1. Give y in function notation.. Describe how to tell if a relationship is a function. 1-4 Eercises FOR EXTRA PRACTICE see page 755 Homework Help Online go.hrw.com Keyword: MP4 1-4 See Eample 1 See Eample GUIDED PRACTICE Make a table and a graph of each function. 1. y 4. y y 3 4. y 1 Determine if each relationship represents a function. 5. y 6. y 7. y Chapter 1 Sequences and Functions

24 See Eample 3 For each function, find f (0), f (3), and f ( 1). 8. y y 10. O y See Eample 1 See Eample INDEPENDENT PRACTICE Make a table and a graph of each function. 11. y 4 1. y 3( 1) 13. y (3 ) 14. y (1 ) Determine if each relationship represents a function. 15. y 16. y 17. y O 7 See Eample 3 For each function, find f(0), f(), and f( 3). 18. y 19. y y O PRACTICE AND PROBLEM SOLVING Give the domain and the range of each function. 1. y. y y y Functions 611

25 Home Economics In 1879, Thomas Edison used a carbonized piece of sewing thread to form a light bulb filament that lasted 13.5 hours before burning out. Today, a typical light bulb lasts more than 50 times that long. 5. HOME ECONOMICS The cost of using a 60-watt light bulb is given by the function f() The cost is in dollars, and represents the number of hours the bulb is lit. a. How much does it cost to use a 60-watt light bulb 8 hours a day for a week? b. What is the domain of the function? c. If the cost of using a 60-watt bulb was $1.98, for how many hours was it used? 6. BUSINESS The function f() gives the daily profit of a company if they manufacture items. The company can manufacture 50, 55, or 60 items per day. How many items should be manufactured to make the most daily profit? 7. SPORTS A speed skater trains by skating 1000 meters at a time. His coach recorded the distance covered by the skater every 0 seconds. The results are presented in the table. Time (s) Distance y (m) a. Does the relationship represent a function? b. What is the domain of the function? What is the range? c. Graph the data points to verify your answer from part a. 8. WHAT S THE QUESTION? The following set of points defines a function: {(3, 6), ( 4, 1), (5, 5), (9, 6), (10, ), (, 10)}. If the answer is 3, 4, 5, 9, 10, and, what is the question? 9. WRITE ABOUT IT Eplain how you can tell if a graph does not represent a function. 30. CHALLENGE Create a table of values for f () 1 using 3,, 1, 0.5, 0.5, 0.5, 0.5, 1,, and 3. Sketch the graph of the function. What happens when 0? Find each percent or number. (Lesson 8-) 31. What percent of 1 is 61? 3. What is 35% of 340? 33. What is 145% of 15? 34. What percent of 1193 is 477.? 35. What percent of 1.5 is 136? 36. What percent of 990 is 3960? 37. TEST PREP Thomas earns a weekly salary of $35 plus 8% commission on sales over $500. What would his weekly pay be if he had $650 in sales? (Lesson 8-6) A $695 B $735 C $640 D $ Chapter 1 Sequences and Functions

26 1-5 Linear Functions Learn to identify linear functions. Vocabulary linear function Elephant seals weigh about 100 pounds at birth. The mother s milk is so rich about 50% fat that the pup gains about 8 pounds per day while nursing. Weight of Elephant Seal Pup Day Weight (lb) Elephant seals are the largest seals. Adult males weigh an average of about 5000 pounds, and adult females weigh an average of about 1100 pounds. Notice that the weights form an arithmetic sequence with a common difference of 8. Also, the data can be plotted on a coordinate plane as a line with slope 8 and y-intercept 100. The graph of a linear function is a line. The linear function f() m b has a slope of m and a y-intercept of b. You can use the equation f() m b to write the equation of a linear function from a graph or table y 4 EXAMPLE 1 Writing the Equation for a Linear Function from a Graph Write the rule for the linear function. Use the equation f() m b. To find b, identify the y-intercept from the graph. O b 3 f() m ( 3) f() m 3 Locate another point on the graph, such as (1, ). Substitute the - and y-values of the point into the equation, and solve for m. f() m 3 m(1) 3 (, y) (1, ) m m The rule is f() 1 3, or f() 3. y 1-5 Linear Functions 613

27 EXAMPLE Writing the Equation for a Linear Function from a Table Write the rule for each linear function. A y The y-intercept can be identified from the table as b f(0) 7. Substitute the - and y-values of the point (1, 6) into the equation f() m 7, and solve for m. f() m 7 6 m(1) 7 6 m m The rule is f() 1 7, or f() 7. B y Use two points, such as (1, 7) and (, 4), to find the slope. y y1 m 4 ( 1 7) Substitute the - and y-values of the point (1, 7) into f() 3 b, and solve for b. f() 3 b 7 3(1) b (, y) (1, 7) 7 3 b b The rule is f() 3 ( 10), or f() EXAMPLE 3 Life Science Application An elephant seal weighs 100 pounds at birth and gains 8 pounds each day while nursing. Find a rule for the linear function that describes the growth of the pup, and use it to find out how much the pup will weigh after 3 days, when it will be weaned. f() m 100 The y-intercept is the birth weight, 100 pounds. 108 m(1) 100 At 1 day old, the pup will weigh 108 pounds. 108 m m The rule for the function is f() After 3 days, the pup s weight will be f(3) 8(3) pounds. Think and Discuss 1. Describe how to use a graph to find the equation of a linear function. 614 Chapter 1 Sequences and Functions

28 1-5 Eercises FOR EXTRA PRACTICE see page 755 Homework Help Online go.hrw.com Keyword: MP4 1-5 GUIDED PRACTICE See Eample 1 Write the rule for each linear function. 1. y. 4 y 4 O 4 O 4 See Eample 3. y 4. y See Eample 3 5. Kim earns $400 per week for 40 hours of work. If she works overtime, she makes $15 per overtime hour. Find a rule for the linear function that describes her weekly salary if she works hours of overtime, and use it to find how much Kim earns if she works 7 hours of overtime. INDEPENDENT PRACTICE See Eample 1 Write the rule for each linear function. 6. y 7. y 4 4 O 4 4 O 4 See Eample 8. y y See Eample A tank contains 100 gallons of water. The tank is being drained at a rate of 45 gallons per minute. Find a rule for the linear function that describes the amount of water in the tank, and use it to determine how much water will be in the tank after 15 minutes. 1-5 Linear Functions 615

29 Recreation PRACTICE AND PROBLEM SOLVING 11. RECREATION A hot air balloon at a height of 145 feet above sea level is ascending at a rate of 5 feet per second. a. Write a linear function that describes the balloon s height after seconds. b. What will the balloon s height be in 5 minutes? How high will it have climbed from its original starting point? 1. ECONOMICS Linear depreciation means that the same amount is subtracted each year from the value of an item. Suppose a car valued at $17,440 depreciates $1375 each year for years. a. Write a linear function for the car s value after years. b. What will the car s value be in 7 years? The volume of a typical hot air balloon is between 65,000 and 105,000 cubic feet. Most hot air balloons fly at altitudes of 1000 to 1500 feet. KEYWORD: MP4 Balloons 13. LIFE SCIENCE Suppose a puppy was born weighing 4 pounds, and it gained about 3 pounds each month during the first year. Find a rule for the linear function that describes the puppy s growth and use it to find out how much the puppy would weigh after 8 months. 14. BUSINESS The table shows a retailer s cost for certain items and the price at which the retailer sells each item. Dealer Cost $15 $ $30.50 $40 Selling Price $19.50 $8.60 $39.65 $5 a. Write a linear function for the selling price of an item that costs the retailer dollars. b. What would the selling price of a television be that costs the retailer $65? 15. WRITE ABOUT IT Eplain how you can determine whether a function is linear. 16. WHAT S THE QUESTION? Consider the function f() 3 9. If the answer is 6, what is the question? 17. CHALLENGE What is the only kind of line on a coordinate plane that is not a linear function? Give an eample of such a line. Find the point-slope form of each equation. (Lesson 11-4) 18. slope 5; passes through point (4, 1) 19. slope ; passes through point (6, 6) 0. slope 4; passes through point (0, 1) 1. slope 1.4; passes through point (1, 3). TEST PREP Which of the following ordered pairs is not a solution of the inequality 5 13y 61? (Lesson 11-6) A (1, 6) B (0, 0) C ( 4, 3) D (6, 10) 616 Chapter 1 Sequences and Functions

30 Learn to identify and graph eponential functions. Vocabulary eponential decay 1-6 Eponential Functions eponential function eponential growth Do you think you will live to be 100? According to U.S. census data, the number of Americans over 100 nearly doubled from about 37,000 in 1990 to more than 70,000 in 000. Number (thousands) Americans Over Year Suppose the number of Americans over 100 doubles each decade. The populations would form a geometric sequence with a common ratio of. An eponential function has the form f() p a, where a 0 and a 1. If the input values are the set of whole numbers, the output values form a geometric sequence. The y-intercept is f(0) p. The epression a is defined for all values of, so the domain of f() p a is all real numbers. 000 Population of Americans over 100 Year Population (thousands) EXAMPLE 1 Graphing an Eponential Function Create a table for each eponential function, and use it to graph the function. A f() 1 B f() 1 y y y y 1 1 O Eponential Functions 617

31 If a 1, the output f() gets larger as the input gets larger. In this case, f is called an eponential growth function. EXAMPLE In algebra, you will learn the meaning of epressions like 9.5. You can use a calculator to evaluate these epressions. Using an Eponential Growth Function The number of Americans over 100 was about 70,000 in 000. If the population of Americans over 100 doubles each decade, estimate the population of Americans over 100 in the year 095. Year Number of Decades Population f() (thousands) f() p a f() 70 a f(0) p f() 70 f(1) 70 a 1 140, so a. The year 095 is 9.5 decades after the year 000, so let 9.5. f(9.5) ,685 Substitute 9.5 for. If the population over 100 doubles each decade, there will be 50,685,000 Americans over 100 in 095. In the eponential function f() p a, if a 1, the output gets smaller as gets larger. In this case, f is called an eponential decay function. EXAMPLE 3 Physical Science Using an Eponential Decay Function Technetium-99m has a half-life of 6 hours, which means it takes 6 hours for half of the substance to decompose. Find the amount of technetium-99m remaining from a 100 mg sample after 90 hours. Hours Number of Half-lives Technetium-99m f() (mg) Technetium-99m is used to diagnose diseases in humans and animals. f() p a f() 100 a f(0) p f() f(1) 100 a 1 50, so a = 1. Divide 90 hours by 6 hours to find the number of half-lives: 15. f(15) Substitute 15 for. There is approimately mg left after 90 hours. Think and Discuss 1. Compare the graphs of eponential growth and decay functions. 618 Chapter 1 Sequences and Functions

32 1-6 Eercises FOR EXTRA PRACTICE see page 755 Homework Help Online go.hrw.com Keyword: MP4 1-6 GUIDED PRACTICE See Eample 1 Create a table for each eponential function, and use it to graph the function. 1. f() 3. f() f() 3 4. f() See Eample See Eample 3 5. At the beginning of an eperiment, a bacteria colony has a mass of 10 6 grams. If the mass of the colony doubles every 10 hours, what will the mass of the colony be after 80 hours? 6. Radioactive glucose is used in cancer detection. It has a half-life of 100 minutes. How much of a 100 mg sample remains after 4 hours? See Eample 1 See Eample See Eample 3 INDEPENDENT PRACTICE Create a table for each eponential function, and use it to graph the function. 7. f() 3 8. f() (0.) 9. f() f() Mariano invested $500 in an account that will double his balance every 8 years. Write an eponential function to calculate his account balance. What will his balance be in 3 years? 1. Cesium-137 is a radioactive element with a half-life of 30 years. It is used to study upland soil erosion. How much of a 50 mg sample of cesium-137 would remain after 180 years? PRACTICE AND PROBLEM SOLVING For each eponential function, find f( 5), f(0), and f(5). 13. f() 14. f() f() f() 00 1 Write the equation of the eponential function that passes through the given points. Use the form f() p a. 17. (0, 3) and (1, 6) 18. (0, 4) and (1, ) 19. (0, 1) and (, 9) Graph the eponential function of the form f() p a. 0. p 6, a 5 1. p 1, a 1. p 100, a Carbon-14 is used by archaeologists to find the approimate age of animal and plant material. It has a half-life of 5730 years. What percent of a sample remains after 34,380 years? 1-6 Eponential Functions 619

33 Health The half-life of a substance in the body is the amount of time it takes for your body to metabolize half of the substance. An eponential decay function can be used to model the amount of the substance in the body. Acetaminophen is the active ingredient in many pain and fever medications. Use the table for Eercises 4 6. Acetaminophen Levels in the Body Elapsed Time (hr) Substance Remaining (mg) How much acetaminophen was present initially? 5. Find the half-life of acetaminophen. Write an eponential function that describes the level of acetaminophen in the body. a. How much acetaminophen will be present after 1 hours? b. How much acetaminophen will be present after 1 day? Vitamin deficiencies can cause serious diseases, such as scurvy, rickets, and beriberi. 6. If you take 500 mg of acetaminophen, what percent of that amount will be in your system after 9 hours? 7. The half-life of vitamin C is about about 6 hours. If you take a 500 mg vitamin C tablet at 9:00 AM, how much of the vitamin will still be present in your system at 9:00 PM? 8. Caffeine has a half-life of about 5 hours in adults. Two 6 oz cups of coffee contain about 00 mg caffeine. If an adult drinks cups of coffee, how much caffeine will be in his system after 1 hours? 9. CHALLENGE In children, the half-life of caffeine is about 3 hours. If a child has a 1 oz soft drink containing 40 mg caffeine at 1:00 PM and another at 6:00 PM, about how much caffeine will be present at 10:00 PM? Sources of caffeine include coffee, tea, and some pain medications. Determine if each sequence could be geometric. If so, give the common ratio. (Lesson 1-) 30. 5, 10, 15, 0, 5, , 6, 1, 4, 48, , 3, 9, 7, 81, , 0., 0.3, 0.4, , 4, 4, 4, 4, , 0.01, 0.001, , TEST PREP The function f() 1, gives the value of a car (in dollars) years after it was purchased. What is the car s value 8 years after it was purchased? (Lesson 1-5) A $600 B $7300 C $4000 D $ Chapter 1 Sequences and Functions

34 1-7 Quadratic Functions Learn to identify and graph quadratic functions. Vocabulary quadratic function parabola A quadratic function contains a variable that is squared. In the quadratic function f() a b c the y-intercept is c. The graphs of all quadratic functions have the same basic shape, called a parabola. The cross section of the large mirror in a telescope is a parabola. Because of a property of parabolas, starlight that hits the mirror is reflected toward a single point, called the focus. The mirror of this telescope is made of liquid mercury that is rotated to form a parabolic shape. EXAMPLE 1 Quadratic Functions of the Form f() a b c Create a table for each quadratic function, and use it to make a graph. A f() f() 3 ( 3) 7 ( ) 1 ( 1) 1 0 (0) 1 (1) 1 () 3 (3) Plot the points and connect them with a smooth curve. y 4 B f() f() 3 ( 3) ( 3) 4 ( ) ( ) 0 1 ( 1) ( 1) 0 (0) 0 4 O 4 1 (1) 1 0 () 4 3 (3) 3 10 Plot the points and connect them with a smooth curve. 4 y 1-7 Quadratic Functions 61

35 You may recall that when a product ab is 0, either a must be 0 or b must be zero. 0( 0) 0 100(0) 0 You can use this knowledge to find intercepts of functions. Eample: f () ( 5)( 8) The product is 0 when 5 or when 8. (5 5)(5 8) 0 (8 5)(8 8) 0 Some quadratic functions can be written in the form f () ( r)( s). Although the variable does not appear to be squared in this form, the is multiplied by itself when the epressions in parentheses are multiplied together. EXAMPLE The -intercepts are where the graph crosses the -ais. Quadratic Functions of the Form f() ( r)( s) Create a table for each quadratic function, and use it to make a graph. A f() ( 3)( 1) The parabola crosses the -ais at 1 and 3. f() ( 3)( 1) 3 ( 3 3)( 3 1) 4 ( 3)( 1) 15 1 ( 1 3)( 1 1) 8 0 (0 3)(0 1) 3 1 (1 3)(1 1) 0 ( 3)( 1) 1 3 (3 3)(3 1) O Plot the points and connect them with a smooth curve. y 4 B f() ( )( 1) The parabola crosses the -ais at 1 and. f() ( )( 1) 3 ( 3 )( 3 1) 10 ( )( 1) 4 1 ( 1 )( 1 1) 0 0 (0 )(0 1) 1 (1 )(1 1) ( )( 1) 0 3 (3 )(3 1) O Plot the points and connect them with a smooth curve. y 4 6 Chapter 1 Sequences and Functions

36 EXAMPLE 3 Astronomy Application In a liquid mirror, a container of liquid mercury is rotated around an ais. Gravity and centrifugal force cause the liquid to form a parabolic shape. The cross section of a liquid mirror that rotates at 10 revolutions per minute is approimated by the graph of f() If the diameter of the mirror is 3 m, about how much higher are the sides than the center? Spinning mercury forms a parabolic surface. First graph the cross section. Create a table of values. f() 0.07( ) ( 1) (0) (1) () y m O 1 The center of the mirror is at 0, and the height is 0 m. If the diameter of the mirror is 3 m, the highest point on the sides is at 1.5, and the height is f(1.5) 0.07(1.5) 0.06 m. The sides are about 0.06 m higher than the center. Think and Discuss 1. Compare the graphs of f() and f() 1.. Describe the shape of a parabola. 1-7 Quadratic Functions 63

37 1-7 Eercises FOR EXTRA PRACTICE see page 755 Homework Help Online go.hrw.com Keyword: MP4 1-7 See Eample 1 See Eample See Eample 3 GUIDED PRACTICE Create a table for each quadratic function, and use it to make a graph. 1. f() 4. f() 3 3. f().5 4. f() f() ( )( 3) 6. f() ( 4)( 1) 7. f() ( 1)( 5) 8. f() ( 6)( ) 9. The function f(t) 0.15t.4t 5.1 gives the height in feet of a baseball t seconds after it was thrown. What was the height of the baseball when it was initially thrown (t 0)? See Eample 1 See Eample See Eample 3 INDEPENDENT PRACTICE Create a table for each quadratic function, and use it to make a graph. 10. f() f() 1. f() f() f() ( 1)( 1) 15. f() ( 1.5)( 3) 16. f() ( ) 17. f() ( 3)( 7) 18. The function f() ,450 gives the cost of manufacturing items per day. Which number of items will give the lowest cost per day, 50, 70, or 85? What will the cost be? PRACTICE AND PROBLEM SOLVING Find f( 3), f(0), and f(3) for each quadratic function. 19. f() 5 0. f() 1 1. f(). f() ( 3)( 3) 3. f() 5 4. f() 1 3 Find the -intercepts of each quadratic function. 5. f() ( 5)( 11) 6. f() ( 1)( 6) 7. f() ( )( 1) 8. f() ( 7) 9. f() ( 1.8)(.6) 30. f() ( )( 7) The sum of two numbers is 10. The sum of their squares is given by the function f() (10 ). Create a table of values for f(), using 3, 4, 5, 6, and 7. Which pair of numbers gives the least sum of squares? What is the sum of their squares? 64 Chapter 1 Sequences and Functions

38 3. PHYSICAL SCIENCE The height of a toy rocket launched straight up with an initial velocity of 48 feet per second is given by the function f(t) 48t 16t. The time t is in seconds. a. Graph the function for t 0, 0.5, 1, 1.5,,.5, and 3. b. When is the rocket at its highest point? What is its height? c. How many seconds does it take for the rocket to land? 33. BUSINESS A store owner can sell 30 digital cameras a week at a price of $150 each. For every $5 drop in price, she can sell more cameras a week. If is the number of $5 price reductions, the weekly sales function is Predicted Sales f() (30 )(150 5). Price $150 $145 $140 a. Find f() for 3, 4, 5, 6, Number Sold and 7. How many $5 price Weekly Sales $4500 $4640 $4760 reductions will result in the highest weekly sales? b. What will the price of a camera be in part a? 34. HOBBIES The height of a model airplane launched from the top of a 4 ft hill is given by the function f(t) 0.08t.6t 4. Find f(40). What does this tell you about t 40 seconds? 35. WRITE ABOUT IT Which will grow faster as gets larger, f() or f()? Check by testing each function for several values of. 36. CHOOSE A STRATEGY Suppose the function f() gives a company s profit for producing items. How many items should be produced to maimize profit? A 0 B 5 C 30 D CHALLENGE Create a table of values for the quadratic function f() ( 1), and then graph it. What are the -intercepts of the function? Write the slope and y-intercept of each equation. (Lesson 11-3) 38. y y y y y y y y 46. TEST PREP The 4th term of an arithmetic sequence is 10. The common difference is 5. What is the 1st term of the sequence? (Lesson 1-1) A 5 B 15 C 0 D TEST PREP The 4th term of a geometric sequence is The common ratio is 1.5. What is the 1st term of the sequence? (Lesson 1-1) F 1.5 G 5.65 H 3 J Quadratic Functions 65

39 Eplore Cubic Functions 1B Use with Lesson 1-7 You can use your graphing calculator to eplore cubic functions. To graph the cubic equation y 3 in the standard graphing calculator window, press Y= ; enter the right side of the equation, X,T,,n 3; and press ZOOM 6:ZStandard. Notice that the graph goes from the lower left to the upper right and crosses the -ais once, at 0. Lab Resources Online go.hrw.com KEYWORD: MP4 Lab1B Activity 1 1 Graph y 3. Describe the graph. Press 3. Y=, and enter the right side of the equation, The graph goes from the upper left to the lower right and crosses the -ais once. Graph y 3 3. Describe the graph. Press Y= ; enter the right side of the equation, X,T,,n 3 3 X,T,,n ; and press ZOOM 6:ZStandard. The graph goes from the lower left to the upper right and crosses the -ais three times. ( ) X,T,,n Think and Discuss 1. How does the sign of the 3 term affect the graph of a cubic function?. How could you find the value of 7 3 from the graph of y 3? Try This Graph each function and describe the graph. 1. y 3. y y ( ) 3 4. y Chapter 1 Sequences and Functions

40 Activity 1 Compare the graphs of y 3 and y 3 3. Graph Y 1 X^3 and Y X^3 3 on the same screen, as shown. Use the TRACE button and the and buttons to trace to any integer value of. Then use the and keys to move from one function to the other to compare the values of y for both functions for the value of. You can also press TABLE nd GRAPH to see a table of values for both functions. The graph of y 3 3 is translated up 3 units from the graph of y 3. Compare the graphs of y 3 and y ( 3) 3. Graph Y 1 X^3 and Y (X 3)^3 on the same screen. Notice that the graph of y ( 3) 3 is the graph of y 3 moved left TABLE 3 units. Press nd GRAPH to see a table of values. The graph of y ( 3) 3 is translated left 3 units from the graph of y 3. 3 Compare the graphs of y 3 and y 3. Graph Y 1 X^3 and Y X^3 on the same screen. Use the TRACE button and the arrow keys to see the values of y for any value TABLE of. Press nd GRAPH to see a table of values. The graph of y 3 is stretched upward from the graph of y 3. The y-value for y 3 increases twice as fast as it does for y 3. The table of values is shown. Think and Discuss 1. What function would translate y 3 right 6 units?. Do you think that the methods shown of translating a cubic function would have the same result on a quadratic function? Eplain. Try This Compare the graph of y 3 to the graph of each function. 1. y 3. y ( 7) 3 3. y y 5 3 Technology Lab 67

41 1-8 Inverse Variation 55Hz Learn to recognize inverse variation by graphing tables of data. Vocabulary inverse variation The frequency of a piano string is related to its length. You can double a string s frequency by placing your finger at the halfway point of the string. The lowest note on the piano is A 1. As you place your finger at various fractions of the string s length, the frequency will vary inversely. 110Hz 0Hz 440Hz Full length: 55 Hz 1 the length: 110 Hz 1 4 the length: 0 Hz The fraction of the string length times the frequency is always 55. INVERSE VARIATION Words Numbers Algebra An inverse variation is a relationship in which one variable quantity increases as another variable quantity decreases. The product of the variables is a constant. y 1 0 y 10 y k y k EXAMPLE 1 To determine if a relationship is an inverse variation, check if the product of and y is always the same number. Identifying Inverse Variation Tell whether each relationship is an inverse variation. A The table shows the number of days needed to construct a building based on the size of the work crew. Crew Size Days of Construction (9) 180; 10(18) 180; 5(36) 180; 3(60) 180; (90) 180 y 180 The product is always the same. The relationship is an inverse variation: y B The table shows the number of chips produced in a given time. Chips Produced Time (min) Chapter 1 Sequences and Functions 36(3) 108; 60(5) 300 The product is not always the same. The relationship is not an inverse variation. INVERSE VARIATION

42 In the inverse variation relationship y k, where k 0, y is a function of. The function is not defined for 0, so the domain is all real numbers ecept 0. EXAMPLE Graphing Inverse Variations Graph each inverse variation function. A f() 1 B f() y y y y O O EXAMPLE 3 Music Application The frequency of a piano string changes according to the fraction of its length that is allowed to vibrate. Find the inverse variation 1 function, and use it to find the resulting frequency when 1 6 of the string A 1 is allowed to vibrate. Frequency of A 1 by Fraction of the Original String Length Frequency (Hz) Fraction of the Length You can see from the table that y 55(1) 55, so y If the string is reduced to 1 of its length, then its frequency will be y Hz. 6 Think and Discuss 1. Identify k in the inverse variation y 3.. Describe how you know if a relationship is an inverse variation. 1-8 Inverse Variation 69

43 1-8 Eercises FOR EXTRA PRACTICE see page 755 Homework Help Online go.hrw.com Keyword: MP4 1-8 See Eample 1 GUIDED PRACTICE Tell whether each relationship is an inverse variation. 1. The table shows the number of CDs produced in a given time. CDs Produced Time (min) The table shows the construction time of a wall based on the number of workers. Construction Time (hr) Number of Workers See Eample See Eample 3 Graph each inverse variation function. 3. f() 3 4. f() 5. f() 1 6. Ohm s law relates the current in a circuit to the resistance. Find the inverse variation function, and use it to find the current in a 1-volt circuit with 9 ohms of resistance. Current (amps) Resistance (ohms) See Eample 1 INDEPENDENT PRACTICE Tell whether each relationship is an inverse variation. 7. The table shows the time it takes to throw a baseball from home plate to first base depending on the speed of the throw. Speed of Throw (ft/s) Time (s) The table shows the number of miles jogged in a given time. Miles Jogged Time (min) See Eample See Eample 3 Graph each inverse variation function. 9. f() f() f() 3 1. According to Boyle s law, when the volume of a gas decreases, the pressure increases. Find the inverse variation function, and use it to find the pressure of the gas if the volume is decreased to 4 liters. Volume (L) Pressure (atm) Chapter 1 Sequences and Functions

44 PRACTICE AND PROBLEM SOLVING Find the inverse variation equation, given that and y vary inversely. 13. y when 14. y 10 when 15. y 8 when If y varies inversely with and y 7 when 3, find the constant of variation. 17. The height of a triangle with area 50 cm varies inversely with the length of its base. If b 5 cm when h 4 cm, find b when h 10 cm. 18. PHYSICAL SCIENCE If a constant force of 30 N is applied to an object, the mass of the object varies inversely with its acceleration. The table contains data for several objects of different sizes. Mass (kg) Acceleration (m/s ) a. Use the table to write an inverse variation function. b. What is the mass of an object if its acceleration is 15 m/s? 19. FINANCE Mr. Anderson wants to earn $15 in interest over a -year period from a savings account. The principal he must deposit varies inversely with the interest rate of the account. If the interest rate is 6.5%, he must deposit $1000. If the interest rate is 5%, how much must he deposit? 0. WRITE ABOUT IT Eplain the difference between direct variation and inverse variation. 1. WRITE A PROBLEM Write a problem that can be solved using inverse variation. Use facts and formulas from your science book.. CHALLENGE The resistance of a 100 ft piece of wire varies inversely with the square of its diameter. If the diameter of the wire is 3 in., it has a resistance of 3 ohms. What is the resistance of a wire with a diameter of 1 in.? For each function, find f( 1), f(0), and f(1). (Lesson 1-4) 3. f() f() f() 3( 9) 6. f() f() ( 5)( 7) 8. f() TEST PREP The half-life of a particular radioactive isotope of thorium is 8 minutes. If 160 grams of the isotope are initially present, how many grams will remain after 40 minutes? (Lesson 1-6) A 10 grams B.5 grams C 5 grams D 1.5 grams 1-8 Inverse Variation 631

45 NASA Alabama NASA Marshall Space Flight Center At NASA s Marshall Space Flight Center in Huntsville, Alabama, scientists work on the development of the International Space Station. One area of research that scientists at the Marshall Center specialize in is microgravity. Microgravity researchers try to minimize the effects of gravity in order to simulate the zero gravity of space. To find the distance d in meters that a free-falling object travels in t seconds with no air resistance, you would use the function d 1 gt. In this distance function, g is the gravitational constant. On Earth, this constant is g 9.8 m/s. 1. What is the domain of the function d 1 gt? What is the range?. Graph d 1 gt. 3. In a microgravity eperiment, NASA scientists recorded that it took 4.5 seconds for an object to fall 100 meters. Find the gravitational constant g in the eperiment. NASA s KC-135 aircraft, referred to as the Weightless Wonder or the Vomit Comet, is used to create a microgravity environment. 4. While the KC-135 is climbing at a 45 angle, the equation of its path is y. While it is descending at a 45 angle, the equation of its path is y. Are these linear or quadratic functions? 5. While the KC-135 is in a microgravity environment, the equation of its path is y. Is this function linear or quadratic? Altitude (ft) 34,000 3,000 30,000 8,000 6,000 4,000 NASA Flight Path of KC Time (s) NASA 63 Chapter 1 Sequences and Functions

46 Muscle Shoals Located on the Tennessee River, Muscle Shoals was once the site of the Muscle Shoals Canal. When the canal was built in the 1830s, its purpose was to connect Colbert and Lauderdale Counties with a passageway that was easy to travel. During later attempts to improve the canal, dams were built to control its water flow. Around 194, two of these dams, Wilson Dam and Wheeler Dam, flooded the canal and created the lakes that are known today as Wilson Lake and Wheeler Lake. When a lake s water level gets too low, a dam s floodgates can be opened to allow water to enter the lake. For 1 3, use the table 1. What kind of sequence is formed by the total amount of water released after each second at 8 A.M. from Wheeler Dam? What is a possible rule for this sequence?. Write a possible rule for the sequence formed by the total amount of water released after each second at 9 A.M. from Wilson Dam. If the pattern continues, what will the total amount of water released be after 6 seconds? 3. Suppose water was released from Wheeler Dam at 3 A.M. at a rate of 1000 cubic feet per second, at 4 A.M. water was released at a rate of 100 cubic feet per second, at 5 A.M. water was released at a rate of 1440 cubic feet per second, and at 6 A.M. water was released at a rate of 178 cubic feet per second. a. What kind of sequence do the rates of water release at each hour appear to form? b. Write a possible rule for the sequence. c. If the pattern continues, at what rate would you epect water to be released at 10 A.M.? Water Release at Wheeler Dam and Wilson Dam on March 15, 00 Total Amount of Water Released (ft 3 ) Number of Wheeler Dam Wilson Dam Seconds 8 A.M. 9 A.M. 8 A.M. 9 A.M ,50 7,310 0,300 1,366 19,040 14,60 40,600 3,049 8,560 1,930 60,900 4,73 38,080 9,40 81,00 The generator hall at Wilson Dam in 00 (above) and 194 (below) Problem Solving on Location 633

47 M A T H - A B L E S Squared Away How many squares can you find in the figure at right? Did you find 30 squares? There are four different-sized squares in the figure. Size of Square Number of Squares 3 3 squares squares Total 30 So the total number of squares is Draw a 5 5 grid and count the number of squares of each size. Can you see a pattern? What is the total number of squares on a 6 6 grid? a 7 7 grid? Can you come up with a general formula for the sum of squares on an n n grid? What s Your Function? One member from the first of two teams draws a function card from the deck, and the other team tries to guess the rule of the function. The guessing team gives a function input, and the card holder must give the corresponding output. Points are awarded based on the type of function and number of inputs required. The first team to reach 0 points wins. Go to go.hrw.com for a complete set of rules and game cards. KEYWORD: MP4 Game1 634 Chapter 1 Sequences and Functions

48 Use with Lesson 1- Generate Arithmetic and Geometric Sequences Graphing calculators can be used to eplore arithmetic and geometric sequences. Lab Resources Online go.hrw.com KEYWORD: MP4 TechLab1 Activity 1 The command seq( is used to generate a sequence. a. Press nd STAT OPS 5:seq. The seq( command is followed by the rule for generating the sequence, the variable used in the rule, and the positions of the first and last terms in the sequence. To find the first 0 terms of the arithmetic sequence generated by the rule 5 ( 1) 3, enter seq(5 ( 1) 3,, 1, 0): 5 X,T,,n 1 3, X,T,,n, 1, 0 b. You can see all 0 terms by pressing the right arrow key repeatedly. From the calculator display, the first term is 5, the second is 8, the third is 11, the fourth is 14, and so on. Consider the geometric sequence whose nth term is n 1. To use a graphing calculator to find the first 15 terms in fraction form, press nd LIST STAT X,T,,n LIST ENTER 5:seq , X,T,,n, MATH ENTER 1, 15 1:Frac. To see all 15 terms, press the right arrow key repeatedly. Think and Discuss 1. Why is the seventh term of the sequence in not displayed as a fraction? Try This Find the first 15 terms of each sequence. Tell if the consecutive terms increase or decrease (n 1) n , 14, 19, 4,... 4., 3, 9,,... 7 Technology Lab 635

49 Chapter 1 Study Guide and Review Vocabulary Study Guide and Review arithmetic sequence common difference common ratio domain eponential decay eponential function eponential growth Fibonacci sequence first differences function function notation geometric sequence input inverse variation linear function output parabola quadratic function range second differences sequence term Complete the sentences below with vocabulary words from the list above. Words may be used more than once. 1. A list of numbers or terms in a certain order is called a(n)?.. A sequence in which there is a common difference is a(n)? ; a sequence in which there is a common ratio is a(n)?. 3. A famous sequence in which you add the two previous terms to find the net term is the?. 4. A rule that relates two quantities so that each input value corresponds to eactly one output value is a(n)?. The set of all input values is the? ; the set of output values is the?. 1-1 Arithmetic Sequences (pp ) EXAMPLE Find the 10th term of the arithmetic sequence: 1, 10, 8, 6,.... d 10 1 a n a 1 (n 1)d a 10 1 (10 1)( ) a a 10 6 EXERCISES Find the given term in each arithmetic sequence. 5. 8th term: 3, 7, 11, th term: 0.05, 0.15, 0.5, th term: 3, 7 6, 5 3, Chapter 1 Sequences and Functions

50 1- Geometric Sequences (pp ) EXAMPLE Find the 10th term of the geometric sequence: 6, 1, 4, 48,.... r 1 6 a n a 1 r n 1 a 10 6() EXERCISES Find the given term in each geometric sequence. 8. 8th term: 5, 10, 0, 40, th term: 1, 1 3, 9, th term: 1, 1, 1, 1, Other Sequences (pp ) EXAMPLE Find the first four terms of the sequence defined by a n ( 1) n 1 1. a 1 ( 1) a ( 1) a 3 ( 1) a 4 ( 1) The first four terms are 3, 1, 3, EXAMPLE Functions (pp ) For the function f() 3 4, find f(0), f(3), and f( ). f(0) 3(0) 4 4 f(3) 3(3) 4 31 f( ) 3( ) 4 16 EXERCISES Find the first four terms of the sequence defined by each rule. 11. a n 3n 1 1. a n n a n 8( 1) n n 14. a n n! EXERCISES For each function, find f(0), f(), and f( 1). 15. f() f() f() f() f() f() 1 Study Guide and Review 1-5 Linear Functions (pp ) EXAMPLE Use the table to write the equation for the linear function. The y-intercept is f(0) 4. f() m 4 f() m b Substitute and solve for m. 11 m(1) 4 (, y) (1, 11) m 7 f() 7 4 y EXERCISES Write the equation for each linear function. 1. y. y Study Guide and Review 637

51 1-6 Eponential Functions (pp ) EXAMPLE Graph the eponential function. f() f() O y EXERCISES Graph each eponential function. 3. f() f() f() 6. f() Quadratic Functions (pp ) EXAMPLE EXERCISES Study Guide and Review Graph the quadratic function. f() 1 f() O y 1 3 Graph each quadratic function. 7. f() 8. f() 4 9. f() 30. f() Inverse Variation (pp ) EXAMPLE Graph the inverse variation function. f() 6 y y EXERCISES Graph each inverse variation function. 31. f() 1 3. f() f() f() Chapter 1 Sequences and Functions

52 Chapter Test Chapter 1 Find the given term in each arithmetic sequence. 1. 8th term: 7, 10, 13, th term: 7, 7 1 5, 7 5, th term: 11, 10.9, 10.8, th term: 75, 6, 49, 36,... Find the given term in each geometric sequence. 5. 7th term: 8, 3, 18, th term: 5, 5, th term: 17, 0.34, , th term: 0.5, 1.5, 6.5, 31.5,... Find the first five terms of each sequence, given its rule. 9. a n 6n 10. a n 3 n 11. a n ( 1) n 5 n Use first and second differences to find the net three terms in each sequence. 1. 7, 17, 3, 5, 77, , 16, 0,,, , 1, 1.05, 1.15, 1.30,... For each function, find f(0), f(4), and f( 3). 15. y y y 5 Chapter Test Write the equation for each linear function. 18. y y Graph each inverse variation function. 0. f() 6 1. f() f() 3. A microbiologist began a bacterial culture with 1000 E. coli bacteria. If the number of bacteria doubles every 0 minutes, find the number of bacteria in the culture after hours. 4. Carbon-14 (C14), a radioactive form of carbon, has a half-life of about 5730 years. C14 is used to date old objects made from plant material. If a wooden cup had 1000 grams of C14 when the tree it came from was cut, about how many grams of C14 would be present 1400 years later? Chapter 1 Test 639

53 Chapter 1 Performance Assessment Show What You Know Create a portfolio of your work from this chapter. Complete this page and include it with your four best pieces of work from Chapter 1. Choose from your homework or lab assignments, mid-chapter quiz, or any journal entries you have done. Put them together using any design you want. Make your portfolio represent what you consider your best work. Performance Assessment Tasks 1. Consider the sequence 1,, 6, 4, 10, 70,... a. Determine whether the sequence is arithmetic, geometric, or neither. b. Find the ratio of each pair of consecutive terms. What patterrn do you notice? c. Write a rule for the sequence. Use your rule to find the net two terms.. Consider the sequence,,,,... a. Write out the net three terms of the sequence. b. Use your calculator to evaluate each term of the sequence. What number do the terms of the sequence seem to be approaching? Problem Solving Choose any strategy to solve each problem. 3. When playing the trombone, you produce different notes by changing the effective length of the tube by moving it in and out. This movement produces a sequence of lengths that form a geometric sequence. a. If the length is inches in the nd position and inches in the 4th position, what is the length in the 3rd position? b. Write a rule that would describe this relationship. 4. A basketball player throws a basketball in a path defined by the function f() , where is the time in seconds and f() is the height in feet. a. Determine the height of the ball after 0.5 seconds. b. Graph the function, and estimate how long it would take the basketball to reach its maimum height. 640 Chapter 1 Sequences and Functions

54 State-Specific Test Practice Online go.hrw.com Keyword: MP4 TestPrep Standardized Test Prep Chapter 1 Cumulative Assessment, Chapters What is the net term in the sequence? 1,, 4, 7, 11,... A 13 C 15 B 14 D 16. A sequence is formed by doubling the preceding number:, 4, 8, 16, 3,.... What is the remainder when the 15th term of the sequence is divided by 6? F 0 H G 1 J 4 is 7. In parallelogram JKLM, KP perpendicular to diagonal. JL Which of the following is true? M y L P z J K A y z 180 B z 90 C y z 90 D y Which equation describes the relationship shown in the graph? Salary ($) s (8, 96) Hours (15,180) h A h 1s C h s 88 B s 1h D s h 88 TEST TAKING TIP! Reworking the given choices: It is sometimes useful to look at a choice in a form different from the given form. 8. If 3 1 8, then what is the value of? F 3 H Standardized Test Prep 4. Which of the following is a solution of the system shown? 3 y F (4, 1) H (5, 1) G (4, 3) J ( 5, 4) t 5. If r 5 and 10r 3, find the value of t. A 64 C 16 B 3 D 8 6. If a 3 and b 4, evaluate b ab a. F 64 H 188 G 8 J 174 G J SHORT RESPONSE The length of a rectangle is 8 ft less than twice its width w. Draw a diagram of the rectangle, and label each side length. What is the perimeter of the rectangle epressed in terms of w? 10. SHORT RESPONSE If two different numbers are selected at random from the set {1,, 3, 4, 5, 6}, what is the probability that their product will be 1? Show your work or eplain in words how you determined your answer. Standardized Test Prep 641