- Patterns and Functions -. Plan What You ll Learn To write a function rule To understand relationships of quantities in a function... And Why To find reasonable domain and range for real-world situations, as in Example Check Skills You ll Need Part Writing Simplifying a Function and Evaluating Rule Expressions and Formulas Lesson - Write an algebraic expression for each phrase.. 0 more than twice a number. a number divided by t x + 0. 8 minus six times a number. twice a number subtracted from 7 8-6n 7 - p New Vocabulary function function rule dependent variable independent variable domain range In Lesson -, the equation C = p + 0.06p shows the relationship between the original price p of a pair of sneakers and the total cost C after the 6% sales tax is included. The relationship between these quantities is a function. A function is a relationship that assigns exactly one output value for each input value. For each input there is only one corresponding output. A function rule, such as C = p + 0.06p, is an equation that describes a functional relationship. for Help Objectives To write a function rule To understand relationships of quantities in a function Examples Writing a Function Rule Writing a Function Rule Identifying Independent and Dependent Quantities Reasonable Domain and Range Math Background Any set of ordered pairs forms a relation. Only a relation in which each input (or x-value) has a unique output (or y-value) is a function. Writing a Function Rule More Math Background: p. C Laundry Suppose you are washing and drying clothes at a self-service laundry. The relationship between the number of loads (input) and the cost (output) is a function. Use the table to write a function rule. Lesson Planning and Resources Loads Cost $.75 $5.50 $8.5 $.00 See p. E for a list of the resources that support this lesson. Relate Total cost is.75? number of loads Describe how the quantities relate. Define Let n = the number of loads. The number of loads is the input. Let c = the total cost. The total cost is the output. Write c =.75? n Translate this relationship to a function rule. The function rule is c =.75n. Write a function rule for the relationship between the number of hours (input) and the number of miles (output). m 60h Hours Total Miles 60 0 80 0 Bell Ringer Practice Check Skills You ll Need For intervention, direct students to: Using Variables Lesson - Extra Skills and Word Problem Practice, Ch. Lesson - Patterns and Functions 7 Special Needs L Help students understand the nature of a functional relationship by explaining that the output depends on the value of the input. Have them rephrase function rules, as in Example : the cost depends on the number of loads. learning style: verbal Below Level L Ask: Is a relationship with input as the student s last name and output as age in years a function? Yes, unless two have the same last name and different ages. learning style: verbal 7
. Teach Writing a Function Rule Guided Instruction One House Two Houses The relationship between the number of houses (input) and the number of toothpicks (output) is a function. Use the table to write a function rule. Special Needs Have a volunteer describe what you do at a self-service laundry, pointing out that you use each machine by inserting coins. Additional Examples Suppose you are buying apples from a vending machine. Use the table to write a function rule. c 0.5n Make a two-story house by adding one toothpick that goes from side to side. Use the table to write a function rule. c 0.5n L of Apples 6 8 0 Cost $.5 $.70 $.05 $.0 of Houses Total Toothpicks 7 9 5 Suppose you want to find out how much gas a trip will take, in a car that uses one gallon of gas to go about 8 miles. What are the independent quantity and the dependent quantity for this situation? independent quantity: miles in the trip; dependent quantity: gallons of gas used Three Houses b. No; you need toothpicks to make houses and 6 toothpicks to make houses. Relationships in a Function Relate Total number of toothpicks is one more than five times the number of houses. Define Let n the number of houses. Let t the total number of toothpicks. Describe how the quantities relate. The number of houses is the input. The total number of toothpicks is the output. Write t 5n Translate this relationship to a function rule. The function rule is t 5n. Houses Total Toothpicks a. How many toothpicks are needed to make seven houses? 6 b. Reasoning Could a set of houses be built with exactly toothpicks? Explain. c. Write a rule for the function represented by the data in the table. y = ± x Input x Output y 5 8 In Example, the total number of toothpicks t depends on the number of houses n. The value of the dependent variable t depends on the value of the independent variable n. In real-world situations, you must determine which varying quantity is dependent and which is independent. 6 Identifying Independent and Dependent Quantities Memory Stick The table and graph model a function relating the storage capacity of a memory stick and its cost. Identify the independent and dependent quantities. Memory (mb) 6 8 56 Cost ($) 9 9 9 Cost ($) 70 60 50 0 0 0 0 9 0 6 8 9 56 0 Memory (megabytes) 6 Cost of Memory Sticks (8, 9) (6, 9) (, 9) (56, 55) The cost is the dependent quantity because it depends on the storage capacity of the memory stick. Memory is the independent quantity. The cooking time for an unstuffed turkey is about 0 minutes per pound. What are the independent quantity and dependent quantity for this situation? independent quantity: weight of turkey; dependent quantity: cooking time 8 Chapter Variables, Function Patterns, and Graphs 8 Advanced Learners L Students may wish to use other formulas they know to make a table and write the matching function rule. learning style: visual English Language Learners ELL Help students learn the special vocabulary of functions by having them write, compare, and discuss their own explanations of the Examples using the words in the New Vocabulary list. learning style: verbal
. independent: the amount of money he can spend; dependent: number of songs he can download. domain: $.00 to $6.00; range: to 8 songs. EXERCISES The possible values for the input, or the independent variable, of a function are the domain of the function. The possible values of the output, or the dependent variable, are the range of the function. Practice and Problem Solving Reasonable Domain and Range Maria earns $7 per hour for baby-sitting after school and on Saturday. She works no more than 6 hours a week. a. Identify the independent and dependent quantities for this situation. The amount Maria earns depends on the number of hours she works in a week. So the amount Maria earns is the dependent variable. The number of hours she works is the independent variable. b. Find reasonable domain and range values for this situation. A reasonable domain is from 0 to 6 hours. If Maria works 0 hours, she earns $0. If she works 6 hours, she earns 7? $6, or $. So, a reasonable range is from $0 to $. Charlie downloads songs for $.75 each. He has between $.00 and $6.00 to spend on songs. Identify the independent and dependent quantities for this situation and find reasonable domain and range values. See above left. For more exercises, see Extra Practice. English Language Learners ELL Some students may not be familiar with the term downloads (in the for Example ). Ask a volunteer to explain. Additional Examples Ken burns 5 calories per hour when he bikes. He bikes from to 7 hours each weekend. Identify the independent and dependent quantities for this situation and find reasonable domain and range values. independent quantity: hours he bikes; dependent quantity: calories he burns; domain: to 7 hours; range:,75 to,975 calories Resources Daily Notetaking Guide - L Daily Notetaking Guide - Adapted Instruction L A Practice by Example for Help Example (page 7) Example (page 8) Example (page 8) The relationships in the tables below are functions. Write a function rule for each... Cans of Soup Servings s c. Hours Cost to Home. a Plumber Works Owner 8 6 $65 $90 $5 $0 Cans of Frozen Orange Juice Time (hours) c 0 ± 5h c ± 6t Identify the independent and dependent quantity in each situation. 5. The cost of a long-distance telephone call increases, with the number of minutes of the call. number of minutes; cost 6. Water pressure increases 0. pounds per square inch (0. psi) with each increase of one foot in depth below sea level. depth; water pressure Cost of Bike Rental $0 $6 $ $8 Total Cost $.5 $.50 $.75 $5.00 c.5n Closure Ask students to group these terms into two related lists: range, domain, x-value, y-value, dependent, independent, abscissa, ordinate. Ask them to explain their choices. Sample: List : domain, x-value, independent, abscissa List : range, y-value, dependent, ordinate Lesson - Patterns and Functions 9 9
. Practice Assignment Guide A B -, 9-,, 5 A B 5-8,, C Challenge 6, 7 Test Prep 8- Mixed Review 5- Homework To check students understanding of key skills and concepts, go over Exercises, 7, 9,,. GPS Enrichment Guided Problem Solving Reteaching Adapted Practice Practice Name Class Date Practice - Adding Real s Simplify each expression.. 6 + (-). - + (-). -8 +. 5 + (-) 5. -7 + (-) 6. 8 + (-) 7. -. +.9 8. -. + (-.9) 9. 9.8 + (-7.) a b a b 7 6 0... () a b.. 5. 6. -6. + 8. 7. -.8 +.8 8. -7.8 + 9 9. u u u u 0. u 6 u. Evaluate each expression for m.5.. -m +.6. -. + m. -.5 + (-m) Simplify. 5. - + (-6) + 6. + (-8) + (-) 7..7 + (-.) +.5 8. -.5 + (-.) + (-.) 9. 0. a b Simplify.. c. 5 d c d.7. The temperature at 5:00 A.M. is-8 F. The temperature rises 0 by :00 A.M. What is the temperature at :00 A.M.?. A football team has possession of the ball on their own 5-yd line. The next two plays result in a loss of 7 yd and a gain of yd, respectively. On what yard line is the ball after the two plays? 5. Suppose your opening checking account balance is $.5. After you write a check for $57.9 and make a deposit of $0.9, what is your new balance? 6. During an emergency exercise, a submarine dives 7 ft, rises 6 ft, and then dives 8 ft. What is the net change in the submarine s position after the second dive? 5...7.5.6 u 7 u a b a b L L L L L Pearson Education, Inc. All rights reserved. Example (page 9) 7. number of gallons of gas; distance traveled; 0 to gallons; 50 to 00 miles B Apply Your Skills a. Yes; there is one range value for each domain value. b. year, number of students nline Homework Help Visit: PHSchool.com Web Code: ate-00 b. No; you cannot have a fractional amount of a book. Yes; books can weigh fractional amounts. Identify the independent and dependent quantities for each situation, and find reasonable domain and range values. 7. Tara s car travels about 5 miles on one gallon of gas. She has between 0 and gallons of gas in the tank. 8. Sal and three friends plan to bowl one or two games each. Each game costs $.50. number of games; total cost; to 8 games; $0 to $0 Copy and complete each table. Then write a function rule for each relationship. 9. 0. Minutes Hours of Yoga Reading Words Read Lessons Total Cost 5 50 75 500 5 65 w 5m. The graph shows the average size of a school s ninth-grade class over several years. a. Does this graph represent a function? Explain. a b. See left. b. If the graph represents a function, identify the independent and dependent quantities.. a. The weight of the books carried in a backpack depends on the number of books in the backpack. Susan s books weigh from pound up to.5 pounds each and she carries at most books. Find a reasonable domain and range for the situation. 0 to books; 0 to.0 pounds b. Critical Thinking Can noninteger numbers be part of the domain? of the range? Explain.. Each side of the first figure is one unit. Copy and complete the table. Then find a GPS function rule for the relationship between a figure s number and its perimeter. Figure Perimeter 5 units 7 units 9 units units. Motorcycles The table below shows the price of several motorcycles and the size of their engines. Identify the independent and dependent quantities. Explain. See margin. Engine (cc) 9 98 65 996 Price ($) 99 Students $5 $7 $9 $5 5 $6 c ± h 0 8 6 0 8 6 Figure Figure Figure Figure p + n 6099 6699 Average Size of a Ninth-Grade Class 998 000 00 00 006 Year 8599 0 Chapter Variables, Function Patterns, and Graphs. size of engine; price. The greater the engine size the greater the cost of the motorcycle. 0
C Challenge 5. a. Copy the table at the right. Evaluate the expression x for x,,, and. Record your results in the second column. b. Critical Thinking Do the numbers in the table represent a function? Explain. Yes, there is one range value (x + ) for each domain value (x). 6. a. Look at the pattern below. Make a table that relates the figure number n to the number of hexagons h. x x 5 0 7. Assess & Reteach Lesson Quiz Copy and complete each table. Then write a function rule for each relationship.. of Lemons Cups of Punch.5.5 Test Prep Multiple Choice Figure Figure Figure b. Predict What will be the number of hexagons in Figure? 7 c. Does your table represent a function? Explain. Yes, there is one range value (h) for each domain value (figure number). 7. a. Complete the table for the side lengths and areas of squares. b. Identify the independent and the Side length Area.5.5 dependent quantities..5 6.5 c. Graph the data. Show the data for the independent quantity on the 9 horizontal axis and the data for the dependent quantity on the vertical axis. b. side length; area c. See margin. 8. The graph at the right represents the temperature of water that was boiled and then left to cool. Which statement is true? B A. Time depends on the water temperature. B. The water temperature depends on time. C. The cooling time depends on how long the water boils. D. The water temperature depends on the boiling time. 9. The function c = (n ) + 7 represents Time the cost c in cents for mailing a letter weighing n ounces. Which statement is true? H F. The weight of a letter depends on its cost. G. The number of letters mailed depends on the total number of ounces of the letters. H. The cost to mail a letter depends on its weight. J. Each letter costs 7 an ounce. 0. Evaluate the expression x 5 for x =, 0,, and 5. Which value of x produces the greatest value from the expression? D A. B. 0 C. D. 5 Water Temperature c.5n. Minutes in Phone Call Cost of Call $.5 $.0 $.5 c.5m ± Identify the independent and dependent quantity in each situation and find reasonable domain and range values.. Gerald rents movies for each day of the -day holiday weekend. Each rental costs between $.50 and $5. independent quantity: number of movies; dependent quantity: cost of rental; domain: 6 movies; range: $5 to $0. The weight of each kitten in the litter of 5 is between 0 and ounces. Jill chooses a basket that can support the total weight of the litter. independent quantity: hours he bikes; dependent quantity: calories he burns; domain: to 7 hours; range:,75 to,975 calories 7. c. 8 y lesson quiz, PHSchool.com, Web Code: ata-00 Lesson - Patterns and Functions Area 6 6. a. Figure n Hexagons h 7 O Side length x 9
Alternative Assessment Have students make a table with 5 sets of values, and write the function rule for their table. Have them identify the dependent and independent variables, and the domain and range. Test Prep Resources For additional practice with a variety of test item formats: Standardized Test Prep, p. 5 Test-Taking Strategies, p. 6 Test-Taking Strategies with Transparencies. Simplify u 7 8 u u 9 u. G F. 6 G. 0 H. 6 J. 8. Jacob recorded the height of a sunflower each week for 5 weeks. Week Height (cm) 5 0 Which function best describes the relationship, if w represents the week and h represents the height? A A. h = 5w B. w = 5h C. h = w + 5 D. w = h + 5. Monica s basketball team is playing an exhibition game against a local college. After the first half of the game, Monica sees that her team has points, and the college team has points. How many more points does Monica s team need to tie the game? G F. 6 G. 8 H. J. 56. A collection of quarters and nickels is worth $.0. There are coins. How many of the coins are quarters? B A. 9 B. C. D. 5 5 60 5 75 Mixed Review for Help Lesson - Are whole numbers, integers, or rational numbers the most reasonable for each situation? 5 7. Answers may vary. Samples are given. 5. your height rational numbers 6. the number of pages in your math text whole numbers 7. the overnight low temperature in degrees Celsius for Barrow, Alaska, on January 5 rational numbers Lesson - Evaluate each expression for c = 5 and d = 8. 8. d - c 6 9. cd 0 0. dc 00. (cd),00. cd - c 5. (c) d 00 Algebra at Work Cartographer A cartographer, or mapmaker, makes measurements of an area being mapped. The cartographer uses these measurements to create the scale of the map showing the ratio of map distance to actual distance. Knowing the scale of a map means that you can use a proportion to calculate any distance on the map. PHSchool.com For: Information about cartographers Web Code: atb-0 Chapter Variables, Function Patterns, and Graphs