Name Date Chapter Evaluate the epression.. Fair Game Review 5 +. 3 3 7 3 8 4 3. 4 4. + 5 0 5 6 5. 3 6. 4 6 5 4 6 3 7. 5 8. 3 9 8 4 3 5 9. You plan to jog 3 4 of a mile tomorrow and 7 8 of a mile the net da. How far will ou jog over the net two das? 0. A recipe makes 9 cups. recipe make? A serving is 3 cup. 4 How man servings does the Copright Big Ideas Learning, LLC Big Ideas Math Algebra 83
Name Date Chapter Fair Game Review (continued) Solve the proportion. d.. 4 8 5 6 m 3. 4 a 9 4. 8 5 k 5. 35 6. 7 c 9 8 7. It costs $36 for 6 students to visit a museum. How much does it cost for 5 students? 84 Big Ideas Math Algebra Copright Big Ideas Learning, LLC
Name Date. Direct and Inverse Variation For use with Activit. Essential Question How can ou recognize when two variables var directl? How can ou recognize when the var inversel? ACTIVITY: Recognizing Direct Variation Work with a partner. You hang different weights from the same spring. Equilibrium 0 kg 0. kg 0. kg 0.3 kg 0.4 kg a. Describe the relationship between the weight and the distance d the spring stretches from equilibrium. Eplain wh the distance is said to var directl with the weight. 0.5 kg 0.6 kg 3 4 5 6 7 8 9 0 centimeters 0.7 kg b. Graph the relationship between and d. What are the characteristics of the graph? 0 9 8 7 6 5 4 3 d 0. 0. 0.4 0.6 0.8.0 Copright Big Ideas Learning, LLC Big Ideas Math Algebra 85
Name Date. Direct and Inverse Variation (continued) c. Write an equation that represents d as a function of. d. In phsics, the relationship between d and is described b Hooke s Law. How would ou describe Hooke s Law? ACTIVITY: Recognizing Inverse Variation Work with a partner. The area of each rectangle is 64 square inches. 64 in. in. 4 in. in. 6 in. 3 in. 3 in. in. 6 in. 4 in. 8 in. 8 in. in. 64 in. a. Describe the relationship between and. Eplain wh is said to var inversel with. 86 Big Ideas Math Algebra Copright Big Ideas Learning, LLC
Name Date. Direct and Inverse Variation (continued) b. Graph the relationship between and. What are the characteristics of the graph? 60 50 40 30 c. Write an equation that represents as a function of. 0 0 0 0 30 40 50 60 What Is Your Answer? 3. IN YOUR OWN WORDS How can ou recognize when two variables var directl? How can ou recognize when the var inversel? 4. Does the flapping rate of a bird s wings var directl or inversel with the length of its wings? Eplain our reasoning. Copright Big Ideas Learning, LLC Big Ideas Math Algebra 87
Name Date. Practice For use after Lesson. Tell whether and show direct variation, inverse variation, or neither. Eplain our reasoning.. 3 4. 0 7 4 8 The variable varies directl with. Write and graph a direct variation equation that relates and. 3. When 3, 5. 4. When 8,. O O The variable varies inversel with. Write and graph an inverse variation equation that relates and. 5. When 5, 8. 6. When 9, 4. O O 7. An airplane makes a 3000 mile flight. The average fling speed r (in miles per hour) 3000 is given b r, where t is the time t (in hours) it takes for the flight. Graph the function. Make a conclusion from the graph. r t 88 Big Ideas Math Algebra Copright Big Ideas Learning, LLC
Name Date. Graphing Rational Functions For use with Activit. Essential Question What are the characteristics of the graph of a rational function? ACTIVITY: Graphing a Rational Function Work with a partner. As a fundraising project, our math club is publishing an optical illusion calendar. The cost of the art, tpesetting, and paper is $850. In addition to this one-time cost, the unit cost of printing each calendar is $3.5. a. Let A represent the average cost of each calendar. Write a rational function that gives the average cost of printing calendars. A b. Make a table showing the average costs for several different production amounts. Then use the table to graph the average cost function. A 0 8 6 4 0 8 6 4 00 400 600 800 000 00 400 Copright Big Ideas Learning, LLC Big Ideas Math Algebra 89
Name Date. Graphing Rational Functions (continued) ACTIVITY: Analzing the Graph of a Rational Function Work with a partner. Use the graph in Activit. a. What is the greatest average cost of a calendar? Eplain our reasoning. b. What is the least average cost of a calendar? Eplain our reasoning. What characteristic of the graph is associated with the least average cost? 3 ACTIVITY: Analzing Profit and Revenue Work with a partner. Consider the calendar project in Activit. Suppose our club sells 400 calendars for $0 each. a. Find the revenue our club earns from the calendars. b. How much profit does our club earn? Eplain our reasoning. 90 Big Ideas Math Algebra Copright Big Ideas Learning, LLC
Name Date. Graphing Rational Functions (continued) What Is Your Answer? 4. IN YOUR OWN WORDS What are the characteristics of the graph of a rational function? Illustrate our answer with the graphs of the following rational functions. a. + b. + c. + 3 Copright Big Ideas Learning, LLC Big Ideas Math Algebra 9
Name Date. Practice For use after Lesson. Find the ecluded value of the function.. 3 8. + 6 Graph the function. Identif the asmptotes of the graph of the function. Describe the domain and range. 3. 5 + 4. 3 + 4 O O Graph the function. Compare the graph to the graph of 5. 6 6.. + 4 + 5 O O 7. A high school football team is taking a trip to an NFL game. Tickets cost $8 and a charter bus costs $350. The function 350 8 + 6 + represents the cost (in dollars) per plaer when plaers and 6 coaches go on the trip. Graph the function. How man plaers must go on the trip for the cost per plaer to be about $90? 9 Big Ideas Math Algebra Copright Big Ideas Learning, LLC
Name Date Etension. Practice For use after Etension. Find the inverse of each relation.. (, ), (, ), ( 0, 5 ), (, 8 ), (, ). ( 3, 8 ), (, 0 ), ( 0, ), (, 0 ), ( 3, 8) 3. (, 3 ), ( 3, 5 ), ( 4, 6 ), ( 6, 7 ), ( 6, 0 ) 4. Input 4 0 4 Output 3 4 5 6 5. Input 3 0 Output 0 4 3 6. Input 8 4 4 0 8 Output 8 4 6 4 Copright Big Ideas Learning, LLC Big Ideas Math Algebra 93
Name Date Etension. Practice (continued) Find the inverse of the function. Graph the inverse function. 7. f( ) + 8. f( ) 4 6 O O 9. f( ) 5, where 0 0. ( ) f, where 0 3 O O. f( ) + 4. f( ) 3 O O 3. Suppose f and g are inverse functions and f () 8 3. What is the value of g( 3? ) 94 Big Ideas Math Algebra Copright Big Ideas Learning, LLC
Name Date.3 Simplifing Rational Epressions For use with Activit.3 Essential Question How can ou simplif a rational epression? What are the ecluded values of a rational epression? ACTIVITY: Simplifing a Rational Epression Work with a partner. Sample: You can see that the rational epressions + 3 + 3 and are equivalent b graphing the related functions + 3 + 3 and. 4 4 + 3 + 3 66 6 6 6 4 Both functions have the same graph. 4 Match each rational epression with its equivalent rational epression. Use a graphing calculator to check our answers. a. + b. + c. + d. + + + e. + + + A. + B. + C. + D. E. + Copright Big Ideas Learning, LLC Big Ideas Math Algebra 95
Name Date.3 Simplifing Rational Epressions (continued) ACTIVITY: Finding Ecluded Values Work with a partner. Are the graphs of + and + eactl the same? Eplain our reasoning. 4 4 + + 6 6 6 6 4 4 3 ACTIVITY: Simplifing and Finding Ecluded Values Work with a partner. Simplif each rational epression, if possible. Then compare the ecluded value(s) of the original epression with the ecluded value(s) of the simplified epression. a. + b. + c. 96 Big Ideas Math Algebra Copright Big Ideas Learning, LLC
Name Date.3 Simplifing Rational Epressions (continued) d. + 4 + 4 + e. 4 f. + What Is Your Answer? 4. IN YOUR OWN WORDS How can ou simplif a rational epression? What are the ecluded values of a rational epression? Include the following rational epressions in our answer. a. ( + ) b. + 3 + + c. + 3 9 Copright Big Ideas Learning, LLC Big Ideas Math Algebra 97
Name Date.3 Practice For use after Lesson.3 Simplif the rational epression, if possible. State the ecluded value(s).. 0 5. 8t t 4 3. d d 4. 7ab 36 3 ab 5. m 9m m 3 5m 45 6. h + 7 6 h 7. 4 6 3 4 8. p 7p 8 p + p + p 3 0 9. The shapes shown have the same area. Write and simplif an epression for the height h of the trapezoid. h cm ( + ) cm ( + 4) cm ( + 4) cm ( + 3) cm 98 Big Ideas Math Algebra Copright Big Ideas Learning, LLC
Name Date.4 Multipling and Dividing Rational Epressions For use with Activit.4 Essential Question How can ou multipl and divide rational epressions? ACTIVITY: Matching Quotients and Products Work with a partner. Match each quotient with a product and then with a simplified epression. Eplain our reasoning. Quotient of Two Rational Epressions Product of Two Rational Epressions Simplified Epression a. 4 A. 5 0 5 0. 4 5 4 b. 0 B. 5 4 0. 5 4 7 3 c. 5 4 C. 0 5 4 3. 0 3 4 5 d. 5 0 D. 4 4 4. 5 0 7 e. + + 4 E. 4 + 5. f. + 4 F. 4 6. 3 + g. 4 G. 4 + 7. + h. + 4 H. 4 + + 8. + 3 + Copright Big Ideas Learning, LLC Big Ideas Math Algebra 99
Name Date.4 Multipling and Dividing Rational Epressions (continued) Quotient of Two Rational Epressions Product of Two Rational Epressions i. ( ) I. ( ) Simplified Epression 9. + j. ( ) J. 0. k. K.. + l.. L. ( ) ( ) ACTIVITY: Solving a Math Crossword Puzzle Work with a partner. Solve the crossword puzzle. Use the clues from the net page. 3 4 5 6 7 8 9 0 3 4 5 6 7 8 9 0 300 Big Ideas Math Algebra Copright Big Ideas Learning, LLC
Name Date.4 Multipling and Dividing Rational Epressions (continued) Across. Inverse of 6. subtraction k Down. 30 8. Graph approaches h 4. or 7. 3 7. Greek 8. 3 ft mathematician 0. Longest side of a right triangle 9.. + ( ) 0. 4 or 3 5. C of this is π r 9. + 3. Dimension. Two numbers of whose product is 5. 0 3. in 6. Is the same as 5., 0,, etc. 4. k What Is Your Answer? 7. About 3.4 9. in ( + ) 3. IN YOUR OWN WORDS How can ou multipl and divide rational epressions? Include the following in our answer. a. + 3 + 3 b. + 3 Copright Big Ideas Learning, LLC Big Ideas Math Algebra 30
Name Date.4 Practice For use after Lesson.4 Find the product.. 8k k 5 4. 7 k w + 8 3w + 8 6 w 3. 3 b b 5 b b 5 6b 4. 4n 8n n 4 n + 9n + 4 4n Find the quotient. g + 3 g + 3 5. 6. 4g g + 5 0 8 a a 64 3 7. ( 3a + 4a ) 8. ( r + r 56) r 7r 3 9. Two distinct prairie dog populations, P and P can be modeled b P 00 and P + 00, where is the number of ears since 000. + 3 a. Write a function that models the ratio of Population to Population, P that is. P b. Find the ratio of Population to Population in 004. 30 Big Ideas Math Algebra Copright Big Ideas Learning, LLC
Name Date.5 Dividing Polnomials For use with Activit.5 Essential Question How can ou divide one polnomial b another polnomial? ACTIVITY: Dividing Polnomials Work with a partner. Si different algebra tiles are shown below. Step : Arrange tiles to model ( + 5 + 4 ) ( + ) in a rectangular pattern. Step : Complete the pattern. Step 3: Use the completed pattern to write ( ) ( ) + 5 + 4 +. Dividend Divisor Quotient Copright Big Ideas Learning, LLC Big Ideas Math Algebra 303
Name Date.5 Dividing Polnomials (continued) Complete the pattern and write the division problem. a. b. ACTIVITY: Dividing Polnomials Work with a partner. Write two different polnomial division problems that can be associated with the given algebra tile pattern. Check our answers b multipling. a. b. c. d. 304 Big Ideas Math Algebra Copright Big Ideas Learning, LLC
Name Date.5 Dividing Polnomials (continued) 3 ACTIVITY: Dividing Polnomials Work with a partner. Solve each polnomial division problem. a. ( 3 8 3) ( 3) b. ( 8 3) ( 4 3) What Is Your Answer? 4. IN YOUR OWN WORDS How can ou divide one polnomial b another polnomial? Include the following in our answer. a. ( 3 + 0 7) ( + 7) b. ( 4 4 3) ( 3) Copright Big Ideas Learning, LLC Big Ideas Math Algebra 305
Name Date.5 Practice For use after Lesson.5 Find the quotient.. ( b + b + 8) ( b + ). ( n n 30 ) ( n + 5 ) 3. ( 4w + 9w 3) ( w + 3) 4. ( 5c 0 c) ( c ) 5. ( + 8 6) ( 4 7) 6. ( 5h 4 + 6h ) ( h + 9) 7. ( r ) ( r + 8 ) 8. ( + 8) ( 3 4) 9. The volume of the triangular prism 3 is + 6 + + 6. Write an epression for the height of the prism. + + 3 306 Big Ideas Math Algebra Copright Big Ideas Learning, LLC
Name Date.6 Adding and Subtracting Rational Epressions For use with Activit.6 Essential Question How can ou add and subtract rational epressions? ACTIVITY: Adding Rational Epressions Work with a partner. You and a friend have a summer job mowing lawns. Working alone it takes ou 40 hours to mow all of the lawns. Working alone it takes our friend 60 hours to mow all of the lawns. a. Write a rational epression that represents the portion of the lawns ou can mow in t hours. Portion ou mow in t hours Time Rate b. Write a rational epression that represents the portion of the lawns our friend can mow in t hours. Portion our friend mows in t hours Time Rate c. Add the two epressions to write a rational epression for the portion of the lawns that the two of ou working together can mow in t hours. + Time Rate Time Rate d. Use the epression in part (c) to find the total time it takes both of ou working together to mow all of the lawns. Eplain our reasoning. Copright Big Ideas Learning, LLC Big Ideas Math Algebra 307
Name Date.6 Adding and Subtracting Rational Epressions (continued) ACTIVITY: Adding Rational Epressions Work with a partner. You are hang gliding. For the first 0,000 feet, ou travel feet per minute. You then enter a valle in which the wind is greater, and for the net 6000 feet, ou travel feet per minute. a. Use the formula d rt to write a rational epression that represents the time it takes ou to travel the first 0,000 feet. Time to travel first 0,000 feet Distance Rate b. Use the formula d rt to write a rational epression that represents the time it takes ou to travel the net 6000 feet. Time to travel net 6000 feet Distance Rate c. Add the two epressions to write a rational epression that represents the total time it takes ou to travel 6,000 feet. + d. Use the epression in part (c) to find the total time it takes ou to travel 6,000 feet when our rate during the first 0,000 feet is 000 feet per minute. 308 Big Ideas Math Algebra Copright Big Ideas Learning, LLC
Name Date.6 Adding and Subtracting Rational Epressions (continued) What Is Your Answer? 3. IN YOUR OWN WORDS How can ou add and subtract rational epressions? Include the following in our answer. a. + b. 5 0 3 4 + c. 9 + 3 d. e. 4 + f. 3 3 Copright Big Ideas Learning, LLC Big Ideas Math Algebra 309
Name Date.6 Practice For use after Lesson.6 Find the sum or difference. 3 6. +. 5g 5g 4 + v + 3 v + 3 3. m 3m + 4m 4m 4. 9 + 6 + 6 5. 3a + a + 6. 4a 6a k k 5 + 8 k 7. + + 3 4 4 4 6 3d 7 8. d + 5 d + d 5 9. You drive 45 miles from home to a relative s house and 45 miles back home. Due to construction, our speed on the wa back is onl 60% of our speed on the wa there. Let r be our speed (in miles per hour) while driving to our relative s house. Write an epression that represents the amount of time ou spend driving on our trip. 30 Big Ideas Math Algebra Copright Big Ideas Learning, LLC
Name Date.7 Solving Rational Equations For use with Activit.7 Essential Question How can ou solve a rational equation? ACTIVITY: Solving Rational Equations Work with a partner. A hocke goalie faces 799 shots and saves 707 of them. a. What is his save percentage? Save Percentage Shots saved Shots faced b. Suppose the goalie has additional consecutive saves. Write an epression for his new save percentage. Save Percentage 707 plus additional saves 799 plus additional shots faced c. Complete the table showing the goalie s save percentage as increases. Additional Saves, 0 0 40 60 80 00 0 40 Save Percentage d. The goalie wants to end the season with a save percentage of.900. How man additional consecutive saves must he have to achieve this? Justif our answer b solving an equation. Copright Big Ideas Learning, LLC Big Ideas Math Algebra 3
Name Date.7 Solving Rational Equations (continued) ACTIVITY: Solving Rational Equations Work with a partner. A baseball plaer has been at bat 47 times and has 8 hits. a. What is his batting average? Batting Average b. Suppose the plaer has additional consecutive hits. Write an epression for his new batting average. Batting Average c. Complete the table showing the plaer s batting average as increases. Additional Hits, 0 3 4 5 6 7 Batting Average d. The plaer wants to end the season with a batting average of.50. How man additional consecutive hits must he have to achieve this? Justif our answer b solving an equation. 3 Big Ideas Math Algebra Copright Big Ideas Learning, LLC
Name Date.7 Solving Rational Equations (continued) What Is Your Answer? 3. IN YOUR OWN WORDS How can ou solve a rational equation? Include the following in our answer. a. 6 b. 6 3 + 56 c. 6 + 4 3 Copright Big Ideas Learning, LLC Big Ideas Math Algebra 33
Name Date.7 Practice For use after Lesson.7 Solve the equation. Check our solution.. h 4 8 3 h. q 6 5 q 3. m 5 m + 3 m + 7 4. c 3 c 5c 6 c 3 5. 6 3 6 z 3 z z 6. 4 4 8 + k k + 5 k + 5 d 4 7. + d + 3 d d + d 3 t 5 3t + 3 8. t 7 t 4 t t + 8 9. An academic challenge team has 4 members, 9 of which are bos. The team is required to have 50% bos and 50% girls for a competition. How man bos does the team need to add to reach this proportion? 34 Big Ideas Math Algebra Copright Big Ideas Learning, LLC