Algebra: Nonlinear Functions and Polynomials 10 • Standard 7AF2.0 Interpret and evaluate expressions involving integer powers and simple roots. • Standard 7NS1.0 Use exponents, powers, and roots and use exponents in working with fractions. Key Vocabulary cube root (p. 554) nonlinear function (p. 522) quadratic function (p. 528) Real-World Link Fountains Many real-world situations, such as this fountain at Paramount’s Great America theme park in Santa Clara California, cannot be modeled by linear functions. These can be modeled using nonlinear functions. Algebra: Nonlinear Functions and Polynomials Make this Foldable to help you organize your notes. Begin with eight sheets of grid paper. 1 Cut off one section of the grid paper along both the long and short edges. 2 Cut off two sections from the second sheet, three sections from the third sheet, and so on to the 8th sheet. 3 Stack the sheets from narrowest to widest. 4 Label each of the right tabs with a lesson number. 520 Chapter 10 Algebra: Nonlinear Functions and Polynomials Richard Cummins/SuperStock GET READY for Chapter 10 Diagnose Readiness You have two options for checking Prerequisite Skills. Option 2 Take the Online Readiness Quiz at ca.gr7math.com. Option 1 Take the Quick Check below. Refer to the Quick Review for help. Graph each equation. (Lesson 11-2) Example 1 1. y = x – 4 Graph y = x + 1. 2. y = 2x First, make a table of values. Then, graph the ordered pairs and connect the points. 3. y = x + 2 x y (x, y) y = 2.54x describes about how many centimeters y are in x inches. Graph the function. 0 1 (0, 1) 1 2 (1, 2) 2 3 (2, 3) (Lessons 11-2) 3 4 (3, 4) 4. MEASUREMENT The equation Write each expression using a positive exponent. (Lesson 2-9) 5. a -9 6. 6 -4 -5 -2 7. x 8. 5 Write each expression using exponents. (Lesson 2-9) 9. 6 · 6 · 6 · 6 10. 3 · 7 · 7 · 3 · 7 11. FUND-RAISER The students at y yx1 O x Example 2 Write n -3 using a positive exponent. 1 n -3 = _ 3 n definition of negative exponent Example 3 Write 5 · 4 · 5 · 4 · 5 using exponents. 5 is multiplied by itself 3 times and 4 is multiplied by itself 2 times. So, 5 · 4 · 5 · 4 · 5 = 5 3 · 4 2. Hampton Middle School raised 8 · 8 · 2 · 8 · 2 dollars to help build a new community center. How much money did they raise? (Lesson 2-9) Chapter 10 Get Ready for Chapter 10 521 Linear and Nonlinear Functions 10-1 Main IDEA ROCKETRY The tables show the flight data for a model rocket launch. The first table gives the rocket’s height at each second of its ascent, or upward flight. The second table gives its height as it descends back to Earth using a parachute. Determine whether a function is linear or nonlinear. Preparation for AF1.5 Represent quantitative relationships graphically and interpret the meaning of a specific part of a graph in the situation represented by the graph. NEW Vocabulary nonlinear function Ascent Descent Time (s) Height (m) Time (s) Height (m) 0 0 7 140 1 38 8 130 2 74 9 120 3 106 10 110 4 128 11 100 5 138 12 90 6 142 13 80 1. During its ascent, did the rocket travel the same distance each second? Justify your answer. 2. During its descent, did the rocket travel the same distance each second? Justify your answer. 3. Graph the ordered pairs (time, height) for the rocket’s ascent and descent on separate axes. Connect the points with a straight line or smooth curve. Then compare the graphs. REVIEW Vocabulary constant rate of change occurs when the rate of change between any two data points is proportional. (Lesson 4-10) In Lesson 9-2, you learned that linear functions have graphs that are straight lines. These graphs represent constant rates of change. Nonlinear functions are functions that do not have constant rates of change. Therefore, their graphs are not straight lines. Identify Functions Using Tables Determine whether each table represents a linear or nonlinear function. Explain. 1 +2 +2 +2 x y 2 50 4 35 6 20 8 5 2 -15 +3 -15 +3 -15 +3 As x increases by 2, y decreases by 15 each time. The rate of change is constant, so this function is linear. 522 Chapter 10 Algebra: Nonlinear Functions and Polynomials Doug Martin x y 1 1 4 16 7 49 10 100 +15 +33 +51 As x increases by 3, y increases by a greater amount each time. The rate of change is not constant, so this function is nonlinear. Determine whether each table represents a linear or nonlinear function. Explain. a. x 0 5 10 15 y 20 16 12 8 b. x 0 2 4 6 y 0 2 8 18 Identify Functions Using Graphs Determine whether each graph represents a linear or nonlinear function. Explain. y 3 y 4 x y2 1 y 0.5x 2 x O x O The graph is a curve, not a straight line. So, it represents a nonlinear function. This graph is also a curve. So, it represents a nonlinear function. Determine whether each graph represents a linear or nonlinear function. Explain. c. d. y O e. y x O y x O x Recall that the equation for a linear function can be written in the form y = mx + b, where m represents the constant rate of change. Identifying Linear Equations Always examine an equation after it has been solved for y to see that the power of x is 1 or 0. Then check to see that x does not appear in the denominator. Identify Functions Using Equations Determine whether each equation represents a linear or nonlinear function. Explain. _ 6 y = 6x 5 y=x+4 Since the equation can be written as y = 1x + 4, this function is linear. f. y = 2x 3 + 1 Extra Examples at ca.gr7math.com g. y = 3x The equation cannot be written in the form y = mx + b. So, this function is nonlinear. h. y = _ x 5 Lesson 10-1 Linear and Nonlinear Functions 523 7 BASKETBALL Use the table to determine whether Round(s) of play Teams 1 32 Examine the differences between the number of teams for each round. 2 16 3 8 16 - 32 = -16 4 - 8 = -4 4 4 5 2 the number of teams is a linear function of the number of rounds of play. 8 - 16 = -8 2 - 4 = -2 While there is a pattern in the differences, they are not the same. Therefore, this function is nonlinear. Graph the data to verify the ordered pairs do not lie on a straight line. Real-World Link The NCAA women’s basketball tournament begins with 64 teams and consists of 6 rounds of play. y 32 24 Teams Check 16 8 0 2 4 6 8 x Rounds of Play i. TICKETS Tickets to the school dance cost $5 per student. Are the ticket sales a linear function of the number of tickets sold? Explain. Number of Tickets Sold 1 Ticket Sales $5 2 3 $10 $15 Personal Tutor at ca.gr7math.com Determine whether each table, graph, or equation or represents a linear or nonlinear function. Explain. Examples 1–6 1. (pp. 522–523) x 0 1 2 3 y 1 3 6 10 3. x 0 3 6 9 y -3 9 21 33 4. y O 2. y x O 5. y = _ x 3 Example 7 (p. 524) Elise Amendola/AP/Wide World Photos 6. y = 2x 2 7. MEASUREMENT The table shows the measures of the sides of several rectangles. Are the widths of the rectangles a linear function of the lengths? Explain. 524 Chapter 10 Algebra: Nonlinear Functions and Polynomials x Length (in.) 1 4 8 10 Width (in.) 64 16 8 6.4 (/-%7/2+ (%,0 For Exercises 8–13 14–19 20–25 26–29 See Examples 1, 2 3, 4 5, 6 7 Determine whether each table, graph, or equation or represents a linear or nonlinear function. Explain. 8. 10. 12. x 3 6 9 12 y 12 10 8 6 x 5 10 15 20 y 13 28 43 58 x 2 4 6 8 y 10 12 16 24 y 14. 9. 11. 13. 15. x 1 2 3 4 y 1 4 9 16 x 1 y -2 x 4 8 12 16 y 3 0 -3 -6 3 5 7 -18 -50 -98 16. y y x O 17. 18. y O x 19. y y x O x O 20. y = x 3 - 1 21. y = 4x 2 + 9 22. y = 0.6x 23. y = _ 24. y = _ x 25. y = _ x +5 3x 2 x O x O 8 4 26. TRAVEL The Guzman family drove from Sacramento to Yreka. Use the table to determine whether the distance driven is a linear function of the hours traveled. Explain. Time (h) 1 2 3 4 Distance (mi) 65 130 195 260 27. BUILDINGS The table shows the height of several buildings in Chicago, Illinois. Use the table to determine whether the height of the building is a linear function of the number of stories. Explain. Stories Height (ft) Harris Bank III 35 510 One Financial Place 40 515 Kluczynski Federal Building 45 545 Mid Continental Plaza 50 582 North Harbor Tower 55 556 Building Source: The World Almanac Lesson 10-1 Linear and Nonlinear Functions 525 MEASUREMENT For Exercises 28 and 29, use the following information. Recall that the circumference of a circle is equal to pi times its diameter and that the area of a circle is equal to pi times the square of its radius. 28. Is the circumference of a circle a linear or nonlinear function of its diameter? Explain your reasoning. 29. Is the area of a circle a linear or nonlinear function of its radius? Explain your reasoning. For Exercises 30–34, determine whether each equation or table represents a linear or nonlinear function. Explain. 30. y - x = 1 33. 32. y = 2 x 31. xy = -9 x 0.5 1 1.5 2 y 15 8 1 -6 34. x -4 0 y 2 1 35. FOOTBALL The graphic shows 36. MEASUREMENT Make a graph 8 -1 -4 :fcc\^\9fnc>Xd\j 8m\iX^\8kk\e[XeZ\ 6ISITORS the decrease in the average attendance at college bowl games from 1983 to 2003. Would you describe the decline as linear or nonlinear? Explain. 4 showing the area of a square as a function of its perimeter. Explain whether the function is linear. 9EAR Source: USA Today %842!02!#4)#% 37. GRAPHING Water is poured at a constant rate into the vase at the right. Draw a graph of See pages 702, 717. the water level as a function of time. Is the water level a linear or nonlinear function Self-Check Quiz at of time? Explain. ca.gr7math.com H.O.T. Problems 38. CHALLENGE True or false? All graphs of straight lines are linear functions. Explain your reasoning or provide a counterexample. 39. Which One Doesn’t Belong? Identify the function that is not linear. Explain your reasoning. y = 2x y = x2 y -2 = x x-y=2 40. OPEN ENDED Give an example of a nonlinear function using a table of values. 41. */ -!4( Describe two methods for determining whether a (*/ 83 *5*/( function is linear given its equation. 526 Chapter 10 Algebra: Nonlinear Functions and Polynomials 42. Which equation describes the data in 43. Which equation represents a nonlinear the table? function? x -7 -5 -3 0 4 y 50 1 17 A 5x + 1 = y B xy = 68 26 10 F y = 3x + 1 x G y=_ 3 H 2xy = 10 C x2 + 1 = y J 2 y = 3(x - 5) D -2x + 8 = y STATISTICS Determine whether a scatter plot of the data for the following might show a positive, negative, or no relationship. (Lesson 9-8) 44. grade on a test and amount of time spent studying 45. age and number of siblings 46. number of Calories burned and length of time exercising 49. 7k + 12 = 8 - 9k 50. 13.4w + 17 = 5w - 4 51. 8.1a + 2.3 = 5.1a - 3.1 !R AB IC 48. 1 - 3c = 9c + 7 %N GLI SH 3P AN ISH (Lesson 8-4) (I ND I Solve each equation. Check your solution. ,ANGUAGES3POKENBY.ATIVE3PEAKERS -A ND ARI N languages spoken by at least 100 million native speakers worldwide. What conclusions can you make about the number of Mandarin native speakers and the number of English native speakers? (Lesson 9-7) .ATIVE3PEAKERSMILLIONS 47. LANGUAGES The graph shows the top five ,ANGUAGES 52. 4.1x - 23 = -3.9x - 1 53. 3.2n + 3 = -4.8n - 29 Source: The World Almanac For Kids 54. PARKS A circular fountain in a park has a diameter of 8 feet. The park director wants to build a fountain that has an area four times that of the current fountain. What will be the diameter of the new fountain? (Lesson 7-1) 55. MEASUREMENT The cylindrical air duct of a large furnace has a diameter of 30 inches and a height of 120 feet. If it takes 15 minutes for the contents of the duct to be expelled into the air, what is the volume of the substances being expelled each hour? (Lesson 7-5) PREREQUISITE SKILL Graph each equation. 56. y = 2x 57. y = x + 3 (Lesson 9-2) 58. y = 3x - 2 59. y = _x + 1 1 3 Lesson 10-1 Linear and Nonlinear Functions 527 10-2 Graphing Quadratic Functions Main IDEA Graph quadratic functions. Standard 7AF1.5 Represent quantitative relationships graphically and interpret the meaning of a specific part of a graph in the situation represented by the graph. Standard 7AF3.1 Graph functions of the form y = nx 2 and y = nx 3 and use in solving problems. You know that the area A of a square is equal to the length of a side s squared, A = s 2. Copy and complete the table. s s2 (s, A) 0 0 (0, 0) 1 1 (1, 1) 2 3 Graph the ordered pairs from the table. Connect them with a smooth curve. 4 5 6 1. Is the relationship between the side length and the area of a square linear or nonlinear? Explain. 2. Describe the shape of the graph. NEW Vocabulary quadratic function A quadratic function, like A = s 2, is a function in which the greatest power of the variable is 2. Its graph is U-shaped, opening upward or downward. The graph opens upward if the number in front of the variable that is squared is positive, downward if it is negative. Graph Quadratic Functions Quadratic Fuctions The graph of a quadratic function is called a parabola. 1 Graph y = x 2. To graph a quadratic function, make a table of values, plot the ordered pairs, and connect the points with a smooth curve. y x x2 y (x, y) -2 (-2) 2 = 4 4 (-2, 4) -1 2 (-1) = 1 1 (-1, 1) 0 (0) 2 = 0 0 (0, 0) 2 1 (1) = 1 1 (1, 1) 2 (2) 2 = 4 4 (2, 4) y x2 x O 2 Graph y = -2x 2. x -2x 2 -2 -2(-2) 2 = -8 2 -1 -2(-1) = -2 0 -2(0) 2 = 0 1 -2(1) 2 = -2 2 -2(2) 2 = -8 y (x, y) -8 (-2, -8) -2 (-1, -2) 4 O ⫺8 4 ⫺4 (0, 0) ⫺4 -2 (1, -2) ⫺8 -8 (2, -8) 0 528 Chapter 10 Algebra: Nonlinear Functions and Polynomials y 8x y 2x 2 ⫺12 Extra Examples at ca.gr7math.com READING in the Content Area For more strategies in reading this lesson, visit ca.gr7math.com. 3 Graph y = x 2 + 2. x x2 + 2 -2 (-2) 2 + 2 = 6 -1 0 y (x, y) 6 (-2, 6) 2 3 (-1, 3) 2 2 (0, 2) 2 (-1) + 2 = 3 (0) + 2 = 2 1 (1) + 2 = 3 3 (1, 3) 2 (2) 2 + 2 = 6 6 (2, 6) y y x2 2 x O 4 Graph y = -x 2 + 4. x -x 2 + 4 y (x, y) -2 -(-2) 2 + 4 = 0 0 (-2, 0) -1 -(-1) 2 + 4 = 3 3 (-1, 3) 2 0 -(0) + 4 = 4 4 (0, 4) 1 -(1) 2 + 4 = 3 3 (1, 3) 2 -(2) 2 + 4 = 0 0 (2, 0) y y x2 4 x O Graph each function. a. y = 6x 2 b. y = x 2 - 2 c. y = -2x 2 - 1 5 MONUMENTS The function h = 0.66d 2 represents the distance d in miles you can see from a height of h feet. Graph this function. Then use your graph and the information at the left to estimate how far you could see from the top of the Eiffel Tower. Distance cannot be negative, so use only positive values of d. (d, h) 0 0.66(0) 2 = 0 (0, 0) 10 0.66(10) 2 = 66 (10, 66) Source: structurae.de 1,000 Height (ft) Real-World Link The Eiffel Tower in Paris, France, opened in 1889 as part of the World Exposition. It is about 986 feet tall. h h = 0.66d 2 d 800 20 2 0.66(20) = 264 (20, 264) 25 2 0.66(25) = 412.5 (25, 412.5) 30 0.66(30) 2 = 594 (30, 594) 400 35 0.66(35) 2 = 808.5 (35, 808.5) 200 40 2 (40, 1,056) 0.66(40) = 1,056 600 0 10 20 30 40 d Distance (mi) At a height of 986 feet, you could see approximately 39 miles. d. TOWERS The outdoor observation deck of the Space Needle in Seattle, Washington, is 520 feet above ground level. Estimate how far you could see from the observation deck. Personal Tutor at ca.gr7math.com Lesson 10-2 Graphing Quadratic Functions Lance Nelson/CORBIS 529 Examples 1–4 (pp. 528–529) Example 5 (p. 529) (/-%7/2+ (%,0 For Exercises 8–11 12–19 20, 21 See Examples 1, 2 3, 4 5 Graph each function. 1. y = 3x 2 2. y = -5x 2 3. y = -4x 2 4. y = -x 2 + 1 5. y = x 2 - 3 6. y = -x 2 + 2 7. CARS The function d = 0.006s 2 represents the braking distance d in meters of a car traveling at a speed s in kilometers per second. Graph this function. Then use your graph to estimate the speed of the car if its braking distance is 12 meters. Graph each function. 8. y = 4x 2 9. y = 5x 2 10. y = -3x 2 11. y = -6x 2 12. y = x 2 + 6 13. y = x 2 - 4 14. y = -x 2 + 2 15. y = -x 2 - 5 16. y = 2x 2 - 1 17. y = 2x 2 + 3 18. y = -4x 2 - 1 19. y = -3x 2 + 2 20. RACING The function d = _at 2 represents the distance d that a race car will 1 2 travel over an amount of time t given the rate of acceleration a. Suppose a car is accelerating at a rate of 5 feet per second every second. Graph this function. Then use your graph to find the time it would take the car to travel 125 feet. 21. WATERFALLS The function d = -16t 2 + 182 models the distance d in feet a drop of water falls t seconds after it begins its descent from the top of the 182-foot high American Falls in New York. Graph this function. Then use your graph to estimate the time it will take the drop of water to reach the river at the base of the falls. Graph each function. 22. y = 0.5x 2 + 1 23. y = 1.5x 2 24. y = 4.5x 2 - 6 25. y = _x 2 - 2 26. y = _x 2 27. y = -_x 2 + 1 1 3 1 2 1 4 MEASUREMENT For Exercises 28 and 29, write a function for each of the following. Then graph the function in the first quadrant. %842!02!#4)#% See pages 702, 717. 28. The surface area of a cube is a function of the edge length a. Use your graph to estimate the edge length of a cube with a surface area of 54 square centimeters. 29. The volume V of a rectangular prism with a square base and a fixed height Self-Check Quiz at ca.gr7math.com of 5 inches is a function of the base edge length s. Use your graph to estimate the base edge length of a prism whose volume is 180 cubic inches. 530 Chapter 10 Algebra: Nonlinear Functions and Polynomials H.O.T. Problems CHALLENGE The graphs of quadratic functions may have exactly one highest point, called a maximum, or exactly one lowest point, called a minimum. Graph each quadratic equation. Determine whether each graph has a maximum or a minimum. If so, give the coordinates of each point. 30. y = 2x 2 + 1 31. y = -x 2 + 5 32. y = x 2 - 3 33. OPEN ENDED Write and graph a quadratic function that opens upward and has its minimum at (0, -3.5). */ -!4( Write a quadratic function of the form y = ax 2 + c and (*/ 83 *5*/( 34. explain how to graph it. 35. Which graph represents the function y = -0.5x 2 - 2? y A O y B y C x O x x Determine whether each equation represents a linear or nonlinear function. 37. y = 3x 3 + 2 x O O 36. y = x - 5 y D 38. x + y = -6 (Lesson 10-1) 39. y = -2x 2 STATISTICS For Exercises 40–42, use the information at the right. (Lesson 9-8) Year Population 40. Draw a scatter plot of the data and draw a line of fit. 2000 172 41. Does the scatter plot show a positive, negative, or no relationship? 42. Use your graph to estimate the population of the whooping crane at the refuge in 2005. Whooping Cranes 2001 171 2002 181 2003 194 2004 197 43. SAVINGS Anna’s parents put $750 into a college savings account. After 6 years, the investment had earned $540. Write an equation that you could use to find the simple interest rate. Then find the simple interest rate. (Lesson 5-9) 44. PREREQUISITE SKILL A section of a theater is arranged so that each row has the same number of seats. You are seated in the 5th row from the front and the 3rd row from the back. If your seat is 6th from the left and 2nd from the right, how many seats are in this section of the theater? Use the draw a diagram strategy. (Lesson 4-4) Lesson 10-2 Graphing Quadratic Functions 531 10-3 Problem-Solving Investigation MAIN IDEA: Solve problems by making a model. Standard 7MR2.5 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. Standard 7AF1.1 Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g. three less than a number, half as large as area A. e-Mail: MAKE A MODEL YOUR MISSION: Make a model to solve the problem. THE PROBLEM: Determine if there are enough tables to make a 10-by-10 square arrangement. EXPLORE PLAN ▲ Tonya: We have 35 square tables. We need to arrange them into a square that is open in the middle and has 10 tables on each side. You know Tonya has 35 square tables. Start by making models of a 4-by-4 square and of a 5-by-5 square. Then look for a pattern. SOLVE {LÞ{ ÃµÕ>Ài CHECK ÓÊ}ÀÕ«ÃÊvÊ{Ê>` ÓÊ}ÀÕ«ÃÊvÊÓ xLÞx ÃµÕ>Ài ÓÊ}ÀÕ«ÃÊvÊxÊ>`Ê ÓÊ}ÀÕ«ÃÊvÊÎ For a 10-by-10 square, Tonya needs 2 · 10 + 2 · 8 or 36 tables. She has 35 tables, so she needs one more. You can estimate that Tonya needs 4 × 10 or 40 tables. But each of the corner tables is counted twice. So, she needs 40 - 4 or 36 tables. 1. Draw a diagram showing another way the students could have grouped the tiles to solve this problem. Use a 4-by-4 square. 2. */ -!4( Write a problem that can be solved by making a (*/ 83 *5*/( model. Describe the model. Then solve the problem. 532 Chapter 10 Algebra: Nonlinear Functions and Polynomials Laura Sifferlin For Exercises 3–5, solve by making a model. 3. STICKERS In how many different ways can three rectangular stickers be torn from a sheet of 3 × 3 stickers so that all three stickers are still attached? Draw each arrangement. 4. MEASUREMENT A 10-inch by 12-inch piece of cardboard has a 2-inch square cut out of each corner. Then the sides are folded up and taped together to make an open box. Find the volume of the box. 8. PETS Mrs. Harper owns both cats and canaries. Altogether, her pets have thirty heads and eighty legs. How many cats does she have? GEOMETRY For Exercises 9 and 10, use the figure at the right. 9. How many cubes would it take to build this tower? 10. How many cubes would it take to build a similar tower that is 12 cubes high? 5. GEOMETRY A computer game requires players to stack arrangements of five squares arranged to form a single shape. One arrangement is shown at the right. How many different arrangements are there if touching squares must border on a full side? 11. CARS Yesterday you noted that the mileage on the family car read 60,094.8 miles. Today it reads 60,099.1 miles. Was the car driven about 4 or 40 miles? Use any strategy to solve Exercises 6–11. Some strategies are shown below. For Exercises 12 and 13, select the appropriate operation(s) to solve the problem. Justify your selections(s) and solve the problem. G STRATEGIES PROBLEM-SOLVIN tep plan. • Use the four-s m. • Draw a diagra eck. • Guess and Ch 12. SCIENCE The light in the circuit will turn on if one or more switches are closed. How many combinations of open and closed switches will result in the light being on? l. • Make a mode 6. CAMP The camp counselor lists 21 chores on separate pieces of paper and places them in a basket. The counselor takes one piece of paper, and each camper takes one as the basket is passed around the circle. There is one piece of paper left when the basket returns to the counselor. How many people could be in the circle if the basket goes around the circle more than once? 7. PARKING Parking space numbers consist of 3 digits. They are typed on a slip of paper and given to students at orientation. Tara accidentally read her number upside-down. The number she read was 795 more than her actual parking space number. What is Tara’s parking space number? a b c d e 13. HOBBIES Lorena says to Angela, “If you give me one of your baseball cards, I will have twice as many baseball cards as you have.” Angela answers, “If you give me one of your cards, we will have the same number of cards.” How many cards does each girl have? Lesson 10-3 Problem-Solving Investigation: Make a Model 533 10-4 Graphing Cubic Functions Main IDEA Standard 7AF3.1 Graph functions of the form y = nx 2 and y = nx 3 and use in solving problems. Standard 7AF3.2 Plot the values from the volumes of three-dimensional shapes for various values of the edge lengths (e.g., cubes with varying edge lengths or a triangle prism with a fixed height and an equilateral triangle base of varying lengths). MEASUREMENT You can find the area A of a square by squaring the length of a side s. This relationship can be represented in different ways. Words and Equation Area A s2 = s Table length of a side squared. equals s s2 (s, A) 2 0 =0 (0, 0) 2 s 0 Graph 1 1 =1 (1, 1) 2 22 = 4 (2, 4) A A s2 Area Graph cubic functions. s O Side 1. The volume V of a cube is found by cubing the length of a side s. Write a formula to represent the volume of a cube as a function of side length. s s s 2. Graph the volume as a function of side length. (Hint: Use values of s such as 0, 0.5, 1, 1.5, 2, and so on.) 3. Would it be reasonable to use negative numbers for x-values in this situation? Explain. You can graph cubic functions such as the formula for the volume of a cube by making a table of values. Graph a Cubic Function 1 Graph y = x 3. x y = x3 (x, y) -1.5 (-1.5) 3 ≈ -3.4 (-1.5, -3.4) -1 Graphing It is often helpful to substitute decimal values of x in order to graph points that are closer together. 3 (-1, -1) 3 (-1) = -1 0 (0) = 0 (0, 0) 1 (1) 3 = 1 (1, 1) 1.5 (1.5) 3 ≈ 3.4 (1.5, 3.4) y O Graph each function. a. y = x 3 - 1 b. y = -4x 3 Personal Tutor at ca.gr7math.com 534 Chapter 10 Algebra: Nonlinear Functions and Polynomials c. y = x 3 + 4 x 2 PACKAGING A packaging company wants to manufacture a cardboard box with a square base of side length x inches and a height of (x – 3) inches as shown. Real-World Link Packaging is the nation’s third largest industry, with over $130 billion in sales each year. Source: San Jose State University (x 3) in. x in. x in. Write the function for the volume V of the box. Graph the function. Then estimate the dimensions of the box that would give a volume of approximately 8 cubic inches. V = lwh Volume of a rectangular prism V = x · x · (x – 3) Replace l with x, w with x, and h with (x – 3). V = x 2(x – 3) x · x = x2 V = x 3 – 3x 2 Distributive Property The function for the volume V of the box is V = x 3 – 3x 2. Make a table of values to graph this function. You do not need to include negative values of x since the side length of the box cannot be negative. x 3 2 (x, V) 0 (0) – 3(0) = 0 (0, 0) 0.5 (0.5) 3 – 3(0.5) 2 ≈ –0.6 (0.5, –0.6) 3 2 1 (1) – 3(1) = –2 (1, –2) 1.5 (1.5) 3 – 3(1.5) 2 ≈ –3.8 (1.5, –3.8) 2 Analyze the Graph Notice that the graph is below the x-axis for values of x < 3. This means that the “volume” of the box is negative for x < 3. To have a box with a positive height and a positive volume, x must be greater than 3. V = x 3 – 3x 2 3 2 (2) – 3(2) = –4 3 2 (2, –4) 2.5 (2.5) – 3(2.5) ≈ –3.1 (2.5, –3.1) 3 (3) 3 – 3(3) 2 = 0 (3, 0) 3 2 3.5 (3.5) – 3(3.5) ≈ 6.1 (3.5, 6.1) 4 (4) 3 – 3(4) 2 = 16 (4, 16) 20 y 18 16 14 12 10 8 6 4 2 0 2 4 y x 3 2x 2 x 2 4 6 8 10 12 Looking at the graph, we see that the volume of the box is approximately 8 cubic inches when x is about 3.6 inches. The dimensions of the box when the volume is about 8 cubic inches are 3.6 inches, 3.6 inches, and 3.6 – 3 or 0.6 inch. d. PACKAGING A packaging company wants to manufacture a cardboard box with a square base of side length x feet and a height of (x – 2) feet. Write the function for the volume V of the box. Graph the function. Then estimate the dimensions of the box that would give a volume of about 1 cubic foot. Extra Examples at ca.gr7math.com Getty Images Lesson 10-4 Graphing Cubic Functions 535 Example 1 (p. 534) Example 2 (p. 535) (/-%7/2+ (%,0 For Exercises 6–17 18, 19 See Examples 1 2 Graph each function. 1. y = -x 3 2. y = 0.5x 3 3. y = x 3 – 2 4. y = 2x 3 + 1 5. MEASUREMENT A rectangular prism with a square base of side length x centimeters has a height of (x + 1) centimeters. Write the function for the volume V of the prism. Graph the function. Then estimate the dimensions of the box that would give a volume of approximately 9 cubic centimeters. Graph each function. 6. y = -2x 3 7. y = -3x 3 8. y = 0.2x 3 9. y = 0.1x 3 10. y = x 3 + 1 11. y = 2x 3 + 1 12. y = x 3 – 3 13. y = 2x 3 – 2 14. y = _ x 3 15. y = _ x 3 + 2 16. y = -x 3 – 2 17. y = -x 3 + 1 1 4 1 3 18. MEASUREMENT Jorge built a scale model of the Great %842!02!#4)#% See pages 703, 717. Self-Check Quiz at ca.gr7math.com Pyramid. The base of the model is a square with side length s and the model’s height is (s – 1) feet. Write the function for the volume V of the model. Graph this function. Then estimate the length of one side of the square base of the model if the model’s volume is approximately 8 cubic feet. 19. MEASUREMENT The formula for the volume V of a tennis ball is given by 4 3 the equation V = _ πr where r represents the radius of the ball. Graph 3 this function. Use 3.14 for π. Then estimate the length of the radius if the volume of the tennis ball is approximately 11 cubic inches. Graph each pair of equations on the same coordinate plane. Describe their similarities and differences. 20. y = x 3 y = 3x 21. y = x 3 3 3 y=x –3 22. y = 0.5x 3 y = 2x 23. y = 2x 3 3 y = -2x 3 FARMING For Exercises 24 and 25, use the following information. A grain silo consists of a cylindrical main section and a hemispherical roof. The cylindrical main section has a radius of r units and a height h equivalent to the radius. The volume V of a cylinder is given by the equation V = πr 2h. 24. Write the function for the volume V of the cylindrical main section of the grain silo in terms of its radius r. 25. Graph this function. Use 3.14 for π. Then estimate the radius and height in meters of the cylindrical main section of the grain silo if the volume is approximately 15.5 cubic meters. 536 Chapter 10 Algebra: Nonlinear Functions and Polynomials r H.O.T. Problems 26. OPEN ENDED Write the equation of a cubic function whose graph in the first quadrant shows faster growth than the function y = x 3. CHALLENGE The zero of a cubic function is the x-coordinate at which the function crosses the x-axis. Find the zeros of each function below. 27. y = x 3 29. 28. y = x 3 + 1 */ -!4( The volume V of a cube with side length s is given by (*/ 83 *5*/( the equation V = s 3. Explain why negative values are not necessary when creating a table or a graph of this function. 30. Which equation could represent the graph shown below? 31. Which equation could represent the graph shown below? y O y O x A y = x3 F y = x3 – 2 B y = -x 3 G y = x3 + 2 C y = 2x 3 H y = -2x 3 D y = -2x 3 J x y = 2x 3 + 1 32. MANUFACTURING A company packages six small books for a children’s collection in a decorated 4-inch cube. They are shipped to bookstores in cartons. Twenty cubes fit in a carton with no extra space. What are the dimensions of the carton? Use the make a model strategy. (Lesson 10-3) Graph each function. (Lesson 10-2) 33. y = -2x 2 34. y = x 2 + 3 35. y = -3x 2 + 1 Estimate each square root to the nearest whole number. 37. √ 54 38. - √ 126 (Lesson 3-2) 39. √ 8.67 PREREQUISITE SKILL Write each expression using exponents. 41. 3 · 3 · 3 · 3 · 3 42. 5 · 4 · 5 · 5 · 4 43. 7 · (7 · 7) 36. y = 4x 2 + 3 40. - √ 19.85 (Lesson 2-9) 44. (2 · 2) · (2 · 2 · 2) Lesson 10-4 Graphing Cubic Functions 537 Extend 10-4 Main IDEA Use a graphing calculator to graph families of nonlinear functions. Standard 7AF3.1 Graph functions of the form y = nx 2 and y = nx 3 and use in solving problems. Standard 7MR3.3 Develop generalizations of the results obtained and the strategies used and apply them to new problem situations. Graphing Calculator Lab Families of Nonlinear Functions Families of nonlinear functions share a common characteristic based on a parent function. The parent function of a family of quadratic functions is y = x 2. You can use a graphing calculator to investigate families of quadratic functions. Graph y = x 2, y = x 2 + 5, and y = x 2 - 3 on the same screen. Clear any existing equations from the Y= list by pressing CLEAR . Enter each equation. Press X,T,,n ENTER , X,T,,n 5 ENTER , and X,T,,n 3 ENTER . Graph the equations in the standard viewing window. Press ZOOM 6. ANALYZE THE RESULTS 1. Compare and contrast the three equations you graphed. 2. Describe how the graphs of the three equations are related. 3. MAKE A CONJECTURE How does changing the value of c in the equation y = x 2 + c affect the graph? 4. Use a graphing calculator to graph y = 0.5x 2, y = x 2, and y = 2x 2. 5. Compare and contrast the three equations you graphed in Exercise 4. 6. Describe how the graphs of the three equations are related. 7. MAKE A CONJECTURE How does changing the value of a in the equation y = ax 2 affect the graph? 8. Use a graphing calculator to graph y = 0.5x 3, y = x 3, and y = 2x 3. 9. Compare and contrast the three equations you graphed in Exercise 8 to the equations you graphed in Exercise 4. 538 Chapter 10 Nonlinear Functions and Polynomials Other Calculator Keystrokes at ca.gr7math.com 10-5 Multiplying Monomials Main IDEA Multiply monomials. Standard 7NS2.3 Multiply, divide, and simplify rational numbers by using exponent rules. Standard 7AF2.1 Interpret positive whole-number powers as repeated multiplication and negative whole-number powers as repeated division or multiplication by the multiplicative inverse. Simplify and evaluate expressions that include exponents. Standard 7AF2.2 Multiply and divide monomials; extend the process of taking powers and extracting roots to monomials when the latter results in a monomial with an integer exponent. SCIENCE The pH of a solution describes its acidity. Neutral water has a pH of 7. Lemon juice has a pH of 2. Each one-unit decrease in the pH means that the solution is 10 times more acidic. So, a pH of 8 is 10 times more acidic than a pH of 9. pH Times More Acidic Than a pH of 9 Written Using Powers 8 10 10 1 7 10 × 10 = 100 10 1 × 10 1 = 10 2 6 10 × 10 × 10 = 1,000 10 1 × 10 2 = 10 3 5 10 × 10 × 10 × 10 = 10,000 10 1 × 10 3 = 10 4 4 10 × 10 × 10 × 10 × 10 = 100,000 10 1 × 10 4 = 10 5 1. Examine the exponents of the factors and the exponents of the products in the last column. What do you observe? A monomial is a number, a variable, or a product of a number and one or more variables. Exponents are used to show repeated multiplication. You can use this fact to find a rule for multiplying monomials. 2 factors 4 factors NEW Vocabulary 3 2 · 3 4 = (3 · 3) · (3 · 3 · 3 · 3) or 3 6 monomial 6 factors Notice that the sum of the original exponents is the exponent in the final product. This relationship is stated in the following rule. +%9#/.#%04 Product of Powers To multiply powers with the same base, add their exponents. Words Examples Numbers 4 3 2 ·2 =2 4+3 Algebra or 2 7 m a · an = am + n Multiply Powers 1 Find 5 2 · 5. Express using exponents. Common Error When multiplying powers, do not multiply the bases. 4 5 · 4 2 = 4 7, not 16 7. 52 · 5 = 52 · 51 = 52 + 1 =5 3 5 = 51 Check 5 2 · 5 = (5 · 5) · 5 The common base is 5. =5·5·5 Add the exponents. = 53 Lesson 10-5 Multiplying Monomials CORBIS 539 2 Find -3x 2(4x 5). Express using exponents. -3x 2(4x 5) = (-3 · 4)(x 2 · x 5) Commutative and Associative Properties = (-12)(x 2 + 5) = -12x The common base is x. 7 Add the exponents. Multiply. Express using exponents. a. 9 3 · 9 2 b. 2 (_35 ) (_35 ) 9 c. -2m(-8m 5) 3 The population of Groveton is 6 5. The population of Putnam is 6 3 times as great. How many people are in Putnam? Real-World Link A census is taken every ten years by the U.S. Census Bureau to determine population. The government uses the data from the census to make many decisions. The population of Putnam is 6 8 or 1,679,616 people. Source: census.gov d. RIVERS The Guadalupe River is 2 8 miles long. The Amazon River is To find out the number of people, multiply 6 5 by 6 3. 6 5 · 6 3 = 6 5+3 or 6 8 Product of Powers almost 2 4 times as long. Find the length of the Amazon River. Personal Tutor at ca.gr7math.com In Lesson 2-9, you learned to evaluate negative exponents. Remember that any nonzero number to the negative n power is the multiplicative inverse of that number to the n th power. The Product of Powers rule can be used to multiply powers with negative exponents. Multiply Negative Powers 4 Find x 4 · x -2. Express using exponents. METHOD 1 METHOD 2 x 4 · x -2 = x 4 + (-2) The common base is x. =x 2 Add the exponents. x 4 · x -2 1 _ 1 1 _ -2 =x·x·x·x·_ x · x x = x2 = x2 Simplify. Simplify. Express using positive exponents. d. 3 8 · 3 -2 e. n 9 · n -4 540 Chapter 10 Algebra: Nonlinear Functions and Polynomials Prisma/SuperStock f. 5 -1 · 5 -2 Extra Examples at ca.gr7math.com Examples 1–4 (pp. 539–540) Example 3 Simplify. Express using exponents. 1. 4 5 · 4 3 2. n 2 · n 9 3. -2a(3a 4) 4. 5 2x 2y 4 · 5 3xy 3 5. r 7 · r -3 6. 6m · 4m 2 7. AGE Angelina is 2 3 years old. Her grandfather is 2 3 times her age. How old (p. 540) (/-%7/2+ (%,0 For Exercises 8–25 26–28 See Examples 1, 2, 4 3 is her grandfather? Simplify. Express using exponents. 8. 6 8 · 6 5 9. 2 9 · 2 10. n · n 7 11. b 13 · b 12. 2g · 7g 6 13. (3x 8)(5x) 14. -4a 5(6a 5) 15. (8w 4)(-w 7) 16. (-p)(-9p 2) 17. -5y 3(-8y 6) 18. 4m -2n 5(3m 4n -2) 19. (-7a 4bc 3)(5ab 4c 2) 20. x 6 · x -3 21. y -1 · y 4 22. z -2 · z -3 23. m 2n -1 · m -3n 3 24. 3f -4 · 5f 2 25. -3ab · 4a -3b 3 26. INSECTS The number of ants in a nest was 5 3. After the eggs hatched, the number of ants increased 5 2 times. How many ants are there after the eggs hatch? 27. COMPUTERS The processing speed of a certain computer is 10 11 insructions per second. Another computer has a processing speed that is 10 3 times as fast. How many instructions per second can the faster computer process? 28. LIFE SCIENCE A cell culture contains 2 6 cells. By the end of the day, there are 2 10 times as many cells in the culture. How many cells are there in the culture by the end of the day? Simplify. Express using exponents. %842!02!#4)#% See pages 703, 717. 29. xy 2(x 3y) 3 32. (_23 ) (_23 ) 35. (_14 ) (_14 ) Self-Check Quiz at ca.gr7math.com 4 30. 2 6 · 2 · 2 3 -4 -5 33. (_78 ) (_78 ) 36. (_25 ) (_25 ) 3 31. 4a 2b 3(7ab 2) 13 -2 4 -7 (_25 ) (_25 ) (_25 ) 2 37. (_) (_72 ) 7 34. -2 6 -3 Lesson 10-5 Multiplying Monomials 541 38. CHALLENGE What is twice 2 30? Write using exponents. H.O.T. Problems 39. OPEN ENDED Write a multiplication expression whose product is 4 15. */ -!4( Determine whether the following statement is true or (*/ 83 *5*/( 40. false. Explain your reasoning or give a counterexample. If you change the order in which you multiply two monomials, the product will be different. 41. Which expression is equivalent to 2 42. Which expression describes the area in 2 square feet of the rectangle below? 2 2 A 64x y z F 11x 10 B 64x 2 yz 2 G 30x 10 C 16x 2 y 2z 2 H 11x 16 D 384x 2 y 2z 2 J 8x y · 8yz ? 2 Graph each function. 2 5x ft 8 6x ft 30x 16 (Lessons 10-2 and 10-4) 43. y = -x 3 44. y = 0.5x 3 45. y = x 3 - 2 46. y = 5x 2 47. y = x 2 + 5 48. y = x 2 – 4 49. BIOLOGY The table shows how long it took for the first 400 bacteria cells to grow in a petri dish. Is the growth of the bacteria a linear function of time? Explain. (Lesson 10-1) Express each number in scientific notation. Time (min) 46 53 57 60 Number of cells 100 200 300 400 (Lesson 2-10) 50. The flow rate of some Antarctic glaciers is 0.00031 mile per hour. 51. A human blinks about 6.25 million times a year. ALGEBRA Solve each equation. Check your solution. 52. k - 4.1 = -9.38 Find each sum or difference. Write in simplest form. 3 7 55. _ - _ 8 10 PREREQUISITE SKILL 59. 3 · 3 · 3 · 3 (Lesson 2-7) 3 1 53. 1_ + p = -6_ 2 4 5 1 56. -_ + _ 12 5 54. 61. 7 · (7 · 7) 542 Chapter 10 Algebra: Nonlinear Functions and Polynomials 10 (Lesson 2-6) 2 1 57. 9_ + _ 3 6 Write each expression using exponents. 60. 5 · 4 · 5 · 5 · 4 c _ = 0.845 58. -2_ - 1_ 3 4 1 8 (Lesson 2-9) 62. (2 · 2) · (2 · 2 · 2) CH APTER Mid-Chapter Quiz 10 Lessons 10-1 through 10-5 Determine whether each equation or table represents a linear or nonlinear function. Explain. (Lesson 10-1) STANDARDS PRACTICE Which graph shows y = x 2 + 1? (Lesson 10-2) 11. A C y y 1. 3y = x 2. y = 5x 3 + 2 3. 4. 1 x 3 5 7 y -5 -6 -7 -8 x -1 0 1 2 y 1 0 1 4 O O B x x D y O y O x x 5. LONG DISTANCE The graph shows the amount of data transferred as a function of time. Is this a linear or nonlinear function? Explain your reasoning. (Lesson 10-1) 12. MEASUREMENT Brenda has a photograph that is 10 inches by 13 inches. She decides $ATA4RANSFER 1 to frame it, using a frame that is 2_ inches 4 wide on each side. Find the total area of the framed photograph. Use the make a model strategy. (Lesson 10-3) 'IGABYTES Graph each function. (Lesson 10-4) 13. y = -2x 3 4IMEMIN Graph each function. 6. y = 2x (Lesson 10-2) 2 2 7. y = -x + 3 2 14. y = 3x 3 15. y = 2x 3 16. y = 0.1x 3 Simplify. Express using exponents. 4 17. 10 · 10 (Lesson 10-4) 7 8. y = 4x - 1 18. 3 -3 · 3 5 · 3 2 9. y = -3x 2 + 1 19. 2 3a 7 · 2a -3 20. (3 2xy 4z 2)(3 5x 3y -2z 3) 10. AMUSEMENT PARK RIDES Your height h feet above the ground t seconds after being released at the top of a free-fall ride is given by the function h = -16t 2 + 200. Graph this function. After about how many seconds will the ride be 60 feet above the ground? (Lesson 10-2) 21. STANDARDS PRACTICE Which expression below has the same value as 5m 2? (Lesson 10-5) F 5m H 5·5·m·m G 5·m·m J 5·m·m·m 10-6 Dividing Monomials Main IDEA Divide monomials. Standard 7NS2.3 Multiply, divide, and simplify rational numbers by using exponent rules. Standard 7AF2.1 Interpret positive whole-number powers as repeated multiplication and negative whole-number powers as repeated division or multiplication by the multiplicative inverse. Simplify and evaluate expressions that include exponents. Standard 7AF2.2 Multiply and divide monomials; extend the process of taking powers and extracting roots to monomials when the latter results in a monomial with an integer exponent. NUMBER SENSE Refer to the table shown that relates division sentences using the numbers 2, 4, 8, and 16, and the same sentences written using powers of 2. 1. Examine the exponents of the divisors Division Sentence Written Using Powers of 2 4÷2=2 22 ÷ 21 = 21 8÷2=4 23 ÷ 21 = 22 8÷4=2 23 ÷ 22 = 21 16 ÷ 2 = 8 24 ÷ 21 = 23 16 ÷ 4 = 4 24 ÷ 22 = 22 16 ÷ 8 = 2 24 ÷ 23 = 21 and dividends. Compare them to the exponents of the quotients. What do you notice? 2. MAKE A CONJECTURE Write the quotient of 2 5 and 2 2 using powers of 2. As you learned in Lesson 10-5, exponents are used to show repeated multiplication. You can use this fact to find a rule for dividing powers with the same base. 7 factors Notice that the difference of the original exponents is the exponent in the final quotient. This relationship is stated in the following rule. 57 5·5·5·5·5·5·5 _ = __ or 5 3 5·5·5·5 54 4 factors +%9#/.#%04 Quotient of Powers To divide powers with the same base, subtract their exponents. Words Examples Numbers Algebra 37 _ = 3 7 – 3 or 3 4 am _ = a m – n, where a ≠ 0 an 33 Divide Powers Simplify. Express using exponents. n9 2 _ 4 48 1 _ 2 4 Common Error When dividing powers, do not divide n 48 _ = 48 – 2 42 =4 6 The common base is 4. Simplify. n9 _ = n9 – 4 n4 =n The common base is n. 5 Simplify. 48 4 the bases. _2 = 4 6, not 1 6. Simplify. Express using exponents. a. 57 _ 54 b. x 10 _ 544 Chapter 10 Algebra: Nonlinear Functions and Polynomials x3 c. 12w 5 _ 2w The Quotient of Powers rule can also be used to divide powers with negative exponents. It is customary to write final answers using positive exponents. Look Back To review adding and subtracting integers, see Lessons 1-4 and 1-5. Use Negative Exponents Simplify. Express using positive exponents. 69 3 _ -3 6 69 _ = 6 9 – (-3) Quotient of Powers 6 -3 = 6 9 + 3 or 6 12 Simplify. w -1 4 _ -4 w w -1 _ = w -1 – (-4) Quotient of Powers w -4 = w -1 + 4 or w 3 Simplify. Simplify. Express using positive exponents. d. 11 -8 _ 11 e. 2 b -4 _ f. b -7 6h 5 _ 3h -5 22 · 45 · 52 5 _ = 5 4 2 2 ·4 ·5 A 2 Remember that the Quotient of Powers Rule allows you to Read the Item 52 _ = 5 2 - 2 = 5 0 = 1. Solve the Item 52 5 simplify _2 . 52 1 C _ B 1 D 0 2 You are asked to divide one monomial by another. ( )( )( ) 22 · 45 · 52 52 22 _ 45 _ _ = _ 25 · 44 · 52 25 44 52 Group by common base. = 2 -3 · 4 1 · 5 0 Subtract the exponents. 1 =_ ·4·1 3 2 -3 = _3 4 1 =_ or _ Simplify. 2 8 2 1 2 The answer is C. Extra Examples at ca.gr7math.com Lesson 10-6 Dividing Monomials 545 (_16 ) × (_16 ) __ g. Simplify . 1 _ (6) -12 4 -3 1 F _ (6) 5 1 G _ H 64 6 J 65 Personal Tutor at ca.gr7math.com 6 SOUND The loudness of a conversation is 10 6 times as intense as the loudness of a pin dropping, while the loudness of a jet engine is 10 12 times as intense. How many times more intense is the loudness of a jet engine than the loudness of a conversation? Real-World Link The decibel measure of the loudness of a sound is the exponent of its relative intensity multiplied by 10. A jet engine has a loudness of 120 decibels. To find how many times more intense, divide 10 12 by 10 6. 10 12 _ = 10 12 – 6 or 10 6 10 6 Quotient of Powers The loudness of a jet engine is 10 6 or 1,000,000 times as intense as the loudness of a conversation. h. SOUND The loudness of a vacuum cleaner is 10 4 times as intense as the loudness of a mosquito buzzing, while the loudness of a jack hammer is 10 9 times as intense. How many times more intense is the loudness of a jack hammer than that of a vacuum cleaner? Personal Tutor at ca.gr7math.com Examples 1–4 Simplify. Express using positive exponents. (pp. 544-545) Example 5 (p. 545) 1. 76 _ 5. 9c 7 _ 7 3c 2 (p. 546) 29 _ 6. 24k 9 _ 2 13 6k 6 y _ 7. 15 -6 _ y 5 15 2 4. z _ 8. 35p _ z2 1 5p -4 22 · 33 · 44 2·3 ·4 B 2 1 C _ 2 1 D _ (2) 2 10. ASTRONOMY Venus is approximately 10 8 kilometers from the Sun. Saturn is more than 10 9 kilometers from the Sun. About how many times farther away from the Sun is Saturn than Venus? 546 Chapter 10 Algebra: Nonlinear Functions and Polynomials Mug Shots/Corbis 8 3. 9. Simplify _ . 3 5 A 22 Example 6 2. (/-%7/2+ (%,0 For Exercises 11–26 27–30 31–34 See Examples 1–4 5 6 Simplify. Express using positive exponents. 11. 8 15 _ 12. 29 _ 15. h7 _ 16. g _ 19. 36d 10 _ 20. 23. 22 -9 _ 27. x y _ 84 h6 6d 5 22 4 13. 43 _ 14. 13 2 _ 17. x8 _ 18. n _ 16t 4 _ 21. 20m 7 _ 22. 75r 6 _ 24. 3 -1 _ 25. 42w -6 _ 26. 12y _ 28. 63 · 66 · 64 _ = 29. (5) (5) __ 30. 3x 4 _ 18 6 14 x 4y 9 2 g 6 8t 3 -5 62 · 63 · 64 47 x 11 5m 5 n8 25r 5 -6 7w -2 _1 2 × _1 (_15 ) 13 5 2y -10 -6 2 3 4x -2 31. POPULATION The continent of North America contains approximately 10 7 square miles of land. If the population doubles, there will be about 10 9 people on the continent. At that point, on average, how many people will occupy each square mile of land? 32. FOOD An apple is 10 3 times as acidic as milk, while a lemon is 10 4 times as acidic. How many times more acidic is a lemon than an apple? 33. ANIMALS A common flea 2 -4 inch long can jump about 2 3 inches high. About how many times its body size can a flea jump? 34. MEDICINE The mass of a molecule of penicillin is 10 -18 kilograms and the mass of a molecule of insulin is 10 -23 kilograms. How many times greater is a molecule of penicillin than a molecule of insulin? Find each missing exponent. 35. 17 _ = 17 8 17 4 36. k6 _ = k2 k 37. 5 _ = 53 5 -9 ANALYZE TABLES For Exercises 39 and 40, use the information below and in the table. %842!02!#4)#% See pages 703, 717. Self-Check Quiz at ca.gr7math.com For each increase of one on the Richter scale, an earthquake’s vibrations, or seismic waves, are 10 times greater. -1 38. Earthquake p _ = p 10 p Richter Scale Magnitude San Francisco, 1906 8.3 Adana, Turkey, 1998 6.3 Source: usgs.gov 39. How many times greater are the seismic waves of an earthquake with a magnitude of 6 than an aftershock with a magnitude of 3? 40. How many times greater were the seismic waves of the 1906 San Francisco earthquake than the 1998 Adana earthquake? Lesson 10-6 Dividing Monomials 547 3 100 3 41. NUMBER SENSE Is _ greater than, less than, or equal to 3? Explain your 99 H.O.T. Problems reasoning. 42. OPEN ENDED Write a division expression with a quotient of 4 15. 43. CHALLENGE What is half of 2 30? Write using exponents. 44. */ -!4( Explain why the Quotient of Powers Rule cannot (*/ 83 *5*/( 5 x be used to simplify the expression _ . 2 y 45. Which expression below is equivalent 8 47. One meter is 10 3 times longer than one millimeter. One kilometer is 10 6 times longer than one millimeter. How many times longer is one kilometer than one meter? 9m to _ ? 2 3m A 6m 4 C 3m 4 B 6m 6 D 3m 6 A 10 9 46. The area of a rectangle is 2 6 square B 10 6 feet. If the length is 2 3 feet, find the width of the rectangle. F 2 feet H 2 3 feet G 2 2 feet J D 10 2 9 feet Simplify. Express using positive exponents. 4 48. 6 · 6 7 Graph each function. 3 52. y = x + 2 C 10 3 3 49. 18 · 18 -5 (Lesson 10-5) 50. (-3x 11)(-6x 3) 51. (-9a 4)(2a -7) – 54. y = -2x 3 55. y = -0.1x 3 (Lesson 10-4) 53. y = _ x 3 1 3 State the slope and the y-intercept for the graph of each equation. 56. y = x – 3 57. 2 y=_ x+7 3 58. 3x + 4y = 12 (Lesson 9-5) 59. x + 2y = 10 60. COIN COLLECTING Jada has 156 coins in her collection. This is 12 more than 8 times the number of nickels in the collection. How many nickels does Jada have in her collection? (Lesson 8-3) Simplify. Express using positive exponents. (Lesson 10-5) 61. 5n · 3n 4 63. (-5b 7)(-2b 4) 62. (-x)(-8x 3) 548 Chapter 10 Algebra: Nonlinear Functions and Polynomials 64. (-4w)(6w -2) 10-7 Powers of Monomials Main IDEA Find powers of monomials. MEASUREMENT Suppose the side length of a cube is 2 2 centimeters. Standard 7AF2.2 Multiply and divide monomials; extend the process of taking powers and extracting roots to monomials when the latter results in a monomial with an integer exponent. 1. Write a multiplication expression for the volume of the cube. 2 2 cm 2. Simplify the expression. Write as a single power of 2. 3. Using 2 2 as the base, write the multiplication expression 2 2 · 2 2 · 2 2 using an exponent. 3 4. Explain why (2 2) = 2 6. You can use the rule for finding the product of a power to discover the rule for finding the power of a power. 5 factors (6 4) 5 = (6 4) (6 4) (6 4) (6 4) (6 4) Apply the rule for the product of powers. = 64 + 4 + 4 + 4 + 4 = 6 20 Notice that the product of the original exponents, 4 and 5, is the final power 20. This relationship is stated in the following rule. +%9#/.#%04 Power of a Power To find the power of a power, multiply the exponents. Words Examples Numbers 2 3 (5 ) = 5 2·3 Algebra or 5 6 m n (a ) = a m · n Find the Power of a Power Common Error When finding the power of a power, do not add the exponents. 1 Simplify (8 4) 3. (8 4) 3 = 8 4 · 3 =8 12 2 Simplify (k 7) 5. Power of a Power (k 7) 5 = k 7 · 5 = k 35 Simplify. Power of a Power Simplify. (8 4) 3 = 8 12, not 8 7. Simplify. Express using exponents. a. (2 5) 2 Extra Examples at ca.gr7math.com b. (w 4) 6 c. [(3 2) 3] 2 Lesson 10-7 Powers of Monomials 549 Extend the power of a power rule to find the power of a product. 5 factors (3a 4) 5 = (3a 4) (3a 4) (3a 4) (3a 4) (3a 4) ssociative Property of = 3 · 3 · 3 · 3 · 3 · a 4 · a 4 · a 4 · a 4 · a 4 AMultiplication = 3 5 · ( a 4) 5 Write using powers. Apply the rule for power of = 243 · a 20 or 243a 20 a power. This example suggests the following rule. +%9#/.#%04 Power of a Product To find the power of a product, find the power of each factor and multiply. Words Examples Numbers 2 3 3 Algebra 2 3 (6x ) = (6) • (x ) or 216x 6 (ab) m = a mb m Power of a Product 3 Simplify (4p 3) 4. 4 Simplify (-2m 7n 6) 5. (4p 3) 4 = 4 4 · p 3 · 4 Alternative Method (4p 3) 4 can also be expressed as (4p 3)(4p 3)(4p 3)(4p 3) or (4 · 4 · 4 · 4) (p · p · p)(p · p · p) (p · p · p)(p · p · p) which is 256p 12. = 256p 12 (-2m 7n 6) 5 = (-2) 5m 7 · 5n 6 · 5 = -32m 35n 30 Simplify. Simplify. Simplify. d. (8b 9) 2 e. (6x 5y 11) 4 f. (-5w 2z 8) 3 5 GEOMETRY Express the area of the square as a monomial. A = s2 Area of a square 4 A = (7a b) 2 Replace s with 7a 4b. A = 7 2(a 4) 2(b 1) 2 Power of a Product A = 49a 8b 2 7a 4b Simplify. The area of the square is 49a 8b 2 square units. g. GEOMETRY Find the volume of a cube with sides of length 8x 3y 5. Express as a monomial. Personal Tutor at ca.gr7math.com 550 Chapter 10 Algebra: Nonlinear Functions and Polynomials Examples 1–4 Simplify. (pp. 549-550) 1. (3 2) Example 5 4. (7w 7) 5 2. (h 6) 4 3 3. [(2 3) 2] 3 12 5. (5g 8k ) 4 6. (-6r 5s 9) 2 (p. 550) 7. MEASUREMENT Express the volume of the cube at the right as a monomial. (/-%7/2+ (%,0 For Exercises 8–27 28–31 See Examples 1–4 3 3c 3d 2 Simplify. 8. (4 2) 3 9. (2 2) 7 10. (5 3) 3 11. (3 4) 2 12. (d 7) 6 13. (m 8) 5 14. (h 4) 9 15. (z 11) 5 16. [(3 2) 2] 2 17. [(4 3) 2] 2 18. [(5 2) 2] 2 19. [(2 3) 3] 2 20. (5j 6) 4 21. (8v 9) 5 22. (11c 4) 3 23. (14y) 4 24. (6a 2b 6) 3 25. (2m 5n 11) 6 26. (-3w 3z 8) 5 27. (-5r 4s 12) 4 GEOMETRY Express the area of each square below as a monomial. 28. 29. 8g 3h 12d 6e 7 GEOMETRY Express the volume of each cube below as a monomial. 30. 31. 5r 2s 3 7m 6n 9 Simplify. 32. (0.5k 5) 2 35. 3 -6 9 2 (_ a b ) 5 1 3 34. (_w 5z ) 2 33. (0.3p 7) 3 36. (3x -2 4 4 6 2 ) (5x ) 37. (-2v 7) 3(-4v -2) 4 38. PHYSICS A ball is dropped from the top of a building. The expression 4.9x 2 %842!02!#4)#% gives the distance in meters the ball has fallen after x seconds. Write and simplify an expression that gives the distance in meters the ball has fallen after x 2 seconds. Then write and simplify an expression that gives the distance the ball has fallen after x 3 seconds. See pages 703, 717. 39. BACTERIA A certain culture of bacteria doubles in population every hour. At Self-Check Quiz at 1 P.M., there are 5 cells. The expression 5(2 x)gives the number of bacteria that are present x hours after 1 P.M. Simplify the expressions [5(2 x)] 2 and [5(2 x)] 3 and describe what they each represent. ca.gr7math.com Lesson 10-7 Powers of Monomials 551 MEASUREMENT For Exercises 40-42, use the table that gives the area and volume of a square and cube, respectively, with side lengths shown. Side Length (units) Area of Square (units 2) Volume of Cube (units 3) x x2 x3 40. Copy and complete the table. 2x 41. Describe how the area and volume are 3x each affected if the side length is doubled. Then describe how they are each affected if the side length is tripled. x2 x3 42. Describe how the area and volume are each affected if the side length is squared. Describe how they are each affected if the side length is cubed. H.O.T. Problems 43. OPEN ENDED A googol is 10 100. Use the Power of a Power rule to write three different expressions that are equivalent to a googol where each expression uses exponents. CHALLENGE Solve each equation for x. 44. (7 x) 3 = 7 15 5 45. (-2m 3n 4) x = -8m 9n 12 3 */ -!4( Compare and contrast how you would correctly (*/ 46. 83 *5*/( simplify the expressions (2a 3)(4a 6) and (2a 3) 6. 47. Which expression is equivalent to 49. Which of the following has the same 4 8 value as 64m 6? (10 ) ? A 10 2 C 10 12 B 10 4 D 10 32 A the area in square units of a square whose side length is 8m 2 B the expression (32m 3) 2 48. Which expression has the same value C the expression (8m 3) 2 as 81h 8k 6? F (9h 6k 4) 2 H (6h 5k 3) 3 G (9h 4k 3) 2 J (3h 2k) 6 Simplify. Express using positive exponents. 15 _ 15 4 (Lesson 10-6) 10 7 50. D the volume in cubic units of a cube whose side length is 4m 3 51. y _ y2 52. 18m 9 _ 6m 4 3 53. 24g _ 3g 8 54. MEASUREMENT Find the area of a rectangle with a length of 9xy 2 and a width of 4x 2y. (Lesson 10-5) Find each square root. 55. √ 49 (Lesson 3-1) 56. √ 121 57. √ 225 552 Chapter 10 Algebra: Nonlinear Functions and Polynomials 58. √ 400 10-8 Roots of Monomials Main IDEA Find roots of monomials. Standard 7AF2.2 Multiply and divide monomials; extend the process of taking powers and extracting roots to monomials when the latter results in a monomial with an integer exponent. NEW Vocabulary cube root REVIEW Vocabulary square root: the opposite of squaring a number (Lesson 3-1) NUMBER THEORY The square root of a number is one of the two equal factors of the number. Some perfect squares can be factored into the product of two other perfect squares. 1. Find two factors of 100 that are also perfect squares. 2. Find the square root of 4 and 25. Then find their product. 3. How does the product relate to 100? 4. Repeat Questions 1–3 using 144. The pattern you discovered about the factors of a perfect square is true for any number. +%9#/.#%04 Product Property of Square Roots For any numbers a and b, where a ≥ 0 and b ≥ 0, the square root of the product ab is equal to the product of each square root. Words Examples Numbers √ 9 · 16 = √ 9 · √ 16 Algebra √ ab = √ a · √ b = 3 · 4 or 12 The square root of a monomial is also one of its two equal factors. You can use the product property of square roots to find the square roots of monomials. √ x 2 = √x · x = ⎪x⎥ Since x represents an unknown value, absolute value is used to indicate the positive value of x. √ x 4 = √ x2 · x2 = x2 Absolute value is not necessary since the value of x 2 will never be negative. Simplify Square Roots 4y 2 . √ 4y 2 = √4 · √ y2 √ 36q 6 . √ 36q 6 = √ 36 · √ q6 √ = √ 6 · 6 · √ q3 · q3 1 Simplify Absolute Value Use absolute value to indicate the positive value of y and q 3. 2 Simplify = √ 2 · 2 · √ y·y = 2⎪y⎥ = 6 ⎪q 3 ⎥ Simplify. a. √ v2 Extra Examples at ca.gr7math.com b. √ c 6d 8 c. √ 121x 4z 10 Lesson 10-8 Roots of Monomials 553 READING Math Cube Root Symbol The cube root of a is shown by 3 the symbol √ a. The process of simplifying expressions involving square roots can be extended to cube roots. The cube root of a monomial is one of the three equal factors of the monomial. 3 3 √ 8 = √ 2·2·2=2 3 3 √ a3 = √ a ·a·a=a +%9#/.#%04 Product Property of Cube Roots For any numbers a and b, the cube root of the product ab is equal to the product of each cube root. Words Examples Numbers 3 Algebra 3 3 = √ √ab a · √ b 3 3 3 √ 216 = √ 8 · √ 27 = 2 · 3 or 6 Simplify Cube Roots 3 3 Simplify √ c3. 3 √ c3 = c (c) 3 = c 3 3 64g 6 . √ 64g 6 = √ 64 · √ g6 √ 4 Simplify 3 3 3 3 = √ 4·4·4· Product Property of Cube Roots 3 g2 · g2 · g2 √ = 4 · g 2 or 4g 2 Absolute Value Because a cube root can be negative, absolute value is not necessary when simplifying cube roots. Simplify. Simplify. d. 3 √ s3 e. 3 27y 3 √ f. 3 √ 216k 12 5 GEOMETRY Express the length of one side of the square whose area is 81y 2z 6 square units as a monomial. A = s 2 Area of a square 81y 2z 6 = s 2 Replace A with 81y 2z 6. y 2z 6 = s Definition of square root. √81 √ 81 · √ y 2 · √ z 6 = s Product Property of Square Roots. 9⎪yz 3⎥ = s Simplify. Add absolute value. The length of one side of the square is 9⎪yz 3⎥ units. g. GEOMETRY Find the length of one side of a square whose volume is is 125a 15 cubic units. Personal Tutor at ca.gr7math.com 554 Chapter 10 Algebra: Nonlinear Functions and Polynomials Examples 1–2 (p. 553) Example 3–4 Simplify. 1. √ d2 5. 3 √ m3 2. √ 25a 2 6. 8p 3 √ (p. 554) Example 5 3 3. 49x 6y 2 √ 7. √ 125r 6s 9 3 4. √ 121h 8k 10 8. 64 x 12y 3 √ 3 9. GEOMETRY Express the length of one side of the square whose area is 256u 2v 6 square units as a monomial. (p. 554) 10. GEOMETRY Express the length of one side of a cube whose volume is 27b 3c 12 cubic units as a monomial. (/-%7/2+ (%,0 For Exercises 11–18 14–26 27–34 See Examples 1–2 3 5 Simplify. 2 11. √n 12. y4 √ 13. g 8k 14 √ 14. √ 64a 2 17. 9p 8q 4 √ 18. 225x 4y 6 √ 22. √ 64k 3 26. √ 216x 12w 15 15. √ 36z 12 16. √ 144k 4m 6 19. 3 √ h3 20. √ v3 21. √ 27b 3 23. √ 125d 9e 3 24. 8q 9r 18 √ 25. √ 343m 3n 21 3 3 3 3 3 3 3 GEOMETRY Express the length of one side of each square whose area is given as a monomial. 27. 28. 29. 30. A 36m 6n 8 A 400x 2y 10 A 121a 2b 2 A 49p 4q 6 GEOMETRY Express the length of one side of each cube whose volume is given as a monomial. 31. 32. 33. 34. V 125k 9m 18 V 27g 24h 3 V 64w 3z 3 V 343c 6d 12 Simplify. %842!02!#4)#% See pages 704, 717. Self-Check Quiz at ca.gr7math.com 35. √ 0.25x 2 36. Simplify each expression if x _ √ 16 2 38. 39. 3 0.08p 9 √ 37. 8 3 6 w x √_ 27 40. 121 √_ h k 3 √a . √_ab = _ √b 81 √_ m 4 8 6 Lesson 10-8 Roots of Monomials 555 H.O.T. Problems 41. OPEN ENDED Write a monomial and its square root. CHALLENGE Solve each equation for x. 42. √ 25a x = 5 ⎪a 3⎥ 43. 3 √ 64a 3b x = 4ab 7 simplifying the expression y 2 and not necessary when simplifying √ y4. √ 46. Which expression is equivalent 48. Which of the following has the same 3 value as √ 27m 3n 6 ? to √ 144g 2 ? A 12g C 12g 2 B 12⎪g⎥ D 12⎪g 2 A the length of the side of a square whose area is 27m 3n 6 ⎥ B the expression 9mn 3 47. Which expression has the same value F 20hk 2 H 20h 2k 4 G 20 ⎪h⎥ k 2 J 49. (6 3) 5 (Lesson 10-7) 50. (n 7) 2 5 9 _ 93 D the length of the side of a cube whose volume is 3mn 2 200 ⎪h⎥ k 2 Simplify. Express using positive exponents. 53. C the expression 3mn 2 √ 400h 2k 4 ? Simplify. √ 81a 4b x = 9a 4 ⎪b 5⎥ */ -!4( Explain why absolute value is necessary when (*/ 83 *5*/( 45. as 44. 54. k 15 _ k6 51. (2a 3b 2) 4 52. (-4p 11q) 3 (Lesson 10-6) 4 55. 24y _ 4y 2 3 56. 45g _ 3g 7 57. RETAIL Find the discount to the nearest cent for a flat-screen television that costs $999 and is on sale at 15% off. (Lesson 5-8) Math and Economics Getting Down to Business It’s time to complete your project. Use the information and data you have gathered about the cost of materials and the feedback from your peers to prepare a video or brochure. Be sure to include a scatter plot with your project. Cross-Curricular Project at ca.gr7math.com 556 Chapter 10 Algebra: Nonlinear Functions and Polynomials CH APTER 10 Study Guide and Review Download Vocabulary Review from ca.gr7math.com Key Vocabulary cube root (p. 554) monomial (p. 539) Be sure the following Key Concepts are noted in your Foldable. Key Concepts Functions (Lessons 10-1, 10-2, and 10-3) • Linear functions have constant rates of change. • Nonlinear functions do not have constant rates of change. • Quadratic functions are functions in which the greatest power of the variable is 2. • Cubic functions are functions in which the greatest power of the variable is 3. Monomials nonlinear function (p. 522) quadratic function (p. 528) Vocabulary Check State whether each sentence is true or false. If false, replace the underlined word or number to make a true sentence. 1. The expression y = x 2 - 3x is an example of a monomial. 2. A nonlinear function has a constant rate of change. (Lessons 10-5 through 10-8) • To multiply powers with the same base, add their exponents. • To divide powers with the same base, subtract their exponents. 3. A quadratic function is a function whose greatest power is 2. 4. The product of 3x and x 2 + 3x will have 3 terms. • To find the power of a power, multiply the exponents. 5. A quadratic function is a nonlinear • To find the power of a product, find the power of each factor and multiply. 6. The graph of a linear function is a curve. function. 7. To divide powers with the same base, subtract the exponents. 8. The Quotient of Powers states when dividing powers with the same base, subtract their exponents. 9. The graph of a cubic function is a straight line. Vocabulary Review at ca.gr7math.com Chapter 10 Study Guide and Review 557 CH APTER 10 Study Guide and Review Lesson-by-Lesson Review 10-1 Linear and Nonlinear Functions (pp. 522–527) Determine whether each equation or table represents a linear or nonlinear function. Explain. 10. y - 4x = 1 12. 10-2 11. y = x 2 + 3 Time (h) 2 Number of Pages 98 3 4 5 147 199 248 Graphing Quadratic Functions Example 1 Determine whether the table represents a linear or nonlinear function. y -2 -3 -1 -1 0 1 1 3 As x increases by 1, y increases by 2. The rate of change is constant, so this function is linear. (pp. 528–531) Graph y = -x 2 - 1. Graph each function. Example 2 13. -4x 2 14. y = x 2 + 4 15. y = -2x 2 + 1 16. y = 3x 2 - 1 Make a table of values. Then plot and connect the ordered pairs with a smooth curve. 17. SCIENCE A ball is dropped from the top of a 36-foot tall building. The quadratic equation d = -16t 2 + 36 models the distance d in feet the ball is from the ground at time t seconds. Graph the function. Then use your graph to find how long it takes for the ball to reach the ground. 10-3 x PSI: Make a Model x y = -x 2 - 1 (x, y) -2 -(-2) 2 - 1 (-2, -5) -1 -(-1) 2 - 1 (-1, -2) 0 2 -(0) - 1 (0, -1) 1 2 -(1) - 1 (1, -2) 2 2 (2, -5) -(2) - 1 y O x y x 2 1 (pp. 532-533) Solve the problem by using the make a model strategy. 18. MEASUREMENT Sydney has a postcard that measures 5 inches by 3 inches. She decides to frame it, using a frame that 3 is 1_ inches wide. What is the 4 Example 3 DISPLAYS Cans of oil are displayed in the shape of a pyramid. The top layer has 2 cans in it. One more can is added to each layer, and there are 4 layers in the pyramid. How many cans are there in the display? perimeter of the framed postcard? 19. MAGAZINES A book store arranges it best-seller magazines in the front window. In how many different ways can five best-seller magazines be arranged in a row? So, based on the model there are 14 cans. 558 Chapter 10 Algebra: Nonlinear Functions and Polynomials Mixed Problem Solving For mixed problem-solving practice, see page 717. 10-4 Graphing Cubic Functions (pp. 534-537) 20. y = 2x 3 – 4 21. y = 0.25x 3 - 2 x y = -x 3 (x, y) 22. y = 2x 3 + 4 23. y = 0.25x 3 + 2 -2 -(-2) 3 (-2, 8) -1 -(-1) 3 (-1, 1) 0 -(0) 3 1 -(1) 3 (1, -1) 2 -(2) 3 (2, -8) 24. MEASUREMENT A rectangular prism with a square base of side length x inches has a height of (x - 1) inches. Write the function for the volume V of the prism. Graph the function. Then estimate the dimensions of the box that would give a volume of approximately 18 cubic inches. 10-5 Multiplying Monomials 25. 4 · 4 5 26. x 6 · x 2 27. -9y 2(-4y 9) 28. _3 -4 3 · _ (7) (7) 2 29. LIFE SCIENCE The number of bacteria after t cycles of reproduction is 2 t. Suppose a bacteria reproduces every 30 minutes. If there are 1,000 bacteria in a dish now, how many will there be in 1 hour? Dividing Monomials n5 31. _ n 21c _ -7c 8 (0, 0) x Example 5 Find 4 · 4 3. Express using exponents. 4 · 43 = 41 · 43 4 = 41 = 41 + 3 The common base is 4. 4 =4 Add the exponents. Example 6 Find 3a 3 · 4a 7. 3a 3 · 4a 7 = (3 · 4)a 3 + 7 Commutative and Associative Properties = 12a 10 33. Example 7 68 6 Simplify_ . Express using exponents. 3 -1 3 11 32. y x 3 (pp. 544-548) Simplify. Express using exponents. 59 30. _ 52 y (pp. 539-542) Simplify. Express using exponents. 10-6 Graph y = -x 3. Example 4 Graph each function. (_47 ) × (_47 ) __ _4 7 34. MEASUREMENT The area of the family room is 3 4 square feet. The area of the kitchen is 4 3 square feet. What is the difference in area between the two rooms? 68 _ = 68 - 3 63 The common base is 6. 5 =6 Example 8 Simplify. -8 s . Express using exponents. Simplify _ -4 s s -8 =_ = s -8 - (-4) s –4 = s -8 + 4 or s -4 Quotient of Powers Simplify. Chapter 10 Study Guide and Review 559 CH APTER 10 Study Guide and Review 10-7 Powers of Monomials (pp. 549-552) Example 9 Simplify. 35. (9 2) 3 36. (d 6f 3) 4 Simplify (7 3) 5. 37. (5y 5) 4 38. (6z 4x 3) 5 39. (_n -1) 2 40. [(p 2) 3] 2 41. (5 -1) 2 42. (-3k 2) 2(4k -3) 2 (7 3) 5 = 7 3 · 5 Power of a Power = 7 15 Simplify. Example 10 3 4 43. GEOMETRY Find the volume of a cube with sides of length 5s 2t 4 as a monomial. Simplify (2x 2y 3) 3. (2x 2y 3) 3 = 2 3 · x 2 · 3 y 3 · 3 Power of a Product = 8 x 6y 9 Simplify. 44. GEOMETRY Find the area of a square with sides of length 6a 3b 5 as a monomial. 10-8 Roots of Monomials Simplify. 2 45. √a 47. 49. 36x 2y 6 √ 3 6 √p 3 51. √ 64c 6d 21 (pp. 553-556) 46. √4 48. 81q 14 √ 50. 49n 3 18 √ 8m 3 52. √ 125r 9s 15 Example 11 16f 8g 6 . Simplify √ 16f 8g 6 = √ 16 · √ f 8 · √ g 6 Product √ Property of Square Roots = 4 · f 4 · ⎪g 3⎥ or 4f 4 ⎪g 3⎥ 53. GEOMETRY Express the length of one side of the square whose area is 64b 16 square units as a monomial. 54. GEOMETRY Express the length of one side of a cube whose volume is 216a 9c 3 cubic units as a monomial. Example 12 3 Simplify √ x9. 3 √ x9 = x3 560 Chapter 10 Algebra: Nonlinear Functions and Polynomials (x) 9 = x 3 Use absolute value to indicate the positive value of g 3. CH APTER Practice Test 10 Determine whether each graph, equation, or table represents a linear or nonlinear function. Explain. 1. 12. CRAFTS Martina is making cube-shaped gift boxes from decorative cardboard. Each side of the cube is to be 6 inches long, and there 1 is a _ -inch overlap on each side. How much 2. 2x = y y 2 cardboard does Martina need to make each box? x O Simplify. Express using exponents. 3. x -3 -1 2 y 10 1 3 18 26 13. 15 3 15 5 15. 3 15 _ 14. -5m 6(-9m 8) 16. 37 -40w 8 _ 8w Graph each function. 4. y = _x 2 1 2 5. y = -2x 2 + 3 6. BUSINESS The function p = 60 + 2d 2 models the profit made by a manufacturer of digital audio players. Graph this function. Then use your graph to estimate the profit earned after making 20 players. Simplify. 17. √ m2 18. √ 144a 2b 6 19. 64x 3y 15 √ 20. STANDARDS PRACTICE Simplify the algebraic expression (3x 3y 2)(7x 3y). 7. A 21x 9y 2 B 21x 6y 2 C 21x 6y 3 D 21x 6y 6 Graph each function. 3 STANDARDS PRACTICE Which (12x 4)(4x 3) expression is equivalent to _ ? 5 F 12x 7 H 6x 4 G 12x 2 J 8x 6x 2 21. MEASUREMENT Find the area of the rectangle at the right. 4s 2t 2 3st 3 3 8. y = x + 4 9. y = x 3 - 4 Simplify. 10. y = _x 3 22. [(x 2) 4] 3 11. MEASUREMENT A neighborhood group 24. (3 -3) 2 1 3 23. (-2b 3) 2(4b 2) 2 would like Jacob to fertilize their lawns. The average area of each lawn is 6 4 square feet. If there are 6 2 lawns in this neighborhood, how many total square feet of lawn does Jacob need to fertilize? Chapter Test at ca.gr7math.com 25. GEOMETRY Express the length of one side of a square with an area of 121x 4y 10 square units as a monomial. Chapter 10 Practice Test 561 CH APTER 10 California Standards Practice Cumulative, Chapters 1–10 Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. 1 3 The equation c = 0.8t represents c, the cost of t tickets on a ferry. Which table contains values that satisfy this equation? A A car used 4.2 gallons of gasoline to travel 126 miles. How many gallons of gasoline would it need to travel 195 miles? t c Cost of Ferry Tickets 1 2 3 $0.80 $1.00 $1.20 4 $1.40 t c Cost of Ferry Tickets 1 2 3 $0.80 $1.60 $2.40 4 $3.20 t c Cost of Ferry Tickets 1 2 3 $0.75 $1.50 $2.25 4 $3.00 t c Cost of Ferry Tickets 1 2 3 $1.80 $2.60 $3.40 4 $4.20 B A 2.7 B 5.0 C 6.5 C D 7.6 2 The scatter plot below shows the cost of computer repairs in relation to the number of hours the repair takes. Based on the information in the scatter plot, which statement is a valid conclusion? Total Cost ($) Cost of Computer Repairs 55 50 45 40 35 30 25 20 15 10 5 y D 4 Shanelle purchased a new computer for $1,099 and a computer desk for $699 including tax. She plans to pay the total amount in 24 equal monthly payments. What is a reasonable amount for each monthly payment? F $50 G $75 1 2 3 4 5 6 7 x Number of Hours H $150 J $1,800 F As the length of time increases, the cost of the repair increases. G As the length of time increases, the cost of the repair stays the same. H As the length of time decreases, the cost of the repair increases. J As the length of time increases, the cost of the repair decreases. Question 4 You can often use estimation to eliminate incorrect answers. In this question, Shanelle’s total spent can be estimated by adding $1,100 and $700, then dividing by 24. The sum of $1,100 and $700 is $1,800 before dividing by 24, so choice J can be eliminated. 562 Chapter 10 Algebra: Nonlinear Functions and Polynomials More California Standards Practice For practice by standard, see pages CA1–CA39. 5 Which of the following is the graph of 2 x 2? y=_ 3 y y C A O O x y B D O 8 9 x The area of a rectangle is 30m 11 square feet. If the length of the rectangle is 6m 4 feet, what is the width of the rectangle? F 5m 7 ft H 36m 15 ft G 24m 7 ft J 180m 15 ft Which expression is equivalent to 5 4 × 5 6? A 5 10 C 25 10 B 5 24 D 25 24 y x Pre-AP O 6 Record your answers on a sheet of paper. Show your work. x 10 An electronics store is having a sale on Simplify the expression shown below. certain models of televisions. Mr. Castillo would like to buy a television that is on sale. This television normally costs $679. (3m 3n 2)(6m 4n) F 18m 12n 2 H 18 m 7n 3 G 18 m 7n 2 J 18 m 7n 6 Last Year’s Models 7 40% off What is the height h of the gutter in the figure below? Wednesday Only Television Sale! Take an additional 10% off a. What price, not including tax, will Mr. Castillo pay if he buys the television on Saturday? 20 ft h b. What price, not including tax, will Mr. Castillo pay if he buys the television on Wednesday? 12 ft A 10 ft C 16 ft B 14 ft D 18 ft c. How much money will Mr. Castillo save if he buys the television on Saturday? NEED EXTRA HELP? If You Missed Question... 1 2 3 4 5 6 7 8 9 10 Go to Lesson... 4-3 9-8 9-1 5-5 10-2 10-4 10-4 10-5 10-5 5-8 For Help with Standard... AF4.2 PS1.2 MR2.5 MR3.1 AF3.1 AF2.2 MG3.3 NS2.3 NS2.3 NS1.7 California Standards Practice at ca.gr7math.com Chapter 10 California Standards Practice 563

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