Chapter 11 X Resource Masters. Course13

Transcription

1 Chapter 11 X Resource Masters Course13

2 DATE PERID Reading to Learn Mathematics Vocabular Builder This is an alphabetical list of new vocabular terms ou will learn in Chapter 11. As ou stud the chapter, complete each term s definition or description. Remember to add the page number where ou found the term. Add this page to our math stud notebook to review vocabular at the end of the chapter. Vocabular Term arithmetic sequence [air-ith-meh-tik] Found on Page Definition/Description/Eample Vocabular Builder best-fit line boundar common difference common ratio dependent variable domain function function table geometric sequence [je-o-met-rik] half plane Glencoe/McGraw-Hill vii Mathematics: Applications and Concepts, Course 3

3 DATE PERID Reading to Learn Mathematics Vocabular Builder (continued) Vocabular Term independent variable Found on Page Definition/Description/Eample linear function range scatter plot sequence slope formula slope-intercept form substitution sstem of equations term -intercept -intercept Glencoe/McGraw-Hill viii Mathematics: Applications and Concepts, Course 3

4 DATE PERID Stud Guide and Intervention Sequences A sequence is an ordered list of numbers. Each number is called a term. An arithmetic sequence is a sequence in which the difference between an two consecutive terms is the same. This difference is called the common difference. To find the net term in the sequence, add the common difference to the last term. 4, 1,, 5, State whether the sequence 4, 1,, 5, 8,... is arithmetic. If it is, state the common difference. Write the net three terms of the sequence. Notice that 1 ( 4) 3, ( 1) 3, and so on. The terms have a common difference of 3, so the sequence is arithmetic , , The net three terms are 11, 14, and 17. A geometric sequence is a sequence in which the ratio between an two consecutive terms is the same. This ratio is called the common ratio. To find the net term in the sequence, multipl the last term b the common ratio. 1,, 4, 8 16 ( ) ( ) ( ) ( ) State whether the sequence 1,, 4, 8, 16,...is geometric. If it is, state the common ratio. Write the net three terms of the sequence. Notice that, 4, and so on. The terms have a 1 common ratio of, so the sequence is geometric. Lesson , 3 64, The net three terms are 3, 64, and 18. Some sequences are neither arithmetic nor geometric. To etend a sequence like this, look for a pattern in the consecutive differences or consecutive ratios. Then appl the pattern to the last term of the sequence. State whether each sequence is arithmetic, geometric, or neither. If it is arithmetic or geometric, state the common difference or common ratio. Write the net three terms of the sequence. 1. 0, 3, 6, 9, 1,.... 3, 6, 1, 4, 48,... arithmetic; 3; 15, 18, 1 geometric; ; 96, 19, , 11, 16, 1, 6, , 1, 3, 6, 10,... arithmetic; 5; 31, 36, 41 neither; 15, 1, , 1,1,3,9, , 6,, 18, 14,... 3 geometric; 3; 7, 81, 43 arithmetic; 4; 10, 6, Glencoe/McGraw-Hill 619 Mathematics: Applications and Concepts, Course 3

5 DATE PERID Practice: Skills Sequences State whether each sequence is arithmetic, geometric, or neither. If it is arithmetic or geometric, state the common difference or common ratio. Write the net three terms of the sequence. 1. 1, 5, 9, 13, 17,.... 3, 6, 1, 4, 48,... arithmetic; 4; 1, 5, 9 geometric; ; 96, 19, , 5, 8, 11, 14, , 6, 16, 31, 51,... arithmetic; 3; 17, 0, 3 neither; 76, 106, , 7, 11, 15, 19, , 9, 14, 19, 4,... arithmetic; 4; 3, 7, 31 arithmetic; 5; 9, 34, , 3, 9, 7, 81, ,, 1,, 1,... geometric; 3; 43, 79,,187 neither;, 1, 9. 7, 4, 1,, 5, , 4, 4, 4, 4,... arithmetic; 3; 8, 11, 14 geometric; 1; 4, 4, , 10, 0, 40, 80,... 1., 5, 8, 31, 34,... geometric; ; 160, 30, 640 arithmetic; 3; 37, 40, ,, 5, 9, 14, , 1, 36, 108, 34,... neither; 0, 7, 35 geometric; 3; 97,,916, 8, , 10, 50, 50, 1,50, , 76, 7, 68, 64,... geometric; 5; 6,50, 31,50, arithmetic; 4; 60, 56, 5 156, , 4, 11, 56, 8, ,, 4, 1, 48,... geometric; 1 ; 14, 7, 7 neither; 40, 1,440, 10, , 6 1, 10, 13 1,17, , 7, 9, 3, 1,... arithmetic; 3 1 ; 0 1, 4, 7 1 geometric; 1 3 ; 1 3, 1 9, 1 7 Glencoe/McGraw-Hill 60 Mathematics: Applications and Concepts, Course 3

6 DATE PERID Practice: Word Problems Sequences GEMETRY For Eercises 1 and, use the sequence of rectangles below. 4 units 5 units 6 units 7 units units 3 units 4 units 5 units 1. Write a sequence for the perimeters of the rectangles. Is the sequence arithmetic, geometric, or neither? Eplain how ou know. If it is arithmetic or geometric, state the common difference or common ratio. Find the net four terms of the sequence. 1, 16, 0, 4,...; Arithmetic; the differences between consecutive terms are all 4; 4; 8, 3, 36, PIZZA A large pizza at Joe s Pizza Shack costs $7 plus $0.80 per topping. Write a sequence of pizza prices consisting of pizzas with no toppings, pizzas with one topping, pizzas with two toppings, and pizzas with three toppings. Is the sequence arithmetic, geometric, or neither? How do ou know? 7.00, 7.80, 8.60, 9.40; Arithmetic; ou add 0.80 to each term to get the net term. 5. PAYMENT PLAN A famil purchased furniture on an interest-free pament plan with a fied monthl pament. Their balances after each of the first four paments were $1,95, $1,750, $1,575, and $1,400. Is the sequence of the balances arithmetic, geometric, or neither? Eplain how ou know. If it is arithmetic or geometric, state the common difference or common ratio. Arithmetic; the differences between consecutive terms are all 175; Write a sequence for the areas of the rectangles. Is the sequence arithmetic, geometric, or neither? If it is arithmetic or geometric, state the common difference or common ratio. Eplain how to find the net four terms of the sequence. Then find the net four terms. 8, 15, 4, 35,...; Neither; the differences between consecutive terms are consecutive odd integers, 7, 9, and 11, so add 13, 15, 17, and 19 to find four more terms; 48, 63, 80, SAVINGS The ending balances in Carissa s savings account for each of the past four ears form the sequence $1,000, $1,100, $1,10, $1,331,... Is the sequence arithmetic, geometric, or neither? Eplain how ou know. Find the net two terms of the sequence. Geometric; the ratios of consecutive terms are all 1.1; $1, and $1, MNEY Continue to find the terms of the sequence of balances in Eercise 5 until ou get a term of 0. After how man paments will the balance be $0? $1,5, $1,050, $875, $700, $55, $350, $175, $0; 1 paments Lesson 11 1 Glencoe/McGraw-Hill 61 Mathematics: Applications and Concepts, Course 3

7 Pre-Activit DATE PERID Reading to Learn Mathematics Sequences Complete the Mini Lab at the top of page 51 in our tetbook. Write our answers below. 1. Continue the pattern for 4, 5, and 6 triangles. How man toothpicks are needed for each case? 9; 11; 13. Stud the pattern of numbers. How man toothpicks will ou need for 7 triangles? Continue the pattern for 4, 5, and 6 squares. How man toothpicks are needed for each case? 13; 16; How man toothpicks will ou need for 7 squares? Reading the Lesson 5. Eplain how to determine whether a sequence is arithmetic. Find the difference between each pair of consecutive terms. For an arithmetic sequence, the differences are all the same. 6. Determine whether the sequence 4, 6, 9, 13, 18,... is arithmetic. Eplain. No; the difference between consecutive terms is not the same, 6 4 but Eplain how to determine whether a sequence is geometric. Find the ratio of consecutive terms. For a geometric sequence, the ratios are all the same. 8. Determine whether the sequence, 6, 18, 54, 16,...is geometric. Eplain. Yes; the ratios of consecutive terms all equal 3. Helping You Remember 9. Work with a partner. Have one partner write three sequences of numbers, one that is arithmetic, one that is geometric, and one that is neither arithmetic nor geometric. Have the other partner identif the tpe of sequence and the common difference or ratio. See students work. Glencoe/McGraw-Hill 6 Mathematics: Applications and Concepts, Course 3

8 DATE PERID Enrichment Convergence and Limits There are man different kinds of sequences. An infinite sequence has no last term. Three dots indicate that a sequence is infinite. 0.3, 0.33, 0.333, , ,... In some infinite sequences, the terms get closer and closer to a given number. The terms of the sequence above are getting closer to the rational number 1 3. When an infinite sequence gets arbitraril close to some number, the sequence is called convergent. The sequence above is converging to the rational number 1 3.And, 1 3 is called the limit of the sequence. The number line below shows a convergent sequence with a limit of 1. The denominators of the fractions are increasing powers of ; each numerator is 1 less than the denominator Lesson 11 1 Write the limit for each convergent sequence. 1. 1, 1, 1 3, 1 4, 1 5, 1 6, 1, , 0.99, 0.999, , , 5 1,5 1 4,5 1 6,5 1 8,5 1 16, , 1, 1 3, 1 4, 1 5, 1 6, , 3 4, 7 8, , 3 1, ,10 1,9 3 4,10 1 3,9 7 8,10 1, , 0.66, 0.666, , ,, 0.4, 0.08, 0.016, 0.003, Create a convergent sequence of our own. Give the limit of our sequence and illustrate it on a number line. Answers will var. Glencoe/McGraw-Hill 63 Mathematics: Applications and Concepts, Course 3

9 DATE PERID Stud Guide and Intervention Functions A function connects an input number,, to an output number, f(), b a rule. To find the value of a function for a certain number, substitute the number into the rule in place of, and simplif. Find f(5) if f() 3. f() 3 f(5) 3(5) or 17 So, f(5) 17. Write the function. Substitute 5 for into the function rule and simplif. You can organize the input, rule and output of a function using a function table. Complete the function table for f() 4. Substitute each value of, or input, into the function rule. Then simplif to find the output. f() 4 f( 1) ( 1) 4 or f(0) (0) 4 or 4 f(1) (1) 4 or 6 f() () 4 or 8 Input Rule 4 utput f() 1 ( 1) 4 0 (0) (1) 4 6 () 4 8 Find each function value. 1. f(1) if f() 3 4. f(6) if f() 1 3. f(4) if f() f(9) if f() f( ) if f() f( 5) if f() Complete each function table. 7. f() f() 6 9. f() 3 10 f() 1 ( 1) f() 3 ( 3) ( 1) 6 4 () (4) f() 3( ) 8 0 3(0) 3 3(3) 7 4 3(4) 10 Glencoe/McGraw-Hill 64 Mathematics: Applications and Concepts, Course 3

10 DATE PERID Practice: Skills Functions Find each function value. 1. f() if f() 4 6. f(9) if f() f(3) if f() 8 4. f(6) if f() f( 7) if f() f(8) if f() f( 5) if f() f( 3) if f() f( 4) if f() Complete each function table. 10. f() f() f() 8 7 f() f() f() 3 ( 3) 8 1 ( 1) (0) (4) f() f() f() f() ( ) 3 7 () (5) (8) f() 4 3( 4) 4 8 3( ) 4 1 3(1) (3) f() 3 7 3( 3) ( 1) (3) 5 7 3(5) 8 Lesson f() f() f() f() 4 4( 4) ( 1) 5 1 4() (6) f() 1 4( ) (0) (3) (5) 19 6 f() 5 6( 5) 3 3 6( 3) 0 6() (7) 40 Glencoe/McGraw-Hill 65 Mathematics: Applications and Concepts, Course 3

11 DATE PERID Practice: Word Problems Functions 1. JBS Strom works as a valet at the Westside Mall. He makes $48 per da plus $1 for each car that he parks. The total amount that Strom earns in one da can be found using the function f() 48, where represents the number of cars that Strom parked. Make a function table to show the total amount that Strom makes in one da if he parks 5 cars, 30 cars, 35 cars, and 40 cars.. PLUMBING Rico s Plumbing Service charges $40 for a service call plus $30 per hour for labor. The total charge can be found using the function f() 30 40, where represents the number of hours of labor. Make a function table to show the total amount that Rico s Plumbing Service charges if a job takes 1 hour, hours, 3 hours, and 4 hours. 3. GEMETRY The perimeter of an equilateral triangle equals 3 times the length of one side. Write a function using two variables for this situation. Sample answer: P 3s 4. GEMETRY Eplain how to use the function that ou wrote in Eercise 3 to find the perimeter of an equilateral triangle with sides 18 inches long. Then find the perimeter. Sample answer: Replace s in the function P 3s with 18 and simplif the right side; 54 in. 5. LIBRARY FINES The amount that Sunrise Librar charges for an overdue book is $0.5 per da plus a $1 service charge. Write a function using two variables for this situation. Sample answer: f 0.5d 1 6. LIBRARY FINES Eplain how to find the amount of the fine the librar in Eercise 5 will charge for a book that is overdue b 1 das. Then find the amount. Sample answer: Replace d in the function f 0.5d 1 with 1 and simplif the right side; $4.00. Glencoe/McGraw-Hill 66 Mathematics: Applications and Concepts, Course 3

12 Pre-Activit DATE PERID Reading to Learn Mathematics Functions Read the introduction at the top of page 517 in our tetbook. Write our answers below. 1. Complete the table at the right.. If a dog is 6 ears old, what is its equivalent human age? 4 ears old 3. Eplain how to find the equivalent human age of a dog that is 10 ears old. Multipl 7 times 10. Dog s Age Equivalent Human Age Reading the Lesson 4. If f() 5, eplain how to find f(). Then find f(). Replace with in f() 5 and simplif the right side; Identif the input value and the output value in Eercise 4. ; 7 6. Define domain.what number in Eercise 4 is part of the domain? the set of input values; 7. Eplain wh the output value is called the dependent variable. What represents the dependent variable in the function f() 5? It depends on the input value; f(). Lesson 11 Helping You Remember 8. When looking at the word domain, ou see the word in located at the end of the word. This is a wa to remember that the domain is the set of input values. Find a wa to remember that the range is the set of output values. Sample answer: When thinking of the word range, ou can think of an area outdoors where cattle graze. Glencoe/McGraw-Hill 67 Mathematics: Applications and Concepts, Course 3

13 DATE PERID Enrichment Going with the Flow A mathematical function results when one or more operations are performed on a number. A function flowchart and ten numbers are given below. Input each number into the flowchart at the place marked START, then follow the number through the flowchart. When ou reach the place marked STP, record the output number. START Add 10 Multipl b.5 No Is the number whole? Yes Divide b Yes Is the number whole? Square the number No Is the number greater than 8? Yes No Add 5 Round to the nearest whole number Multipl b Find the square root Is the number greater than 5? No Divide b 0.5 Subtract Subtract 10 No Is the number less than 10? Yes Subtract 10 Multipl b 0.5 Add 3 Yes STP Glencoe/McGraw-Hill 68 Mathematics: Applications and Concepts, Course 3

14 DATE PERID Stud Guide and Intervention Graphing Linear Functions A function in which the graph of the solutions forms a line is called a linear function.a linear function can be represented b an equation, a table, a set of ordered pairs, or a graph. Graph. Step 1 Choose some values for. Use these values to make a function table. (, ) 0 0 (0, ) (1, 1) 0 (, 0) (3, 1) Step Graph each ordered pair on a coordinate plane. Draw a line that passes through the points. The line is the graph of the linear function. The value of where the graph crosses the -ais is called the -intercept. The value of where the graph crosses the -ais is called the -intercept. For the graph in Eample 1, the -intercept is and the -intercept is. (3, 1) (, 0) (1, 1) (0, ) Complete the function table. Then graph the function (, ) 3 1 (, 1) (0, 3) (1, 4) 3 5 (, 5) Graph each function Lesson 11 3 Glencoe/McGraw-Hill 69 Mathematics: Applications and Concepts, Course 3

15 Complete the function table. Then graph the function NAME DATE PERID Practice: Skills Graphing Linear Functions 4 (, ) 4 (, ) ( 1, 3) (0, 4) (1, 5). 1 1 (, ) 1 ( 1) 1 3 ( 1, 3) 0 (0) 1 1 (0, 1) 1 (1) 1 1 (1, 1) () 1 3 (, 3) Graph each function Glencoe/McGraw-Hill 630 Mathematics: Applications and Concepts, Course 3

16 DATE PERID Practice: Word Problems Graphing Linear Functions 1. FUEL CNSUMPTIN The function d 18g describes the distance d that Rick can drive his truck on g gallons of gasoline. Graph this function. Eplain wh it is sufficient to graph this function in the upper right quadrant onl. Use the graph to determine how far Rick can drive on.5 gallons of gasoline. 100 Neither the d distance nor 80 the number 60 of gallons of 40 gasoline can 0 be negative; g 45 mi. Distance (mi) Gasoline (gal) 3. GIFTS Jonah received $300 in cash gifts for his fourteenth birthda. The function describes the amount remaining after weeks if Jonah spends $5 each week. Graph the function and determine the amount remaining after 9 weeks. Amount Remaining ($) Week $75 5. GIFTS What is the -intercept of a graph? Find the -intercept of the graph in Eercise 3 and interpret its meaning. The value of where the graph crosses the -ais; 300; there was $300 before the first week.. HTELS The function c 0.5m 1 describes the cost c in dollars of a phone call that lasts m minutes made from a room at the Shad Tree Hotel. Graph the function. Use the graph to determine how much a 7-minute call will cost. $4.50 Cost ($) $5.00 $4.00 $3.00 $.00 $1.00 d Length of Call (min) 4. GIFTS What is the -intercept of a graph? Find the -intercept of the graph in Eercise 3 and interpret its meaning. The value of where the graph crosses the -ais; 1; after 1 weeks the mone is gone. 6. GIFTS Eplain how ou can use our graph in Eercise 3 to determine during which week the amount remaining will fall below $190. Then find the week. Find the point on the graph corresponding to 190 on the vertical ais. This point lies between 4 and 5 on the horizontal ais. The amount falls below $190 during the fifth week. m Lesson 11 3 Glencoe/McGraw-Hill 631 Mathematics: Applications and Concepts, Course 3

17 Pre-Activit DATE PERID Read the introduction at the top of page 5 in our tetbook. Write our answers below. 1. Complete the following function table. Input Reading to Learn Mathematics Graphing Linear Functions Rule 1.5 utput 1 1.5(1) 1.5 (1, 1.5) 1.5() 3.0 (, 3.0) 3 1.5(3) 4.5 (3, 4.5) 4 1.5(4) 6.0 (4, 6.0) (Input, utput) (, ). Graph the ordered pairs on a coordinate plane. 3. What do ou notice about the points on our graph? The appear to be in a straight line. Reading the Lesson 4. In our own words, eplain how to graph a function. Sample answer: Choose some values for and substitute them in the equation to find the values of. Graph each resulting ordered pair on a coordinate plane. Draw a line that passes through all the points. 5. Eplain how to find the -intercept of the graph of a linear function. Then find the -intercept of 8. Replace with 0 and solve for ; Eplain how to find the -intercept of the graph of a linear function. Then find the -intercept of 8. Replace with 0 and solve for ; 8. Helping You Remember 7. Think of a gas pump with prices for regular and super gasoline. When the same amount of gas is being pumped into a tank, how does the price per gallon affect the total cost of the gas? Sample answer: A few cents per gallon can make a big difference in the total costs. Glencoe/McGraw-Hill 63 Mathematics: Applications and Concepts, Course 3

18 DATE PERID Enrichment Graphing Functions Depending on the scale of a graph, the graphed shape of a function can be made to appear ver different from one graph to another. Given the function f(n) = n, find the values of f(n) for each value in the table. Write the ordered pairs, then draw the graph of the function on each grid. 8 f (n) n n n f(n) (n, f(n)) ( ) 8 (, 8) ,4 1 1 ( 1) ( 1, ) , 1 0 (0) 0 (0, 0) , 1 1 (1) (1, ) ,4 1 () 8 (, 8) 8 f (n) n Lesson Eamine each graph of the function f(n) = n.what do ou notice about the graphs? The graphs are not duplicates of each other.. Eplain wh the graphs are different. The graphs are drawn with different scales. Glencoe/McGraw-Hill 633 Mathematics: Applications and Concepts, Course 3

19 The slope m of a line passing through points ( 1, 1 ) and (, ) is the ratio of the difference in the coordinates to the corresponding difference in the coordinates. As an equation, the slope is given b m 1, where 1. 1 DATE PERID Stud Guide and Intervention The Slope Formula m 1 1 Find the slope of the line that passes through A( 1, 1) and B(, 3). Definition of slope m 3 ( 1) ( 1, 1 ) ( 1, 1), ( 1) (, ) (, 3) ( 1, 1) A B (, 3) m 4 3 Simplif. Check When going from left to right, the graph of the line slants upward. This is consistent with a positive slope. m 1 1 Find the slope of the line that passes through C(1, 4) and D(3, ). Definition of slope m 4 ( 1, 1 ) (1, 4), 3 1 (, ) (3, ) m 6 or 3 Simplif. (1, 4) C (3, ) D Check When going from left to right, the graph of the line slants downward. This is consistent with a negative slope. The slope of an horizontal line is zero. The slope of an vertical line is undefined. Find the slope of the line that passes through each pair of points. 1. A(0, 1), B(3, 4) 1. C(1, ), D(3, ) 3. E(4, 4), F(, ) 3 4. G(3, 1), H(6, 3) 3 5. I(4, 3), J(, 4) 1 6. K( 4, 4), L(5, 4) 0 Glencoe/McGraw-Hill 634 Mathematics: Applications and Concepts, Course 3

20 DATE PERID Practice: Skills The Slope Formula Find the slope of the line that passes through each pair of points. 1. A(, 4), B(, 4). C(0, ), D(, 0) 1 3. E(3, 4), F(4, ) 6 4. G( 3, 1), H(, ) 1 5. I(0, 6), J( 1, 1) 5 6. K(0, ), L(, 4) 3 7. (1, 3), P(, 5) 8 8. Q(1, 0), R(3, 0) 0 9. S(0, 4), T(1, 0) U(1, 3), V(1, 5) 11. W(, ), X( 1, 1) 1 1. Y( 5, 0), Z(, 4) 4 3 undefined 13. A(, 1), B( 4, 4) C(, ), D( 4, ) E( 1, 4), F( 3, 0) 16. G(7, 4), H(, 0) K(, ), L(, 3) 18. M( 1, 1), N( 4, 5) 4 3 undefined 19. (5, 3), P( 3, 4) Q( 1, 3), R(1, ) 5 1. W(3, 5), X(1, 1) 3. Y(, ), Z( 5, 4) 6 3. C(0, ), D(3, ) 0 4. G( 3, 5), H( 3, ) 7 undefined Lesson 11 4 Glencoe/McGraw-Hill 635 Mathematics: Applications and Concepts, Course 3

21 DATE PERID Practice: Word Problems The Slope Formula 1. MVIES B the end of its first week, a movie had grossed $.3 million. B the end of its sith week, it had grossed $6.8 million. Graph the data with the week on the horizontal ais and the revenue on the vertical ais, and draw a line through the points. Then find and interpret the slope of the line. Revenue (millions of dollars) ; The film earned an average of $0.9 million dollars per week for 10 weeks 6. Week. BASKETBALL After Game 1, Felicia had scored 14 points. After Game 5, she had scored a total of 8 points for the season. After Game 10, she had scored 19 points. Graph the data with the game number on the horizontal ais and the number of points on the vertical ais. Connect the points using two different line segments. Number of Points Game 3. BASKETBALL Find the slope of each line segment in our graph from Eercise and interpret it. Which part of the graph shows the greater rate of change? Eplain. 17; Felicia scored an average of 17 points per game for Games 5; 9.4; she scored an average of 9.4 points per game for Games 6 10; the first part; it has a greater slope. 4. GEMETRY The figure shows triangle ABC plotted on a coordinate sstem. Eplain how to find the slope of the line through points A and B. Then find the slope. Use the slope B(, 4) formula; 6 5. A( 3, ) C(, ) 5. Use the figure in Eercise 4. What is the slope of the line through points A and C? How do ou know? 0; The line is horizontal. 6. Use the figure in Eercise 4. What is the slope of the line through points B and C? How do ou know? Undefined; the line is vertical. Glencoe/McGraw-Hill 636 Mathematics: Applications and Concepts, Course 3

22 Pre-Activit DATE PERID Reading to Learn Mathematics The Slope Formula Do the Mini Lab at the top of page 56 in our tetbook. Write our answers below. 1. Find the slope of the line b counting units of vertical and horizontal change. 3. Subtract the -coordinate of A from the -coordinate of B. Call this value t. t 3 3. Subtract the -coordinate of A from the -coordinate of B. Call this value s. s t 4. Write the ratio. Compare the slope of the line with t 3 s. s ; The are the same. Reading the Lesson 5. A line passes through the points A( 1, 5), B(0, 1), C(1, 3), and D(, 7). Does it matter which two points ou use to find the slope using the slope formula? Eplain. No; ou can use an two points on a line to find its slope. 6. Suppose ou choose to find the slope of the line in Eercise 5 using points C(1, 3) and D(, 7). If our numerator after substitution into the slope formula is 3 7, what should be our denominator? Eplain. 1 ; the numbers from each ordered pair in the denominator should appear in the same order as those in the numerator. 7. Eplain the difference between 0 3 and is 0, but 3 is undefined. 0 Helping You Remember 8. Fill in the table with the appropriate term, positive, negative, zero, or undefined. Description of Line When going from left to right, the line slants upward. The line is horizontal. The line is vertical. When going left to right, the line slants downward. Slope Positive Zero Undefined Negative Lesson 11 4 Glencoe/McGraw-Hill 637 Mathematics: Applications and Concepts, Course 3

23 DATE PERID Enrichment lga Taussk-Todd lga Taussk-Todd ( ) had a rich and varied career as a research mathematician, mathematics professor, and author and editor of mathematical tets. Born in eastern Europe, she lived and worked in Austria, German, England, and the United States. She served as the consultant in mathematics for the National Bureau of Standards in Washington, D.C. for ten ears. In 1957, she became the first woman appointed to the mathematics department of the California Institute of Technolog. Dr. Taussk-Todd made contributions in man areas of mathematics and phsics. The eercises below will help ou learn some more about her life. Find the slope of the line that passes through each pair of points. The phrase following the slope will complete the statement correctl. 1. A(0, 4), B(, 10) Her paper on sums of squares won the Ford Prize of the Mathematical Association of America in?. 3: : C(, 1), D(3, 9) In 1975, she was elected? of the Austrian Academ of Sciences. 1 : Correspondent : Corresponding Member 3. E( 1, 3), F(8, 4) In 1978, the government of Austria awarded her the?. 1 : Cross of Honor in Science and Arts, First Class 9 9: Purple Cross 4. G(8, 3), H(5, 3) In 1988, the Universit of Southern California awarded her an?. 3: honorar science degree 0: honorar Doctor of Science degree Glencoe/McGraw-Hill 638 Mathematics: Applications and Concepts, Course 3

24 DATE PERID Stud Guide and Intervention Slope-Intercept Form Linear equations are often written in the form m b. This is called the slope-intercept form. When an equation is written in this form, m is the slope and b is the -intercept. State the slope and -intercept of the graph of 3. 3 Write the original equation. 1 ( 3) Write the equation in the form m b. m b m 1, b 3 The slope of the graph is 1, and the -intercept is 3. Lesson 11 5 You can use the slope-intercept form of an equation to graph the equation. Graph 1 using the slope and -intercept. Step 1 Find the slope and -intercept. The slope is, and the -intercept is 1. Step Graph the -intercept (0, 1). Step 3 Write the slope as. Use 1 it to locate a second point on the line. m 1 change in : up units change in :right 1 unit Step 4 Draw a line through the two points. right 1 up 1 State the slope and -intercept of the graph of each equation ; 1. 4 ; ; 1 Graph each equation using the slope and -intercept Glencoe/McGraw-Hill 639 Mathematics: Applications and Concepts, Course 3

25 DATE PERID Practice: Skills Slope-Intercept Form State the slope and -intercept of the graph of each equation ; 4. ; ; ; ; ; ; ; ; Graph a line with a 11. Graph a line with a 1. Graph a line with a slope of 1 and a slope of and a slope of 1 and a 3 -intercept of 4. -intercept of 3. -intercept of 1. Graph each equation using the slope and -intercept Glencoe/McGraw-Hill 640 Mathematics: Applications and Concepts, Course 3

26 DATE PERID Practice: Word Problems Slope-Intercept Form CAR RENTAL For Eercises 1 and, use the following information. Ace Car Rentals charges $0 per da plus a $10 service charge to rent one of its compact cars. The total cost can be represented b the equation 0 10, where is the number of das and is the total cost. 1. Graph the equation. What do the slope and -intercept represent? The slope is the charge per da, and the -intercept is the service 10 charge. Cost ($) Number of Das. Eplain how to use our graph to find the total cost of renting a compact car for 7 das. Then find this cost. Locate 7 on the -ais. Find the -coordinate on the graph where the -coordinate is 7. This value is 150; $150 Lesson 11 5 TRAVEL For Eercises 3 and 4, use the following information. Thomas is driving from ak Ridge to Lakeview, a distance of 300 miles. He drives at a constant 60 miles per hour. The equation for the distance et to go is , where is the number of hours since he left. 3. What is the slope and -intercept? Eplain how to use the slope and -intercept to graph the equation. Then graph the equation. 6 0 ; ; P l o t 300 the point 00 (0, 300). Then locate 100 another point down 60 and right 1. Draw Time (h) the lines through the points. Distance (mi) 5. WEATHER The equation can be used to find the amount of accumulated snow in inches hours after 5 P.M. on a certain da. Identif the slope and -intercept of the graph of the equation and eplain what each represents. 0.; 3.5; the snow is falling at an average rate of 0. inch per hour; there were 3.5 inches of accumulated snow at 5 P.M. 4. What is the -intercept? What does it represent? 5; the total travel time, 5 hours 6. SALARY Janette s weekl salar can be represented b the equation , where is the dollar total of her sales for the week. Identif the slope and -intercept of the graph of the equation and eplain what each represents. 0.4; 500; Janette s salar increases $0.40 for each dollar of sales; Janette earns $500 even if she sells nothing. Glencoe/McGraw-Hill 641 Mathematics: Applications and Concepts, Course 3

27 Pre-Activit Complete the Mini Lab at the top of page 533 in our tetbook. Write our answers below. 1. Use the graphs to find the slope and -intercept of each line. Complete the table.. Compare each equation with the value of its slope. What do ou notice? The slope is the same as the coefficient of. 3. Compare each equation with its -intercept. What do ou notice? The -intercept is the same as the constant. Reading the Lesson DATE PERID Reading to Learn Mathematics Slope-Intercept Form Equation Slope -intercept ( 1) In the formula m b, what do the letters m and b represent? slope; -intercept Identif the slope and the -intercept of the graph of each equation slope 3; -intercept slope ; -intercept How can ou find the slope and the -intercept of the graph of 8? Sample answer: Write the equation in slope-intercept form m b b subtracting from each side, 8. The slope is 1, and the -intercept is If ou know the -intercept of a line is 4 and that the slope is 3, how do ou graph the line? Sample answer: Graph the point (0, 4) then move down 3 units and right units. Helping You Remember 9. Work with a partner. Using a coordinate grid, take turns graphing lines and identifing the slope and -intercept of each graph. See students work. Glencoe/McGraw-Hill 64 Mathematics: Applications and Concepts, Course 3

28 DATE PERID Enrichment Point-Slope Form You have learned to write the equation of a line given the slope and the -intercept. You can use what ou know to write the equation of a line given the slope and an point on the line. To do so, use the point-slope form of a linear equation. The point-slope form of a linear equation is 1 = m( 1 ), where ( 1, 1 ) is a given point on a nonvertical line and m is the slope of the line. ( 1, 1 ) (, ) Lesson 11 5 Write the point-slope form of an equation for the line that passes through each point with the given slope m (1, ) m 3 (1, ) m 3 (1, ) 3( 1) ( 1) 3 ( 1) 4. (6, ), m 3 5. (4, 1), m 5 6. (3, 1), m 3( 6) 1 5( 4) 1 ( 3) 7. ( 4, 1), m 1 8. ( 9, ), m 1 9. ( 5, 6), m ( 4) 1 ( 9) 6 3 ( 5) (, 0), m (1, 3), m 0 1. (0, ), m 3 0 3( ) 3 0( 1) 3( 0) Glencoe/McGraw-Hill 643 Mathematics: Applications and Concepts, Course 3

29 DATE PERID Stud Guide and Intervention Scatter Plots When ou graph two sets of data as ordered pairs, ou make a scatter plot. The pattern of the data points determines the relationship between the two sets of data. Data points that go generall upward show a positive relationship. Data points that go generall downward show a negative relationship. Data points with no clear pattern show no relationship between the data sets. Determine whether a scatter plot of the data might show a positive, negative, or no relationship. miles driven and gallons of gas used As the number of miles driven increases, the amount of gas used increases. Therefore, the scatter plot will show a positive relationship. Gallons of Gas Used Miles Driven number of minutes a candle burns and a candle s height As the number of minutes increases, the height of the candle will decrease. Therefore, the scatter plot will show a negative relationship. Height of Candle (in.) Minutes Burned Determine whether a scatter plot of the data for the following might show a positive, negative, or no relationship. 1. a student s age and the student s grade level in school positive. number of words written and amount of ink remaining in a pen negative 3. square feet of floor space and the cost of carpet for the entire floor positive 4. a person s height and the number of siblings the person has no relationship 5. length of time for a shower and the amount of hot water remaining negative 6. number of sides of a polgon and the area of the polgon no relationship Glencoe/McGraw-Hill 644 Mathematics: Applications and Concepts, Course 3

30 DATE PERID Practice: Skills Scatter Plots Determine whether a scatter plot of the data for the following might show a positive, negative, or no relationship. 1. rotations of a biccle tire and distance traveled on the biccle positive. number of pages printed b an inkjet printer and the amount of ink in the cartridge negative 3. age of a child and the child s shoe size positive 4. number of letters in a person s first name and the person s height no relationship 5. shots attempted and points made in a basketball game positive Lesson ear and winning time in the 100-meter dash in the lmpics negative 7. diameter of the trunk of a tree and the height of the tree positive 8. number of a bank account and the amount of mone in the bank account no relationship 9. length of a tai ride and the amount of the fare positive 10. dail high temperature and the amount of clothing a person wears negative 11. a person s age and a person s street address no relationship 1. outside temperature and the cost of air conditioning positive 13. the age of a car and how man people fit inside of it no relationship 14. inches of rainfall in the last 30 das and the water level in a reservoir positive 15. miles ridden on a biccle tire and thickness of the tire tread negative 16. population of a U.S. state and the number of U.S. senators a state has no relationship Glencoe/McGraw-Hill 645 Mathematics: Applications and Concepts, Course 3

31 WAGES For Eercises 1 and, use the table at the right. DATE PERID Practice: Word Problems Scatter Plots Year Average Hourl Wage 1998 $ $ $ $ $ $ Eplain how to draw a scatter plot for the data. Then draw one. Wage ($) Year 003 For each ear, plot the point (ear, wage) on a coordinate plane. Do not connect the points.. Does the scatter plot show a positive, negative, or no relationship? Eplain. Positive; the data points go up and to the right. RESALE VALUE For Eercises 3 6, use the scatter plot at the right. It shows the resale value of 6 SUVs plotted against the age of the vehicle. Value (thousands) Age (ears) 3. Does the scatter plot show a positive, negative, or no relationship? Eplain what this means in terms of the resale value of a SUV. Negative; as the age of an SUV increases, its resale value decreases. 5. Find the slope and -intercept of the best-fit line and eplain what each represents.,000; 5,000; on average, an SUV is worth $,000 less each ear; the cost of a new SUV is $5, The equation,000 5,000 is an equation of a best-fit line for the data. Eplain what a best-fit line is. a line that is ver close to most of the data points 6. Eplain how to use the equation in Eercise 4 to estimate the resale value of an 8-ear-old SUV. Find the value. Replace in the equation with 8 and simplif; $9,000 Glencoe/McGraw-Hill 646 Mathematics: Applications and Concepts, Course 3

32 Pre-Activit DATE PERID Reading to Learn Mathematics Scatter Plots Complete the Mini Lab at the top of page 539 in our tetbook. Write our answers below. 1. Graph each of the ordered pairs listed on the chalkboard. See students work.. Eamine the graph. Do ou think there is a relationship between height and arm span? Eplain. es; the greater the height, the greater the arm span Reading the Lesson 3. How is a scatter plot different from the graph of a linear function? Sample answer: A scatter plot consists of unconnected points, while the graph of a linear function is a continuous line. Lesson What pattern would ou epect to see in a scatter plot that shows a positive relationship? data points that appear to go upward to the right 5. What pattern would ou epect to see in a scatter plot that shows a negative relationship? data points that appear to go downward to the right 6. Would ou epect a scatter plot to show a positive, negative, or no relationship between the population of a state and its number of representatives in the U.S. Congress? Eplain. Positive; states with larger populations have more representatives in Congress. Helping You Remember 7. Using a newspaper or magazine, find an article with data given. Plot the data on a coordinate plane and identif whether the data has a positive, negative, or no relationship. See students work. Glencoe/McGraw-Hill 647 Mathematics: Applications and Concepts, Course 3

33 DATE PERID Enrichment Latin Squares Suppose that an eperimenter was comparing four new kinds of biccle tires. The eperimenter might choose 4 different kinds of biccles and 4 different riders. Then, 64 test rides would be needed to check all the possible combinations. ne wa to reduce the number of needed trials is to use a Latin square. In this eample, the four tires are labeled T 1, T, T 3, and T 4. The four tpes of tires must be arranged in the Latin square so that each tpe appears onl once in a row or a column. Now the number of test rides is just 16, one for each cell of the Latin square. R 1 R R 3 R 4 B 1 T 1 T T 3 T 4 B T T 1 T 4 T 3 B 3 T 3 T 4 T 1 T B 4 T 4 T 3 T T 1 Make two 4-b-4 Latin squares that are different from the eample. 1. R 1 R R 3 R 4. R 1 R R 3 R 4 See students work. B 1 B 1 B B B 3 B 3 B 4 B 4 Make three different 3-b-3 Latin squares. 3. R 1 R R 3 4. R 1 R R 3 5. R 1 R R 3 B 1 B 1 B 1 B B B B 3 B 3 B 3 Glencoe/McGraw-Hill 648 Mathematics: Applications and Concepts, Course 3

34 DATE PERID Stud Guide and Intervention Graphing Sstems of Equations A set of two or more equations is called a sstem of equations.solving a sstem of equations means finding an ordered pair that is a solution of all the equations. You can solve a sstem of equations b graphing. If ou graph the equations on the same coordinate plane, the point where the graphs intersect is the solution of the sstem of equations. Solve the sstem 1 and 5 b graphing. Both equations are in slope-intercept form. Use the slope and -intercept of each equation to graph the two equations. The graphs appear to intersect at (, 1). Check this b substituting the coordinates into each equation. Check () The solution of the sstem of equations is (, 1). The sstem of equations in Eample 1 had one solution. It is also possible for a sstem of equations to have no solution. If the lines in the graph are parallel, there is no intersection point, and the sstem of equations will have no solution. A sstem of equations can also be solved b a method called substitution. 1 5 (, 1) Lesson 11 7 Solve the sstem 4 and 1 b substitution. 4 Write the first equation. 1 4 Replace with Simplif. The solution of the sstem is ( 5, 1). Subtract 4 from each side. Solve each sstem of equations b graphing (1, ) (3, 1) Solve each sstem of equations b substitution (1, 6) (, 3) 6 3 Glencoe/McGraw-Hill 649 Mathematics: Applications and Concepts, Course 3

35 DATE PERID Practice: Skills Graphing Sstems of Equations Solve each sstem of equations b graphing (, 1) ( 1, 0) (, 1) (, 3) Solve each sstem of equations b substitution. 5. (4, 6) 6. 3 (, 5) 7. 6 (, ) ( 3, 1) (, 1) (5, 13) (1, 7) 1. 5 (, 8) (, 7) 7 7 Glencoe/McGraw-Hill 650 Mathematics: Applications and Concepts, Course 3

36 DATE PERID Practice: Word Problems Graphing Sstems of Equations TAXI SERVICE For Eercises 1 4, use the following information. A-1 Tai service charges $5 for pickup plus $1 per mile for a tai ride. All-About-Town Tai service charge $1 for pickup plus $ per mile. 1. Write an equation for the total charge for a ride that covers miles in an A-1 Tai. 5. Write an equation for the total charge for a ride that covers miles in an All-About-Town Tai Eplain how to solve a sstem of equations b graphing. Then solve the sstem b graphing. Cost of Tai Ride ($) Graph both lines 4 on the same coordinate grid and Miles identif the point of intersection; (4, 9). 4. For what distance is the charge the same for both companies? What is the charge for a ride of this distance? Eplain how ou know this. 4 miles; $9; the point (4, 9) is a solution of both equations. Lesson INCME Robert and Leta each work at a biccle shop selling biccles. Leta makes $150 per week plus $0 for each biccle she sells, and Robert makes $50 per week. The equations and 50 can be used to represent their weekl salaries. Eplain how to solve the sstem of equations b substitution. Then solve the sstem b substitution. What does our solution represent? Substitute 50 for in the first equation and solve for ; (5, 50); Leta and Robert have the same weekl salar, $50, when Leta sells 5 biccles. 6. FD Antonio s Pizza charges $8.00 for a large pizza and $1.50 for each topping. Zina s Pizzaria charges $10.00 for a large pizza and $1.00 for each topping. Write and solve a sstem of equations to determine the number of toppings for which the pizzas would cost the same. What is that cost? , ; (4, 14); 4 toppings; $14 Glencoe/McGraw-Hill 651 Mathematics: Applications and Concepts, Course 3

37 Pre-Activit Read the introduction at the top of page 544 in our tetbook. Write our answers below. 1. Graph both of the equations on a coordinate plane.. What are the coordinates of the point where the two lines intersect? What does this point represent? (, 1); The storm and the ship will meet in hours. At that time the storm will have traveled 1 miles. Reading the Lesson DATE PERID Reading to Learn Mathematics Graphing Sstems of Equations 3. Eplain how to find the solution of a sstem of two equations b graphing. Graph both equations on the same coordinate plane and locate the point(s) of intersection. 4. Under what conditions will a sstem of two linear equations have infinitel man solutions? when the graphs of the two equations are the same 5. Under what conditions will a sstem of two linear equations have no solution? when the lines are parallel 6. What is the first step in solving the sstem 6 and 1 b substitution? Replace with 1 in the first equation How do ou check a solution of a sstem of two equations? Replace the variables in both equations with the solution and verif that both are true sentences Helping You Remember 8. Use a clean sheet of paper and Eamples 3 and 4 on pages 545 in our tetbook. Starting with Eample 3, cover everthing up in the eample with our paper ecept the title and its question. Now tr to work the problem without looking at the book. Then compare our work to the work in the book. Repeat this with Eample 4. See students work. Glencoe/McGraw-Hill 65 Mathematics: Applications and Concepts, Course 3

38 DATE PERID Enrichment Graphing Sstems of Equations Because checking accounts var from one financial institution to another, educated consumers should carefull weigh the options offered b various accounts when choosing a checking account. The four equations below describe the cost of four different checking accounts. In each equation, C represents the monthl cost in dollars, and n represents the number of checks written. Find the cost for the number of checks written in each account. Account A Account B Account C Account D C 0.1 n 1.50 C 0. n 1.00 C 0.5 n 0.75 C 0.05 n 1.75 n C 0 $1.50 $ $.00 8 $.30 1 $ $3.00 n C 0 $1.00 $ $ $.00 8 $ $3.00 n C 0 $ $ $.00 6 $.5 8 $.75 9 $3.00 n C 0 $1.75 $ $.00 9 $.0 14 $.45 0 $.75 Lesson 11 7 Graph and label each account on the grid. n 0 Number of Checks Cost in Dollars c 1. The break-even point of the graph is the point at which the costs of the accounts are the same. What is the break-even point for these accounts? five checks and $.00. Which account should be chosen b a consumer who writes fewer than five checks per month? account C 3. Which account should be chosen b a consumer who writes more than five checks per month? account D Glencoe/McGraw-Hill 653 Mathematics: Applications and Concepts, Course 3

39 DATE PERID Stud Guide and Intervention Graphing Linear Inequalities Graphing a linear inequalit takes several steps. First, ou must graph the related equation. The related equation is obtained b replacing the inequalit smbol with an equals sign. If the inequalit smbol is or, the related equation should be graphed as a solid line. If the inequalit smbol is or, the related equation should be graphed as a dashed line. This solid or dashed line is the boundar of the solution. Net, test an point above or below the line to determine which region is the solution of the inequalit. Shade the region that contains the solution. All points in this region are solutions to the inequalit. Graph. Step 1 Graph the boundar line. Since is used in the inequalit, make the boundar line a dashed line. Step Test a point not on the boundar line, such as (0, 0). Write the inequalit. 0? (0) Replace with 0 and with 0. 0 Simplif. Step 3 Since (0, 0) is a solution of, shade the region that contains (0, 0). Graph each inequalit Glencoe/McGraw-Hill 654 Mathematics: Applications and Concepts, Course 3

40 DATE PERID Practice: Skills Graphing Linear Inequalities Graph each inequalit Lesson Glencoe/McGraw-Hill 655 Mathematics: Applications and Concepts, Course 3

41 NUTRITIN For Eercises 1 and, use the following information. Carl is making his own sports drink b miing orange juice and water in a 40 ounce container. 1. Make a graph showing all the different amounts of orange juice and water that Carl can use in his drink. DATE PERID Practice: Word Problems Graphing Linear Inequalities J (oz) Water (oz). Give three possible amounts of orange juice and water that Carl can use. Sample answers: water 10 oz, J 10 oz; water 0 oz, J 15 oz; water 30 oz, J 10 oz GEMETRY For Eercises 3 and 4, use the following information. The formula for the perimeter, P, of a rectangle of length and width is P. 3. Make a graph for all rectangles that have a perimeter of less than or equal to 0 units Give three possible measurements for the length and width of a rectangle that has a perimeter of less than or equal to 0 units. Sample answers: length 8 units, width units; length units, width units; length 4 units, width 5 unit FD For Eercises 5 and 6, use the following information. At the local farmer s market, apples are $ per pound and blueberries are $3 per pound. Rene wants to bu at least $1 worth of apples and blueberries. 5. Make a graph for all the weights of apples and blueberries that Rene can bu. Blueberries (lb) Give three possible was that Rene can bu the amount of fruit she wants. Sample answers: 7 pounds of apples; 5 pounds of apples and pounds of blueberries; 4 pounds of apples and 4 pounds of blueberries Apples (lb) Glencoe/McGraw-Hill 656 Mathematics: Applications and Concepts, Course 3

42 Pre-Activit DATE PERID Reading to Learn Mathematics Graphing Linear Inequalities Read the introduction at the top of page 548 in our tetbook. Write our answers below. 1. Use the graph to list three different combinations of CDs and books that Sabrina can purchase for $16. Sample answer: CDs, 1 books; 4 CDs, 8 books; 6 CDs, 4 books. Suppose Sabrina wants to spend less than $16. Substitute (, 7), (4, ), (5, 1), and (7, 5) in 16. Which values make the inequalit true? (, 7), (4, ) 3. Which region do ou think represents 16? the ellow region 4. Suppose Sabrina can spend more than $16. Substitute (, 7), (4, ), (5, 1), and (7, 5) in 16. Which values make the inequalit true? (5, 1), (7, 5) 5. Which region do ou think represents 16? the blue region Reading the Lesson 6. Wh is graphing a line an important skill to have in order to graph a linear inequalit? The first step of solving a linear inequalit is to graph the boundar line. 7. What does it mean when ou have a dashed boundar line? a solid boundar line? A dashed boundar line means the points are not included in the solution; a solid boundar line means the points are included in the solution. 8. Eplain how ou know which half plane should be shaded. Sample answer: Test a point in one half plane. If it is a solution, shade the half plane containing it. If not, shade the other half plane. Lesson 11 8 Helping You Remember 9. Complete the table to help ou remember the tpes of lines and where to shade when graphing inequalities. Inequalit Smbol Dashed or Solid Line dashed solid solid dashed Glencoe/McGraw-Hill 657 Mathematics: Applications and Concepts, Course 3