Chapter Chapter Maintaining Mathematical Proficienc (p.). C A B E F D. This point is in Quadrant I.. This point is in Quadrant II.. This point is on the positive -ais.. This point is in Quadrant III. 5. This point is on the negative -ais.. This point is in Quadrant IV. 7. = (7) = = 7. 5 + = 5() + = 5 + = 7 9. + = (5) + = 5 + =. 9 = 9() = =. = ( ) = ( ) = + =. 5 + 9 = 5( ) + 9 = 5 + 9 =. (a, b): Start at the origin. Move a units right and b units up. Then plot the point. ( a, b): Start at the origin. Move a units left and b units up. Then plot the point. (a, b): Start at the origin. Move a units right and b units down. Then plot the point. ( a, b): Start at the origin. Move a units left and b units down. Then plot the point. Chapter Mathematical Practices (p. ). es; The range of the -ais is 7 ( ) = 5 units, and the range of the -ais is 7 ( ) = units. The ratio of height to width is 5 =, which matches a tpical graphing calculator screen.. no; The range of the -ais is ( ) = units, and the range of the -ais is 9 ( 9) = units. The ratio of height to width is =, which does not match a tpical graphing calculator screen.. es; The range of the -ais is ( ) = units, and the range of the -ais is ( ) = units. The ratio of height to width is =, which matches a tpical graphing calculator screen.... = + = 5 = 7 5. 7. 9. = = + 5 = +. The slope of a line appears less steep when ou use a standard viewing window as opposed to a square viewing window because the tic marks on the -ais are closer together than the tic marks on the -ais. 5 Copright Big Ideas Learning, LLC Algebra
Chapter. Eplorations (p. ). a. Sample answer: Each -coordinate is paired with eactl one -coordinate. b. Sample answer: {(, ), (, ), (, ), (, ), (, ) }. a. es; Each input value is paired with eactl one output value. b. no; The input value of is paired with more than one output value. c. no; The input value of is paired with more than one output value. d. no; The input values of 5 and 7 are paired with more than one output value. e. es; Each input value is paired with eactl one output value. f. no; The input value of is paired with more than one output value. g. es; Each input value is paired with eactl one output value. h. no; Each input television station is paired with more than one output channel. i. no; The input value of is paired with ever possible output value of. j. es; Each input value of is paired with eactl one output value of.. A function is a relation that pairs each input with eactl one output. a. Sample answer: {(, ), ( 5, ), (, ), (5, ), (, ) }; Each student in class is assigned a different random number; = 7 b. Sample answer: {(, ), (, ), (, ), (, ), (, ), (, ), (, ) }; 7 7 5 5 5 5 7. no; Infinitel man vertical lines can be drawn so that it passes through two points, such as (, ) and (, ).. es; No vertical line can be drawn through more than one point on the graph. 9. The domain is,,,, and. The range is,, and.. The domain is 5. The range is.. a. The number a of avocados ou have left depends on the number b of batches ou made. So, a is the dependent variable and b is the independent variable. b. The range is,,, and.. a. The temperature t of an oven depends on the number of minutes m that is has been preheating. So, t is the dependent variable and m is the independent variable. b. If m =, then t = 9() + 5 = 5. If t = 5, ou can solve the equation for m. 5 = 9m + 5 5 = 9m 5 = m So, the domain is m 5 and the range is 5 F t 5 F. In other words, over a span of 5 minutes, the temperature will steadil increase from 5 F to 5 F.. Eercises (pp. ) Vocabular and Core Concept Check. Sample answer: The independent variable represents the input values of a function and can be an value in the domain. The dependent variable represents the output values of the function and depends on the value of the independent variable.. Find the range of the function represented b the table; range: 7, 5, and ; domain:,, and Monitoring Progress and Modeling with Mathematics. es; Each input has eactl one output.. Monitoring Progress (pp. 7). no; The input 5 has two outputs, and.. es; Each input has eactl one output.. es; Each input has eactl one output.. no; The input has three outputs,,, and. 5. es; No vertical line can be drawn through more than one point on the graph.. es; No vertical line can be drawn through more than one point on the graph.. no; The input has two outputs, and. 5. no; The input has two outputs, and.. es; Each input has eactl one output. 7. no; The input has two outputs, and, and the input has two outputs, and.. es; Each input has eactl one output. 9. es; No vertical line can be drawn through more than one point on the graph.. no; Three vertical lines can be drawn through more than one point on the graph, including one through (, ) and (, 5). Algebra Copright Big Ideas Learning, LLC
Chapter. no; A vertical line can be drawn through more than one point on the graph, including one through (, ) and (, ).. es; No vertical line can be drawn through more than one point on the graph.. Write the ordered pairs. Identif the inputs and outputs. (, ), (, ), (, ), (, ), (, ) The domain is,,,, and. The range is,, and.. Write the ordered pairs. Identif the inputs and outputs. (, ), (, ), (, ), (, ) The domain is,,, and. The range is. 5. Identif the - and -values represented b the graph. The domain is. The range is.. Identif the - and -values represented b the graph. The domain is < < 7. The range is < <. 7. a. The amount of our monthl rent depends on how man das late it is when ou pa. So, is the dependent variable and is the independent variable. b. Make an input-output table to find the range. Input, 5 + 5 Output, 5() + 5 5 5() + 5 55 5() + 5 55 5() + 5 575 5() + 5 5 5(5) + 5 5 The range is 5, 55, 55, 575,, and 5.. a. The cost of a tai ride depends on the distance in miles of the ride. So, is the dependent variable and is the independent variable. b. The domain is. If =, then =.5() +. =., and if =, then = 5() +. = 7.. So, the range is. 7.. So, if ou have enough mone to travel at most miles in the tai, then ou will spend between $. and $7. on our tai ride. 9. Sample answer: A function can have the same output paired with more than one input, but a relation is not a function if the same input is paired with more than one output; The relation is a function. Each input is paired with eactl one output.. The output values should be used for the range. The relation is a function. The range is, 7,, and 9.. The amount of time ou have on a meter depends on the amount of quarters ou put into the meter. So, the independent variable is the number of quarters and the dependent variable is the amount of time.. The amount of batter power remaining on our MP plaer depends on the amount of time ou listen to it. So, the independent variable is the amount of time and the dependent variable is the amount of batter power remaining.. a. Sample answer: You started a bank account with $, and then ou add $5 each month through month. b. (, ), (, 5), (, 5), (, 75), (, ) c. Balance (dollars) 5 5 5 Month. a..5 +.5 =.5 =.5 = b. Input, Output, () () () () 5 () 5 (5) 9 () 7 (7) () The ordered pairs are (, ), (, ), (, ), (, 5), (, ), (5, 9), (, ), (7, ), and (, ). c. Softcover Hardcover Copright Big Ideas Learning, LLC Algebra 5
Chapter 5. =. Sample answer: t v 7. a. Each input (letter-number) combination is paired with eactl one output location (food or drink item). b. The food or drink item selected depends on the letternumber combination that is entered. So, the letter-number combination is the independent variable, and the food or drink item is the dependent variable. c. The domain is A, A, A, B, B, B, B, C, C, C, and C. The range is popcorn, nuts, pretzels, protein bar, granola bar, cereal, energ bar, orange juice, water, and milk.. a. No vertical line can be drawn through more than one point on the graph. So, the graph represents a function. b. The approimate height after.5 second is feet and after.5 seconds is 9 feet. c. Sample answer: The approimate domain is t.. d. no; Sample answer: There are man values of h that have two corresponding values of t. 9. no; A vertical line does not represent a function.. Sample answer: Each ninth grade student is paired with eactl one eleventh or twelfth grade mentor when he or she starts high school. Because the name of the mentor depends on the name of the ninth grader, the independent variable is the name of the ninth grader, and the dependent variable is the name of the mentor. The domain of the function is a list of the ninth graders, and the range of the function is the list of the mentors.. no; Sample answer: Items that cost the same to make could be sold for different prices.. es; Sample answer: Each input (selling price) is paired with eactl one output (amount of sales ta).. es; Sample answer: Each student has eactl one homeroom teacher.. no; Sample answer: Each input (chaperone) is paired with more than one output (student) on a school trip. 7. false; Sample answer: A function ma have more than one output paired with the same input, and it is still a function, but if this is switched, then there will be at least one input paired with more than one output, and it will no longer be a function.. false; Sample answer: Because more than one input can be paired with the same output, the outputs could be selected from a finite list. 9. a. Let p be the perimeter of the given triangle. Then, p = h + + = h +. b. The perimeter of the triangle depends on the side lengths. So, the side length h is the independent variable, and the perimeter p is the dependent variable. c. Because the sum of an two side lengths must be greater than the third side length, h has to be greater than and less than. So, the domain is < h <. If h =, then p = + =, and if h =, then p = + =. So, the range is < p <.. Because an value can go into an absolute value function, the domain is all real numbers. Because onl positive values and zero come out of an absolute value, the range is.. Because an value can go into an absolute value function, the domain is all real numbers. Because onl positive values and zero come out of an absolute value, and this function takes the opposite after taking the absolute value, the range is.. Because an value can go into an absolute value function, the domain is all real numbers. Because onl positive values and zero come out of an absolute value, and this function subtracts after taking the absolute value of the input, the range is.. Because an value can go into an absolute value function, the domain is all real numbers. Because onl positive values and zero come out of an absolute value, and this function subtracts what comes out from, the range is. Maintaining Mathematical Proficienc. < 5.. 7 d 5 7. w + >. = = 9. ( ) = ( ) ( ) ( ) ( ) = 5. 5 = 5 5 = 5 5. 5 = = 5. true. false; Sample answer: A relation is not a function when an input value has more than one output value. Algebra Copright Big Ideas Learning, LLC
Chapter. Eplorations (p. ). a. 5 P P d. r 5 A... 5. 7.5 A b. This pattern is linear. Sample answer: The graph is a line. 5 A 5 A This pattern is nonlinear. Sample answer: The graph is not a line. c. r 5 C... 5.. C r This pattern is linear. Sample answer: The graph is a line. r This pattern is nonlinear. Sample answer: The graph is not a line.. Sample answer: No vertical line can be drawn through more than one point on an of the graphs.. Sample answer: Sketch the graph of the function. When the graph is a line, the pattern is linear. When the graph is not a line, the pattern is not linear.. Sample answer: A linear function would describe the distance a car travels when it is traveling at 5 miles per hour. A nonlinear function would describe the height of a baseball from when it leaves the outfielder s hand until it reaches the catcher s glove at home plate.. Monitoring Progress (pp. ). The graph is a line. So, the function is linear.. The graph is not a line. So, the function is nonlinear.. As increases b, increases b. The rate of change is constant. So, the function is linear.. As increases b, decreases b different amounts. The rate of change is not constant. So, the function is nonlinear. 5. You can rewrite the equation = + 9 as = + 9. So, it represents a linear function.. You can rewrite the equation = 5 as = +. So, it 5 represents a linear function. 7. You cannot rewrite the equation = 5 in the form = m + b. So, this equation represents a nonlinear function. Copright Big Ideas Learning, LLC Algebra 7
Chapter. a. You cannot bu part of a DVD, onl a certain number of DVDs. Because d represents the number of DVDs, it must be a whole number. Also, if d =, then m = 5 9() =, and ou cannot have a negative amount of mone left. So, the maimum number of DVDs, ou can bu is 5, and minimum is since ou cannot bu a negative number of DVDs. So, the domain is,,,,, and 5, and it is discrete. b. Input, d 5 9d Output, m (d, m) 5 9() 5 (, 5) 5 9() (, ) 5 9() (, ) 5 9() (, ) 5 9() (, ) 5 5 9(5) 5 (5, 5) Plot the ordered pairs. The domain is discrete. So, the graph consists of individual points. Mone left (dollars) m 5 (, 5) (, ) (, ) (, ) (, ) (5, 5) d DVDs purchased 9. You cannot have part of a stor on a building, onl a certain number of stories. So, the domain is discrete.. a. es; As m increases b, the value of g decreases b.5. The bathtub is draining at a constant rate. So, this situation represents a linear function. b. You can measure time b parts of a minute. So, the domain is continuous. Since =, it will take.5 minutes for the bathtub to empt. So, the amount of time, m, could be an number between and, or m. c. Input, m Output, g (m, g) (, ) 5 (, 5) (, ) 5 (, 5) (, ) Plot the ordered pairs. The domain is continuous. So, the points should be connected to show the possible values in between. Gallons remaining g (, ) (, 5) (, ) (, 5) (, ) Minutes. Sample answer: One possibilit is the number of tokens that ou earn in a game based on the number of times that ou land on the Take a Token space, where ever plaer starts the game with tokens, and the first plaer to get tokens wins. Because it is not possible to take part of a token, the domain is discrete.. Sample answer: One possibilit is the cost of a tai ride that is $.5 per mile plus a $. fee and onl runs along a -mile portion of the cit. Because it is possible to travel an portion of a mile, the domain is continuous.. Eercises (pp. 7 ) Vocabular and Core Concept Check. A linear equation in two variables is an equation that can be written in the form = m + b, where m and b are constants.. Sample answer: Both linear and nonlinear functions describe a relation in which each input is paired with eactl one output. The graph of a linear function is a nonvertical line, but the graph of a nonlinear function is not a line. Linear functions have a constant rate of change, but nonlinear functions do not. A linear function is represented b an equation that can be written in the form = m + b, where m and b are constants, but nonlinear functions cannot. m. Sample answer: Both discrete and continuous domains are sets of input values in an interval, but a discrete domain consists of onl certain numbers in the interval, while a continuous domain consists of all numbers in the interval.. Sample answer: The graph of a function with a discrete domain will have separate points, but the graph of a function with a continuous domain will show a line or curve connecting all of the possible values. Algebra Copright Big Ideas Learning, LLC
Chapter Monitoring Progress and Modeling with Mathematics 5. The graph is not a line. So, the function is nonlinear.. The graph is a line. So, the function is linear. 7. The graph is a line. So, the function is linear.. The graph is a line. So, the function is linear. 9. The graph is not a line. So, the function is nonlinear.. The graph is not a line. So, the function is nonlinear.. As increases b, increases b 5. The rate of change is constant. So, the function is linear.. As increases b, increases b different amounts. The rate of change is not constant. So, the function is nonlinear.. As increases b, increases b different amounts. The rate of change is not constant. So, the function is nonlinear.. As increases b, decreases b 5. The rate of change is constant. So, the function is linear. 5. A linear function should have the same amount added to or subtracted from the output values when the input values have the same amount added to or subtracted from them; As increases b, increases b different amounts. Since the rate of change is not constant, the function is nonlinear.. B definition, the graph of a linear function must be a nonvertical line; The graph is a vertical line. So, the graph does not represent a linear function. 7. You cannot rewrite the equation = + in the form = m + b. So, this equation represents a nonlinear function.. You can rewrite the equation = 7 as = + 7. So, it represents a linear function. 9. You can rewrite the equation = as = +. So, it represents a linear function.. You cannot rewrite the equation = ( ) which simplifies to =, in the form = m + b. So, this equation represents a nonlinear function.. You can rewrite the equation + = + as = + b subtracting from both sides, and then = + b multipling both sides b. So, it represents a linear function.. You can rewrite the equation = as = b adding to both sides, and then 5 = b adding to both sides, and then = 9 5 + b multipling both sides b and adding. 5 So, it represents a linear function.. You can rewrite the function = as = + b adding to both sides, and then = 9 b dividing both sides b. So, it represents a linear function.. You cannot rewrite the equation + = 9 in the form = m + b. So, this equation represents a nonlinear function. 5. A, C, F; Sample answer: You cannot rewrite the equations = +, =, and = + in the form = m + b. So, these equations do not represent linear functions. However, ou can rewrite + = as =, the equation = 9 as = +, and the equation = 5 as = 5 +. So, the represent linear functions.. Because the change in is ( ) =, and ou need to make four equal increases, add = to each -value in order to get the net one. 5 5 5 5 7. The domain is,, and. It is discrete because the graph consists of individual points.. The domain is 7. It is continuous because the graph consists of all numbers in an interval. 9. The domain is discrete because ou cannot have part of a bag, onl a certain number of bags.. The domain is continuous because ou could measure the height of the tree after an part of a ear.. The domain is continuous because ou could find the distance after an part of an hour.. The domain is discrete because ou can onl have a certain number of teams.. Sample answer: The domain should list all of the input values. The domain is,,,, 5, and.. Sample answer: The graph shows a set of input values that consists of all numbers in an interval, so it is continuous. The graph ends at =, so the domain is. Copright Big Ideas Learning, LLC Algebra 9
Chapter 5. a. You cannot bu part of a book, onl a certain number of books. So, the domain is discrete. Because 55.5, ou.5 cannot bu more than books. So, the domain is,,,,, 5, and. b. Input, b Output, m (b, m) 55 (, 55).5 (,.5) (, ) 9.5 (, 9.5) (, ) 5.5 (5,.5) (, ) Plot the ordered pairs. The domain is discrete. So, the graph consists of individual points. Mone left (dollars) m 5 (, 55) (,.5) (, ) (, 9.5) (, ) Books bought (5,.5) (, ) b. a. Because ou can do rock climbing for an part of an hour, the domain is continuous. So, the domain is. b. Input, Output, (, ) (, ) 5 (, 5) (, ) 95 (, 95) Plot the ordered pairs. Draw a line through the points. The line should start at (, ) and continue to the right. Calories burned Hours (, 95) (, ) (, 5) (, ) 7. a. As t increases b, d increases b.. The rate of change is constant. So, the function is linear. b. The travel time t of the sound wave could be an number greater than or equal to. So, the domain is t, and it is continuous. c. d Distance (miles) (,.7) (,.7) (,.) (,.) (,.) Time (seconds). a. You can rewrite the equation = + 5 as = 5 +. So, it represents a linear function. b. Since there are five services to choose from, and ou cannot bu part of a service, the domain is,,,,, and 5, and it is discrete. c. Input, Output, (, ) (, ) 5 (, 5) (, ) 5 (, 5) 5 (, 5) 5 55 (5, 55) Plot the ordered pairs. The domain is discrete. So, the graph consists of individual points. Cost (dollars) 5 (5, 55) (, 5) (, 5) (, ) (, 5) (, ) Etra services bought 9. Sample answer: The number of dogs that can be fed depends on how man packs of food were purchased, where the food onl comes in -packs. The domain is discrete because packs can onl be purchases in multiples of.. Sample answer: Over a -minute span, there is a temperature drop before a snow storm. The temperature in degrees Celsius depends on the number of minutes that have passed. The domain is continuous because the drop in temperature happens over a span of minutes, and the temperature could be recorded after an part of a minute. Algebra Copright Big Ideas Learning, LLC
Chapter. Sample answer: The elevation of a submarine depends on how long it has been ascending. The domain is continuous because ou could find the elevation of the submarine after an amount of time.. Sample answer: You have started a new babsitting job. You go over for hours on Sunda afternoon without getting paid so that ou can get to know the children. Then, ou babsit for hours ever Thursda and get paid $ each time. The amount of mone ou have earned after hours of babsitting in the first month is represented b. The domain is discrete because ou get paid each time ou do our -hour babsitting job.. a. 5.; Because 7.. =., ou can find the missing value b adding. to.. b. $. per hour; As the hours worked increases b, the earnings increases b $... a. nonlinear; As t increases b, the distance d increases b different amounts. Because the rate of change is not constant, the function is nonlinear. b. The average speed of Car B in the first hours is = miles per hour, and in the last hours, the average speed is = = 5 miles per hour. Because Car A is traveling at 5 miles per hour, Car B is traveling faster. 5. The volume of the rectangular prism can be found using the equation V = s(s)(9), or V = 9s. Because there is an eponent on the variable, this equation cannot be written in the form = m + b. So, this equation represents a nonlinear function.. The volume of the rectangular prism can be found using the equation V = (b)()(), or V = b. This equation can be rewritten as V = b +. So, this equation represents a linear function. 7. The volume of the clinder can be found using the equation V = π () (h), or V = πh. This equation can be rewritten as V = π h +. So, this equation represents a linear function.. The volume of the cone can be found using the equation V = π (r) (5), or V = 5πr. Because there is an eponent on the variable, this equation cannot be written in the form = m + b. So, this equation represents a nonlinear function. 9. a. The epression represents the combined amount of water used to fill one of each kind of jug. Because a jog could hold an amount of water, the domain is continuous. b. The epression represents the total number of jugs the compan fills. Because the number of jugs must be a whole number, the domain is discrete. c. The epression represents the total amount of water used to fill all the jugs of the first size. Because a jug could hold an amount of water, the domain is continuous. d. The epression represents the total amount of water used to fill all of the jugs. Because a jug could hold an amount of water, the domain is continuous. 5. Sample answer: The cost c at a farmer s market is most likel based on weight w. Assume tomatoes cost $.9 per pound. Because the tomatoes could weigh an part of a pound, the domain is w, and it is continuous. Input, w Output, c (w, c) (, ).9 (,.9) 5.7 (, 5.7).7 (,.7).5 (,.5) 5.5 (5,.5) Draw a line through the points. The line should start at (, ) and continue to the right. Cost (dollars) c (5,.5) (,.5) (,.7) (, 5.7) (,.9) (, ) w Weight (pounds) 5. When increases b, increases b ; when increases b, increases b ; when increases b, increases b. Because = = =, it must be that when increases b, increases b, and this is a constant rate of change. So, the function is linear. 5. a. Sample answer: You run at a constant rate because our graph is a line, which means that the function representing our run is linear. Your friend starts out slow, increases in speed and then slows down at the end. A person ma not run at a constant rate because of fatigue. b. The domain of both functions is. Both ou and our friend run for a total of minutes. 5. Sample answer: When ou wake up in the morning, the temperature is below C, but the temperature goes up as the sun comes out. 5. Sample answer: a professional golfer s score on holes as it compares to par; So, for eample, if the golfer gets a on a par, then the score is. Copright Big Ideas Learning, LLC Algebra
Chapter Maintaining Mathematical Proficienc 55. Because the graph does not pass through the origin, and do not show direct variation. 5. Because the graph passes through the origin and is a line, and show direct variation. 57. Because the graph is not a line, and do not show direct variation. 5. + = () + = + = 59. + = () + = + = + =. ( + 5) = ( + 5()) = ( + ) = ( ) = ( ) =. + 5 7 = + 5() 7 = + 5() 7 = + 7 = 7 =. Eplorations (p. ). a. B; Because f () = is in the form = m + b, the function is linear, and the graph is a line with a positive slope and a -intercept of. b. D; Because g() = + is in the form = m + b, the function is linear, and the graph is a line with a negative slope and a -intercept of. c. A; Because h() = has an eponent of on the independent variable, the function is a parabola with a -intercept of. d. C; Because h() = has an eponent of on the independent variable, the function is a parabola with a -intercept of.. a. The location of the point is (, ). b. The location of the point is (, ). c. The location of the point is (, ). d. The location of the point is (, ). Sample answer: Locate the -coordinate, trace it up to the line, and then trace over to the -ais to determine the -coordinate.. Sample answer: Both notations are used to represent functions. If f is a function, and is in its domain, then f () represents the output of f corresponding to the input. Similarl, when standard notation is used to represent a function, if is in the range and is in the domain, then represents the output corresponding to the input. For both standard and function notation, other letters can be used to represent the variables. Equations in standard notation can have man different forms, and variables can be on both sides of the equal sign, but functions in function notation are usuall written with the f () on one side of the equal sign, and an epression containing on the other side.. Monitoring Progress (pp. ). f () = 5. g() = f () = () 5 g() = () = 5 = = = f () = 5 g() = f () = () 5 g() = () = 5 = = 5 = f () = 5 g() = f () = () 5 g() = () = 5 = = =. a. The temperature after hours is 75 F. So, the temperature at p.m. is 75 F. b. The temperature m hours after 9 a.m. is 7 F. c. The output of f when t = is the same as the output of f when t = 9. So, the temperature at a.m. ( hours after 9 a.m.) is the same as the temperature at p.m. (9 hours after 9 a.m.). d. The output of f when t = is greater than the initial value of the function. So, the temperature at p.m. ( hours after 9 a.m.) is greater than the temperature at 9 a.m.. f () = + 9 = + 9 9 9 = = = When =, f () =. Algebra Copright Big Ideas Learning, LLC
Chapter 5. g() = + = + = () = ( ) = When =, f () =. 9. Graph f () = 5 75. Make an input-output table to find the ordered pairs. f () 5 75 5 Plot the ordered pairs. Draw a line through the points. Note that the function onl makes sense when and f () are positive. So, onl draw the line in the first quadrant.. f () 5 5 f() = 5 75 7. f() = g() 5 g() = + 5 5 From the graph of the first flight, ou can see that when f () =, =. From the graph of the second flight, ou can see that when f () =, is slightl greater than. So, the second flight takes more time. You can check that our answer is correct b finding the value of for which f () =. f () = 5 75 = 5 75 5 = 75 = So, the second flight takes hours, or hours and minutes, which is more than.. h() 5 7 h() =. Eercises (pp. 5 ) Vocabular and Core Concept Check. When ou write the function = + as f () = +, ou are using function notation.. b() represents our height at age. Monitoring Progress and Modeling with Mathematics. f () = +. g() = f ( ) = + g( ) = ( ) = = f () = + g() = f () = + g() = () = = f () = + g() = f (5) = 5 + g(5) = (5) = = 5 Copright Big Ideas Learning, LLC Algebra
Chapter 5. h() = + 9. r () = 7 h( ) = ( ) + 9 r ( ) = ( ) 7 = + 9 = 7 = = 5 h() = + 9 r () = 7 h() = () + 9 r () = () 7 = + 9 = 7 = 9 = 7 h() = + 9 r () = 7 h(5) = (5) + 9 r (5) = (5) 7 = + 9 = 5 7 = = 7. p() = + p( ) = + ( ) = + ( ) = p() = + p() = + () = + = p() = + p(5) = + (5) = + = 7. b() =.5 b( ) =.5( ) = ( ) = + = 9 b() =.5 b() =.5() = = b() =.5 b(5) =.5(5) =.5 = 5.5 9. v() = 5 v( ) = ( ) 5 = () 5 = + 5 = 5 = v() = 5 v() = () 5 = 5 = 5 = 7 v() = 5 v(5) = (5) 5 = 5 = 5 =. n() = + n( ) = ( ) + = + + = + = 5 n() = + n() = + = + = n() = + n(5) = 5 + = + =. a. The initial value of the function is. So, there were no customers in the restaurant at a.m. b. The output of c when t = is the same as the output of c when t =. So, the number of customers in the restaurant at a.m. ( hours after a.m.) is the same as the number of customers at p.m. ( hours after a.m.). c. The output when t = n is 9. So, there are 9 customers in the restaurant n hours after a.m. d. The output of c when t = is less than the output of c when t =. So, there are fewer people in the restaurant at 9 p.m. ( hours after a.m.) than there are at p.m. ( hours after a.m.). Algebra Copright Big Ideas Learning, LLC
Chapter. a. The output when = is 55. So, 55% of U.S. households had Internet use in ( ears after 9). b. The output when = is k. So, k% of U.S. households had Internet use in 9 ( ears after 9). c. The output of H when = 7 is greater than or equal to %. So, at least % of U.S. households had Internet use in 7 (7 ears after 9). d. The sum of the output of H when = 7 and the output of H when = is approimatel equal to the output of H when = 9. So, the sum of the percents of U.S. households with Internet use in 997 and is about the same as the percent of U.S. households with Internet use in 9.. h() = 7 = 7 7 = 7 7 9 = When = 9, h() =.. t() = = = = When =, t() =. 5. m() = + 5 7 = + 5 5 5 = = = When =, m() = 7.. k() = = + + + = = 5 = When = 5, k() =. 7. q() = = + + = ( ) = ( ) = When =, q() =.. j() = 5 + 7 5 = 5 + 7 7 7 = 5 5 ( ) = 5 ( 5 ) 5 = When = 5, j() = 5. 9. = 5; The point (5, 7) is on the line. So, f (5) = 7.. = ; The point (, 7) is on the line. So, f ( ) = 7.. a. C() = 7.5 C(5) = 7.5(5) = 7.5 = 77.5 It costs $77.5 to bu five tickets. b. C() = 7.5 = 7.5 + + = 7.5 7.5 = 7.5 7.5 = You can bu tickets with $.. a. d(t) =,t d(5) =,(5) =,5, Light travels,5, kilometers in 5 seconds. b. d(t) =,t,, =,t,,, =,t, = t It takes light seconds to travel million kilometers. Copright Big Ideas Learning, LLC Algebra 5
Chapter. 7. p() g() p() = g() = + 7. h() 5 5 5 5 5. f () 5 9 9 h() = 5 f() = 5. d() 5. d() = 5 5 w() 5 w() = + 5 9. Graph p =.75.5t. Make an input-output table to find the ordered pairs. t p.75.5.5.75.5 Plot the ordered pairs. Draw a line through the points. Note that the function onl makes sense when t and p are positive. So, onl draw the line in the first quadrant. p (,.75). (,.5) (,.5). (,.75) (,.5) Hours t Algebra Copright Big Ideas Learning, LLC Power remaining From the graph of the laptop, ou can see that when p =, t = 5. From the graph of the tablet computer, ou can see that when p =, t =. So, the tablet computer s batter will last longer. You can check that our answer is correct b finding the value of t for which p =. p =.75.5t =.75.5t.75 =.5t = t So, the tablet computer s batter will last hours, which is more than 5.
Chapter. To graph C() = 5 + 5, make an input-output table to find the ordered pairs for Certified Remodeling. C() 5 75 5 5 Plot the ordered pairs for both remodeling companies. Draw a line through each set of points. Note that the functions onl make sense when the input and output values are positive. Cost (dollars) 5 5 5 Certified Remodeling Master Remodeling Hours From the graph, ou can see that when =, the cost for Certified Remodeling appears to $5, and the cost for Master Remodeling is a little more than $, but definitel less than $5. So, ou should hire Master Remodeling because the will charge less.. no; Sample answer: Because the function rule is unknown, it is unknown whether an increase in would result in an increase or decrease in the value of the function.. Sample answer: The output of B when = is greater than the initial value of the function but less than the output of B when =. So, after months, ou have more mone in our bank account than ou did initiall, but less than ou did after months. You must have deposited more mone than ou withdrew in the first months of having the bank account, but then ou withdrew more than ou deposited over the net months.. a. d = r d(r) = r d(5) = (5) = When the radius of a circle is 5 feet, the diameter is feet. b. A = π r A(r) = π r A(5) = π 5 = 5π When the radius of a circle is 5 feet, the area of the circle is 5π, or about 7.5 square feet. c. C = π r C(r) = π r C(5) = π (5) = π When the radius of a circle is 5 feet, the circumference of the circle is π, or about. feet.. a. Sample answer: Attendance decreases immediatel following the flu outbreak, but then it steadil increases again as some students get better and return to school. b. A() 5 students; The school attendance was about 5 students weeks after the flu outbreak. c. Sample answer: When and, A() =. About week after the flu outbreak, there are about students in attendance, and then again about weeks after the flu outbreak, the attendance has climbed back up to students. d. Sample answer: The least attendance was 5 students, and this occurred about weeks after the flu outbreak. e. Sample answer: There are about students enrolled at this high school. The graph does not show an attendance values greater than. 5. a. Because f (5) = 9, the coordinates of this point on the graph are (5, 9). b. Because n = 5 is a solution of the equation f (n) =, the coordinates of this point on the graph are (5, ).. f (a + b) does not alwas equal f (a) + f (b). Sample answer: For eample, if f () = + 5, and a =, and b =, then f (a + b) = f ( + ) = f () = + 5 =, but f (a) + f (b) = f () + f () = ( + 5) + ( + 5) = + 5 =. Because, f (a + b) f (a) + f (b). Maintaining Mathematical Proficienc 7. + + + 9 7 The solution is 9 7. 9 7. 5a < 5 or a > 5a 5 < 5 5 + + a < 7 a > 5 The solution is a < 7 or a > 5. 7 5 Copright Big Ideas Learning, LLC Algebra 7
Chapter 9. < k + < < k < k < < < k < The solution is < k <.. d + 7 < 9 or d > 7 7 + + d < d < d > d > d < d > The solution is d < or d >.. 5 + < 7 < 9 < 9 < The solution is <.. v + 9 5 or v 9 9 v v v The solution is v. v v.. What Did You Learn? (p. 7). Sample answer: You can graph each of the functions on a graphing calculator and use minimum and maimum features to verif the limits of the domain and range.. Sample answer: Look for a pattern in the change in earnings.. Sample answer: Because the cost is increasing b $7.5 for each additional ticket purchased, this is the rate of change, or the cost per ticket. Then ou subtract $ in order to get the total since ou have a $ coupon... Quiz (p. ). Each input is paired with eactl one output. So, the relation is a function.. The input has two outputs, and, and the input has two outputs, and. So, the relation is not a function.. The domain is,,,, and. The range is 5,,, and.. The domain is all real numbers. The range is. 5. The domain is <. The range is.. The graph is a line. So, the rate of change is constant, and the function is linear. 7. As increases b 5, increases b different amounts. So, the rate of change is not constant, and the function in nonlinear.. The equation = ( ) can be written as =, but because it has an eponent on the variable, it cannot be written in the form = m + b. So, it is a nonlinear function. 9. The depth could be an part of a foot. So, the domain is continuous.. You cannot bu part of a hat, onl a certain number of hats. So, the domain is discrete.. w() = + 7 = + 7 7 7 = = 5 = When = 5, f () =. Algebra Copright Big Ideas Learning, LLC
Chapter... g() 5 g() = + p() 5 7 p() = m() m() = 5. a. The amount of mone m ou have left depends on the number r of video games ou rent. So, r is the independent variable, and m is the dependent variable. b. You cannot rent part of a video game, onl a certain number of them. So, the domain is discrete. If r =, then m =, which means that ou are out of mone and cannot rent an more video games. So, the domain is,,,,, 5,, 7,, 9, and. Create an input-output table to determine the range. r 5 7 9 m 7 5 9 The range is,,, 9,, 5,,,, 7, and. c. m (, ) (, 7) (, ) (, ) (, ) (5, 5) (, ) (7, 9) (, ) (9, ) (, ) r Videos rented Mone left (dollars). a. d() = 75 d() = 75 () = 95 The train is 95 miles from its destination after hours. b. d() = 75 = 75 75 75 75 = 75 =.5 = The train travels for.5 hours before reaching its destination. Copright Big Ideas Learning, LLC Algebra 9
Chapter. Eplorations (p. 9). a. adult $ Number of adult tickets + $ Number of child child tickets = $ + = b. Let =. + = () + = = = = c. Let =. Let =. + = + = () + = () + = + = + = = = = = = = Let =. Let =. + = + = () + = () + = + = + = = = = = = = The points create a line with a slope of. For each adult ticket sold, the number of child tickets sold decreases b. d. es; Sample answer: You can put the number of adult tickets in for and then solve the equation for, which is the number of child tickets.. a. pound $ Pounds of $ Pounds of + = $ Swiss pound cheddar + = b. + = + = = + + = + = = + The function onl makes sense when and are positive. So, the domain is, and the range is. c. The -intercept is. The -intercept is. d. Sample answer: Because the -intercept is the -coordinate of a point where the graph crosses the -ais, it occurs when =. So, ou can substitute for and solve the equation for. Similarl, because the -intercept is the -coordinate of a point where the graph crosses the -ais, it occurs when =. So, ou can substitute for and solve the equation for. e. Sample answer: In the contet of this problem, the -intercept represents how man pounds of Swiss cheese ou could bu with $ if ou did not bu an cheddar cheese. Similarl, the -intercept represents how man pounds of cheddar cheese ou could bu with $ if ou did not bu an Swiss cheese.. Sample answer: The graph of an equation in the form A + B = C is a straight line.. Sample answer: For our dance school s upcoming performance, ou are selling t-shirts that cost $ each and sweatshirts that cost $ each. You have collected $ so far, but ou cannot find the order form that tells how man of each ou have sold.. Monitoring Progress (pp. ). Algebra Copright Big Ideas Learning, LLC
Chapter. 5. + = + = () + = + () = = = = = = = 5. To find the -intercept, substitute for and solve for. = = = = = To find the -intercept, substitute for and solve for. = () = = = = = (, ) (, ). To find the -intercept, substitute for and solve for. + = 9 + () = 9 = 9 To find the -intercept, substitute for and solve for. + = 9 + = 9 = 9 = 9 = ( 9, ) (, ) (5, ) 5 The -intercept shows that ou can rent 5 small tables when ou do not rent an large tables. The -intercept shows that ou can rent large tables when ou do not rent an small tables.. Eercises (pp. ) Vocabular and Core Concept Check. Sample answer: Both -intercepts and -intercepts are the value of a coordinate of a point where the graph crosses an ais, but the -intercept is the -value when the graph crosses the -ais, and the -intercept is the -value when the graph crosses the -ais. Both occur when the other variable is zero. To find the -intercept, ou let = and solve for, and similarl, to find the -intercept, let = and solve for.. The point (, ) does not belong because it is not on either ais. The other points are on the -ais or the -ais or both, and the could be the coordinates of the -intercept, the -intercept, or both. Monitoring Progress and Modeling with Mathematics.. 5.. (, ) Copright Big Ideas Learning, LLC Algebra
Chapter 7. + = + = + () = () + = = = = = = = The -intercept is. The -intercept is.. + = + = + () = () + = = = = = = = The -intercept is. The -intercept is. 9. + = + = + () = = = = () + = = = = The -intercept is. The -intercept is.. + 9 = + 9 = + 9() = = = = () + 9 = 9 = 9 9 = 9 = The -intercept is. The -intercept is.. = = () = () = = = = = = = The -intercept is. The -intercept is.. + = + = + () = () + = = = = = = = The -intercept is. The -intercept is.. 5 + = 5 + = 5 + () = 5() + = 5 = = 5 5 = 5 = = = (, ) (, ). + = + = + () = () + = = = = = = = (, ) (, ) 5. + = + = + () = () + = (, ) = = = = = = (, ) Algebra Copright Big Ideas Learning, LLC
Chapter. + = + = + () = () + = ( 9, ) = = = = = 9 = (, ) 7. + = + = + () = = = = 5 5 (, ) ) () + = = = = 9. + = 7 + = 7 + () = 7 () + = 7 ( 7, ) = 7 = 7 = 7 = 7 = 7 = 7 (, 7 ) ). = 5 = 5 () = 5 () = 5 = 5 = 5 = 5 = 5 (, 5) = 5 = 5 (, ) ( ) ( 5, ). + 7 = + 7 = + 7() = = = = () + 7 = 7 = 7 7 = 7 = ( ) (, ) (, ) Copright Big Ideas Learning, LLC Algebra
Chapter. 5 + = 5 + = 5 + = 5 () + = 5 = = 5 ( 5 ) = 5 = (, ) (, ). + = + = + = () + = = = = () = (, ) (, ). a. + = + = + () = () + = = = = = = = The -intercept shows that ou can fit all the football plaers into cars if ou do not use an vans. The -intercept shows that ou can fit all plaers in vans if ou do not use an cars. b. Onl whole-number values of and make sense in the contet of the problem. Besides the intercepts, it appears that the line passes through the points (, ), (, ), and (9, ). To verif that these points are solutions, check them in the equation. + = () + () =? + =? = + = () + () =? + =? = + = (9) + () =? + =? = So, four possible combinations of cars and vans that will hold football plaers are: cars and vans, cars and vans, cars and vans, or 9 cars and vans. Algebra Copright Big Ideas Learning, LLC
Chapter. a. + = + = + () = () + = = = = = = = 5 The -intercept shows that ou can order short-sleeve shirts if ou do not order an long-sleeve shirts. The -intercept shows that ou can order 5 long-sleeve shirts if ou do not order an short-sleeve shirts. b. Let =, solve for. + = () + = + = = = = 5 So, if students order short-sleeve shirts, ou have enough mone to order 5 long-sleeve shirts. 5. The intercepts should be two different points, one on each ais. + = + = + () = () + = = = = = = = The -intercept is at (, ). The -intercept is at (, ).. The -intercept should be at a point on the -ais where =. The -intercept should be at a point on the -ais where =. + = + = + () = () + = = = = = = 5 = The -intercept is at (5, ). The -intercept is at (, ). 7. Your friend is incorrect. Sample answer: Instead he should sa, When ou want to find the -intercept, just substitute for and continue to solve the equation.. a. + = 5 + = 5 + () = 5 () + = 5 = 5 = 5 = 5 = 5 = 7 = The -intercept shows that ou can make 7 two-point baskets if ou do not make an three-point baskets. The -intercept shows that ou can make three-point baskets if ou do not make an two-point baskets. b. no; If the team makes an odd number of three-point baskets, then there will be an odd number of points scored. Because ou can onl make an even number of points with two-point baskets, in order to score an even number of points in total, ou must also score an even number of points with three-point baskets. c. Sample answer: Onl whole number values of and make sense in the contet of the problem. Besides the intercepts, it appears that the line passes through the points (9, ) and (, ). To verif that these points are solutions, check them in the equation. + = 5 + = 5 (9) + () =? 5 () + () =? 5 + =? 5 + =? 5 5 = 5 5 = 5 So, two more possible combinations of baskets made are: 9 two-point baskets and three-point baskets, or two-point baskets and three-point baskets. 9. A; When =, =, and when =, =. So, the -intercept is and the -intercept is. 5 + = 5 + = 5 + () = 5() + = 5 = = 5 5 = 5 = = = Copright Big Ideas Learning, LLC Algebra 5
Chapter. C; When =, =, and when =, =. So, the -intercept is and the -intercept is. 5 + = 5 + = 5 + () = 5() + = 5 = = 5 5 = 5 = = =. D; When =, =, and when =, =. So, the -intercept is and the -intercept is. 5 = 5 = 5 () = 5() = 5 = = 5 5 = 5 = = =. B; When =, =, and when =, =. So, the -intercept is and the -intercept is. 5 = 5 = 5 () = 5() = 5 = =. 5 5 = 5 = = = = = 5 = = The lines form a square. Sample answer: The length of each side is units, and the sides are horizontal and vertical lines. So, the are either parallel or perpendicular and form right angles at each of the four corners.. a. The -intercept shows that 9 students can go on the class trip if no one chooses to get the meal plan. The -intercept shows that students can go on the class trip if everone chooses to get the meal plan. b. The domain is to 9 students (whole numbers onl) not choosing the meal plan. The range is to students (whole numbers onl) choosing the meal plan. 5. + = ; To check these values, find the - and -intercepts. + = + = + () = () + = = = = = = = 5. Sample answer: + = ; The intercepts will be whole numbers because both A and B are factors of C. 7. es; Sample answer: = a can be written as + = a, and = b can be written as + = b.. If the -and -intercepts are integers, then both and 5 must be factors of k. So, k must be a positive or negative multiple of 5. Maintaining Mathematical Proficienc 9. ( ) () = + + = =. = =. 9 ( 7) = 9 + 7 = 5 = 5 7. 5 ( ) = 5 5 + = 5 = 5.5 Eplorations (p. 5) change in. a. slope = change in = The -intercept is. change in b. slope = change in = = The -intercept is.. a. = + 7 5 Algebra Copright Big Ideas Learning, LLC
Chapter b. = The graph is a line with a slope of = and a -intercept of. c. = +. The graph of the equation = m + b is a line with slope m and -intercept b. a. Sample answer: The value of m affects the steepness of the graph and whether the graph rises or falls from left to right. b. Sample answer: The value of b affects the location of the -intercept. c. Sample answer: () Using the equation = and changing m onl, one possible equation is =. (, ) The graph of a line with a slope of = and a -intercept of. d. = 5 (, ) (, ) = (, ) (, ) Because m = in = has a greater absolute value than m = in =, this graph is steeper than the graph of =. Also, because the slope, m =, is negative, the graph falls from left to right as opposed to when the slope, m =, is positive, and the graph rises from left to right; Both graph have a -intercept of. () Using the equation = and changing b onl, one possible equation is = +. The graph is a line with a slope of = and a -intercept of. Equation Description of graph Slope of graph a. = + Line -intercept b. = Line c. = + Line d. = Line = + (, ) (, ) (, ) (, ) (, ) This graph has the same slope as the graph of =, but this one crosses the -ais at, and the graph of = crosses the -ais at. When a linear equation is in the form = m + b, m is the slope of the graph and b is the -intercept. Copright Big Ideas Learning, LLC Algebra 7
Chapter.5 Monitoring Progress (pp. ). The line falls from left to right. So, the slope is negative. Let (, ) = (, ) and (, ) = (, ). m = = () = + = 5 The slope is 5.. The line rises from left to right. So, the slope is positive. Let (, ) = (, ) and (, ) = (, ). m = = ( ) ( ) = + + = = The slope is.. The line rises from left to right. So, the slope is positive. Let (, ) = (, ) and (, ) = (5, ). m = = ( ) 5 The slope is 7. = + 5 = 7. Choose an two points from the table and use the slope formula. Use the points (, ) = (, ) and (, ) = (, ). m = = The slope is 5. = = 5 5. Note that there is no change in. Choose an two points from the table and use the slope formula. Use the points (, ) = (5, ) and (, ) = (5, ). m = ( ) = = + 5 5 5 5 = Because division b zero is undefined, the slope of the line is undefined.. = m + b = + The slope is and the -intercept is. 7. = m + b = = + The slope is and the -intercept is.. + = = = = 5 The slope is and the -intercept is 5, or.5. 9. = m = and b = The line crosses the -ais at (, ). So, the -intercept is.. + = = m = and b = The line crosses the -ais at (, ). So, the -intercept is.. + = = + = + = + m = and b = The line crosses the -ais at (, ). So, the -intercept is. Algebra Copright Big Ideas Learning, LLC
Chapter. The change in is. The change in is 5. So, the slope is 5. Since h() =, the -intercept is. The line crosses the -ais at (, ). So, the -intercept is.. a. t Time (minutes) Elevation (feet), h The domain is t and the range is, h. b. The slope is 5. So, the submersible ascends at a rate of 5 feet per minute. The h-intercept is,. So the elevation of the submersible after minutes, or when the ascent begins, is, feet. The t-intercept is. So, the submersible takes minutes to reach an elevation of feet, or sea level..5 Eercises (pp. ) Vocabular and Core Concept Check. The slope of a nonvertical line passing through two points is the ratio of the rise to the run.. A constant function is a linear equation written in the form = + b, or = b. The graph of a constant function is a horizontal line. The slope of a constant function is.. The slope-intercept form of a linear equation is written in the form = m + b. This form is called slope-intercept form because the slope of the line is m and the -intercept is b.. = ; The other three equations are in slope-intercept form, but this one is in standard form. Monitoring Progress and Modeling with Mathematics 5. The line falls from left to right. So, the slope is negative. Let (, ) = (, ) and (, ) = (, ). m = = ( ) = + 5 = 5 = 5 The slope is 5.. The line rises from left to right. So, the slope is positive. Let (, ) = (, ) and (, ) = (, ). m = = ( ) = + = The slope is. 7. The line is horizontal. So, the slope is. Let (, ) = (, ) and (, ) = (, ). m = = The slope is. ( ) ( ) = + + = =. The line falls from left to right. So, the slope is negative. Let (, ) = (, ) and (, ) = (5, ). m = = 5 = 5 = 5 The slope is 5. 9. m = = The slope is.. m = = The slope is. ( 5) = + 5 = = ( ) 5. m = = = The slope is undefined. = + 5 = =. m = 5 = () = 5 + = 7 = 7 The slope is 7.. m = = = = The slope is, which means that the distance was increasing b miles for ever increase of hour. So, the bus was traveling at miles per hour.. m = 5.99 5.99 = = 7 = Since there is no change in, the slope is, which means that ou are charged the same admission fee regardless of how long ou sta. Copright Big Ideas Learning, LLC Algebra 9