Selected Answers. Topic 1

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1 Topic 1 Lesson 1-1. Recall that the sum (or difference) of two rational numbers is alwas rational. Assume that the sum of a rational number and an irrational number is rational. That is, rational number + irrational number = rational number. Then, irrational number = rational number rational number. This is alwas a false statement, so the assumption is also false. Therefore, the sum of a rational number and an irrational number is irrational.. Jacinta did not consider the case when the rational number is equal to zero. If the rational number is zero, then zero times an irrational number will be zero, which is rational.. es. 1, 1, It is irrational. 1. No; an irrational square root times itself gives a rational square in. < < 1 1 in. 1. rational, real 1. whole, integers, rational, real 0. irrational, real. 10,.5, 1. C. B. Irrational; is irrational 0. Rational; can be epressed as a ratio. Yes; the difference of two rational numbers can be rewritten as a single ratio, so the difference is alwas rational.. no. You do not know the shape of the shed, so length could be real, rational, irrational, integer, a whole number, or some combination of those. If each piece is square, the length is B Lesson 1-. Use the Addition Propert of Equalit on both sides to get 9 alone on the left side of the equation, and then use the Division Propert of Equalit on both sides to get alone on the left side of the equation. Substitute the value of into the original equation, and then evaluate the epressions on both sides to see that the are equal.. Venetta should have multiplied b 1. km to convert to 1 mi kilometers or Subtraction Propert of Equalit. Subtracting from both sides isolates the variable term. Answers ma var. Sample: A first step could be multipling terms b to remove fractions, then using the Distributive Propert to epand the epression ( 1)( + ). That wa, ou can combine like terms on the left side before isolating. 1. The student made a sign error when distributing in the first step.; = should be = +, and the correct solution is 1 =. 1. Recall that an number divided b 0 is undefined. To see wh, consider the equation a =, where a and 0 are non-zero real numbers. If a could be divided b 0 to get, then times 0 equals a, which is impossible because an number multiplied b 0 is 0. Therefore, c cannot be , or 1.. 7, or 0. 1, or a. d d 1 = 1 b. 5 mi b + (b 1) = 7 ; 1 bottles. 0 drops; about 0.17%. A 1 envision Algebra 1 1 Selected Answers

2 Topic 1 Lesson 1-. Both sides simplif to show that the two epressions are eactl the same.. No. The side on which ou choose to isolate the variable does not affect the solution. However, the side ou choose can affect the number of steps needed to obtain the solution.. h =. no solution 10. The variable can be on onl one side of the equation. For the equation = 5, the equation simplifies to 0 = 5, so there is no solution. 1. The student made a math error in the first step when distributing 0.15 over the quantit 0.. The epression should be The solution is = cm, 1 cm, and 1 cm; cm identit 0. identit. no solution. no solution smaller boes. About 1. seconds; the heights in feet of the washers can be modeled b 1 + t and t, where 1 t is time in seconds. The washers will reach the same height when 1 + t = t, or when t 1. 1 seconds.. No; although t = 11 is a solution to an equation relating the costs of the two gms, the membership is prepaid for 1 months. So the membership fees are never equal.. C, D, E 50. Part A.75 minutes; 7.15 feet Part B No; Benito will finish first. At.75 minutes, the will have painted the same number of feet. After that, Benito will paint more fence in less time because he paints at a faster rate than Tler. So, Benito will reach the end first. Part C Benito will finish first and get to rest for about minutes. Solve the equation 15t = 150 to find that it takes Benito 10 minutes to finish. Solve the equation t = 150 to find that it takes Tler about 1 minutes to finish. So Benito will get to rest for about 1 10 = minutes. Lesson 1-. Answers ma var. Sample: Similar: You use the same properties of equalit to solve each. Different: There is a numerical solution for + 1 = 9, =. d c The solution for + c = d is =. 9 1 Substitute 1 and 9 for c and d: =.. Dani should have added 1 k instead of 1, then multiplied each side b k. Dani could also begin b multipling each side b k, then adding 1 to each side.. m = 5 n 7. a. = 1 ( ) ( q + q + q ) 1 b. q = q q = (90) 5 = s = P ;.5 ft b.5 ft;.75 ft 1. sometimes, sometimes, alwas 1. = a k 1. = w(a b), or = wa wb 1. n = + 0. u = 5( 5). m = 11. h = V πr + 1. = m + b. l = 117 w 0. b = ,000 h, 1,000 h = (b 1) ; 1,791 ft 1.7. The formula to place in cell A is, = D / (B * C).. D envision Algebra 1 Selected Answers

3 Selected Answers Topic 1 Lesson 1-5. Since > 0, the quotient of and a positive quantit is positive and the inequalit smbol stas the same, but the quotient of and a negative quantit is negative, so the inequalit smbol must be reversed.. Rachel reversed the inequalit smbol when multipling b a positive quantit > a. a > b a b > 0 Subtract b from each side. c(a b) < 0 Since c is negative, c(a b) is negative, and the inequalit smbol is reversed. ca cb < 0 Distributive Propert ca < cb b. a > b Add cb to each side. a b > 0 Subtract b from each side. a b c < 0 c Since c is negative, a b c is negative, and the inequalit smbol is reversed. a c b c < 0 c Distributive Propert a c b c < 0 Simplif. a c < b c Add b c to each side. 1. The student reversed the inequalit smbol when dividing b. The inequalit smbol need onl be reversed when multipling or dividing b a negative number; The correct solution is >. 1. a. a < b b. No, if a b, then c(a b) < 50; > 0; < 5; ; > ; 0. < 9; ; D. A. >. >. 0.. >. Etend the graph to include the entire number line to represent the solution all real numbers.. + < ; > ; The office will have to bu more than jugs each month for Best Water to cost less than Acme.. A. I B. II C. III D. IV 50. Part A.5 1 ; ; Steve would have to walk at least hours. Part B. <. + ; <.5 ; Elijah is behind Mercedes for up to.5 hours. envision Algebra 1 Selected Answers

4 Topic 1 Part C. + > +.5; >.5 ; Elijah is ahead of Aubre after.5 hours. Lesson 1-. The graph of > a and < b is similar to the graph of > a because it ma include some of the same values, but it is different because it will include onl values that are also less than b.. Kona should have graphed > because the inequalities are joined b or, not and.. and < 0. < or ; If a > b, then c = a ; if a < b, then c = b ; if a = b, then c = a = b. 1. > a or < b 1. Let Set A be the solution set for < <. Then Set A = {5,, 7}. Let Set B be the solution set for < < 11. Then Set B = {,, 5,, 7,, 9, 10}. Since each element in Set A is also an element of Set B, Set A is a subset of Set B and or >. all real numbers. < 0 or >.. < 10 or > 0 0. at least 1 and at most 1 pencils ( + 7.5) 00 ; 1.5 ft.5 ft. D 0 Lesson 1-7. The methods are similar because the same strategies and properties are used. The methods are different because equations for two cases must be solved when absolute value is involved.. Yumiko should have solved < 5 or > 5.. =, = Subtract from each side to isolate the absolute value epression on one side of the equation. 1. The student should have rewritten the absolute value inequalit with and instead of or. 1. a. When solving a + b = c, b has to be subtracted from each side of the equation first to isolate the absolute value epression. b. When solving a + b c, two inequalities are written with and, while when solving a + b c, two inequalities are written with or. 1. = 1, = 1 1. no solution 0. = 5, = 1. =, = 1. =, = = ; minimum:.7 s; maimum:. s. > and < 0 0. > 1. and < < < 7 0 envision Algebra 1 Selected Answers

5 Topic 1. < < 0. D. C 0. The minimum number of guests is. The maimum number of guests is. Ashton should reserve tables so that ever guest is seated.. B, C, E, D, A. Part A 0 = 5 ; = 5, = 5 ; minimum speed: 5 mph; maimum speed: 5 mph Part B 5 ; 7 ; The sign blinks when a vehicle is traveling between and 7 mph, inclusive. Part C Use the formula a b, with a = 0, and b = 5; The absolute value formula to use is the one that represents a compound inequalit using or because the message should flash at cars traveling under 15 mph or over 5 mph. Topic Review. formula. element. set. identit 10. real number, rational number 1. real number, rational number, integer ,., Propert of equalit; First ou need to add to each side of the equation to isolate the term 1. t = s = 15. In the third step, the student multiplied the right side b 10 instead of ( 0.) = 0.( 1) = (0. 0.1) = 100(. 0. ) 0 1 = = = =.15. b = 1. No, he is not correct; For 10 months, the first option will cost $19.99(10) + $1.0 = $1.70. The second option will cost $1.59(10) = $ The first option is less epensive.. = k 0. d = c. 1 and The graph shows 1, not > or < > First isolate the absolute value epression b adding 5 to each side of the equation.. = 11, = envision Algebra 1 5 Selected Answers

6 Topic Lesson -1. Both graphs are lines. The both include the point (0, 1). The graph of = + 1 has a slope of, and the -intercept is 1. The graph of = + 1 has a slope of, and the -intercept is 1.. Substitute the coordinates of the two points into the formula m = slope: 1 ; -intercept: = 1. = = = = = +. = = 1 +. =. = + 5 ; The -intercept represents the total distance of 5 miles.. = +. D, E 0. Part A = Part B The -intercept gives the level when the dispenser is full, which is 15 inches. Part C No, the level will be at inches after 1 hours. Substituting 1 for in the equation gives =. Lesson -. Sample: Write the equation of the line in point-slope form. Then substitute a value for in the equation to find a value for for the same point.. You can use the known point and the slope to write the equation of the line in point-slope form. You can use the known slope, but ou will need to find the -intercept to write the equation of the line in slopeintercept form.. = ( + ). = ( + ) or + 1 = ( + ) 10. a. + 1 = ( ) b. 1 = ( ) 1. The - and -coordinates are reversed. The point in Step 1 should be plotted at (, 5), not ( 5, ). Step should plot a point units down and units right from the point (, 5). envision Algebra 1 Selected Answers

7 Topic 1. 1 = ( ) 1. + = ( ) 1. = ( + 1 ) 0. = ( ). = ( ) or = ( ). + 5 = ( ) or + = ( 1). + = 5 ( + ) or = ( 1) Answers ma var. Sample: 100 = 1.( 5) ; = ; The cost per invitation is the slope, $1.0. Slope-intercept form gives information about the set-up fee of $0 as the -intercept. 0. A Lesson -. Both are forms of a linear equation. Standard form does not give the -intercept as slope-intercept form does, but ou can find the -intercept in standard form b setting to 0 and solving for. It is easier to find slope using slope-intercept form. Standard form identifies the constraint.. Answers ma var. Sample: when ou are creating a miture of two items and need to create a given total = ( 15) ; = = 1.50( 5.5) ; Liam can write the equation in slope-intercept form: = The -intercept models the service fee of $ a. A =, B = ; = b. A =, B = 1 ; + 1 = envision Algebra 1 7 Selected Answers

8 Topic 1. The student forgot to keep the negative sign for in step. The -intercept is. 1. Find slope b solving the linear equation in standard form for and writing it in slopeintercept form. Slope is A, or 5 B. 1. -intercept: ; -intercept: 1. -intercept: ; -intercept: 0... A. B. The graph of = is a graph of the form A + B = C when A = =. = = 0. =. + = 1, where is pounds of wheat flour and is pounds of re flour.. A, B, D. Part A mango and pineapple: = ; mango and strawberr: = ; pineapple and strawberr: = Part B mango and pineapple: 10 1 cups; mango and strawberr: 1 cups; pineapple and strawberr: 10 cups Part C mango and pineapple or mango and strawberr; Since Fatima will be using liquid to double the volume of the smoothies, she needs at least 1 cups of fruit to make at least cups of smoothies. The ranges in cups of mango and pineapple and mango and strawberr include totals of at least 1 cups. Lesson Dwane made the slope opposite but did not write the opposite reciprocal of, which is 1.. No; In order for the lines to be parallel, the slopes must be the same. The two given lines do not have the same slope.. = envision Algebra 1 Selected Answers

9 Topic. neither The coefficients of and are the same, so ou know that when the equations are converted to slope-intercept form, the slopes will be equal. When the slopes are equal, the lines are parallel. 1. = = + 0. = = 5. parallel. perpendicular. Check students work = ( ). C 9. Part A = 5 Part B a. = + 1 b. = + 1 c. = + 0 d. = + 5 Part C Check students work. Topic Review. reciprocals. parallel. point-slope form. 10. = = 0.5( ) 1. 1 = ( ) =.5( 5), $ = intercept: ; -intercept: = 5. = = ( 1). neither envision Algebra 1 9 Selected Answers

10 Topic Lesson -1. Both the domain and range are continuous positive real numbers and 0; the amount of time and of rain can be fractions of whole numbers.. Ever function is a relation, because an function can be written as a set of ordered pairs. Not ever relation is a function, because some relations could have domain values for which there is more than one range value.. D: {1,,, 5, 7, }; R: {,,,, 7,, 9}; not a function. a. domain: 1 ; range: 0 b. domain: 0; range: n can be an value ecept 1,, or. 1. a. The relation is not a function because one input value maps to more than one output value. The mapping from to ma or ma not be a function, depending on whether each value of maps to more than one value of. b. The relation ma or ma not be a function, depending on whether an -value maps to more than one -value. The mapping from to is not a function because at least one -value maps to more than one -value. c. The relation represents a function because each input maps to eactl one output. The mapping from to is also a function because each -value maps to eactl one -value. 1. D: {A, B, C, D}; R: {Rockets, Birds, Pups, Cats, Hawks}; The relation is not a function because the input A maps to more than one output. 1. domain: whole numbers; range: multiples of function; one-to-one 0. not a function. whole numbers less than or equal to 0. a. D: {English, Math, Histor, Biolog, Biolog Lab} R: {0, 5, 0, 90} It is a function. b. D: {English, Math, Histor, Biolog, Biolog Lab} R: {5, 0} It is a function. c. No; Answers ma var. Sample: The element 5 in Week 1 is matched with more than one class time in Week. So, class times for Week are not a function of class times for Week 1.. a. D: { t 1 t }; R: { s 0 s 5 } b. Answers ma var. Sample: a table c. Answers ma var. Sample: The train increases its speed at half an hour and then remains constant, a table is the simplest wa to model this information.. E Lesson -. Answers ma var. Sample: A person jogging at a constant rate can be modeled b a linear function. A group of runners jogging at different speeds cannot be modeled b a linear function.. Answers ma var. Sample: Talisa did not convert from minutes to hours correctl. The function should be written as f() = 5.. f() = 0; f() =. f(t) = t + ; 1 ft 10. a. The numbers describing the range of h() should each be one greater than the numbers describing the range of g(). b. The range over 0 < < 5 is 1 < < 11 for g() and < < 1 for h(). envision Algebra 1 10 Selected Answers

11 Topic 1. Answers ma var. Sample: When = 0, = 10 9, so = 1 ; = f(5) = 1 1. f(5) = f(5) = 0. = I. f() + f() A.. II. f(5) B.. III. f(7) f() C. 7. IV. f() D Part A c(n) = 5,50 + 5n, r(n) = 159n ; Subtract costs from revenues: p(n) = 159n (5,50 + 5n) = 105n 5,50. Part B Profit ($) p 15 0 Backpacks Sold n 5 0. Yes, for ever unit increase in -values, the -values increase at a constant rate. Also, the points in the graph can be connected b a line. So the function shown is linear.. f(w) = w f(w) 10 w 0 0 D: 0 < w < ; an value outside of this interval would mean the length or width was negative or zero, which is not possible. 0. a. f(t) = 0 + 0t b. $15 Part C $75; He does not make a profit. Lesson - envision Algebra 1 11 Selected Answers. The line has the same slope; it is just shifted verticall or horizontall.. Multipling the output of a linear function b a constant k changes both the slope and the -intercept because both the -term and the constant term are multiplied b k. When multipling the input b a constant k, onl the -term is multiplied b k, so the -intercept does not change.. shifted up units. The slope is scaled b a factor of ; the -intercept is not changed. 10. f() 1. The student graphed g units to the left of f. The graph of g should be units to the right of f. 1. a. the graph of f shifted 5 units down; h() =

12 Topic b. the graph of g shifted units up; h() = 1 + c. the graph of g with the slope and -intercept scaled b a factor of ; h() = + d. the graph of f with the slope and -intercept scaled b a factor of 1 ; h() = shifted down units 1. shifted right 1 unit 0. vertical stretch b a factor of 5. vertical stretch b a factor of. slope scaled b a factor of 0.5, -intercept unchanged. slope scaled b a factor of 1, -intercept unchanged. slope unchanged, -intercept shifted up 1 unit 0. k = ; horizontal translation right. Both are correct.. A. Answers ma var. Sample: Use the line that passes through the points (1, 7) and (.5, 1). This line has the equation = Transformation shift down 5 units vertical stretch b shift units to the left Lesson - Equation = + = 1 + = +. The student confused the values of a 1 and d when writing the formula. The correct formula is a n = + 5(n 1) or a n = + 5n.. A recursive formula defines a term in relation to the previous term in the sequence. An eplicit formula defines a term in relation to the term number. Both formulas include the common difference.. no. a n = a n 1 and a 1 = Answers ma var. Sample: It tells ou that as the term number increases, the terms values decrease. 1. Answers ma var. Sample: When finding the common difference, the student subtracted the numbers in the wrong order. Instead of subtracting a higher term from a lower term, the student should have subtracted a lower term from a higher term: 9 = 7, 15 = 7, 15 = 7, and 1 = 7. So, the common difference is a n = a n 1 + ; a 1 =. 1. a. 1, 1,,, 5, b. No, the Fibonacci sequence does not have a common difference. 1. es; 1 0. es;. no. es; 9. a n = a n and a 1 = ; a n = 1 + 9n. a n = a n and a 1 = 15 ; a n = + 9n 0. a n = a n and a 1 = ; a n = n. a n = + n. a n = 77 1n. a n = 7n. a n = a n 1 1; a 1 = a n = a n ; a 1 = 7. a n = a n ; a 1 = 7 1. a n = 500 n ; 79 tickets. n; 5; 9. Part A a n = a n 1 + ; a 1 = 10 Part B a n = + n Part C envision Algebra 1 1 Selected Answers

13 Topic Part D = + ; one of the formulas found for the sequence; Answer ma var. Sample: Because there are onl whole numbers of rows, a formula for a sequence is a better representation. The linear model includes man etra points that do not represent rows. Lesson -5. The data ma show a negative association, but it could also show no association.. No; a trend line just shows the general direction of the data. A good trend line will have about as man data points above the line as below.. positive. Answers ma var. Sample: = ; the increase in the number of points scored on the quiz for each additional hour of stud 10. The slope is positive; The slope is negative. 1. No; the data show no association. A trend line should show the general direction of the data in a scatter plot. If a scatter plot shows no association, there is no general direction of the data, so a trend line is not a good choice for modeling the data. 1. The -intercept tells ou the value at = 0, or the initial value. Interpreting this value in contet can help ou decide whether a linear model makes sense. 1. positive 1. negative correlation = = Answers ma var. Sample: = 7 + 1,10 ; The slope represents the change in the number of trees planted per acre as the planting densit increases. The -intercept represents the number of trees that would be planted per acre if there were no spacing.. decrease, increase envision Algebra 1 1 Selected Answers

14 Topic. Part A Part B Answers ma var. Sample: = ; The slope represents the change in the number of kites sold for ever dollar increase in price. Part C Yes; Answers ma var. Sample: People might prefer the design of a certain kite, and so more of that kite might sell than less epensive kites. Yes, ou can make scatter plots of man variables, including number of a given design sold vs. price per kite. Lesson -. Interpolation is making a prediction about a value within a known range of values. Etrapolation is making a prediction about a value outside of a known range of values.. Yes; Answers ma var. Sample: There ma be some rare eceptions... No; Answers ma var. Sample: B definition, data that have strong positive correlation have a strong linear relationship, so the correlation coefficient would not be as weak as 0.5. However, the data ma have strong positive association and have a correlation coefficient of 0.5 because the best model might be nonlinear. 10. Interpolation; Answers ma var. Sample: Because interpolation is making a prediction within a known range of data, it is more likel that this data will fit the pattern alread observed. Etrapolation is riskier because the pattern ma not continue beond the known range of data. 1. Answers ma var. Sample: Locate the point on the graph that has the -coordinate ou are using to make the prediction. The -coordinate of this point is the number ou are tring to predict. 1. = ; strong negative correlation 1. weak positive correlation 0. The model is not a good fit The linear model is a good fit for the data. envision Algebra 1 1 Selected Answers

15 Topic. No, to determine whether heating bills and number of pets have a causal relationship, ou have to carr out an eperiment that can control for other variables that might influence the relationship.. Answers ma var. Sample: The temperature increases and then decreases as the time after midnight increases. A linear model would not be a good fit because a linear model is best for data that have a constant rate of change, not for data that have a rate of change that is positive and then negative The domain should be between zero and the value of the new automobile. 1. {1, 1,, 0, } 1. = 5(5.5) + 10 = $ It moves it up 5 units. 0. The graph of f is verticall stretched b a factor of.. g() = es; 11. a 1 = 5; a n = a n 1.5. Gabriela will have 1 bracelets left to sell after the fifth da. 0. negative correlation Because the residuals are all 0, the data are perfectl linear.. C Topic Review. positive association. sequence. correlation coefficient. domain: { 5, 1, 0,, }; range { 5,, 0,, }; The relation is a function because each input has onl one output. 10. All real numbers would not be a reasonable domain because the number of pages printed cannot be negative Answers ma var. Sample: = = 0 1 ; The slope of the trend line represents the change in recommended distance of a light source for ever 1-foot increase in ceiling height.. = ;.15. = ; The slope tells ou that the winning time decreases b about seconds per ear. The intercept tells ou what the winning time would have been at ear 0. According to the model, the winning time in 010 was seconds, and the winning time in 00 is predicted to be seconds. envision Algebra 1 15 Selected Answers

16 Topic Lesson -1. The graph of a sstem with one solution will be two distinct lines that intersect at one point, while the graph of a sstem with infinitel man solutions will be one line, and the graph of a sstem with no solution will be two parallel lines that do not intersect.. Reese did not recognize that, while the slopes are the same, the intercepts are also the same, so the lines are coincident and there are infinitel man solutions to the sstem.. infinitel man solutions. a. no solution, b. (, ), c. infinitel man solutions 10. Number of solutions Slopes -intercepts ne solution Different Can be an value Infinitel man solutions Same Same No solution Same Different 1. Answers ma var. Sample: = (0, 0) 1. (, ) 1. no solution 0. approimatel (1.7, 9.). ( 0.5,.50). a. The solution of the sstem is the point where the graphs intersect. The graph shows that the -coordinate of the intersection point is between = and =. b. Answers ma var. Sample: (.5,.5). The solution is precise when it can be substituted back into the equation and ield an eact result. When this cannot happen, the solution is an approimate answer.. D Lesson -. Graphing is more useful when none of the equations in the sstem can be easil solved for one variable, or when ou have a graphing tool readil available.. If the equations can be rewritten to show that the are equivalent, then there will be infinitel man solutions. If the equations are in the same form, cannot be rewritten to show that the are equivalent, and the coefficients of are different, then there will be one solution. If the equations are in the same form, are not equivalent, and the coefficients of are the same and the constants are different, then there will be no solutions. envision Algebra 1 1 Selected Answers

17 Selected Answers Topic. (1, ). infinitel man solutions. 10. substitution, because both equations have isolated, so the can be set equal to each other 1. The student substituted back into the same equation instead of the second equation. 5( ) = = 1 7 = 1 = Then substitute = in the first equation to get =, so the solution is (, ). 1. width: 1 cm and length: cm 1. a. Answers ma var. Sample: 5 = 5 b. Answers ma var. Sample: 10 = 1. (, 1) 0. (5,.5). no solution. (1, ). no solution. no solution 0. a. 5 min, b. 90 m. and. a. 15 classes b. $10.,. Part A The sstem with solution (, ) is: = 1 7 = The sstem with solution (, 1) is: = + = The sstem with solution (0, ) is: + = 7 = 1 Part B (, 1), (, ), (0, ) Part C Yes; Sample: The slopes of the lines represented b equations 1 and are negative reciprocals. Lines with slopes that are negative reciprocals are perpendicular and meet at a right angle. Lesson -. He should have multiplied the second equation b a number that would result in a variable being eliminated if the equations were added. For eample, if he multiplied b instead, he could have eliminated the term b because + = 0.. Let be the width of the rectangle, and be the length. 5 + = 15 = The width is units. The length is units.. (, 7). no solution 10. The structure helps ou to determine the most effective method for solving the sstem. For instance, if one of the equations is in -intercept form, then it would be appropriate to use substitution to solve. If the variables in both equations have the same or opposite coefficients, then elimination might be more straightforward. = ; = (, 5 ) 1. Using the substitution method is helpful because it takes fewer steps to solve than the elimination method. The first equation is alread solved for, so ou can substitute + for in the second equation and solve for. 1. (7,.5) 1. (, ) 0. (, ). (5, 1). Yes; 1 = 1 is equivalent to two times =. envision Algebra 1 17 Selected Answers

18 Topic. Let = cost of a pizza Let = cost of a sub sandwich + = + 10 = 10 Cost of a pizza: $15 Cost of a sub sandwich: $. Elimination; Multipl the first equation b 1 and then add equations to eliminate the -terms; ( 1, 1) 0. Elimination; Multipl the first equation b and then add the equations to eliminate the -terms; (, ). Both methods will work. B addition, the -terms will cancel each other out. Multipling the first equation b and then adding the equations will cancel out the -terms. Either method will result in a solution of (, ).. (.50,.75), which means that a drink costs $.50 and a pretzel costs $.75.. $70. B Lesson -. infinitel man. The graph of < 1 can be graphed on a coordinate grid. Draw a dashed horizontal line at = 1 and shade below the line.. es. 1. Answers ma var. Sample: The graph of the inequalit < + is the same as the graph of the inequalit >. Both this inequalit and < + have a dashed border line with a slope of 1. The border line for the inequalit > passes through the point (0, ), and the border line for the inequalit < + passes through the point (0, ). The shaded region is above the border line in the inequalit > and below the border line in the inequalit < The graph of a linear inequalit in one variable of < is on a number line. There is an open circle over and an arrow pointing to the left. The graph of a linear inequalit in two variables of < is a vertical dashed line at = with the half plane to the left of the line shaded Both; the graph can be represented b the first inequalit. Because the second inequalit can be rewritten as the first inequalit, both inequalities are shown in the graph. 1. envision Algebra 1 1 Selected Answers

19 Topic 0.. line is dashed, so points that lie on it are not part of the solution.. Answers ma var. Sample: Show it using a graph; it is easier to use a graph because all the solutions are indicated b the graph. Attempting to describe the solutions of a sstem of inequalities in words is much more complicated.. = and = > + 5. a ; b. 11 or fewer ; 1. C Lesson ; es. (0, 1) is not a solution of the inequalit > +1, so it is not a solution of the sstem of inequalities. The point lies along the border line of the inequalit > +1, but the border 10. Answers ma var. Sample: A real-world situation best described b a sstem of linear inequalities has multiple constraints. A real-world situation best described b one linear inequalit has one constraint. 1. Answers ma var. Sample: A sstem of two linear inequalities is similar to a sstem of two linear equations because their solutions are determined b where the graphs of the inequalities or equations in the sstem intersect or overlap. The are different because a sstem of linear equations has infinitel man solutions when the two equations in the sstem are equivalent. A sstem of linear inequalities can have infinitel man solutions even when the inequalities are not equivalent. 1. Yes; for eample, there is no region where all planes overlap for the sstem > +, <, <. So, there is no solution to the sstem. envision Algebra 1 19 Selected Answers

20 Topic < + > 0. > and ; Answers ma var. Sample: in section A and in section B, 5 in section A and 5 in section B, in section A and in section B , 0, 0, and + 5 ; The sstem is defined b an additional inequalit. This will reduce the number of possible solutions envision Algebra 1 0 Selected Answers

21 Topic. >,,. Part A + 1, <,, and 5 Part B Yes; Answers ma var. Sample: The image on the watch shows that he wants to do cardio between and 5 hours each week, inclusive. So, the minimum hours he will be doing cardio is large gift boes. Topic Review. sstem of linear inequalities. solution of a sstem of linear inequalities. ( 1.5,.5). no solution 10. ( 11, 7 11 ) 1. (1, ) 1. no solution 1. 1 feet long and feet wide 1. (, 1) 0. Yes; The equation = is times the equation = 1, and the equation 15 + = is times the equation 5 =.. No; If either the - or -value is the same in both equations, then ou can eliminate one variable using addition or subtraction.. es. es The maimum is $,00 for 0 necklaces and 0 bracelets envision Algebra 1 1 Selected Answers

22 Topic 5 Lesson 5-1. The domain and range of g() = a when 0 < a < 1 is the same as the domain and range of f() =, because the graph is compressed toward the -ais. This compression does not change the domain, and does not change the range in this case because the range goes to infinit.. If a is negative, then the verte represents the maimum value of the function g.. Domain: all real numbers; Range: The domain remains all real numbers. The range changes from 0 to The range of the function f() = 10 is not 10 times the range of f() =. The range is the same for both functions, g() = a b 1 a a b b.. 1 Domain: all real numbers; Range: Domain: all real numbers; Range: 0. mi; he walked miles, because he was 1.5 miles from the water stop at the start and at the finish of the race.. ft/s 0. g() =. 0; the rate means that the bicclist is traveling toward the sandwich shop at a speed of 0 mi/h.. Downward; 100 or 100. Part A = when < 0 Part B The verte is (0, 0); increasing for > 0, decreasing for < 0, min. value 0 1. es; ( 11, 11) 1. no 0. es; (, ) envision Algebra 1 Selected Answers

23 Selected Answers Topic 5 Part C (0, 0); the lizard is at ( 100, 00) and the mosquito is at (00, 00). To determine where the new path intersects the -ais, consider the triangles formed b the new path, the vertical line from the lizard to the -ais, the vertical line from the mosquito to the -ais, and the -ais. The triangles would be similar. If d is the horizontal distance from the lizard to the ais of smmetr, then = d. Solving for d, d = 10, so 50 d the coordinates of the verte is are (0, 0). Part D Sample answer: The new slope is = 10. Since it hits 7 the -ais at (0, 0), a new function must be = Lesson 5-7. No; f could have pieces like the absolute value function, leaving some of the real numbers out of the range of f.. ; The interval < 0 has the rule f() =. The interval has the rule f() =.. f() =, < 0 {, You can write the function using the absolute value smbol as f() = or as the piecewise-defined function f() = {, 0, < The student swapped the intervals for the two rules. The interval should be 0 for, and < 0 for. 1. a. < b. Yes; Answers ma var. Sample: If n =, the range of f is <. 1. f() =, 0 {, < 0 1, 0 1. f() = { 1, < increasing: < 1 ; decreasing: 1 increasing: ; decreasing: >. f() = { 5 5, < 5 7,. f() = { 1.5 +, 1 5, > 5 Yes; if she bus a card for rides, it would cost her $0.50, but if she bus a card for rides, it would cost her $.. $ D envision Algebra 1 Selected Answers

24 Selected Answers Topic 5 Lesson 5-. Both can be represented as piecewise-defined functions that round to integer values. Ceiling functions round up to the nearest integer, while floor functions round down to the nearest integer < 0.5, 0.5 < 1.5, 1.5 <.5,.5 <.5, The INT function is eactl the same as the floor function; there is no difference between them. 1. Kenji used the net lower multiple of for the value of the function in each piece. The function values,, 9, and 1 should be replaced b, 9, 1, and 15, respectivel. 1. a. f(.) = ; f(5) = 5 ; f(.5) = 7, <, <, < 1 b. f() = 1, 1 < 0 0, 0 < 1 1, 1 <, <, < f() = floor ( ) Nan will pa $5 more. Amit will pa $175 and Nan will pa $00 because the parking time of 1 hours is in a different interval from 15 hours. Hours () 0 < < < 7 7 < 9 9 < < 1 1 < Cost (f()) envision Algebra 1 Selected Answers

25 Topic 5 0. f() 0 < 1 1 < < < < 5 5 < , = 0 1.5, 0 < 0. Part A t() = 1.75, 0 < , 75 < 5.5, 5 < 10.50, 10 < 150 Part B g() = 7 Part C $1.5 0 Lesson 5-. The value of h does not affect the domain or the range. If the value of a is positive, the value of k changes the range to k. If the value of a is negative, the range is k.. Multipl the function rule b 1, so f() = Answers ma var. Sample: f() = ; g() = Y = Y 1 1. Y = ; 1 Y = + ; Y = ; Y = ; The graph of Y is the translation of the graph of Y 1 up units; The graph of Y is the vertical stretch of the graph of Y 1 b a factor of ; The graph of Y is a reflection of the graph of Y across the -ais (0, ) 1. ( 0.5, 0). (.5, 0) 0. ( 7, ). ( 1, 5) 1 envision Algebra 1 5 Selected Answers

26 Topic 5. The graph of g is a vertical compression of the graph of f b a factor of 1 that is then translated units left and 1 unit down. 10. The graph of g is a reflection of the graph of f across the -ais that is then translated left.5 units and up units.. f() = g() = g() = +. a = 5 + ; D: A Topic Review. ais of smmetr. absolute value function. es; (, ). no 10. Domain: all real numbers; Range: 0 1. (0, 0); minimum 1. f() =, 0 {, < The signs were reversed, and the point = 0 was not included in the domain; f() = 5, f() = 5, 0 { 5, < 0 0. f() = 5 + 5, 10 { 0, > b. $1 envision Algebra 1 Selected Answers

27 Topic 5. ; f() = 1 1. ( 1, ). 0. (0, ). The graph of g is a vertical stretch of the graph of f b a factor of and translated units left and 1 unit down.. The graph of g is a vertical compression of the graph of f b a factor of 0.5, reflected across the -ais, and translated units up.. g() = Answers ma var. Sample: f() = 1; f() = 1. f() = + + envision Algebra 1 7 Selected Answers

28 Topic Lesson -1. 1_ 15, 15. Core reversed the numbers in the numerator and denominator of the eponent. It _ should be.. Rational eponents can be written as radical epressions. Properties of eponents appl to both rational and whole number eponents. 1_. 15 _ = 0 1. = 1. = The rational eponent increases as m increases.. Yes; 1 1 ( ) 1_ 1 = 1_ = 1_ =. a. 5 b. You could find the square root of 5, which is 5, then find the square root of 5, which is 5. r, ou could evaluate 1_ 5 1_. c. 5 = ( 5 1_ ) = 5. 7 a_ 0. b. = 9 0. =. = 1 5 1_ 7. 5_. =. =. =. Quotient of Powers Propert. 9. I. 5 1_ A. 5 II. 5 III. 5 5_ B. _ C. 5 5 IV. D. 5 1_ 50. Part A a = 0, b = 1, c = 1, d = 1_ 0 Part B _ + 1 _ + 1 = 75 Part C The equation is true for an integer values of that can be written as a power with a base of. If =, where is a rational number not 1_ equal to zero, then =. You can 1_ substitute for in the equation = 75 and simplif to get the required equation. Lesson -. If b > 1, the graph will increase from left to right. If 0 < b < 1, the graph will decrease from left to right.. Martin reversed a and b in f() = ab. The function has a -intercept of and has a constant ratio of f() = ( ) 10. Yes, the graph of f() = ab is alwas getting closer and closer to the -ais but never reaches it. 1. Answers ma var. Sample: The -intercept will increase for a > and decrease for 1 < a <. The descent of the graph will be steeper for a > and less steep for 1 < a <. 1. Answers ma var. Sample: The value of the function is negative for all values of. The asmptote remains at = 0, but the graph approaches = 0 from below the -ais instead of above it. 1. -intercept: 1, domain: all real numbers, range: > 0, asmptote: = 0 ; The function decreases as increases envision Algebra 1 Selected Answers

29 Topic 0.. f() = ( 1_ ). linear; The function increases b a constant value of.. f() = 10 ; Find the ratio of f() for =.1 to f() for = 7. ; The intensit of the New Madrid earthquake is about twice the intensit of the San Francisco earthquake.. f() =,000 (5) ; das. 0. E Lesson -. Simple interest increases the account value b a fied amount over time, so it is modeled b a linear function. Compound interest increases the account value b a fied ratio, so it is modeled b an eponential function.. The growth factor (1 + r) is the b term in the eponential function. For there to be growth, b has to be greater than 1. For eample, if a population grows b 5% per ear, then each ear the population is 15% of the previous ear s population. Therefore, ou multipl b 1.5, or 1 + r.. f() = 1,50(1.5 ). f() = 10,000(0. ) 10. The value of a is the initial amount and b is equal to the growth factor ( 1 + r ). 1. The original equation should be A = 1000 ( t ) because the interest is compounded times per ear. 1. a. As n increases, the growth factor decreases because the value of the fraction added to 1 decreases as its denominator increases. b.,9.5;,75.;,75.0;,75.7;,75.7 c. Although the growth factor decreases as n increases, the value of the function increases because the value of the eponent increases. 1. f() = 100(1.5 ) 1. After 5 ears, the investment compounded semiannuall will be worth $17.5 more than the investment compounded annuall. 0. f() = 5,000 (0.7) ; is about 11 when f() = 100. f() = 5 (0.) ; The average rate of change over is about. and the average rate of change over is about 1.1. The rate of change decreases as increases.. f() = 100 (1.1) ; growth factor: 1.1. f() = 0 (1.5) ; g() = 10,000 (0.) ; The functions are equal at about = 1.. Joshua: f(t) = 500 (1.0) t ; Feli: f(t) = 1,000 (1.0) t ; Joshua s mone will double first. For Joshua, f(10) 1,000. For Feli, f(1), growth, growth, deca, deca, growth. Part A A: f() = 10,000(1.0 ) ; B: f() = 10,000(1.01 ) ; C: f() = 10,000(1.0 ) Part B Investment C Part C Investment C will reach $1,000,000 in 5 ears, while Investment B will take 5 ears and Investment A will take 59 ears. envision Algebra 1 9 Selected Answers

30 Topic Lesson -. Both geometric and arithmetic sequences have the natural numbers as their domains, but geometric sequences have a common ratio while arithmetic sequences have a common difference.. Yes, there is a common ratio of.. The sequence is neither arithmetic or geometric.. a =.5( a ). The initial n n 1 condition is a =. 10. f(n) = 5(1. ) n Answers ma var. Sample: Geometric sequences with common ratio r > 1 are similar to eponential growth functions but with the domain limited to natural numbers. Sequences with a common ratio of 0 < r < 1 are similar to eponential deca functions, also with the domain limited to natural numbers. 1. You can divide each term b the preceding term to see if there is a common ratio. 1. a. a = 0(0.95 ) n 1 n b. a = 0(0.95 ) 10 1 = 50. cm 1. es; 10 a = 1 n ( a ), a = 0. no. es; n 1 1 a = n ( a ), a =. es; a = ( a ), n 1 1 n n 1 a = 1 1. es; a = 5( a ), a = n n a = ( a ), a = a = ( a ), n n 1 1 n n 1 a 1 = 0.. a n = 1 () n 1. a n = 7 () n 1. f(n) = 7 () n 1. f(n) = 99 ( ) n 1 0. f(n) = 1( ) n 1. a n = 1(1.5 ) n 1 ; a = 1.5( a ), a = 1 ; The number of n n 1 1 participants will not reach 1,000 in 10 ears: a = 1(1.5 ) n = 1(1.5 ) = n no; es; es; no. Part A Pattern A: a = 5( a ). n n 1 The initial condition is a = 1. 1 Pattern B: a = 5( a ). n n 1 The initial condition is a =. 1 Part B Because the sequences are the same ecept for their initial condition, the value of a in Pattern B is times n the value of a in Pattern A. Part C n ; Pattern B alwas has times as man dots as Pattern A when the are at the same loop number. Lesson -5. In both cases, the effect is a translation, but k translates the graph verticall, and h translates the graph horizontall.. The graph is below the graph of f() = rather than above it.. The graph is translated up 1 unit.. The graph is translated down units. 10. The graph of g is a translation down units from the graph of the general eponential function f() =, and the graph of j is a translation up 1 unit from the graph of the general eponential function f. So the graph of g is a translation down units from the graph of j. 1. The graph is translated right 1 unit. 1. The graph is translated down 1 unit. 1. In order to determine whether the translation is left or right, the student needs to know the sign of h. 1. a. 1 f() = g() = f is a horizontal compression b 1. envision Algebra 1 0 Selected Answers

31 Topic b. (0,1) c. Both graphs have an asmpote of = translation units left. translation up units. 1. f g. f: -intercept: 1; asmptote: = 0 ; range: > 0 ; g: -intercept: ; asmptote: = 0 ; range: > 0 0. The graph of f is a horizontal shift of the parent function, while the graph of g is a vertical shift of the parent function.. a. The new graph will be a vertical stretch of the original graph b a factor of 1.5. The starting value of a in f() = ab is the same as k in the form kf(). b. The new graph will be a vertical stretch of the original graph b a factor of.. D Topic Review. eponential growth. geometric 1_ sequence. asmptote Biccle =, Car = 1 = + = = Biccle = 500 () Month 9: 9,07, f() = 50 (1.15) 0. $19,.0, Annual: $19,5.7. a n = 5 ( 1 ) n 1 a n = 1 ( a n 1 ), a 1 = 5. a n = () n 1 a n = ( a n 1 ), a 1 =. a n = ( a n 1 ), a 1 =.. a n = 0 (1.5) n 1 a n = 1.5( a n 1 ), a 1 = 0 Yes; in two weeks, or 1 das, there will be about 7,75 signatures. 0. translated up 10 units. translated left units. f g 1 1 envision Algebra 1 1 Selected Answers

32 Topic 7 Lesson 7-1. The prefi mono- means one; bimeans two; tri- means three. Therefore, a monomial has one term; binomial has two terms; trinomial has three terms.. + is similar to combining two of the same item, which is. The term,, can be interpreted as, which refers to the area of a square.. linear binomial The student did not put the polnomial in standard form before naming it and named it b the first term. In standard form, the polnomial is 5, so it is a fourth degree trinomial. 1. a. ( + 7) + ( ) = + 1 b. (a + a + 1) ( a + 5a + ( )) 1. a. Yes; the integers are closed under addition. b. No; the positive integers are not closed for subtraction cubic trinomial. fourth degree polnomial a a in sq. units. D Lesson 7-. When using the Distributive Propert, each term of a second polnomial is multiplied b the terms in the first polnomial. This is similar to using a table since each term in a column is multiplied b ever term in a row.. The eponents are added when multipling, resulting in a degree of the product that is different from either of the factors The first term of the product is the product of the first terms in each binomial. The coefficient of the second term of the product is the sum of the coefficients of second terms in the binomials. The last term of the product is the product of the last terms in each binomial The product is The like terms are on the diagonals of the table from bottom left to top right The product is ( 1 + ) ( + ) = Part A Part B 5 cm Lesson 7- envision Algebra 1 Selected Answers. She did not multipl and and and correctl so the their sum would be zero. The correct answer is 9.

33 Topic 7. When the terms are distributed, the middle term will be zero. When the first term of each binomial is multiplied b the second term of each binomial the products are opposites. This eliminates one of the terms so instead of a trinomial, the product is a binomial cm 1. a. m = 9, b. m = ; n = 1. (9 95) = 1, so ou will multipl the sum of the two numbers b 1, which will equal the sum a + a p + 0p , a 1 b a. ( + ), b. 10 cm. A = 1π + π 0. a b.,7 +,70 +, + 51 ft. C Lesson 7-. The variable with the least eponent is a factor of an terms with common variables that have greater eponents.. Andrew is removing a from each term and putting the remainder as the second factor. Multipling ( ) will not give the third term of the original equation. The correct answer is ( + 1 ).. 5. a 10. a b 1. ab(5a + b) 1. 5 ( ) 1. 1( 9 z + z) 1. The answer has a coefficient that is a multiple of but not 1. It must have both an and variable. The eponent for must be 1; however the eponent for can be an whole number. 0. The student did not take out the correct GCF. All terms of the trinomial also have a b. The correct answer is 5ab( a ab ).. Sometimes; an eample of when it would not be a multiple of is n n ; n n ( + 1) ( 5). 1( m 10m + ). Width:, Length: and. 5ab and a b + b + a. V = π r h [ ( π r ) ] V = π r h [π r ] V = π r (h πr) 0. π ; es there is enough dough. D Lesson 7-5. When c is negative in + b + c, then factors have opposite signs.. Answers ma var. Sample: Look at the binomial factors of the polnomial ( + n) and ( + m). When ou multipl them ou get + m + n + mn, so c in the trinomial + b + c is equivalent to mn.. 1 and 1, 1 and 1, and 7, and 7. opposite signs 10. Factoring a trinomial is like factoring a number because in both cases ou are writing an epression as a product. The are different because the factors of a trinomial are variable epressions and the factors of a number are numbers factors into ( + 9)( ) and 7 1 factors into ( 9)( + ). The signs of the binomial factors are opposite. 1., 9,, 9 envision Algebra 1 Selected Answers

34 Topic 7 1. If the sign of the last term is positive ou are looking for either two positive or two negative factors. If the sign of the last term is negative ou are looking for one positive and one negative factor. 1. ( + 1)( + ) 0. Factors Sum of Factors ( + 11)( + ). ( + 5)( ). ( + )( + ). ( + )( + ) 0. ( + )( + ). ( 11)( + ). ( 1)( + )., ( + 1 ), ( + ); ne dimension is units. Another dimension is one unit greater and the third dimension is two units greater.. Sarah cuts in. from one side and in. from the other. 0. A Lesson 7-. For the trinomial to be factorable, b must equal the sum of two of the factors of ac; sometimes this is a + c, but not alwas.. Yes, ou can rewrite b and then factor out common factors of the first two terms and the last two terms. This will give ou the factors of the trinomial.. 1 and, and. es; and + 1. In both situations, the common factor must be a factor of all parts. In factoring a trinomial, the parts are the terms of the trinomial. In factoring a fraction, the parts are the numerator and denominator. 1. ac 1. No, in this trinomial, ac =. The factors of are: 1 and, 1 and, and 1, and 1, and 1, and 1, and 9, and 9, and. None of these pairs of factors sum to 7, the value of b. 0. (p + n)(q + m). ( + 1)( + 1). ( 15)( 1). ( + )( ). 1 and 1 0. and 15. and. ( 1)( + 1). ( + )( + 1). ( + 1)( ) 0. ( + 1)( + ). ( 1) ( + ). ( )( + 1). ( + 5)( + ). ( + ) ( + ) 50., ( + 1 ), ( + ); b 7 b 5; m 5. ac; b 5. Part A + 9 inches, Part B 9 i n., Part C 1 in. Lesson 7-7. This trinomial fits the pattern of a perfect square trinomial, not a difference of two squares.. In both patterns, two of the terms must be perfect squares. In a perfect square trinomial there are three terms and in a difference of two squares there are two terms.. difference of two squares. perfect-square trinomial 10. difference of two squares 1. (7 + 5)(7 5) 1. ( ) No; Answers ma var. Sample: Because the second term of 50 is not a perfect square, a difference of two squares pattern does not appl. Additionall, the two terms have no common factors. envision Algebra 1 Selected Answers