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1 Reteaching Masters To jump to a location in this book. Click a bookmark on the left. To print a part of the book. Click the Print button.. When the Print window opens, tpe in a range of pages to print. The page numbers are displaed in the bar at the bottom of the document. In the eample below, of 5 means that the current page is page in a file of 5 pages.

2 Reteaching. Tables and Graphs of Linear Equations Skill A Identifing a linear relationship Recall In a linear relationship, for each constant difference in the -values, there is also a constant difference in the -values. Eample You can use the formula F.8C to change temperature readings from degrees Celsius to degrees Fahrenheit. C F Is this an eample of a linear relationship? Yes; notice that for each difference of 0 in Celsius values, there is a corresponding difference of 8 in Fahrenheit values. Eample Make a table of - and -values for 5with -values of 0,,,, and. Is this a linear relationship? Copright b Holt, Rinehart and Winston. All rights reserved. For each increase of in -values, there is a decrease of in -values. This is a linear relationship. Eample Is an eample of a linear relationship? No; for each change of in -values, corresponding changes in are not constant. State whether each equation is linear Algebra Reteaching.

3 Skill B Graphing a linear equation Recall If and are linearl related, this relationship can be epressed as a linear equation in the form m b. Eample Graph the linear equation. Since it takes two points to determine a line, pick two -values and use the equation to calculate the corresponding -values. Graph the ordered pairs (, ) and draw a line. Use a third point to check that all three points are on the same line. 0 = Graph each pair of linear equations on the same set of aes.. a.. a. 5. a. b. b. b. For each table, write a linear equation that represents the relationship between and Copright b Holt, Rinehart and Winston. All rights reserved. Reteaching. Algebra

4 ANSWERS Reteaching Chapter 5. Lesson.. linear relationship. linear relationship. linear relationship = + =. linear relationship 5. linear relationship 6. not a linear relationship 7. linear relationship not a linear relationship 9. linear relationship 0. linear relationship Lesson.. slope: ; -intercept:. linear relationship. not a linear relationship. = + Copright b Holt, Rinehart and Winston. All rights reserved.. = + = = =. slope: ; -intercept: 0 =. slope: ; -intercept: + = Algebra Answers 87

5 Reteaching. Slopes and Intercepts Skill A Graphing a linear equation using the slope and -intercept Recall The slope-intercept form of a line is m b, where m is the slope and b is the -intercept. Eample Graph the line with the equation. Solve for. Write the equation in the form. When = 0, then = ; the -intercept is. Graph the point (0, ). rise The slope of indicates that. run From the point (0, ) rise units and run to the right unit to locate the point (, ). = (, ) (0, ) Find the slope, m, and -intercept, b, for each line. Then graph.... m: b: m: b: m: b: Copright b Holt, Rinehart and Winston. All rights reserved m: b: m: b: m: b: Algebra Reteaching.

6 Skill B Writing the equation of a graphed line in the slope-intercept form Recall The slope of the line containing (, ) and (, ) is given b. Eample Write an equation for the line shown at right. The line crosses the -ais at (0, ). Therefore, (, ) the -intercept is. ( ) The slope is. 0 The equation is (0, ). Write an equation in slope-intercept form for each line (0, ) (0, 0) (, ) (, 0) (, ) (0, ) Skill C Finding the - and -intercepts of the graph of a linear equation Recall A line crosses the -ais when = 0; a line crosses the -ais when = 0. Eample Find the - and -intercepts for the line given b 6. Let 0. Let 0. (0) The -intercept is 6 and the -intercept is. Find the - and -intercepts. Copright b Holt, Rinehart and Winston. All rights reserved Reteaching. Algebra

7 ANSWERS Reteaching Chapter 5. Lesson.. linear relationship. linear relationship. linear relationship = + =. linear relationship 5. linear relationship 6. not a linear relationship 7. linear relationship not a linear relationship 9. linear relationship 0. linear relationship Lesson.. slope: ; -intercept:. linear relationship. not a linear relationship. = + Copright b Holt, Rinehart and Winston. All rights reserved.. = + = = =. slope: ; -intercept: 0 =. slope: ; -intercept: + = Algebra Answers 87

8 ANSWERS. slope: ; -intercept: Lesson.. ; m ; 0 + = 6. 5; m ; 5.. ; m ; 0 7 ; m 7 ; 5. ; m 0; 5. slope: ; -intercept: ; m ; = slope: ; -intercept: = intercept: 8; -intercept:. -intercept: 5; -intercept: 5. -intercept: 6; -intercept:. -intercept: 0; -intercept: 0. -intercept: 8; -intercept: 5. -intercept: none; -intercept: Lesson k 8; 8 k ; k.5;.5 k ; k ; k 5; 5 7. \$0 8. \$70 9. \$ \$ a. t 8. n \$ minutes Copright b Holt, Rinehart and Winston. All rights reserved. 88 Answers Algebra

9 Reteaching. Linear Equations in Two Variables Skill A Writing a specified linear equation in slope-intercept form Recall The point-slope form of the equation of a line with slope m that contains (, ) is m( ). Eample Write an equation in slope-intercept form for the line containing the points (, ) and (, ). Find the slope of the line; m ( ). Use the form m( ). Replace m with, with, and with. ( ) 8 5 This is the slope-intercept form. The slope is and the -intercept is 5. Notice that if ou had used the point (, ) rather than (, ), then ou would have the following: ( ) ( ) 5 Copright b Holt, Rinehart and Winston. All rights reserved. Write an equation for the line containing each pair of points. Then state the slope and the -intercept of each line.. (0, 0) and (, ). (6, ) and (, 6). (, ) and (5, 5) slope: slope: slope: -intercept: -intercept: -intercept:. (7, 0) and (0, ) 5. (, ) and (5, ) 6. (6, ) and (0, ) slope: slope: slope: -intercept: -intercept: -intercept: Algebra Reteaching. 5

10 Skill B Writing equations for parallel or perpendicular lines Recall Parallel lines have the same slope; perpendicular lines have slopes that are negative reciprocals of each other. Eample Write an equation in slope-intercept form for the line that contains the point (, ) and is parallel to the line whose equation is 5. The slope of the line with equation 5 is. An line parallel to this line will have a slope of. Use the point-slope form. m( ) ( ) You can use a graphics calculator to check that these two lines are parallel. Eample Write an equation in slope-intercept form for the line that contains the point (, ) and is perpendicular to the line whose equation is 5. Since the slope of the line with equation 5 is, the slope of the perpendicular line will be the negative reciprocal of, which is. ( ) Write an equation in slope-intercept form for the line that contains the given point and is parallel to the given line. Then write the equation for the line that contains the same point and is perpendicular to the given line. (0, ); 7. parallel to: 8. perpendicular to: (, ); 5 9. parallel to: 0. perpendicular to: (, 5); 6 Copright b Holt, Rinehart and Winston. All rights reserved.. parallel to:. perpendicular to: (0, 0); 5 0. parallel to:. perpendicular to: 6 Reteaching. Algebra

11 ANSWERS. slope: ; -intercept: Lesson.. ; m ; 0 + = 6. 5; m ; 5.. ; m ; 0 7 ; m 7 ; 5. ; m 0; 5. slope: ; -intercept: ; m ; = slope: ; -intercept: = intercept: 8; -intercept:. -intercept: 5; -intercept: 5. -intercept: 6; -intercept:. -intercept: 0; -intercept: 0. -intercept: 8; -intercept: 5. -intercept: none; -intercept: Lesson k 8; 8 k ; k.5;.5 k ; k ; k 5; 5 7. \$0 8. \$70 9. \$ \$ a. t 8. n \$ minutes Copright b Holt, Rinehart and Winston. All rights reserved. 88 Answers Algebra

12 Reteaching. Direct Variation and Proportion Skill A Writing a direct variation equation Recall An equation in the form k is called a direct variation equation where k is the constant of variation. Eample The total cost of tickets to a local concert varies directl as the number of tickets ou bu. If tickets cost \$7, find the constant of variation, k, and write an equation to show this direct variation. Then find the cost of tickets. Let represent the number of tickets and represent the total cost. Then for some value of k, = k. 7 k() 7 k k 8 Each ticket costs \$8. An equation for the direct variation is 8. To find the cost of tickets, substitute for. Then find. 8() 98 Thus, tickets will cost \$98. Copright b Holt, Rinehart and Winston. All rights reserved. In Eercises 6, varies directl as. Find the constant of variation, and write an equation of direct variation that relates the two variables.. 0 when 5. 6 when. 5 when 0 k k k equation: equation: equation:. when 5. when 8 6. when 0. k k k equation: equation: equation: Use a direct variation equation to solve each problem. 7. If 5 tickets cost \$65, find the cost of 8 tickets. 8. If cassette tapes on sale cost \$8, find the cost of 0 tapes. 9. If 6 colas cost \$.8, find the cost of 0 colas. 0. If 5 pens cost \$.75, find the cost of 8 pens. Algebra Reteaching. 7

13 Skill B Writing and solving proportions a Recall If, then ad bc. (Cross Product Propert) b c d Eample If T-shirts cost \$8.50, write and solve a proportion to find the cost of 5 T-shirts. number of T- shirts total cost (8.50) Cross Product Propert Five T-shirts will cost \$7.50. Eample a 5 Solve a. (a 5) a 9a 5 a 5a 5 a Solve each proportion for the indicated variable. Check our answers a t 50 n n Use a proportion to solve each problem. 7. If eactl out of students voted for candidate A, how man votes out of the 0 cast were for candidate A? 8. If 8 gallons of gasoline cost \$9., what is the cost for gallons? 9. If Monica jogs at a constant speed and covers miles in minutes, how long will it take her to jog 5 miles? 0. In the smaller of similar right triangles, the hpotenuse measures 0 centimeters and the shorter leg measures centimeters. In the larger triangle the shorter leg measures 8 centimeters. How long is the 6 Copright b Holt, Rinehart and Winston. All rights reserved. hpotenuse of the larger triangle? 8 Reteaching. Algebra

14 ANSWERS. slope: ; -intercept: Lesson.. ; m ; 0 + = 6. 5; m ; 5.. ; m ; 0 7 ; m 7 ; 5. ; m 0; 5. slope: ; -intercept: ; m ; = slope: ; -intercept: = intercept: 8; -intercept:. -intercept: 5; -intercept: 5. -intercept: 6; -intercept:. -intercept: 0; -intercept: 0. -intercept: 8; -intercept: 5. -intercept: none; -intercept: Lesson k 8; 8 k ; k.5;.5 k ; k ; k 5; 5 7. \$0 8. \$70 9. \$ \$ a. t 8. n \$ minutes Copright b Holt, Rinehart and Winston. All rights reserved. 88 Answers Algebra

15 Copright b Holt, Rinehart and Winston. All rights reserved. ANSWERS 0. 6 centimeters Lesson r 0.65; moderate positive correlation r 0.995; strong negative correlation Lesson h A b a p b r C π ; 7.; ; 7; ;.97; ; 5; a 96. t r 8. c a Lesson b (c d) a P h S πr r m E c h V πr c a b 6 a 6 6 t 6 m 6 c a I rt a b c Algebra Answers 89

16 Reteaching.5 Scatter Plots and Least-Square Lines Skill A Using a scatter plot, least-squares line, and correlation coefficient to analze data Recall The least-squares line for a positive correlation will have a positive slope. Eample Draw a scatter plot for the following data. Find an equation for the least-squares line. Draw this line and describe the correlation. X Y Enter the data in our calculator to find a least-squares line of and a correlation coefficient of r The graph and this value of r indicate a strong positive correlation Copright b Holt, Rinehart and Winston. All rights reserved. Create a scatter plot for the data in each table below. Describe the correlation. Then find an equation for the least-squares line. Draw this line on the scatter plot correlation correlation Algebra Reteaching.5 9

17 Skill B Using a least-squares line to predict or estimate values of a variable Recall nce ou have found an equation for the least-squares line, ou can use substitution to estimate or make a prediction of the value of the second variable. Eample Rand used the following table to record miles he had driven and the amount of gas used. miles driven () gallons of gas () Estimate the amount of gas needed for a trip of 75 miles and for a trip of 800 miles. Use a graphics calculator to find the equation of the least-squares line for this data Evaluate with 75. Then 0.. The 75-mile trip will require approimatel 0. gallons of gas. Evaluate with 800. Then 9.8. The 800-mile trip will require approimatel 9.8 gallons of gas. Use a graphics calculator to find the equation of the leastsquares line for each set of data. Then find each value of. Round answers to the nearest hundredth If 6, then If 0, then If 0, then If 00, then If, then If 0, then If 00, then If 750, then Copright b Holt, Rinehart and Winston. All rights reserved. 0 Reteaching.5 Algebra

18 Copright b Holt, Rinehart and Winston. All rights reserved. ANSWERS 0. 6 centimeters Lesson r 0.65; moderate positive correlation r 0.995; strong negative correlation Lesson h A b a p b r C π ; 7.; ; 7; ;.97; ; 5; a 96. t r 8. c a Lesson b (c d) a P h S πr r m E c h V πr c a b 6 a 6 6 t 6 m 6 c a I rt a b c Algebra Answers 89

19 Reteaching.6 Introduction to Solving Equations Skill A Solving linear equations in one variable Recall An equation is solved b using inverse operations. Eample The total cost for a set of CDs, including shipping and handling charges, is \$9.50. If the shipping and handling charges are \$.50, what is the cost of each CD? n where n is the cost of CD n Subtract.5 from each side of the equation. n 6 n 6 Divide each side b. n.5 Each CD costs \$.50. To check our answer, show that (.5) +.5 = 9.5. Copright b Holt, Rinehart and Winston. All rights reserved. Solve each equation. Eample Solve Thus, 7, or. Subtract from each side of the equation. Combine like terms t a 5. ( ) 6. ( ) ( 5) 7. r r 5 r 8. (c ) c 9. 0.( 5) 5 0. a ( a) a. ( ). (5 ) 5( ) 7 5 Algebra Reteaching.6

20 Skill B Solving literal equations for a specified variable Recall Man formulas are literal equations that contain two or more variables. Eample The formula for changing Celsius to Fahrenheit temperature is F 9. 5 C To find a formula for changing Fahrenheit to Celsius, solve this equation for C. F 9 5 C F 9 C 5 Subtract from each side of the equation. F 9 5 C 5 9 Multipl each side of the equation b. 9 (F ) C 5 5 (F ) C 9 Thus, the equation for changing Fahrenheit temperature to Celsius is C 5 (F ). 9 Eample Solve a abc for a. a abc a( bc) Distributive Propert a( bc) Divide each side of the equation b bc. bc bc bc a Solve each literal equation for the indicated variable.. A bh for h. p a b for a 5. C π r for r 6. a b c d for b 7. I P Prt for P 8. S πr(r h) for h Copright b Holt, Rinehart and Winston. All rights reserved. 9. a b c for 0. E mc for m. V. a b πr h for h c for c Reteaching.6 Algebra

21 Copright b Holt, Rinehart and Winston. All rights reserved. ANSWERS 0. 6 centimeters Lesson r 0.65; moderate positive correlation r 0.995; strong negative correlation Lesson h A b a p b r C π ; 7.; ; 7; ;.97; ; 5; a 96. t r 8. c a Lesson b (c d) a P h S πr r m E c h V πr c a b 6 a 6 6 t 6 m 6 c a I rt a b c Algebra Answers 89

22 Reteaching.7 Introduction to Solving Inequalities Skill A Solving linear inequalities in one variable and graphing solutions on a number line Recall When ou multipl or divide each side of an inequalit b a positive number, the inequalit sign remains the same. Recall Eample Solve 5 7. Graph the solution on a number line Notice the open circle at 6 because the number 6 is not in the solution set. When ou multipl or divide each side of an inequalit b a negative number, reverse the inequalit smbol. Copright b Holt, Rinehart and Winston. All rights reserved. Eample Solve a 0 0. Graph the solution on a number line. a 0 0 a a 0 a a 7.5 Notice the filled circle at 7.5 because the number 7.5 is in the solution set. Solve each inequalit and graph the solution on the number line... a 6a m 6. c c t (a ) a Algebra Reteaching.7

23 Skill B Solving and graphing compound linear inequalities in one variable Recall An inequalit involving and is true onl if both parts of the inequalit are true. Eample Solve and Graph the solution on a number line and Eample Graph the solution for and Recall Both inequalities are true onl if. An inequalit involving or is true if at least one part of the inequalit is true. Eample Solve and graph the solution for 7 or or Graph the solution of each compound inequalit on a number line. 0. and 8. 5 and and 5. and z 5 or z or Copright b Holt, Rinehart and Winston. All rights reserved. 6. or 7. a or a Reteaching.7 Algebra

24 Copright b Holt, Rinehart and Winston. All rights reserved. ANSWERS 0. 6 centimeters Lesson r 0.65; moderate positive correlation r 0.995; strong negative correlation Lesson h A b a p b r C π ; 7.; ; 7; ;.97; ; 5; a 96. t r 8. c a Lesson b (c d) a P h S πr r m E c h V πr c a b 6 a 6 6 t 6 m 6 c a I rt a b c Algebra Answers 89

25 ANSWERS 0. and 5. 5 or and 6. t or t no solution 7. a and 5 8. or z or z 5 9. t 6 or t or 0. 0 or m or m a or a. no solution Lesson or a or a b 0 or b or and a or a a and a Copright b Holt, Rinehart and Winston. All rights reserved.. or b or b Answers Algebra

26 Reteaching.8 Solving Absolute-Value Equations and Inequalities Skill A Solving absolute-value equations and graphing solutions on a number line Recall The absolute value of is the distance between and 0 on the number line. Eample Solve. Graph the solution on a number line. or 6 Eample Solve 5. Graph the solution on a number line. 5 8 or 5 6 or Solve each equation. Graph the solution on a number line... a b. 5 Copright b Holt, Rinehart and Winston. All rights reserved a t m t r Algebra Reteaching.8 5

27 Skill B Solving absolute-value inequalities and graphing solutions on a number line a Recall The solution of, where a is nonnegative, is all real numbers less than a and greater than a. Eample Solve 0. Graph the solution on a number line. 0 and 0 8 and a Recall The solution of, where a is nonnegative, is all real numbers less than a or greater than a. Eample Solve a. Graph the solution on a number line. a a 5 a 5 a or a 5 a 8 a or a Solve each inequalit. Graph the solution on the number line a 6. a b c Copright b Holt, Rinehart and Winston. All rights reserved. 6 Reteaching.8 Algebra

28 ANSWERS 0. and 5. 5 or and 6. t or t no solution 7. a and 5 8. or z or z 5 9. t 6 or t or 0. 0 or m or m a or a. no solution Lesson or a or a b 0 or b or and a or a a and a Copright b Holt, Rinehart and Winston. All rights reserved.. or b or b Answers Algebra

29 ANSWERS 8. and c and c not possible a 9 Copright b Holt, Rinehart and Winston. All rights reserved. 0. Reteaching Chapter Lesson or Rational. True; Associative Propert of Multiplication. True; Inverse Propert of Addition. False. False 5. True; Commutative Propert of Multiplication 6. True; Identit for Multiplication 7. True; Distributive Propert 8. True; Commutative Propert of Addition Lesson..6 Integers 8 5 Whole 0 5 Natural Real Numbers Irrational π a b 5. c 8 d z z Lesson.. no; domain: {, }; range: {,,, }. es; domain: {0,,, }; range: {0,,, }. es; domain: {,,, }; range: {,,, }. no; domain: {, 9}; range: {,,, } 5. no; domain: 0; range: all real numbers 6. es; domain: { 5,,, 0,,, }; range: {,, 0,,, } f(t) 60t 8. f(h) h 9. f(e) 6e π 0. f(r) r Lesson Algebra Answers 9

30 Reteaching. perations With Numbers Skill A Classifing real numbers Recall You can classif a real number as belonging to the natural numbers, whole numbers, integers, rational numbers, or irrational numbers. A real number can belong to more than one set of numbers. Eample Classif in as man was as possible. is not a natural number because natural numbers are positive whole numbers. is not a whole number because whole numbers are either positive or 0. is an integer because integers are all the whole numbers and their opposites. is a rational number because it can be written as the terminating decimal.0. is a real number. The number is an integer, a rational number, and a real number. Copright b Holt, Rinehart and Winston. All rights reserved. Use the diagram to classif each number in as man was as possible b writing it in the smallest rectangle in which it belongs. For eample, is placed in the rectangle labeled rational π.6 Skill B Identifing properties of real numbers Recall 0 5 Rational Integers Whole Natural Real Numbers Irrational The real numbers are characterized b the Commutative and Associative Properties of Addition and Multiplication and b the Distributive Propert. Eample Tell if the statement is true or false. Justif our response. a. ab b a b. ( z) ( ) z c. (a b) a b a. True Commutative Propert of Multiplication b. False Subtraction is not associative. c. True Distributive Propert Algebra Reteaching. 7

31 Tell whether each statement is true or false. State the propert that is illustrated in each true statement. All variables represent real numbers.. (6a)b 6(ab).. 7( 7) abd adb 6. 5 ( 5) a a 7. ( w) w 8. 5( ) 5( ) Skill C Simplifing numerical epressions b using the order of operations Recall The order of operations can be remembered b using the following sentence. Please Ecuse M Dear Aunt Sall Parentheses, Eponents, Multiplication and Division, Addition and Subtraction Eample Simplif. 6 ( ) 5 Work inside parentheses first. 6 ( ) Simplif each epression. Use a calculator to check ( ) {6 [6 (6 )] } The fraction bar is a grouping smbol. Perform eponentiation. Check: Enter /(6 ( )) 5 into a calculator. Note the use of parentheses around 6 ( ). The displa will show 9. (0 ) ( ) (7 ) 7 Copright b Holt, Rinehart and Winston. All rights reserved. 8 Reteaching. Algebra

32 ANSWERS 8. and c and c not possible a 9 Copright b Holt, Rinehart and Winston. All rights reserved. 0. Reteaching Chapter Lesson or Rational. True; Associative Propert of Multiplication. True; Inverse Propert of Addition. False. False 5. True; Commutative Propert of Multiplication 6. True; Identit for Multiplication 7. True; Distributive Propert 8. True; Commutative Propert of Addition Lesson..6 Integers 8 5 Whole 0 5 Natural Real Numbers Irrational π a b 5. c 8 d z z Lesson.. no; domain: {, }; range: {,,, }. es; domain: {0,,, }; range: {0,,, }. es; domain: {,,, }; range: {,,, }. no; domain: {, 9}; range: {,,, } 5. no; domain: 0; range: all real numbers 6. es; domain: { 5,,, 0,,, }; range: {,, 0,,, } f(t) 60t 8. f(h) h 9. f(e) 6e π 0. f(r) r Lesson Algebra Answers 9

33 Reteaching. Properties of Eponents Skill A Evaluating numerical epressions with eponents Recall You can associate a negative eponent with a reciprocal. Copright b Holt, Rinehart and Winston. All rights reserved. Eample Without using a calculator, evaluate each epression. a. b. 0 c. d. e. f. a. b. 6 0 For an nonzero number a, a 0. c. A negative eponent indicates a reciprocal. d. e. Notice that the result is not negative. 6 a n a n f. ( ) 8 m a ( a) n m Eample Epress each number as a power of. a. 8 b. c. 9 d. 7 e. f. 7 a. 8 b. 0 c. 9 ( ) 6 d. e. f. 7 7 Evaluate each epression (6) 0 5. ( ) 6. ( ) ( ) 5) ( ) ( ( 5) Epress each number as a power of, if possible Algebra Reteaching. 9

34 Skill B Simplifing algebraic epressions involving eponents Recall When ou simplif an algebraic epression involving eponents, use Properties of Eponents. Eample Simplif ( 5. Write our answer using positive eponents onl. ) ( Power of a Quotient: ( b) 5 ( 5 ) a n an ) ( ) b n 0 Power of a Power: (a m ) n a mn ( ) Quotient of Powers: a n a m n Eample Simplif 5( ). Write our answer using positive eponents onl. a m 5( ) 5( ) 5 5 Power of a Product: (ab) n a n b n Power of Powers: (a m ) n a mn Definition of negative eponent: a n a n Simplif each epression, assuming that no variable equals zero. Write our answer using positive eponents onl. 9. a a a 0.. ( )( ).. (c d) (cd ). ( z ) (5) (a b)( ab )(ab) ( ) ( 0 ) ( ) Copright b Holt, Rinehart and Winston. All rights reserved ( ( 5 0 z ( 5 ) ( ) ) ) 0 Reteaching. Algebra

35 ANSWERS 8. and c and c not possible a 9 Copright b Holt, Rinehart and Winston. All rights reserved. 0. Reteaching Chapter Lesson or Rational. True; Associative Propert of Multiplication. True; Inverse Propert of Addition. False. False 5. True; Commutative Propert of Multiplication 6. True; Identit for Multiplication 7. True; Distributive Propert 8. True; Commutative Propert of Addition Lesson..6 Integers 8 5 Whole 0 5 Natural Real Numbers Irrational π a b 5. c 8 d z z Lesson.. no; domain: {, }; range: {,,, }. es; domain: {0,,, }; range: {0,,, }. es; domain: {,,, }; range: {,,, }. no; domain: {, 9}; range: {,,, } 5. no; domain: 0; range: all real numbers 6. es; domain: { 5,,, 0,,, }; range: {,, 0,,, } f(t) 60t 8. f(h) h 9. f(e) 6e π 0. f(r) r Lesson Algebra Answers 9

36 Reteaching. Introduction to Functions Skill A Identifing that a given relation is a function Recall A function is a relation in which each value in the domain is paired with eactl one value in the range. Eample 0 Does the table at right represent a function? 6 0 Yes; for each value of there is onl one value of. Notice that the two -values of and have the same -value,. This is allowed in the definition of function. Eample Does the solid-line graph shown represent a function? State the domain and range. Yes; notice that an vertical line will intersect the graph in no more than point. domain: range: Copright b Holt, Rinehart and Winston. All rights reserved. State whether each relation represents a function and give the domain and range.. {(, ), (, ), (, ), (, )}. domain: range: domain: range:.. domain: range: domain: range: {(0, 0), (, ), (, ), (, )} 9 9 domain: range: domain: range: Algebra Reteaching.

37 Skill B Writing and evaluating functions Recall The value of f() 5 depends on the value of. Eample Sarah uses an internet server which charges \$.50 per month plus \$0.60 for each hour over 0 hours that she uses it during the month. Write this relation in function notation. How much will she be charged for using the service for 8 hours in April? Let h number of hours over 0. Thus, the function is as follows. f(h) h f(8) (8) where h 8 f(8).0 The charge for April will be \$5.0. Eample If g(), find g( 5). g( 5) means replace with the value 5 and evaluate g(). g( 5) ( 5) ( 5) Thus, g( 5) 0. Let f() 5 and g(). Evaluate each function. 7. f(6) 8. f(0) f( ) g( ).. f() g(0). 5. f(0) g(0) 6. Write each situation in function notation. 7. Driving at 60 miles per hour, the distance ou travel depends on the number of hours spent driving. g() g( ) 8. The charge for electric service is \$0.00 plus \$0.60 for each kilowatt-hour of electricit that ou use each month. g() f(5) g( 6) f( 6) Copright b Holt, Rinehart and Winston. All rights reserved. 9. The surface area of a cube is 6 times the square of the length of one edge. π 0. The volume of a sphere is times the radius cubed. Reteaching. Algebra

38 ANSWERS 8. and c and c not possible a 9 Copright b Holt, Rinehart and Winston. All rights reserved. 0. Reteaching Chapter Lesson or Rational. True; Associative Propert of Multiplication. True; Inverse Propert of Addition. False. False 5. True; Commutative Propert of Multiplication 6. True; Identit for Multiplication 7. True; Distributive Propert 8. True; Commutative Propert of Addition Lesson..6 Integers 8 5 Whole 0 5 Natural Real Numbers Irrational π a b 5. c 8 d z z Lesson.. no; domain: {, }; range: {,,, }. es; domain: {0,,, }; range: {0,,, }. es; domain: {,,, }; range: {,,, }. no; domain: {, 9}; range: {,,, } 5. no; domain: 0; range: all real numbers 6. es; domain: { 5,,, 0,,, }; range: {,, 0,,, } f(t) 60t 8. f(h) h 9. f(e) 6e π 0. f(r) r Lesson Algebra Answers 9

39 Reteaching. perations With Functions Skill A Using the four basic operations on functions to write new functions Recall To write the sum, difference, product, or quotient of two functions, f and g, write the sum, difference, product, or quotient of the epressions that define f and g. Then simplif. Copright b Holt, Rinehart and Winston. All rights reserved. Eample Let f() and g() 5. Write an epression for each function. a. (f g)() b. (f g)() c. (fg)() d. a. (f g)() f() g() ( ) (5 ) 8 b. c. (f g)() f() g() ( ) (5 ) 5 (fg)() f() g() ( )(5 ) ( )(5) ( )( ) ( f g) () d. f(), where g() 0 g(), where 5 5 ( f g) () Combine like terms. Combine like terms. Let f(), g(), and h() 5. Find each new function, and state an domain restrictions.. (f g)().. (h g)(). 5. (hg)() 6. (f h)() (gh)() (f h)() Distributive Propert 7. ( 8. g) f () ( h g) () Algebra Reteaching.

40 Skill B Finding the composite of two functions Recall To write an epression for the composite function (f g) (), replace each in the epression for f with the epression defining g. Then simplif the result. Eample Let f() 5 and g(). Find (f g) () and (g f )(). Then write epressions for (f g) () and (g f )(). (f g) (): g() () 5 f(g()) f(5) 5(5) 5 Thus, (f g) () 5. (g f )(): f() 5() 0 g(f()) g(0) (0) 97 Thus, (g f )() 97. To write epressions for (f g) () and (g f )(), use the variable instead of a particular number. (f g) () f(g()) (g f )() g(f()) f( ) g(5) 5( ) (5) Let f(), g(), and h() 5. Find each composite function. 9. (f g) () 0. (g f )(). (h f )(). (h g) (). (g g) (). (h h) () 5. (g h)() 6. (f f )( ) 7. (f (g h))() 8. (g (g g))(5) Copright b Holt, Rinehart and Winston. All rights reserved. Reteaching. Algebra

41 ANSWERS 8. and c and c not possible a 9 Copright b Holt, Rinehart and Winston. All rights reserved. 0. Reteaching Chapter Lesson or Rational. True; Associative Propert of Multiplication. True; Inverse Propert of Addition. False. False 5. True; Commutative Propert of Multiplication 6. True; Identit for Multiplication 7. True; Distributive Propert 8. True; Commutative Propert of Addition Lesson..6 Integers 8 5 Whole 0 5 Natural Real Numbers Irrational π a b 5. c 8 d z z Lesson.. no; domain: {, }; range: {,,, }. es; domain: {0,,, }; range: {0,,, }. es; domain: {,,, }; range: {,,, }. no; domain: {, 9}; range: {,,, } 5. no; domain: 0; range: all real numbers 6. es; domain: { 5,,, 0,,, }; range: {,, 0,,, } f(t) 60t 8. f(h) h 9. f(e) 6e π 0. f(r) r Lesson Algebra Answers 9

42 ANSWERS , 8. 5, Lesson f(m) 0.[m]; \$.9. f(h) 50 5 h ; \$ Lesson.5. f () {(, 5), (, 0), (5, ), (6, 5)}; 5; 5. f () 6; 5; f () ; 5; f 5 () ; 5; 5 5. {(, ), (, ), (0, ), (, 5), (, 8)} f() f () The inverse is a function. 6. {(, ), (, ), (0, ), (, ), (, )} f() Copright b Holt, Rinehart and Winston. All rights reserved. f () The inverse is not a function. 9 Answers Algebra

43 Reteaching.5 Inverses of Functions Skill A Finding inverses of functions Recall The inverse of a relation is found b interchanging and and then solving for. Eample Find the inverse of the function given b {(, 7), (5, ), (7, 9), (9, 5)}. Tell whether the inverse relation is a function. Interchange and. {(7, ), (, 5), (9, 7), (5, 9)} The inverse is a function because there is onl one -value for each -value. Eample Find the inverse, f, of f() 5. The find f(f ()) and f (f()). Replace f() with. Interchange and. Solve for f(f ()) f( 5 ) ( 5 ) 5 5 f (f()) f ( 5) ( 5) 5 Copright b Holt, Rinehart and Winston. All rights reserved. 5 5 Find the inverse, f (), of each function. Then find f(f (5)) and f (f(5)).. f() {( 5, ), (0, ), (, 5), (5, 6)}. f () f(f (5)). f() 7. f () f (f(5)) f() f () f(f (5)) f() 5 f () 5 5 f (f(5)) f(f (5)) f (f(5)) f(f (5)) f (f(5)) Algebra Reteaching.5 5

44 Skill B Using the horizontal-line test and graphing the inverse of a function Recall The inverse of a function is also a function if and onl if ever horizontal line intersects the graph of the given function in no more than one point. Eample a. Graph the function given b f(). Then use the horizontal-line test to find out if its inverse relation is also a function. b. Sketch the line and the inverse relation on the same set of aes. a. Use several values of to find ordered pairs. {(, 5), (, ), (0, ), (, ), (, 5)} Graph these points and connect them with a smooth curve. The horizontal line drawn shows that this function does not pass the horizontal-line test. Therefore, the inverse relation is not a function. b. Use {(5, ), (, ), (, 0), (, ), (5, )} to graph the inverse relation. Notice that the inverse is the reflection of the graph of f() across the line. Also observe that the inverse relation will not pass the vertical-line test. This confirms that the inverse relation is not a function. Use -values of,, 0,, and to find ordered pairs and graph each function. n the same coordinate grid, sketch the inverse relation and determine whether it is a function. 5. f() 6. f() = Copright b Holt, Rinehart and Winston. All rights reserved. 6 Reteaching.5 Algebra

45 ANSWERS , 8. 5, Lesson f(m) 0.[m]; \$.9. f(h) 50 5 h ; \$ Lesson.5. f () {(, 5), (, 0), (5, ), (6, 5)}; 5; 5. f () 6; 5; f () ; 5; f 5 () ; 5; 5 5. {(, ), (, ), (0, ), (, 5), (, 8)} f() f () The inverse is a function. 6. {(, ), (, ), (0, ), (, ), (, )} f() Copright b Holt, Rinehart and Winston. All rights reserved. f () The inverse is not a function. 9 Answers Algebra

46 Reteaching.6 Special Functions Skill A Evaluating and appling rounding-up, rounding-down, and absolute value functions Recall The rounding-up function rounds a decimal value to the net highest integer.. Recall Eample A long distance telephone compan advertises that weekend calls cost \$0.0 per minute. Each fraction of a minute is rounded up to the net whole minute. Write this as a rounding-up function. Then find the cost of a.5-minute call. f(m) 0. m where m number of minutes f(.5) () is the net highest integer after.5.0 The call will cost \$.0. The rounding-down function rounds a decimal value to the net lowest integer. [.] Eample Evaluate [.7]. [.7] is the net integer to the left of.7 Recall Absolute value means distance from 0... Copright b Holt, Rinehart and Winston. All rights reserved. Eample Evaluate..... is at a distance. units from 0 Evaluate.... [.] [.8] [.] [.8] [ ]. 8.5 [.7] Write a function for each problem. Solve the problem.. Another phone compan charges \$0. per minute, but does not charge for the net minute unless ou use the full minute. What is the charge for a 6.5 minute call?. A local plumber charges \$50 for a house call plus \$5 per hour or an fraction of an hour. What is the charge for a.75 hour house call? Algebra Reteaching.6 7

47 Skill B Graphing piecewise, step, and absolute-value functions Recall A piecewise function in is a function defined b different epressions in on different intervals for. Eample Graph this piecewise function., if 5 f() [], if 5, if [] Graph each function., if 0 5. f() 6. 5 if 0 if f() if 7. if f() if 8. [] if f() if Copright b Holt, Rinehart and Winston. All rights reserved. 8 Reteaching.6 Algebra

48 ANSWERS , 8. 5, Lesson f(m) 0.[m]; \$.9. f(h) 50 5 h ; \$ Lesson.5. f () {(, 5), (, 0), (5, ), (6, 5)}; 5; 5. f () 6; 5; f () ; 5; f 5 () ; 5; 5 5. {(, ), (, ), (0, ), (, 5), (, 8)} f() f () The inverse is a function. 6. {(, ), (, ), (0, ), (, ), (, )} f() Copright b Holt, Rinehart and Winston. All rights reserved. f () The inverse is not a function. 9 Answers Algebra

49 Reteaching.7 A Preview of Transformations Skill A Identifing and describing transformations of a function Recall The functions f(), f(), and f() are called parent functions. If ou have forgotten what their graphs look like, use a calculator to refresh our memor. Eample Graph f(), g(), and h() on the same set of aes. Describe the transformations of f that give g and h. Function g: translation units down Function h: translation units to the right h() f() g() Eample Graph f(), g(), and h(), where 0, on the same set of aes. Describe the transformations of f that give g and h. Function g: reflection across the -ais Function h: reflection across the -ais Eample h() f() g() Copright b Holt, Rinehart and Winston. All rights reserved. Graph f(), g() (), and h() on the same set of aes. Describe the transformations of f that give g and h. Function g: vertical stretch b a factor of Function h: horizontal stretch b a factor of followed b reflection across the -ais Identif the parent function of f and describe the transformations needed to graph f..... f() 5 f() f() ( ) f(), 0 f() h() g() 5. f() 6. f(), Algebra Reteaching.7 9

50 Skill B Writing an equation for a function whose graph is given Recall The transformations below show how to transform the graph of a function horizontall. h k where h and k are numbers These transformations show how to transform the graph of a function f verticall. f() f() h f() k(f()) where h and k are numbers Eample Given the graph of f, write an equation for the graph of g. translation units to the right: f( ) ( ) translation units up: f() reflection across the -ais: f() The transformations above give g() ( ). f() g() Use the graph of f to help ou write an equation for the graph of g. 7. f() 8. f() 9. f() g() g() g() f() f() g() g() 0. f(). f(). g() g() g() f() g() f() g() f() g() f() f() g() Copright b Holt, Rinehart and Winston. All rights reserved. 0 Reteaching.7 Algebra

51 ANSWERS Lesson.7.. f() ; reflection across the -ais and vertical translation 5 units up. f() ; horizontal translation units to the right Copright b Holt, Rinehart and Winston. All rights reserved.. f() ; horizontal translation units to the left and vertical translation units up. f() ; reflection across the -ais and vertical stretch b a factor of 5. f() ; horizontal compression b a factor of 6. f() ; reflection across the -ais, vertical stretch b a factor of, and a horizontal translation units to the left 7. g() ( ) 8. g() 9. g() 0. g(), or g() =. g(). g() Reteaching Chapter Lesson.. (, ) intersecting, consistent, independent (, ). same line, consistent, dependent all (, ) such that parallel lines, inconsistent, independent no solution. (, ) 5. (, ) 6. (5, 7) 7. (, ) 8. (, ) 9. (, 5) 0. (, ). (5, ). (6, ) Lesson.. (, ). (, ). (5, ). (, ) 5. (, ) 6. (, ) 7. (, ) 8. (, ) 9. inconsistent 0. dependent. consistent, (, 0). consistent, (, ). inconsistent. dependent 5. consistent, (, ) 6. dependent Algebra Answers 9

52 Reteaching. Solving Sstems b Graphing or Substitution Skill A Solving a sstem of linear equations b graphing Recall A sstem of equations ma be consistent or inconsistent. If it is consistent, the equations ma be dependent or independent. Eample Graph each sstem. Classif the graphs as intersecting lines, parallel lines, or the same line. Then classif the sstem as consistent or inconsistent. If it is consistent, classif it as dependent or independent and find the solution. a. b. c. The first equation in each sstem is alread solved for. Solve the second equation in each sstem for and graph. a. intersecting lines; consistent; independent; (, ) b. parallel lines; inconsistent; no solution c. same line; consistent; dependent; all ordered pairs = + + = = (, ) such that Copright b Holt, Rinehart and Winston. All rights reserved. Graph and classif each sstem. Then find the solution from the graph Algebra Reteaching.

53 Skill B Solving a sstem b substitution Recall Your objective is to obtain a single equation in one variable which is easil solved. Eample Use substitution to solve this sstem. Solve for In the second equation, substitute for. ( ) The solution is (5, 8). Check to see that the ordered pair (5, 8) satisfies each of the original equations. Eample 7 Use substitution to solve this sstem. It is easier to solve the second equation for, and then substitute the result into the first equation. Now, use to find. 7 ( ) Thus, the solution is (, ). () Use substitution to solve each sstem of equations. Check our solution Copright b Holt, Rinehart and Winston. All rights reserved Reteaching. Algebra

54 ANSWERS Lesson.7.. f() ; reflection across the -ais and vertical translation 5 units up. f() ; horizontal translation units to the right Copright b Holt, Rinehart and Winston. All rights reserved.. f() ; horizontal translation units to the left and vertical translation units up. f() ; reflection across the -ais and vertical stretch b a factor of 5. f() ; horizontal compression b a factor of 6. f() ; reflection across the -ais, vertical stretch b a factor of, and a horizontal translation units to the left 7. g() ( ) 8. g() 9. g() 0. g(), or g() =. g(). g() Reteaching Chapter Lesson.. (, ) intersecting, consistent, independent (, ). same line, consistent, dependent all (, ) such that parallel lines, inconsistent, independent no solution. (, ) 5. (, ) 6. (5, 7) 7. (, ) 8. (, ) 9. (, 5) 0. (, ). (5, ). (6, ) Lesson.. (, ). (, ). (5, ). (, ) 5. (, ) 6. (, ) 7. (, ) 8. (, ) 9. inconsistent 0. dependent. consistent, (, 0). consistent, (, ). inconsistent. dependent 5. consistent, (, ) 6. dependent Algebra Answers 9

55 Reteaching. Solving Sstems b Elimination Skill A Solving a consistent and independent sstem of equations b elimination Recall Two lines with different slopes represent a consistent and independent sstem of equations. Since these lines intersect in one point, there is one solution to the sstem. Copright b Holt, Rinehart and Winston. All rights reserved. Eample Solve b using the elimination method. 6 To eliminate the terms multipl each side of the first equation b. ( ) () Then multipl each side of the second equation b. 6 ( ) (6) 8 6 The sstem that results is shown below Addition Propert of Equalit To find, replace with in the first equation. () Check the ordered pair (, ) in the second original equation. 6 () () Use elimination to solve each sstem of equations. Check our solution Algebra Reteaching.

56 Skill AB Tet Classifing a sstem as dependent or inconsistent Recall Tet If the same line represents two different equations, the sstem is dependent. If parallel lines represent two different equations, the sstem is inconsistent. Eample SBHCT Eample Classif the sstem as consistent or inconsistent, independent or dependent. ( )( ) ( )( ) 0 7 Since 0 7 is a false statement, this sstem represents a pair of parallel lines. There is no solution because this sstem is inconsistent. Eample Classif the sstem as consistent or inconsistent, independent 6 or dependent. 6 6 ( ) () Since 0 0 is a true statement, these are both the same line. This sstem is a consistent and dependent sstem. An solution to the first equation will also be a solution to the second equation. An solution to the second equation will also be a solution to the first equation. Classif each sstem as consistent or inconsistent, independent or dependent. If the sstem is consistent, find the solution Copright b Holt, Rinehart and Winston. All rights reserved. Reteaching. Algebra

57 ANSWERS Lesson.7.. f() ; reflection across the -ais and vertical translation 5 units up. f() ; horizontal translation units to the right Copright b Holt, Rinehart and Winston. All rights reserved.. f() ; horizontal translation units to the left and vertical translation units up. f() ; reflection across the -ais and vertical stretch b a factor of 5. f() ; horizontal compression b a factor of 6. f() ; reflection across the -ais, vertical stretch b a factor of, and a horizontal translation units to the left 7. g() ( ) 8. g() 9. g() 0. g(), or g() =. g(). g() Reteaching Chapter Lesson.. (, ) intersecting, consistent, independent (, ). same line, consistent, dependent all (, ) such that parallel lines, inconsistent, independent no solution. (, ) 5. (, ) 6. (5, 7) 7. (, ) 8. (, ) 9. (, 5) 0. (, ). (5, ). (6, ) Lesson.. (, ). (, ). (5, ). (, ) 5. (, ) 6. (, ) 7. (, ) 8. (, ) 9. inconsistent 0. dependent. consistent, (, 0). consistent, (, ). inconsistent. dependent 5. consistent, (, ) 6. dependent Algebra Answers 9

59 Skill B Graphing linear inequalities in variables Recall When ou multipl or divide each member of an inequalit b a negative number, ou must reverse the inequalit sign. Eample Graph. The indicates that ou shade above the solid boundar line. Eample Graph 6. 6 Shade the half plane where the -coordinate of each point is less than. Notice that the -coordinate of each point can be an real number. Graph each linear inequalit The broken line indicates that points on the line are not in the solution set. Copright b Holt, Rinehart and Winston. All rights reserved. 6 Reteaching. Algebra

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