Name Class Date. Additional Vocabulary Support

Transcription

1 - Additional Vocabular Support Rate of Change and Slope Concept List negative slope positive slope rate of change rise run slope slope formula slope of horizontal line slope of vertical line Choose the concept from the list above that best represents the item in each bo.... slope formula run positive slope. vertical change horizontal change.. slope or rate of change slope of horizontal line rise 7.. change in the dependent variable change in the independent variable 9. negative slope rate of change or slope slope of vertical line Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

2 - Think About a Plan Rate of Change and Slope Profit John s business made $ in Januar and $ in March. What is the rate of change in his profit for this time period? Understanding the Problem. What is the formula for finding rate of change? change in dependent variable change in independent variable. What are the two changing quantities that affect rate of change in this problem? What are the units of each quantit? profit, time; dollars, months. Will the rate of change be positive or negative? Eplain. positive; profit increase over time Planning the Solution. Which quantit is the dependent variable? Which quantit is the independent variable? Eplain. Profit depends on time, so profit is dependent and time is independent. What is the general equation that represents the rate of change? rate change in change in Getting an Answer. Substitute values into our general equation and simplif. Show our work. r $ per month 7. If ou were to graph this relationship, what would the rate of change be in relation to our graph? the slope Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

3 - Practice Form G Rate of Change and Slope Determine whether each rate of change is constant. If it is, find the rate of change and eplain what it represents.. Hocke Team s. Miles Per Gallon. ffense Gallons Miles Games Goals 7 9 Cars Washed Hours Cars es; ; goals per games plaed Find the slope of each line. es; ; miles per gallon es; ; cars washed per hour... Find the slope of the line that passes through each pair of points. 7. (, ), (, ). (, ), (, ) 9. (, ), (7, ). (, ), (, ). (, ), (, ). (, ), (, ) Find the slope of each line.... undefined Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

4 - Practice (continued) Form G Rate of Change and Slope Without graphing, tell whether the slope of a line that models each situation is positive, negative, zero, or undefined. Then find the slope.. The cost of tickets to the amusement park is $9. for ticket and $7 for tickets. positive; The late fee is $ regardless of the number of das the movie is late.. n the trip, Jerr had his cruise control set at mi/h for hours. zero; zero; 9. The contract states that ever da past the agreed upon completion date the project is not finished, the price is reduced b $. negative; State the independent variable and the dependent variable in each situation. Then find the rate of change for each situation.. Shell delivered newspapers after minutes and papers after minutes. ind: time; dep: number of papers delivered;. papers/min. Two pounds of apples cost $.9. Si pounds cost $.9. ind: weight; dep: cost; $.99/lb. An airplane ascended feet in minutes and feet in minutes. ind: time; dep: height; ft/min Find the slope of the line that passes through each pair of points.. (, ), (, ). (, ), (.,.). (.7,.7), (.,.) undefined. Q, R, Q, R 7. Q, 7 R, Q, 7 9. Writing Eplain wh the slope of a horizontal line is alwas zero. The change in the dependent variable is and a. R. (.,.), (.,.). Writing Describe how to draw a line that passes through the origin and has a slope of. n a coordinate grid, plot (, ). Move down and right and plot the point (, ). Draw a line through the points. Each pair of points lies on a line with the given slope. Find or. 7. (7, ), (, ); slope. (, ), (, ); slope. (, ), (, ); slope. (, 9), (, ); slope 9 Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

5 - Practice Form K Rate of Change and Slope Each rate of change is constant. Find the rate of change and eplain what it represents.. Fences Painted. Hours Fences. fences ; hour ne-third of a fence is painted each 9 hour. Miles Per Hour Hours Miles 7 miles ; The are hour travelling at miles per hour. Find the slope of each line..... Find the slope of the line that passes through each pair of points. 7. (, ), (, ). (, ), (, ) 9. (, ), (, ). (, ), (, ) Find the slope of each line... undefined Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

6 - Practice (continued) Form K Rate of Change and Slope Without graphing, tell whether the slope of a line that models each linear relationship is positive, negative, zero, or undefined. Then find the slope.. The cost of a pair of jeans is $. for pair and $7. for pairs. positive;.. An emploee earns $. after hours and $7. after hours. positive; 9. State the independent variable and the dependent variable in each situation. Then find the rate of change for each situation.. The cost of three gallons of milk is $. and five gallons of milk is $.7. independent: gallons of milk; dependent: cost; rate of change. Jacques filled envelopes in minute and envelopes in minutes. Find the slope of the line that passes through each pair of points. 7. (7, ), (7, ) undefined. (, ), (., 9).9 dollars gallon independent: minutes; dependent: envelopes stuffed; rate of change envelopes minute 9. Q, R, Q, R. Q, R, Q, R undefined. Writing Eplain wh the slope of a vertical line is alwas undefined. The slope is alwas undefined because an two points will have the same -coordinates which means the run will alwas be zero. Since the denominator is zero, the slope is undefined.. Writing Describe how to draw a line that passes through the origin and has a slope of. Answers ma var. Sample: Plot a point at the origin. Since the slope is, move up units and to the right units and plot a point. From this point, go up units and to the right units and plot another point. Draw a line through these points. Each pair of points lies on a line with the given slope. Find or.. (, ), (, ); slope. (9, ), (, ); slope Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

7 - Standardized Test Prep Rate of Change and Slope Multiple Choice For Eercises, choose the correct letter.. What is the slope of the line that passes through the points (, ) and (, )? A. B. C. D. C. A line has slope. Through which two points could this line pass? F. (, ), (7, ) H. (, 7), (, ) G. (, ), (, ) I. (, ), (, ) I. The pair of points (, ) and (, ) lie on a line with slope. What is the value of? B A. B. C. D.. What is the slope of a vertical line? I F. G. H. I. undefined. Shawn needs to read a book that is 7 pages long. The graph shown at the right shows his progress over the first hours of reading. If he continues to read at the same rate, how man hours total will it take for Shawn to read the entire book? B A. hours C. 9 hours B. 7 hours D. hours Short Response. Robi has run the first miles of a race in minutes. She reached the mile point after minutes. Without graphing, is the slope of the line that represents this situation positive, negative, zero, or undefined? What is the slope? positive; mi/min or mi/h [] Both parts answered correctl. [] ne part answered correctl. [] Neither part answered correctl. Pages Read 7 (, 7) 7 (, ) Hours Reading Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 7

8 - Enrichment Rate of Change and Slope A wildlife biologist is observing interactions among animals in a forest. She notices a fo sniffing around, in search of lunch. She also notices a rabbit happil chewing on some grass about meters awa. The fo looks up, notices the rabbit and starts heading towards it at a constant rate. There are a number of different possible outcomes regarding this situation and some are modeled in the graphs shown below. The fo s path is shown in black and the rabbit s is shown in gra. Distance Distance Graph A 7 9 Time Graph C 7 9 Time Distance Distance Graph B 7 9 Time Graph D Time 7 9. Eplain what is happening regarding rate of change and slope of each line. A: Fo and rabbit run at the same rate, m/s, so the slopes are the same. B: Fo runs at m/s and rabbit runs at m/s, so the rabbit s graph is steeper. C: Rabbit remains still, and fo runs at m/s, so the rabbit s graph is horizontal. D: Fo runs at m/s and rabbit runs at about m/s, so the fo s graph is steeper.. In each graph, what will happen regarding the fo and rabbit? How does the slope assure this outcome? In A and B the rabbit gets awa; in C and D the fo catches the rabbit. When the slope of the fo s line is greater than the slope of the rabbit s line, then the lines intersect at a time t S and the fo catches the rabbit. When the slope of the fo s line is less than the slope of the rabbit s line, the lines do not intersect for t S and the rabbit escapes.. Assume the rabbit was a bit confused and wound up running towards the fo at a rate of meters per second. Sketch the graph. Note the slope. Determine the approimate time when the fo would catch the rabbit. Distance (m) Rabbit Fo Time (sec) slope:, s Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

9 - Reteaching Rate of Change and Slope The rate of the vertical change to the horizontal change between two points on a line is called the slope of the line. slope There are two special cases for slopes. A horizontal line has a slope of. A vertical line has an undefined slope. vertical change horizontal change rise run Problem What is the slope of the line? slope vertical change horizontal change rise run The slope of the line is. (, ) (, ) rise run In general, a line that slants upward from left to right has a positive slope. Problem What is the slope of the line? vertical change slope horizontal change rise run The slope of the line is. (, ) rise run (, ) In general, a line that slants downward from left to right has a negative slope. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 9

10 - Reteaching (continued) Rate of Change and Slope Eercises Find the slope of each line.... (, ) (, ) (, ) (, ) (, ) (, ) Suppose one point on a line has the coordinates (, ) and another point on the same line has the coordinates (, ). You can use the following formula to find the slope of the line. slope rise run, where Problem What is the slope of the line through R(, ) and S(, 7)? slope 7 Let 7 and. Let and. Eercises Find the slope of the line that passes through each pair of points.. (, ), (, ). (, ), (7, ). (, ), (, ) 7. (, ), (, ). (, ), (,) 7 9. (, ), (, ) Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

11 - Additional Vocabular Support Direct Variation There are two sets of note cards below that show how Latoa finds a direct variation equation relating and. Suppose varies directl with, and when 9. She also wants to find the value of when 7. The set on the left eplains her thinking. The set on the right shows the steps. Write the thinking and the steps in the correct order. Think Cards Start with the function form of a direct variation. Divide each side b 9 to solve for k. Find the value of when 7. Substitute 9 for and for. Write an equation. Substitute for k in k. Write Cards k(9) (7) k k Think Write First, she should start with the function form of a direct variation. Step k Second, she should substitute 9 for and for. Step k(9) Net, she should divide each side b 9 to solve for k. Step k Then, she should write an equation. Substitute for k in = k. Step Finall, she should find the value of when = 7. Step (7) or Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

12 - Think About a Plan Direct Variation Electricit hm s Law V I R relates the voltage, current, and resistance of a circuit. V is the voltage measured in volts. I is the current measured in amperes. R is the resistance measured in ohms. a. Find the voltage of a circuit with a current of amperes and a resistance of ohms. b. Find the resistance of a circuit with a current of amperes and a voltage of volts. Understanding the Problem. Does hm s Law represent a direct variation? Eplain. es; the ratio of V to R is constant, and the ratio of V to I is constant.. If the formula is rearranged to solve for R or I, is it still a direct variation? Eplain. R and I would be in direct variation with V but not with each other. Planning the Solution. For part (a), does hm s Law need to be rearranged to answer the question? Eplain. If it does, how should the formula be rearranged? no; ou want to find V. For part (b), does hm s Law need to be rearranged to answer the question? Eplain. If it does, how should the formula be rearranged? es; ou want to find R; R V I Getting an Answer. For part (a), substitute the given values into the formula and simplif. volts. For part (b), substitute the given values into the formula and simplif..7 ohms Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

13 - Practice Form G Direct Variation Determine whether each equation represents a direct variation. If it does, find the constant of variation.... es; no es; es; no es; Suppose varies directl with. Write a direct variation equation that relates and. Then find the value of when. 7. when.. when. ; ; 9. when when.. ;.77;... when when. ; Graph each direct variation equation. ;.... An equilateral triangle is a triangle with three equal sides. The perimeter of an equilateral triangle varies directl with the length of one side. What is an equation that relates the perimeter p and length l of a side? What is the graph of the equation? p l Perimeter p Side length 7. The amount a ou fill a tub varies directl with the amount of time t ou fill it. Suppose ou fill gallons in minutes. What is an equation that relates a and t? What is the graph of the equation? a t Amount (sal) Time (min) Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

14 - Practice (continued) Form G Direct Variation For the data in each table, tell whether varies directl with. If it does, write an equation for the direct variation es;. no es; Suppose varies directl with. Write and graph a direct variation equation that relates and.. when.. when.. when. Tell whether the two quantities var directl. Eplain our reasoning.. the total number of miles run and the number of miles ou run per da when training for a race es; the total will be the number of das times the miles run per da.. Jackson s age and Dlan s age no; the difference in their age is constant, but the ratio is not.. a recipe that calls for cups of sugar for each cup of flour es; for ever cup of flour, use cups of sugar. 7. Writing In a direct variation equation, describe how the slope of the graph of the line is related to the constant of variation. The are equal.. Janine gets paid $.7 per hour at her job. Write a direct variation equation where h represents the number of hours she works and d represents the amount of mone she earns. Graph the equation. d.7 h Mone earned ($) Hours h Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

15 - Practice Form K Direct Variation Determine whether each equation represents a direct variation. If it does, find the constant of variation.. no. es;. 7 es;. es; Suppose varies directl with. Write a direct variation equation that relates and. Then find the value of when.. when. when ; ; 7. when ;. 7.9 when..;. Graph each direct variation equation The perimeter of a square varies directl with the length of one side. What is an equation that relates the perimeter p and length l of the side? What is the graph of the equation? p l Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

16 - Practice (continued) Form K Direct Variation For the data in each table, tell whether varies directl with. If it does, write an equation for the direct variation.. es; no Write a direct variation equation that relates and. Then graph the equation.. when 7. when Tell whether the two quantities var directl. Eplain our reasoning. 7. Sara makes $. more per hour than Pasco. no; Sara s hourl wages are Pasco s hourl wages plus. The cafeteria provides three meals per da. es; To find the total number of meals provided, ou multipl the number of das b. 9. Jasmine scores points per game. Yes; To find the total number of points Jasmine has scored, multipl the number of games plaed b.. Reasoning How can ou tell, b eamining the graph, if a line represents a direct variation? If the line passes through the origin, then it is a direct variation. Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

17 Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 7 Name Class Date - Standardized Test Prep Direct Variation Gridded Response Solve each eercise and enter our answer on the grid provided.. Suppose varies directl with and when. What is the value of when?. Suppose varies directl with and when. What is the value of when?. The point (, 9) is included in a direct variation. What is the constant of variation?. The equation of the line on the graph at the right is a direct variation equation. What is the constant of variation?. The distance d a train travels varies directl with the amount of time t that has elapsed since departure. If the train travels 7 miles in 9. hours, how man miles did the train travel after hours? / /

18 - Enrichment Direct Variation A rubber ball is dropped out a window. The height that the ball bounces varies directl with the height of that window. This relation is modeled b the equation., in which represents the height from which the ball is dropped in meters and represents the height the ball bounces in meters.. The ball is dropped from a height of m above the ground. How high does it bounce? m. The ball bounces. cm in the air. From what height was it dropped?.7 cm. List at least three factors other than starting height that could affect the final height of the ball. Answers ma var. Sample: tpe of ball; whether ou drop the ball or throw the ball down; wind resistance. How might each of these factors change the equation? Would the new equation be a direct variation? If so, come up with a new constant of variation that makes sense given the factor(s) ou have identified, and answer Eercises and for our new direct variation. Answers ma var. Sample: Suppose a ball bounced to 7% of its starting height..7, 7. m, 9. cm. Draw the graph of the original ball. Graph at least one of our new functions on the same set of aes. Answers ma var. Sample: Bounce Height Bouncer ball riginal Start Height. What can ou sa regarding the largest possible value for the constant of variation given the situation described? Wh does this limit eist? (What would happen if the limit didn t eist?) The maimum constant for variation is ; if it were more than, the ball would have bounced higher. 7. The original rubber ball is dropped from the same distance, but hits the roof of a building ft off the ground. Is the resulting function still a direct variation? Eplain. It will rebound.( ) feet above that roof, or.( ) feet above the ground. This equation,., is not direct variation. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

19 - Reteaching Direct Variation A direct variation is a relationship that can be represented b a function in the form k where k. The constant of variation for a direct variation k is the coefficient of. The equation k can also be written as k. Problem Does the equation 9 represent a direct variation? If so, find the constant of variation. If the equation represents a direct variation, the equation can be rewritten in the form k. So, solve the equation for to determine whether the equation can be written in this form. 9 9 Subtract from each side. Divide each side b. You cannot write the equation in the form k. So 9 does not represent a direct variation. Problem Does the equation represent a direct variation? If so, find the constant of variation. Again, if the equation represents a direct variation, the equation can be rewritten in the form k. So, solve the equation for to determine whether the equation can be written in this form. Divide each side b. The equation has the form k, so the equation represents a direct variation. The coefficient of is, so the constant of variation is. Eercises Determine whether each equation represents a direct variation. If it does, find the constant of variation.... es; no es;... no es; es; Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 9

20 - Reteaching (continued) Direct Variation To write an equation for direct variation, find the constant of variation k using an ordered pair. Then use the value of k to write an equation. Problem Suppose varies directl with, and when. What direct variation equation relates and? What is the value of when? You are given that and var directl. This means that the relationship between and can be written in the form k, where k is a constant. k Start with the direct variation equation. k() Substitute the given values: for and for. k Divide each side b to solve for k. Write the direct variation equation that relates and b substituting for k in k. The equation relates and. When, ()or. Eercises Suppose varies directl with. Write a direct variation equation that relates and. Then find the value of when. 7. when.. when 9. 7; ; 9. when.. when 9. ; 9;. when.. when. ; ;. Consider the direct variation. a. List three ordered pairs that satisf the equation. Answers ma var. Sample: (,), (, ), (. ) b. Plot our three ordered pairs from part (a) on a coordinate grid. Sample: c. Complete the graph of on the grid. (, ) Sample: (, ) (, ) Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

21 - Additional Vocabular Support Slope - Intercept Form Complete the vocabular chart b filling in the missing information. Word or Word Phrase linear equation Definition An equation that models a linear function Picture or Eample linear parent function or f (). parent function slope-intercept form. A famil of functions is a group of functions with common characteristics. A parent function is the simplest function with these characteristics. An equation of the form m b, where m is the slope and b is the -intercept of,, and. m b -intercept. -coordinate of a point where the graph crosses the -ais -intercept Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

22 - Think About a Plan Slope-Intercept Form Hobbies Suppose ou are doing a -piece puzzle. You have alread placed 7 pieces. Ever minute ou place more pieces. a. Write an equation in slope-intercept form to model the number of pieces placed. Graph the equation. b. After more minutes, how man pieces will ou have placed? Understanding the Problem. Is this relationship linear? How do ou know? es; the rate of change ( pieces/min) is constant Planning the Solution. How man pieces have ou alread placed? What does this represent in the slope-intercept form? 7; the -intercept. What two quantities are used to find the rate of change or slope? What is the slope of this relationship? number of pieces placed and change in time; Getting an Answer. Use our answers in Steps and to write an equation in slope-intercept form to model the number of pieces placed. 7. Graph the equation on a coordinate grid.. How man pieces will ou have placed after more minutes? 7 pieces Time (min) Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

23 - Practice Form G Slope-Intercept Form Find the slope and -intercept of the graph of each equation.... ;... ; ; ; 9...7;. ;. ; 9 Write an equation of a line with the given slope m and -intercept b.. m, b. m, b. m, b. m., b. m, b. m, b. Write an equation in slope-intercept form of each line Write an equation in slope-intercept form of the line that passes through the given points. 9. (, ) and (, ). (, ) and (, ). (, ) and (, ). (7, ) and (, ). (, ) and (, ). (, ) and (, ) ;. (, ) and (, ). (, ) and (, ) 7. (, ) and (, ) ; 7 Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

24 - Practice (continued) Form G Slope-Intercept Form Graph each equation Hudson is alread miles awa from home on his drive back to college. He is driving mi/h. Write an equation that models the total distance d travelled after h hours. What is the graph of the equation? d h d h. When Phil started his new job, he owed the compan $ for his uniforms. He is earning $ per hour. The cost of his uniforms is withheld from his earnings. Write an equation that models the total mone he has m after h hours of work. What is the graph of the equation? m h m n Find the slope and the -intercept of the graph of each equation m ; b m ; b m ; b 9. ( 9).. a m ; b m ; b m ; b a Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

25 - Practice Form K Slope-Intercept Form Find the slope and -intercept of the graph of each equation.. 7 m ; b 7. m ; b. 7 m 7; b... m.7; b.. 9 m ; b 9. 7 m ; b 7 Write an equation of a line with the given slope m and -intercept b. 7. m, b. m, b 9. m., b m, b Write an equation in slope-intercept form of each line... Write an equation in slope-intercept form of the line that passes through the given points.. (, ) and (, ). (, 9) and (, ) 7. (, ) and (, ). (, ) and (, 7) 9 Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

26 - Practice (continued) Form K Slope-Intercept Form Graph each equation A car is traveling at mi/h. Write an equation that models the total distance d traveled after h hours. What is the graph of the equation? d h Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

27 - Standardized Test Prep Slope-Intercept Form Multiple Choice For Eercises, choose the correct letter.. What is an equation of the line shown in the graph at the right? C A. C. B. D.. What is an equation of the line that has slope and passes through the point (, )? G F. G. H. I.. What is an equation of the line that passes through the points (, ) and (, )? D A. 7 B. C. 7 D. 7. The data shown in the table is linear. Which equation models the data? F. H. 9 G. I. F 7. Karissa earns $ per week plus $ per item she sells. Which equation models the relationship between her pa p per week and the number of items n she sells? B A. p n C. n p B. p n D. n p Short Response. What is an equation of the line that passes through (, ) and has slope? What is the graph of the equation? [] Both parts answered correctl. [] ne part answered correctl. [] Neither part answered correctl. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 7

28 - Enrichment Slope-Intercept Form Four sisters, Jenn, Julie, Jolie, and Jada, are saving mone for summer vacation. Their mom gets them started b giving the older sisters, Jenn and Jada, $ each and the ounger sisters $ each. Each sister will put part of the pa from their after school job in the account each week. Jenn and Jolie will each deposit $ per week. Julie and Jada will each deposit $ per week.. Make a table for the total savings for each sister. Week Jenn Julie Jolie Jada Graph each sister s savings on the same set of aes. Week. Find the slope and -intercept of the graph of each sister. Jenn:, ; Jolie:, ; Julie:, ; Jada:,. Model each sister s savings plan using an equation in the form m b. Jenn: ; Jolie: ; Julie: ; Jada:. How much will each sister have after weeks? After 7 weeks? Jenn: 9, ; Jolie: 9, 7; Julie:, ; Jada:,. Will Jolie ever have more than Jenn? Eplain. no; the slopes are the same and Jolie starts with less. 7. Will Jolie ever have more than Jada? Eplain. Yes; Jolie saves more than Jada each week and b week, Jolie has more. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

29 - Reteaching Slope-Intercept Form The slope-intercept form of a linear equation is m b. In this equation, m is the slope and b is the -intercept. Problem What are the slope and -intercept of the graph of? The equation is solved for, but it is easier to determine the -intercept if the right side is written as a sum instead of a difference. () Write the subtraction as addition. The slope is and the -intercept is. Problem What is an equation for the line with slope and -intercept 9? When the slope and -intercept are given, substitute the values into the slopeintercept form of a linear equation. m b 9 Substitute for m and 9 for b. Problem What is an equation in slope-intercept form for the line that passes through the points (, ) and (, )? Substitute the two given points into the slope formula to find the slope of the line. m () Then substitute the slope and the coordinates of one of the points into the slope-intercept form to find b. m b Use slope-intercept form. () b Substitute for m, for, and for. b Solve for b. Substitute the slope and -intercept into the slope-intercept form. m b Use slope-intercept form. () Substitute for m and for b. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 9

30 - Reteaching (continued) Slope-Intercept Form Eercises Find the slope and -intercept of the graph of each equation m ; b 7 m ; b m ; b... m ; b m ; b m ; b Write an equation for the line with the given slope m and -intercept b. 7. m, b 7. m, b 9. m, b 7. m, b. m, b. m 7, b 7 Write an equation in slope-intercept form for the line that passes through the given points.. (, ) and (, ). (,) and (, ). (, ) and (,). (,) and (,) 7. (, ) and (, ). (,) and (,) 9. Consider the equation. a. What is the -intercept of the graph of the equation? b. Graph the -intercept. (, ) c. What is the slope of the graph of the equation? d. Use the point ou graphed in part (b) and the slope to find another point on the graph of the equation. Answers ma var. Sample: (, ) e. Graph the equation. See graph in part (b). Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

31 - Additional Vocabular Support Point - Slope Form P r o b l e m A line passes through (, ) and has slope. What is an equation of the line? Justif our steps. m( ) Use the point-slope form. f ()g Substitute (, ) for (, ) and for m. ( ) Simplif inside grouping smbols. Eercises A line passes through (, ) and has slope. What is an equation of the line? Justif our steps. m( ) Use the point-slope form. f ()g _ ( ) Substitute (, ) for (, ) and for m. Simplif inside grouping smbols. A line passes through (, ) and has slope. What is an equation of the line? Justif our steps. Use the point slope form. m( ) () ( ) Substitute (, ) for (, ) and for m. ( ) Simplif inside grouping smbols. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

32 - Think About a Plan Point-Slope Form Boiling Point The relationship between altitude and the boiling point of water is linear. When the altitude is ft, water boils at 97. F. When the altitude is ft, water boils at.9 F. Write an equation giving the boiling point b of water (in degrees Fahrenheit) in terms of the altitude a (in feet). What is the boiling point of water at ft? Understanding the Problem. What are ou given? You know that the relationship between altitude and boiling point is linear and are given points on that line.. In general, how can this information be used to answer the question? Use the two points to find an equation for the boiling point as a function of altitude and then substitute for the altitude to find the boiling point. Planning the Solution. What is the slope formula? m. Substitute given values into the slope formula and simplif. Show our work... Which point can be used to write the point-slope form of the equation? either (, 97.) or (,.9). What strateg can ou use to solve this problem? substitution 7. How can ou determine the boiling point of water at ft? Substitute for a and solve for b. Getting an Answer. Write an equation giving the boiling point b of water (in degrees Fahrenheit) in terms of the altitude a. Answers ma var. Sample: b.a 9. What is the boiling point of water at ft? Show our work. 7. F Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

33 - Practice Form G Point-Slope Form Write an equation of the line in point-slope form through the given point and with the given slope m.. (, ); m. (, ); m. (, ); m. (, ); m Graph each equation.. ( ). ( ) 7. ( ) Write an equation in point-slope form for each line (, ) (, ) (, ) (, ) (, 9) (, ) ( ) or ( ) ( ) or ( ) ( ) or 9 ( ) Write an equation in point-slope form of the line through the given points. Then write the equation in slope-intercept form.. (, ), (, ). (, ), (, ). (, ), (, ) ( ); ( ); ( ); 7. pen-ended Write an equation of a line that has a slope of in each form. a. point-slope form b. slope-intercept form Answers ma var. Sample: ( ) Answers ma var. Sample: Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

34 - Practice (continued) Form G Point-Slope Form Model the data in each table with a linear equation in slope-intercept form. What do the slope and -intercept represent?. Time Washing Cars washed. (hr) Time Fling (hr) Distance from Airport (mi) 7 9 ; cars washed per hour, cars washed at start ; speed in mi/h, distance from airport at start Graph the line that passes through the given point and has the given slope m. 7. (, ); m. (, ), m 9. (, ); m. Writing Describe what ou know about the graph of a line represented b the equation ( ). The slope is and it passes through the point (, ).. Writing Describe how ou would use the point-slope form to write the equation of a line that passes through the points (, ) and (, ) in slope-intercept form. First find the slope: 9. Then use one point in the point-slope form of the equation and simplif: 9 7. Writing Describe how linear data given in a table can help ou write an equation of a line in slope-intercept form. Find the slope using m for an pair of rows in the table. Then substitute a point (a, b) from an row into b m( a) and simplif. 7. A sign sas that tickets cost $. and that 7 tickets cost $.. Write an equation in point-slope form that represents the cost of tickets. What is the graph of the equation?. 7.( ) Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. Cost ($) 7 Tickets

35 - Practice Form K Point-Slope Form Write an equation in point-slope form of the line that passes through the given point and has the given slope.. (, ); m. (, ); m. (, 7); m ( ) ( ) 7 ( ) Graph each equation.. ( ). ( ). ( ) Graph the line that passes through the given point and has the given slope m. 7. (, ); m. (, ); m 9. (, ); m. pen-ended Write an equation in each of the following forms that has a slope of. Answers ma var. Sample: a. point-slope form b. slope-intercept form ( ). Writing Describe what ou know about the graph of a line represented b the equation ( ). The line passes through the point (, ) and has a slope of. Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

36 - Practice (continued) Form K Point-Slope Form Model the data in each table with a linear equation in slope-intercept form. What do the slope and -intercept represent?. Hours Mone. Worked Earned ($) Time Running (min) Distance (mi).; The slope represents how ; The slope represents the much mone ou are paid per hour and number of miles ou run per hour. The the -intercept represents how much -intercept represents the number of mone ou were paid for zero hours. miles ou run at the time ou start. Write an equation in point-slope form of each line... ( ) ( ) Write an equation in point-slope form of the line that passes through the given points. Then write the equation in slope-intercept form.. (, ), (, ) 7. (, ), (, ) ( );. (, ), (, ) 9. (, ), (, 7) ( ); ( ); ( ); 9. Writing Describe how ou would use the point-slope form to write the equation of a line that passes through the points (, ) and (, ) in slope-intercept form. First, use the slope formula to find the slope: m. Then, select one of the points to write the point-slope form of the equation: ( ). Simplif the parentheses b multipling. Add to each side.. A restaurant s goal is to serve customers in hours and 9 customers in hours. Write an equation in point-slope form that represents the number of customers served per hour. What is the graph of the equation? 7( ) Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. Customer Served Hours

37 - Standardized Test Prep Point-Slope Form Multiple Choice For Eercises, choose the correct letter.. Which equation is equivalent to ( )? C A. C. B. D.. Which point is located on the line represented b the equation ( )? H F. (, ) G. (, ) H. (, ) I. (, ). Which equation represents the line that passes through the points (, )and (, 9)? D A. ( 9) C. ( ) B. ( 9) D. ( ). Which equation represents the line shown in the graph? F. G. H. ( ) I. ( ). The population of a cit increases b people each ear. In, the population is projected to be, people. What is an equation that gives the cit s population p (in thousands of people) ears after? B A. p C. p ( ) B. p ( ) D. p Short Response H. The table shows the cost of a large cheese pizza with additional toppings on it. a. What is an equation in point-slope form that represents the relationship between the number of toppings and the cost of the pizza?. b. What is the graph of the equation? Toppings Cost ($)..7. [] Both parts answered correctl. [] ne part answered correctl. [] Neither part answered correctl. Cost ($) Toppings Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 7

38 - Enrichment Point-Slope Form Point-slope form can be used to quickl write the equation of a line when given one point on the line and the slope of the line. If two points are given, there is an additional step of determining the slope between those two points. There are special cases for which the equation of a line can be written quickl.. Write an equation in point-slope form for the line with slope m that has -intercept (a, ). m( a). Write an equation in point-slope form for the line with slope m that has -intercept (, b). b m. Use our result from Eercise to write an equation in point-slope form for the line with slope that has -intercept (, ). ( ). Use our result from Eercise to write an equation in point-slope form for the line with slope that has -intercept (, ).. A line has -intercept (a, ) and -intercept (, b). a. Write an equation in point-slope form for the line using the -intercept. b a ( a) b. Write an equation in point-slope form for the line using the -intercept. b b a? c. Show that the equations ou wrote in parts (a) and (b) are equivalent. b a ( a) b a? b a (a) b a? b b b a? Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

39 - Reteaching Point-Slope Form The point-slope form of a nonvertical linear equation is m( ). In this equation, m is the slope and (, ) is a point on the graph of the equation. Problem A line passes through (, ) and has a slope. What is an equation for this line in point-slope form? m( ) Use point-slope form. () ( ) Substitute (, ) for (, ) and for m. ( ) Simplif. Problem A line passes through (, ) and (, 9). What is an equation for this line in point-slope form? What is an equation for this line in slope-intercept form? First use the two given points to find the slope. m 9 Use the slope and one point to write an equation in point-slope form. m( ) Use point-slope form ( ) Substitute (, ) for (, ) and for m. Distributive Propert Add to each side. An equation in point-slope form is ( ). An equation in slopeintercept form is. Eercises Write an equation for the line through the given point and with the given slope m.. (, ); m. (7, ); m. (, ); m ( ) ( 7) ( ) Write an equation in point-slope form of the line through the given points. Then write the equation in slope-intercept form.. (, ) and (, 7). (, ) and (, ). (, ) and (, ) ( ) or 7 ( ); ( ) or ( ); ( ) or ( ); Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 9

40 - Reteaching (continued) Point-Slope Form You can use the point-slope form of an equation to help graph the equation. The point given b the point-slope form provides a place to start on the graph. Plot a point there. Then use the slope from the point-slope form to locate another point in either direction. Then draw a line through the points ou have plotted. Problem What is the graph of the equation ( )? The equation is in point-slope form, so the line passes through (, ) and has a slope of. Plot the point (, ). Use the slope,. From (, ), go up unit and then right units. Draw a point. Draw a line through the two points. (, ) (, ) rise run Because, ou can start at (, ) and go down unit and to the left units to locate a third point on the line. Eercises Graph each equation. 7. ( ). ( ) 9. ( ) Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

41 - Additional Vocabular Support Standard Form David is going to a fun center and it costs $ to ride bumper boats and $ to ride go-karts. He has $ to spend. He wants to write and graph an equation to find three combinations of rides he can ride. He wrote steps to solve the problem on note cards. Write the equation. Graph the equation. Use the graph to find a total of three combinations. ne possibilit (, ), (, ), and (, ). Define the variables. Find the intercepts. The intercepts give David two combinations of rides. However, onl points with integer coordinates are solutions. Use the note cards to write the steps in order.. First, define the variables.. Second, write the equation.. Third, graph the equation.. Net, find the intercepts. The intercepts give David two combinations of rides. However, onl points with integer coordinates are solutions.. Then, use the graph to find a total of three combinations.. Finall, one possibilit = (, ), (, ), and (, ). Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

42 - Think About a Plan Standard Form Sports A football team scores points. All of the points come from field goals worth points and touchdowns (with successful etra-point attempts) worth 7 points. Write and graph a linear equation that represents this situation. List ever possible combination of field goals and touchdowns the team could have scored. Understanding the Problem. What are ou given? the total score, the value of a field goal,, and the value of a touchdown, 7.. How can touchdowns and field goals be represented? How can these be written as terms to represent the point value of each? t, f ; 7t, f Planning the Solution. What is the equation in standard form that models the situation? 7t f ; 7. How can ou find the -intercept? Solve for. 7 m 9. How can ou find the -intercept? Solve for. m. How can ou use the intercepts to graph the line? Plot (, 9) and (, ) and connect the points. Getting an Answer 7. Graph the relation on a coordinate grid.. Use the graph to determine and list all of the combinations. (, 9); (7, ); (, ); (, ) Touchdowns Field goals Fieldgoals 7 Touchdowns 9 Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

43 - Practice Form G Standard Form Find the - and -intercepts of the graph of each equation.. 7 7; ;. ;.. ;. 7 ; ; 7 Draw a line with the given intercepts. 7. -intercept:. -intercept: 9. -intercept: -intercept: -intercept: -intercept: Graph each equation using - and -intercepts.... For each equation, tell whether its graph is a horizontal or a vertical line horizontal vertical horizontal vertical Graph each equation Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

44 - Practice (continued) Form G Standard Form Write each equation in standard form using integers... ( ). ( ) ( ) You have onl nickels and dimes in our pigg bank. When ou ran the coins through a change counter, it indicated ou have 9 cents. Write and graph an equation that represents this situation. What are three combinations of nickels and dimes ou could have? n n d 9 Nickels Dimes d Answers ma var. Sample: nickels and dimes; nickels and 9 dimes; nickels and 7 dimes For each graph, find the - and -intercepts. Then write an equation in standard form using integers.. 9. ; ; ; ; Find the - and -intercepts of the line that passes through the given points.. (, ), (, ). (, ), (, 7). (, ), (, ) ; ; ; Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

45 - Practice Form K Standard Form Find the - and -intercepts of the graph of each equation.. -intercept: ;. -intercept: ; -intercept: -intercept:. -intercept: ;. -intercept: ; -intercept: -intercept: Draw a line with the given intercepts.. -intercept:. -intercept: -intercept: -intercept: Graph each equation using - and -intercepts For each equation, tell whether its graph is a horizontal or a vertical line.. vertical. horizontal Graph each equation... Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

46 - Practice (continued) Form K Standard Form 7. Writing Eplain how ( ) can be rewritten into standard form. Then show our work in transforming the equation to standard form. Simplif the parentheses and move and to the right side of the equation and the constants to the right side so that it is in the form A B C. Steps ma var; Write each equation in standard form using integers.. 9. ( ). ( ).. You work two jobs. At the first job, ou earn $ per hour. At the second job, ou earn $ per hour. You earned $ last week. Write and graph an equation that represents this situation. What are three combinations of hours ou could have worked at each job? Answers ma var. Sample: hours at st job, hours at nd job; hours at st job, hours at nd job; hours at st job, hours at nd job.. Mike was the kicker for the football team. He scored points during the season kicking field goals ( points) and etra points ( point). Write and graph an equation that represents this situation. What are three combinations of field goals and etra points he could have made? Answers ma var. Sample: field goals, etra points; field goals, etra points; field goals, 7 etra points. For each graph, find the - and -intercepts. Then write an equation in standard form using integers. Second Job ($/hr) Etra points (-pt) First Job ($/hr) Field Goals (-pts).. -intercept: ; -intercept: ; -intercept: ; -intercept: ; Find the - and -intercepts of the line that passes through the given points.. (, ), (, ) 7. (, ), (, ) -intercept: ; -intercept: -intercept: ; -intercept: Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

47 - Standardized Test Prep Standard Form Multiple Choice For Eercises, choose the correct letter.. What is written in standard form using integers? C A. B. C. D.. Which of the following is an equation of a vertical line? G F. G. H. 9 I.. What are the - and -intercepts of the graph of 7 A. -intercept: 7 C. -intercept: -intercept: -intercept:. B. -intercept: 7 D. -intercept: -intercept: -intercept:. D. Cherl is planning to spend $7 on a Christmas gift for her father. He needs new socks and ties. A store has socks s and ties t on sale for $ and $, respectivel. Which equation models this situation? F F. s t 7 H. s t 7 G. s t 7 I. t s Etended Response. The grocer store is selling eggs for $ per dozen and bacon for $ per pound. You plan to spend $ in food for the benefit breakfast. Write and graph an equation that represents this situation. What are three combinations of dozens of eggs and pounds of bacon ou can purchase? e b ; b Answers ma var. Sample: dozen eggs and lb bacon; dozen eggs and lb bacon; dozen eggs and lb bacon; Bacon Eggs e [] All parts answered correctl. [] ne or two parts answered correctl. [] No parts answered correctl. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 7

48 - Enrichment Standard Form Each of the following equations is in standard form. Solve for in terms of and use the resulting equation to determine the slope of the graph of the equation... ; ;.. ; ; Net, eplore the relationship between the standard form of the equation and the concept of slope. 7. Solve the equation A B C for. A B C B. What is the slope of the graph of the equation in terms of A and B? A B 9. Use the answer to Eercise to find the values for A and B and the slope of the graph of the equation. A, B, slope Use the general standard form, A B C, to answer the following questions.. For what values of A, B, and C is the slope positive? negative? The slope is positive if A is positive and B is negative, or if A is negative and B is positive. If both have the same sign, the slope is negative. C has no effect.. For what values of A, B, and C is the absolute value of the slope greater than, less than, or equal to? If u A u S u B u, then u m u S. If u A u R u B u, then u m u R. If u A u u B u, then u m u. C has no effect.. What is the general epression for the -intercept? C B ; ; Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

49 - Reteaching Standard Form The standard form of a linear equation is A B C, where A, B, and C are real numbers, and A and B are not both zero. You can easil determine the - and -intercepts of the graph from this form of the equation. Each intercept occurs when one coordinate is. When substituting for either of or, one of the terms on the left side of the standard form equation disappears. This leaves a linear equation in one variable, with a variable term on the left and a constant on the right. Determining the other coordinate of the intercept requires onl multiplication or division. Problem What are the - and -intercepts of the graph of 9? First find the -intercept. 9 9() Substitute for. Simplif. Divide each side b. Then find the -intercept. 9 () 9 Substitute for. 9 Simplif. Divide each side b 9. The -intercept is and the -intercept is. Eercises Find the - and -intercepts of the graph of each equation ;... ; ; ; ; ; ; 7, ;. Write an equation in standard form with an -intercept of and a -intercept of. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 9

50 - Reteaching (continued) Standard Form You can graph linear equations in standard form b plotting the - and -intercepts. Problem What is the graph of? Find the intercepts. () () The -intercept is, and the -intercept is. Plot the - and -intercepts and draw a line through the points. (, ) (, ) Eercises Graph each equation using - and -intercepts Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

51 - Additional Vocabular Support Parallel and Perpendicular Lines Use the list below to complete the Venn diagram. A vertical line and a horizontal line Lines that intersect to form right angles Two nonvertical lines that have a product of for their slopes Equation can be written in slope-intercept form Nonvertical lines that have the same slope and different -intercepts You can determine the relationship between two lines b comparing their slopes and -intercepts. Lines in the same plane that never intersect Slopes are opposite reciprocals. Vertical lines that have different -intercepts Parallel Lines You Perpendicular Lines Lines in the same plane that never intersect; Nonvertical lines that have the same slope and different -intercepts; Vertical lines that have different -intercepts can determine the relationship between two lines b comparing their slopes and -intercepts. Equation can be written in slopeintercept form Lines that intersect to form right angles; Two nonvertical lines that have a product of for their slopes; A vertical line and a horizontal line; Slopes are opposite reciprocals. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

52 - Think About a Plan Parallel and Perpendicular Lines Agriculture Two farmers use combines to harvest corn from their fields. ne farmer has acres of corn, and the other has acres of corn. Each farmer s combine can harvest acres per da. Write two equations for the number of acres of corn not harvested after das. Are the graphs of the equations parallel, perpendicular, or neither? How do ou know? Understanding the Problem. What is the difference between the two farms? What is the same? The number of acres of corn are different; the number of acres a combine can harvest each da is the same.. How can ou determine if the graphs of two equations are parallel, perpendicular, or neither? The are parallel if the have the same slope, perpendicular if the slopes are opposite reciprocals. Planning the Solution. What is an algebraic epression that represents the amount of corn each farmer can harvest per da?. Write an equation representing the number of acres of corn not harvested after das on the farm with acres.. Write an equation representing the number of acres of corn not harvested after das on the farm with acres. Getting an Answer. Write the equations in Steps and in slope-intercept form. ; 7. What are the slopes of the equations? both are. Are the graphs of the equations parallel, perpendicular, or neither? Eplain. parallel, because the have the same slope Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

53 - Practice Form G Parallel and Perpendicular Lines Write an equation of the line that passes through the given point and is parallel to the graph of the given equation.. (, );. (, ); 7 7. (, );. (, ); 9. (, ); 7. (, ); Determine whether the graphs of the given equations are parallel, perpendicular, or neither. Eplain parallel; same slope perpendicular; one is horizontal, one is vertical.. neither; different slopes that aren t opposite reciprocals perpendicular; slopes are opposite reciprocals ( ) parallel; same slope perpendicular; slopes are opposite reciprocals Determine whether each statement is alwas, sometimes, or never true. Eplain.. Two lines with different slopes are perpendicular. sometimes; when the different slopes are negative inverses the are perpendicular. The slopes of vertical lines and horizontal lines are opposite reciprocals. never; although vertical and horizontal lines are alwas perpendicular, the slopes are undefined and, which are not negative reciprocals.. A vertical line is perpendicular to the -ais. alwas; vertical lines are perpendicular to all horizontal lines, including the -ais. Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

54 - Practice (continued) Form G Parallel and Perpendicular Lines Write an equation of the line that passes through the given point and is perpendicular to the graph of the given equation.. (, ); 7. (, 7);. (, ); 9. (9, ); 9. (, ); ( ). (, );. pen-ended Write the equations of three lines whose graphs are parallel to one another.. pen-ended Write the equations of two lines whose graphs are perpendicular to one another. Answers ma var. Sample:,. What is the slope of a line that is parallel to the -ais?. What is the slope of a line that is perpendicular to the -ais?. What is the slope of a line that is parallel to the -ais? Answers ma var. Sample:,, undefined undefined 7. What is the slope of a line that is perpendicular to the -ais?. n a map, Sandusk St. passes through coordinates (, ) and (, ). Pennslvania Ave. intersects Sandusk St. and passes through coordinates (, ) and (, ). Are these streets perpendicular? Eplain. Sandusk Street has slope 9 ; Pennslvania Ave. has slope ; The slopes are not opposite reciprocals, so the streets are not perpendicular. 9. Writing Eplain how ou can determine if the graphs of two lines are parallel or perpendicular without graphing the lines. Find the slopes. If the are equal, the lines are parallel. If the are negative inverses, the graphs are perpendicular. Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

55 - Practice Form K Parallel and Perpendicular Lines Write an equation in slope-intercept form of the line that passes through the given point and is parallel to the graph of the given equation.. (, );. (, );. (, ); 7. (, ); Determine whether the graphs of the given equations are parallel, perpendicular, or neither. Eplain.. 9 parallel. perpendicular 7. perpendicular. 7 7 neither Determine whether each statement is alwas, sometimes, or never true. Eplain. 9. Two lines with different slopes are parallel. never. Two lines with the same -intercept are perpendicular. sometimes. Two lines whose slopes are opposites of each other are perpendicular. sometimes Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

56 - Practice (continued) Form K Parallel and Perpendicular Lines Write an equation of the line that passes through the given point and is perpendicular to the graph of the given equation.. (, );. (, 7);. (, );. (, ); 9. pen-ended Write the equations of three lines whose graphs are parallel to. Answers ma var. Sample:,, 7. pen-ended Write the equations of two lines whose graphs are perpendicular to 9. Answers ma var. Sample:,. What is the slope of a line that is parallel to? 9. What is the slope of a line that is perpendicular to? undefined. What is the slope of a line that is parallel to? undefined. What is the slope of a line that is perpendicular to?. n a map, Center St. passes through coordinates (, ) and (, 7). Merrie Rd. intersects Center St. and passes through coordinates (, ) and (, ). Are these streets perpendicular? Eplain. es; the slopes are and Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

57 - Standardized Test Prep Parallel and Perpendicular Lines Multiple Choice For Eercises, choose the correct letter.. Which equation has a graph parallel to the graph of 9? B A. B. C. D.. Which equation has a graph perpendicular to the graph of 7? F F. 7 G. H. I. 9. Which equation is the equation of a line that passes through (, ) and is perpendicular to 7? C A. B. 7 C. D. * ). Which of the following * ) coordinates for P will make MN perpendicular to P in the diagram at the right? H F. (, ) H. (, ) G. (, ) I. (, ). Segment XY represents the path of an airplane that passes through the coordinates (, ) and (, ). What is the slope of a line that represents the path of another airplane that is traveling parallel to the first airplane? D A. C. B. D. N(, 7) (, ) M(, ) Short Response. A cit designer is drawing the road map for a new housing development. Palm St. runs through the coordinates (, ) and (, ) on the map. Pepperdine St. is going to run perpendicular to Palm St. The coordinates of Pepperdine St. are (, 7) and (7, ). What is the value of? What is the equation for the line representing Pepperdine St. in slope-intercept form? ; 9 [] Both parts answered correctl. [] ne part answered correctl. [] Neither part answered correctl. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 7

58 - Enrichment Parallel and Perpendicular Lines You have seen that when two lines have the same slope, the are parallel. You also have seen how to quickl recognize slope in equations of the form m b. So, ou should be able to quickl recognize that the graphs of 7 and are parallel. Another wa to think about two lines being parallel is that the functions describing the two lines have no common solution. There are three possibilities for the number of points that the graphs of two equations have in common. If the equations are identical or equivalent (can be rewritten to be identical), the graphs are the same, and the lines will have infinitel man points in common. If the equations are not equivalent and the graphs do not have the same slope, then the graphs will have eactl one point in common. nl if the lines are parallel (same slope, but different -intercept) will the lines have no points in common.. Show algebraicall that the graphs of 7 and are parallel, that is, show that the do not have a common solution no; 7 u. Show algebraicall that the graphs of and are parallel. no; u Consider how to quickl determine whether graphs of equations written in standard form are parallel.. Graph.. Find a point that is not on the graph. Draw a line parallel to that goes through our point. Answers ma var. Sample: (, ). Write an equation of our line in standard form. Answers ma var. Sample:. Repeat Eercises and. Answers ma var. Sample: (, ); (, ) 7 (, ) 7. Graph on the same coordinate plane. See graph in Eercise.. Develop a conjecture based on our work on Eercises to 7 that, if shown to be correct, would allow ou to quickl determine whether two equations written in standard form produce parallel lines. For two equations A B C and a b c, if A ka and B kb, the lines are parallel. If C also equals kc, then the equations represent the same line. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

59 - Reteaching Parallel and Perpendicular Lines Nonvertical lines are parallel if the have the same slope and different -intercepts. The graphs of and are parallel because the have the same slope,, but different -intercepts, and. Problem What is an equation in slope-intercept form of the line that passes through (, 7) and is parallel to the graph of? The slope of is. Because the desired equation is for a line parallel to a line with slope, the slope of the parallel line must also be. Use the slope and the given point in the point-slope form of a linear equation and then solve for to write the equation in slope-intercept form. m( ) Start with the point-slope form. 7 ( ) Substitute (, 7) for (, ) and for m. 7 Distributive Propert Add 7 to each side. The graph of passes through (, 7) and is parallel to the graph of. Eercises. Writing Are the graphs of and parallel? Eplain how ou know. No, because the slopes and are not equal. Write an equation in slope-intercept form of the line that passes through the given point and is parallel to the graph of the given equation.. (, );. (, ); 7. (, ); (, );. (, 7); 7. (, ); Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 9

60 - Reteaching (continued) Parallel and Perpendicular Lines Two lines that are neither horizontal nor vertical are perpendicular if the product of their slopes is. The graphs of and are perpendicular because Q R. Problem What is an equation in slope-intercept form of the line that passes through (, ) and is perpendicular to the graph of? The slope of is. Since (), the slope of the line perpendicular to the given line is. Use this slope and the given point to write an equation in point-slope form. Then solve for to write the equation in slope-intercept form. m( ) Start with the point-slope form. ( ) Substitute (, ) for (, ) and for m. Distributive Propert 9 Add to each side. The graph of 9 passes through (, ) and is perpendicular to the graph of. Eercises. Writing Are the graphs of and parallel, perpendicular, or neither? Eplain how ou know. perpendicular; the slopes and have a product of Write an equation in slope-intercept form of the line that passes through the given point and is perpendicular to the graph of the given equation. 9. (, );. (, );. (, ); 7. (, );. (, );. (7, ); 7 7 Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

61 -7 Additional Vocabular Support Scatter Plots and Trend Lines Concept List causal relationship correlation coefficient etrapolation interpolation line of best fit negative correlation no correlation positive correlation trend line Choose the concept from the list above that best represents the item in each bo.. LinReg (a b).. Practice Time (hours) Grade in Band Class C B A line of best fit positive correlation.. r r r. Strong Negative Correlation No Correlation Strong Positive Correlation causal relationship trend line correlation coefficient no correlation Estimate the value for when.. Estimate the value for when.. etrapolation interpolation negative correlation Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

62 -7 Think About a Plan Scatter Plots and Trend Lines U.S. Population Use the data below. Estimated Population of the United States (thousands) Year Male,,79,9,97,7,97 7, Female,7,7, 7, 9,7,, SURCE: U.S. Census Bureau. a. Make a scatter plot of the data pairs (male population, female population). b. Draw a trend line and write its equation. c. Use our equation to predict the U.S. female population if the U.S. male population increases to,,. d. Reasoning Consider a scatter plot of the data pairs (ear, male population). Would it be reasonable to use this scatter plot to predict the U.S. male population in? Eplain our reasoning.. Make a scatter plot of the data pairs using the male population for the -coordinates and the female population for the -coordinates for each ear. See points at right for scatter plot.. Draw the trend line onto the scatter plot. See graph in Eercise.. How do ou determine the equation of a trend line? What is the equation of this trend line? Show our work. Choose points on the line. Find the slope and use point-slope form; Answers ma var. Sample:.9 Females (Millions) Males (Millions). Substitute for to predict the female population. Answers var depending on equation used:,, females. Make a scatter plot of the data pairs (ear, male population).. Would it be reasonable to use this scatter plot to predict the U.S. male population in? Eplain our reasoning. The data go from to ; projecting to is a ver large etrapolation. Males (millions) Years after Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

63 -7 Practice Form G Scatter Plots and Trend Lines For each table, make a scatter plot of the data. Describe the tpe of correlation the scatter plot shows... Test Scores Tickets Sold Test Score Adult Tickets Stud Time (min) 9 Children Tickets 7 Test score Stud Time (min) Positive Use the table below and a graphing calculator for Eercises through. Child Tickets 7 Adult Tickets Positive Florida Resident Population Year Population (in thousands) ,9,,9,,9 7,7 7,7,9 SURCE: U.S. Census Bureau.. Make a scatter plot of the data pairs (ears since 9, population). See points in graph below.. Draw the line of best fit for the data. See graph in Eercise.. Write an equation for the trend line. 97. According to the data, what will the estimated resident population in Florida be in?, thousands, or,, people Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

64 -7 Practice (continued) Form G Scatter Plots and Trend Lines Use the table below and a graphing calculator for Eercises 7 through. Total Bo ffice Gross Year Gross Revenue (in million $) SURCE: 7. Make a scatter plot of the data pairs (ears since 999, revenue). See points in graph. Draw the line of best fit for the data. See graph in Eercise Write an equation for the line of best fit According to the data, what will the estimated gross revenue be in? about $,7,, In each situation, tell whether a correlation is likel. If it is, tell whether the correlation reflects a causal relationship. Eplain our reasoning.. the number of practice free throws ou take and the number of free throws ou make in a game es; es; practice should improve our game. the height of a mountain and the average elevation of the state it is in es; es; when calculating average elevation of a state, the mountanin s elevation is taken into account.. the number of hours worked and an emploee s wages es; es; wages often depend on hours worked. a drop in the price of a barrel of oil and the amount of gasoline sold es; es; when gasoline is cheaper, people will bu more.. pen-ended Describe a real world situation that would show a strong negative correlation. Eplain our reasoning. Answers ma var. Sample: As the price of a product increases, sales of the product decrease.. Writing Describe the difference between interpolation and etrapolation. Eplain how both could be useful. Interpolation estimates values between two known values; etrapolation estimates values smaller or larger than all known values. Both help ou estimate unknown values. 7. Writing Describe how the slope of a line relates to a trend line. What does the -intercept represent? The slope of a trend line is the average rate of change for the data. The -intercept is the estimated value for. Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

65 -7 Practice Form K Scatter Plots and Trend Lines For each table, make a scatter plot of the data. Describe the tpe of correlation the scatter plot shows.. Tips Earned b Waiter. Hours Worked 9 Tips ($) Tips Earned ($). Use the table below and a graphing calculator. Population (thousands) Hours Worked SURCE: U.S. Census Bureau positive correlation hio Resident Population a. Make a scatter plot of the data pairs (ears since 9, population). Foots Size and Height Foot Size (in.) Height (in.) Year ,,79,7 Height (in.), 7,7, Foot size (in.) positive correlation b. Draw a line of best fit for the data. c. Write an equation for the line of best fit.., d. According to the data, what will the estimated resident population in hio be in?,99, Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

66 -7 Practice (continued) Form K Scatter Plots and Trend Lines In each situation, tell whether a correlation is likel. If it is, tell whether the correlation reflects a causal relationship. Eplain our reasoning.. the number of cabinets mar assembles and the amount of time it takes him to assemble one A correlation is likel that reflects a causal relationship. The more cabinets he assembles, the faster he will be.. the number of darts thrown at the dart board and Jackie s average score in the game of darts A correlation is likel that reflects a causal relationship. The more she throws darts at the target, the better she will be at scoring points.. the dates of the summer beach vacation and the weather no correlation 7. pen-ended Describe a real world situation that would show a strong positive correlation. Eplain our reasoning. Answers ma var. Sample: the number of hours spent studing and the grade earned on a test. Writing Describe how etrapolation could be useful in a business application. Etrapolation would be useful in making predictions of how the business might do in the future based on past patterns. 9. Use the table below and a graphing calculator. Cars Sold (thousands) Sales of Hbrid Cars in the U.S. Year 7 SURCE: hbridcars.com a. Make a scatter plot of the data pairs (ears since, cars sold). b. Draw a line of best fit for the data. c. Write an equation for the line of best fit d. According to the data, about how man hbrid cars will be sold in? 9, Cars Sold (thousands) Cars Sold (thousands) 7 7 Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

67 -7 Standardized Test Prep Scatter Plots and Trend Lines Multiple Choice For Eercises, choose the correct letter.. For the following situation, determine if there is a correlation. If there is a correlation, is it a causal relationship? the number of hours practicing at the batting cages and our batting average A. negative correlation and a causal relationship B. positive correlation but not a causal relationship C. positive correlation and a causal relationship D. no correlation C. When evaluating data on a scatter plot, what can be used to make predictions about the future? G F. interpolation H. correlation coefficient G. etrapolation I. causation. Mr. Bolton has worked for the same compan for 7 ears. What relationship would ou epect between the number of ears he has been with the compan and his annual salar? A A. positive correlation C. no correlation B. negative correlation D. none of the above. A cit had a population of, people in 99. The population growth of the cit is represented b the equation p t where p is the population in thousands and t is the time in ears since 99. In what ear will the population have doubled? H F. 99 G. H. I.. What tpe of correlation is represented b the data in the scatter plot? C A. positive correlation C. no correlation B. negative correlation D. none of the above Short Response. Use the scatter plot to answer the following questions. a. What is an equation of the trend line for the data? 9 b. What would the earnings be for hours worked? [] Both parts answered correctl. [] ne part answered correctl. [] Neither part answered correctl. Height (in.) Earnings ($) Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved Number of Siblings 7 7 Hours Worked

68 -7 Enrichment Scatter Plots and Trend Lines As ou continue to eplore mathematics in this and other courses ou will learn to model situations where the relationship between variables is not linear. In this assignment ou will get a sneak preview of two other tpes of functions.. An eperiment is done to discover a rule that would allow ou to know the height of an object launched from a given building in terms of the time since launch. Let be the time in seconds and be the height in meters. Data from the eperiment is shown in the table below. Plot the points See points in graph for E... Does it seem that there is a trend line that could be drawn that would accuratel describe this data? Wh or wh not? no; the data show a curve that increases and then decreases. Enter the data in a graphing calculator, putting the -values in L and the -values in L. Plot the points in our graphing calculator. Check students work.. Do a linear regression. Note the correlation coefficient. What does it tell ou? Graph the line. Does it seem to match the data? 9. ; r.7; The line is a poor fit.. Do a quadratic regression b going back to STAT, to CALC and to. Write down the information ou are given. 79; r Graph the quadratic equation from Step. Describe the graph and its relationship to our data. The graph is a good fit for the data. 7. Population data often does not produce linear graphs. The population of the U.S. since is given in the table below, where is the number of ears since and is the population in millions. Plot the points. Check students work. Height (m) Height (m) 7 Time (sec) 7 Time (sec) Enter the data in a graphing calculator. Do linear, quadratic, and eponential regressions on the data. Which tpe of the equation is best fit? Give the equation. eponential; ().7 Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

69 -7 Reteaching Scatter Plots and Trend Lines A scatter plot is a graph that relates two different sets of data b displaing them as ordered pairs. A scatter plot can show a trend or correlation, which ma be either positive or negative. r the scatter plot ma show no trend or correlation. It is often easier to determine whether there is a correlation b looking at a scatter plot than it is to determine b looking at the numerical data. If the points on a scatter plot generall slope up to the right, the two sets of data have a positive correlation. If the points on a scatter plot generall slope down to the right, the two sets of data have a negative correlation. If the points on a scatter plot do not seem to generall rise or fall in the same direction, the two sets of data have no correlation. P r o b l e m The table below compares the average height of girls at different ages. Make a scatter plot of the data. What tpe of correlation does the scatter plot indicate? Age in ears 7 9 Height in Inches 7 Treat the data as ordered pairs. The average height of a Girls Growth chart -ear-old girl is inches, so one ordered pair is (, ). Plot this point. Then plot (, 7), (, ), (, ), (, ), (7, ), (, ), (9, ), and HSM_ATR_7_T9 (, ). Notice that the height increases as the age increases. There is a positive correlation for this data. A trend line is a line on a scatter plot that is drawn near the points. You can use a trend line to estimate other values. Age in ears Height (in.) HSM_ATR_7_T9 Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 9

70 -7 Reteaching (continued) Scatter Plots and Trend Lines P r o b l e m Draw a trend line for the scatter plot in the previous problem. What is the equation for our trend line? What would ou estimate to be the average height of a girl who is ears old? Draw a line that seems to fit the data. The line drawn for this data goes through (, ) and (, ). Use these points to write an equation. m. Girls Growth chart Use the point-slope form of the line. m( ).( ).. Use this equation to estimate the average height of -ear-old girls..() Age in ears Eercises HSM_ATR_7_T Ran practices throwing darts. From each distance listed below, he throws darts and records how man times he hits the center. Distance (in feet) Number of Center Hits 7 9. Use the space at the right to make a scatter plot of the data. See points on graph in E... Describe the tpe of correlation that is shown in the scatter plot. negative correlation; as the distance from the target increases, the number of center hits out HSM_ATR_7_T of decreases. Draw a trend line.. What equation represents our trend line?. How man hits do ou estimate Ran would make from feet? about 7 hits Number of Center Hits Distance (in feet) Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 7 HSMATR_7_ANT

71 - Additional Vocabular Support Graphing Absolute Value Functions Use the chart below to review vocabular. These vocabular words will help ou complete this page. Related Words Eplanation Eample Translation trans LAY shun Conversion kun VUR zhun Paraphrase PA ruh fraz Version VUR zhun Ever point in a figure is moved the same distance in the same direction To change in form To reword An interpretation of a matter from a particular point of view The line was a translation of the original line. The figure was multiplied b a conversion factor of. He paraphrased the teacher s directions to the new student. The homework had several versions of similar tpes of problems. Use the vocabular from above to fill in the blanks. The first one is done for ou. To paraphrase means to reword what was said. Ale s notes from class paraphrased what the teacher said.. Each version had the same questions, but in different order.. To convert from decimals to percents, ou multipl b the conversion factor of one hundred.. Jenna moved her game piece b a translation of units back. For Eercises, match each translation for» in Column A with its corresponding equation in Column B. Column A Column B. units up u u. units left u u. units down u u 7. units left u u. units up u u 9. units right u u. units down u u Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 7

72 - Think About a Plan Graphing Absolute Value Functions What point(s) do the graphs of 7 and have in common? Understanding the Problem. What is the parent function of both equations? u u. What shape is the graph of the parent function of both equations? the shape of a V. What transformations occur in 7? reflection over the -ais, and vertical translation up 7. What translations occur in? horizontal translation right Planning the Solution. Could a table help ou answer the question? Eplain. es; make a table of for u u and the translated equation and figure out the change in -values. Could a graph help ou answer the question? Eplain. es; Draw a graph and find the points of intersection. 7. Which method is better? Wh? graphing; the points of intersection ma not be in the table. Getting an Answer. Graph both equations on the same coordinate grid. 9. What point(s) do the graphs have in common? (, ) and (, ) Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 7

73 - Practice Form G Graphing Absolute Value Functions Describe how each graph is related to the graph of... translated down translated up.. translated right translated left HSM_ATR T HSM_ATR T Graph each function b translating HSM_ATR T HSM_ATR T Write an equation for each translation of.. units down 9. unit up.. units up HSMATR ANT u u HSMATR ANT u u u u. HSMATR ANT Graph each function b translating..... Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 7 HSMATR ANTHSMATR ANT9 HSMATR ANT

74 - Practice (continued) Form G Graphing Absolute Value Functions Write an equation for each translation of.. left 7 units. left unit. right unit u 7 u u. u At the right is the graph of. Graph each function b translating The table at the right shows the dosage of a certain medication for a child s weight, w. Graph the step function for this information. HSMATR ANT Medication Dosage HSMATR ANT7 Medication (mg) 7 Weight (lbs) w u u If Weight Is Dosage Is 9 w HSM_ATR T < lbs mg. Reasoning Eamine the epressions m n and n m. Substitute m and n in each epression and simplif. Now, substitute m and n in each epression and simplif. Repeat this process with other sets of numbers for m and n. What is our conclusion? u m n u un m u Medication Dosage w < 7 lbs 7 w < lbs w < 7 lbs mg 7 mg mg. Writing Can the absolute value of a number equal a negative number? Eplain our reasoning. no; the absolute value of a number must be greater than or equal to zero. Graph each translation of. Describe how the graph is related to the graph of... moved left and down moved right and up Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. HSMATR ANT 7 HSMATR ANT9

75 - Practice Form K Graphing Absolute Value Functions Describe how each graph is related to the graph of u u... translated unit up translated unit to the left Graph each function b translating u u.. u u. u u. u u HSM_ATR T7 HSM_ATR T7 Write an equation for each translation of u u.. units up 7. units down HSMATR ANT HSMATR ANT HSMATR ANT.. units down units up.9 Graph each function b translating u u.. u 7 u. u u. u u HSMATR ANT HSMATR ANT HSMATR ANT7 Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 7

76 - Practice (continued) Form K Graphing Absolute Value Functions Write an equation for each translation of u u.. left units. right units. left units. right units At the right is the graph of u u. Graph each function b translating u u. 7. u u. u u Write an equation for each translation of». 9. units down. units left.. units up...7 units right.7. Writing Describe the difference between adding a constant k inside the absolute value ( u k u) and outside the absolute value ( u u k). Adding a constant on the inside of the absolute value shifts the verte to the left or right. Adding a constant on the outside of the absolute value shifts the verte up or down. Graph each translation of u u. Describe how the graph is related to the graph of u u.. u u. u u moved left and down moved right and up Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 7

77 - Standardized Test Prep Graphing Absolute Value Functions Multiple Choice For Eercises, choose the correct letter.. Which equation represents a translation of units right of? B A. B. C. D.. How is the graph shown at the right related to the graph of? I F. translated units left G. translated units right H. translated units up I. translated units down. What is the -intercept of? A A. C. B. D. HSM_ATR T7. Which equation represents translated units up? H F. G. H. I.. Which equation represents the graph shown at the right? C A. C. B. D.. What is the -intercept of? F. H. G. I. I Short Response 7. Let f (). a. What is the graph of the function? b. How is the graph related to the graph of? translated right and up [] Both parts answered correctl. [] ne part answered correctl. [] Neither part answered correctl. HSM_ATR T7 HSMATR ANT Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 77

78 - Enrichment Graphing Absolute Value Functions. Make a table of values for the equation. Consider points in all four quadrants. Answers ma var. Sample:,,,. Use the table of values to graph the equation. Describe the graph in words. How is it similar to and different from the absolute value graphs ou have been working with? It is like the graphs of u u and u u it is not a function and has limits for eample, 9 has no -value.. Predict what ou think the graph of would look like. Sketch our prediction. Answer ma var. sample: The parts of the graph in Eercise resulting from etending out beond w. Make a table of values and then use the table to graph. How accurate was our prediction? - -,, Make predictions, tables of values, and then graph each of the following equations...,,,,,,, The parts of the graph that would etend out beond w.,,,,,,,, Parallel lines and. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 7

79 - Reteaching Graphing Absolute Value Functions The graph of is shown at the right. The graph of k has the same shape as the graph of but is a translation of up or down b k units. If k is positive the translation is up. If k is negative, the translation is down. Problem (, ) (, ) (, ) What is the graph of? can be rewritten as (). The equation is now in the form k. In this case, k, so translate the graph of four units down. zz zz translate units down Eercises Graph each function b translating..... Writing Compare and contrast the graphs of and. If ou translate the first graph down units, ou will make the second graph. The have the same shape. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 79

80 - Reteaching (continued) Graphing Absolute Value Functions The graph of h has the same shape as the graph of but is a translation of right or left b h units. If h is positive, the translation is to the left. If h is negative, the translation is to the right. Problem What is the graph of? can be rewritten as (). The equation is now in the form h. In this case, h, so translate the graph of two units to the right. zz zz translate units right Eercises Graph each function b translating Which equation is the translation units up of? A A. B. C. D. 9. Which equation is the translation 7 units to the right of? I F. 7 G. 7 H. 7 I. 7 Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

81 Chapter Quiz Lessons - through - Form G Do ou know HW? Find the slope of the line that passes through each pair of points.. (, ), (, ) 9. (, 7), (, ). (, ), (, 9). (, ), (, 7). At : a.m., there were, gallons of water remaining in a reservoir. After hours of irrigation, there were, gallons of water remaining. Write a linear equation that describes the number of gallons of water remaining as a function of the time the field had been irrigated.,, Write an equation for the line that passes through the given point with the given slope m.. (, ); m 7. (9, ); m. (, ); undefined slope 7 9. Write an equation for the line that passes through the points (, ) and (, ). Do ou UNDERSTAND?. Writing Eplain the difference between a rate of change that is positive and one that is negative. Give an eample of each. When rate of change is positive, increases as increases. When rate of change is negative, decreases as increases.. Reasoning If varies directl with and increases b, is it possible to determine b how much increases or decreases? Eplain. no; ou need to know the constant of variation Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

82 Chapter Quiz Lessons - through - Form G Do ou know HW? Graph each equation Write an equation of the line that passes through (, ) and is parallel to the graph of. HSMATR ANT. Write an equation of the line HSMATR ANT that passes through (, ) and HSMATR ANT is perpendicular to the graph of Find the equation for the line of best fit for the data shown below. The data represents the amount of lead emissions from fuel combustion for the ears 9 ( ) to 997 ( 9) SURCE: World Almanac, p. 9. Do ou UNDERSTAND? 7. Writing Eplain how the values of h and k in u hu k affect the HSMATR T7 graph of u u. The graph of u u is the parent graph. The graph is shifted u h u units horizontall. The graph moves right if h S and left if h R. It is shifted u k u units verticall, up if k S and down if k R.. The table at the right shows an Internet compan s standard deliver charge for the total cost of an order. Graph the step function for this information. Standard Deliver < < < < < 9 Deliver Charge ($) 7 rder Total ($) Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

83 Chapter Test Form G Do ou know HW? Tell whether each statement is true or false. Eplain.. A rate of change must be negative. false; the rate of change can be positive, negative, or zero. The rate of change for a vertical line is zero. false; the change in would be, so the slope is undefined Find the slope of the line that passes through each pair of points.. (, ), (, ). Q, R, Q, R Graph each equation ( ) Write each equation in slope-intercept form HSMATR ANT HSMATR ANT7 HSMATR ANT. In, a Caribbean nation produced.7 million tons of cane sugar. Annual production was projected to decrease b. million tons each ear for the net five ears. Write a linear function that models this situation..7. Find the - and -intercepts of the graph of each equation.. ;. ; Write an equation in point-slope form for the line that has the given slope m and that passes through the given point.. m ; (, ). m ; (, ) Write an equation in slope-intercept form for the line that passes through the given points.. (, ), (, ) 7. (, ), (, ) 7 7 Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

84 Chapter Test (continued) Form G Write an equation in slope-intercept form for the line that passes through the given point and is parallel to the given line. 7. (, );. (7, );. 7 Write an equation in slope-intercept form for the line that passes through the given point and is perpendicular to the given line. 9. (, ); 7. (, );. The debate club needs $. to attend a debate tournament. The club decides to sell cups of iced tea and lemonade at baseball games. Iced tea will be sold for $. per cup and lemonade will be sold for $. per cup. a. Write an equation to find how man cups of each beverage must be sold to raise $.... b. Graph the equation. What are the - and -intercepts?, c. pen-ended Use our graph to find three different combinations of cups of iced tea sold and cups of lemonade sold that will raise $.. Answers ma var. Sample: cups of lemonade and cups of tea; cups of lemonade and cups of tea; cups of lemonade and cups of tea. Write an equation for each translation of u u.. units up. left units u u Do ou UNDERSTAND? u u. Writing For a direct variation, describe how the constant of variation affects whether increases or decreases as increases. If the constant is positive, increases as increases. If negative, decreases as increases.. Reasoning For what value of k are the graphs of and k 9 parallel? HSMATR ANT9. Writing Eplain wh the absolute value function u u is an eample of a piecewise function. A piecewise function is a function that has different rules for different parts of its domain. For eample, when L,»» and when R,»». Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

85 Chapter Quiz Lessons - through - Form K Do ou know HW? Find the slope of the line that passes through each pair of points.. (7, ), (, ). (, ), (, 9). (, ), (, ). Shell has a gift card for her favorite restaurant for $. She wants to treat a group of her friends to sandwiches at the restaurant. Sandwiches cost $. each. Write an equation that models the total mone Shell owes if she bus sandwiches after using the gift card. Graph the equation.. Write an equation for the line that passes through the given point with the given HSMATR ANT slope m.. (, ); m. (, ); m 7. (, 7); m 7. Write an equation in slope-intercept form of the line that passes through the points (, ) and (, ). 9. Jerem s pa started at $ per week. Ever ear his pa will increase b $ per week. Write an equation that models his weekl pa after ears.. Suppose varies directl with. Write a direct variation equation that relates and when when. Then find the value of when..; Do ou UNDERSTAND?. Writing Eplain how to write the equation of a line when given a point on the line and the slope of the line. Use the point-slope form. Substitute the coordinates of the given point for (, ) and the given slope for m. Then simplif inside the grouping smbols.. a. pen-ended Write an equation of a line that passes through (, ). Answers ma var. For eample,. b. Reasoning Is there onl one line that passes through this point? Eplain. No, there are an infinite number of lines that could pass through this point. Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

86 Chapter Quiz (continued) Lessons - through - Form K Do ou know HW? Graph each equation... Write an equation in slope-intercept form of the line that passes through the given point and is parallel to the graph of the given equation.. (, ); HSMATR ANT. (, ); HSMATR ANT7 Write an equation in slope-intercept form of the line that passes through the given point and is perpendicular to the graph of the given equation.. (, 7); 9. (, ); 7 7. Use the table below. Per Capita Milk Consumption in the U.S. Year Milk Consumption (gal) a. Make a scatter plot of the data. b. Describe the tpe of correlation the scatter plot shows. negative correlation HSM_ATR T7 Do ou UNDERSTAND?. pen-ended Write one equation of a line that is parallel to and one equation of a line that is perpendicular to the graph of the line. HSMATR_7_ANT Answers ma var. For eample, parallel: ; perpendicular: Milk Consumption Years since 9 9. Reasoning For what value of k are the graphs of and k perpendicular? k Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

87 Chapter Test Form K Do ou know HW? Find the slope of the line that passes through each pair of points.. (, ), (, ). (7, ), (, ). Write the equation in slope-intercept form.. Write an equation in point-slope form of the line that passes through the point (, ) and has a slope of. ( ) Find the slope and -intercept of the graph of each equation.. 7. Slope:, -intercept 7. Slope:, -intercept. 7. Hector has a $ off coupon at Brenda s Bagels. Bagels cost $.99 per dozen. Write an equation that models the total amount of mone Hector owes if he bus dozens of bagels..99 Suppose varies directl with. Write a direct variation equation that relates and. Then find the value of when.. when 9. when ; ;. Write an equation in point-slope form for the line that passes through the points (, ) and (, ). Answers ma var. Sample answer ( ) ( ) Graph each equation... 9 HSMATR ANT Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 7HSMATR ANT9

88 Chapter Test (continued) Form K Find the -and -intercepts of the graph of each equation.. ;. 7 7; 7 Write an equation for each translation of».. units up». units left» 7.. units right».. units down» 9. Write an equation in slope-intercept form of the line that passes through (, ) and is parallel to the graph of.. Write an equation in slope-intercept form of the line that passes through (7, ) and is perpendicular the graph of. Do ou UNDERSTAND?. Compare and Contrast How are the graphs of u u and u u the same? How are the different? The graphs have the same shape. Adding to the inside of the absolute value shifts the verte units to the left. Adding on the outside of the absolute value shifts the verte up units.. Writing Consider the number of zip codes in a count and the number of parking meters in the count. Would ou epect a positive correlation, a negative correlation, or no correlation between the two sets of data? Eplain our reasoning. Answers ma var. Sample: I would epect a positive correlation because a count with a large number of zip codes would have a high population densit and probabl have a need for metered parking.. Writing Describe how ou can determine from their equations if two lines are parallel or perpendicular without graphing the lines. Write the equations in slope-intercept form and eamine the slopes of both equations. If the slopes are the same, the lines are parallel. If the slopes are negative reciprocals of each other, the lines are perpendicular. Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

89 Performance Tasks Chapter TASK Suppose ou are the diving officer on board a submarine conducting diving operations. As ou conduct our operations, ou realize that ou can relate the submarine s changes in depth over time to the slope of a line. a. For each situation below, graph the relationship between depth and time. positive slope negative slope slope equal to zero undefined slope d b. For each situation in part (a), calculate the slope on the basis of our graph. Show our work. Answers ma var. Sample: m/min; m/min; m/min; undefined c. What would the graph representing a d submarine descending from the surface at a rate of. ft/s look like? d. Use a graphing calculator to find the equation of the line of best fit for the data at right. d.t 9 d [] Student shows understanding of the task, completes all portions of the task appropriatel with no errors in computation, and full supports work with appropriate graphs. [] Student shows understanding of the task, completes all portions of the task appropriatel, with one error in computation, and supports work with appropriate graphs. [] Student shows understanding of the task, but makes errors resulting in incorrect answers. [] Student shows minimal understanding of the task. [] Student shows no understanding of the task. TASK Computer repair companies use a fied cost equation to determine how much to charge for an in-house repair. For eample, a technician charges $ for a house call and $ per hour. The fied cost equation is. a. How are the slope and -intercept of the graph related to the equation? slope hourl rate, $/h; -intercept fied cost for a house call, $. b. What is the rate of change in the equation? How is the rate of change related to the slope of the line represented b the equation? Eplain. ; the are equal. The slope is the rate of change. [] Student shows understanding of the task, completes all portions of the task appropriatel with no errors in computation. [] Student shows understanding of the task, completes all portions of the task appropriatel, with one error in computation. [] Student shows understanding of the task, but makes errors resulting in incorrect answers. [] Student shows minimal understanding of the task. [] Student shows no understanding of the task. depth (ft) Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 9 d time (sec) Time (seconds) 9 d Depth (feet)

90 Performance Tasks (continued) Chapter TASK You and our friend are having trouble using the different methods for finding the equation of a line. With the help of the graph at the right, provide a detailed step-b-step procedure that demonstrates how each method works. a. Using slope and -intercept Given: slope and -intercept Use slope-intercept form: b. Using slope and a point Given: slope and (, ) is on the line Use point-slope form: ( ()) Simpl to get c. Using two points Given: (, ) and (, ) are on the line. Find the slope: m () Use point-slope form: ( ()) Simplif to get d. Using - and -intercepts Given -intercept and -intercept Multipl the intercepts: () Write the equation in standard form: e. Write the equation of the line at right in the form A B C. ; ; f. Th ink of a real-world situation that can be modeled using the equation ou wrote in part (e). Check students work. [] Student shows understanding of the task, completes all portions of the task appropriatel with no errors in computation. [] Student shows understanding of the task, completes all portions of the task appropriatel, with one error in computation. [] Student shows understanding of the task, but makes errors resulting in incorrect answers. [] Student shows minimal understanding of the task. [] Student shows no understanding of the task. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 9

91 Cumulative Review Chapters - Multiple Choice For Eercises, choose the correct letter.. What is true about the graphs of and? C A. The are parallel. C. The are perpendicular. B. The have the same slope. D. The do not intersect.. A mail order compan sells boes of fishing lures for $.9 per bo. A charge of $.9 is added to orders, regardless of the order size. Which of the following equations models the relationship between the number of boes ordered and the total cost of the order? I F. c.9w; w $ H. c (.9.9)w; w $ G..9c.9w; w $ I. c.9w.9; w $. Which of the following is the correct simplification of the epression? B A. B. C. 9 D.. Which is not a solution of, 7? I F. G. H. 7 I.. Which of the following statements is not true for the graph of the equation? B A. The -intercept is. C. The -intercept is. B. The line has a positive slope. D. The line contains the point Q, R.. What is the constant of variation for? F. G. H. I. 7. What is f () when f ()? C A. B. C. D.. The Girl Scouts hoped to raise $ selling cookies. Instead, the raised $. What percent of their goal did the achieve? G F. % G. % H. % I. 9% 9. Which of the following is the solution of ( )? A A. B. C. D.. What is when and? I F. G. H. 7. I. G Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 9

92 Cumulative Review (continued) Chapters -. What is an equation of the graph at the right? A A. C B. 7 D.. Evaluate each epression. a. for b. for. Find. a. % of is. b. % of is.. What are the coordinates of the verte of the graph of u u? (, ). Find the -intercept of the graph of each equation. a. 7 b.. Find the slope of each line. a. The line containing points (, ) and (, ) b. The line described b the equation 9 7. Find the slope of each line. a. The line that is perpendicular to the graph b. The line that is parallel to the graph Find each answer. 7. A long-distance compan charges $.9 per month plus $. per minute for all in-state long distance calls. Calculate the cost in dollars to make minutes of in-state long distance calls over one month. $. 9. The ratio of black walnut to red oak trees at a tree farm is :. The tree farm has black walnut trees. How man black walnut and red oak trees does the tree farm have altogether? 7 trees. A train moving at a constant speed travels mi in hr. At this rate, how man miles does the train travel in hr? 7. mi. You invest $7. in advertising ourself as a math tutor. You charge $/hr. How man hours do ou have to tutor before ou break even? 7 h. Line A passes through points (, ) and (, 7). Line B is perpendicular to Line A. Line C is perpendicular to Line B. What is the slope of Line C?. pen-ended Make up a problem that could be solved using a rate of change. Answers ma var. Sample: A water tank contains gallons of water. You use gallons per week to water the garden during a dr spell. How much water remains in the water tank after weeks?. What is the -intercept of the graph of the equation 7? Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 9

93 TEACHER INSTRUCTINS Chapter Project Teacher Notes: The Choice Is Yours About the Project The project will have students make comparisons about their first jobs. The will be asked to write, model, solve, and graph equations. Students will also interview adults about their job eperiences as teenagers. Introducing the Project Ask students to brainstorm about jobs the might like to have now. From their brainstorming, have students select two jobs that interest them. Students will research these jobs and then construct a graph that shows income for each job. Elicit from students possible scales to be used for their graphs. After the complete their graphs, have students compare the income difference between jobs for an eight-hour workda. Activit : Graphing Students research the starting hourl wage for two jobs. Then the graph and compare the results. Activit : Modeling Students are asked to correctl model income earned b writing an equation. Have students solve their equations based on the parameters given. Activit : Interviewing Students interview adults regarding jobs the had as teenagers. Have students write equations that describe each adult s weekl earnings after epenses. Finishing the Project You ma wish to plan a project da on which students share their completed projects. Encourage groups to eplain their processes as well as their results. Have students review their project work and update their folders. Have students review the equations, graphs, and eplanations that the needed for the project. Ask groups to share their insights that resulted from completing the project, such as an shortcuts the found for writing equations or making graphs. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 9

94 Chapter Project: The Choice Is Yours Beginning the Chapter Project Do ou have a job? If not, what will our first job be? What epenses will ou have? How much mone will ou actuall earn? How can ou compare earnings between two jobs? Linear equations can help answer all of these questions. As ou work through the activities, ou will make graphs and write equations that model earnings for different jobs. After interviewing someone about his or her first job, ou will choose a job that ou might like and will eplain wh ou made that choice. List of Materials Calculator Graph paper Activities Activit : Graphing Find the starting hourl wage for two jobs that interest ou. For each job, write an equation that gives the income for working hours. With hours worked ( to ) on the horizontal ais and income on the vertical ais, graph our two equations. Check students equations and graphs. Suppose ou work eight hours in one da. Eplain how our graph shows the difference in income for the two jobs. Look at the -values for Activit : Modeling Suppose ou earn $./h at a baker. From our first pacheck ou determine that $./h is withheld for taes and benefits. You work hours during a five-da week and ou spend $.7 each da for lunch. Write an equation for our weekl earnings after taes, benefits, and lunch epenses...7 In this situation, what do the slope and the -intercept represent? slope net hourl wage after taes and benefits; -intercept mone spent on lunches How man hours must ou work to earn $ after taes, benefits, and lunch epenses? about h Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 9

95 Chapter Project: The Choice Is Yours (contined) Activit : Interviewing Interview an adult about a job he or she had as a teenager. Ask about positive and negative aspects of the job, salar, and epenses. Write an equation that describes the person s weekl earnings after epenses. Check students work. Finishing the Project The answers to the three activities should help ou complete our project. Work with several classmates. Share what ou learned about jobs for teenagers. List positive and negative aspects of income and epenses for several jobs. n our own, organize our graphs, equations, and job information in a folder. Write a final paragraph that eplains what job ou would like and wh. Reflect and Revise You should present job-comparison information in a well-organized format. Your graphs should be eas to understand. Be sure ou have eplained clearl how our equations model each income. Make an revisions necessar to improve our project. Etending the Project Some emploers will onl hire people with eperience in the job for which the appl. You can gain eperience b working independentl. Find out how ou might go about starting our own business. Research potential epenses and income. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 9

96 Chapter Project Manager: The Choice Is Yours Getting Started Read the project. As ou work on the project, ou will need a calculator, graph paper, materials on which ou will record our calculations, and materials to make accurate and attractive graphs. Keep all of our work for the project in a folder. Checklist Activit : writing and graphing equations Activit : interpreting equations Activit : interviewing an adult Project displa Suggestions Eplain how ou used our graph to find the difference in pa for an eight-hour da. Review slope and -intercept in our tetbook. Find out about an adult s teenage job eperiences and write an equation that describes the person s weekl earnings after epenses. What have ou learned about jobs for teenagers? How might our graphs differ from those that show income and epenses for adults? When looking for a job, what factors should ou consider when making a decision? Scoring Rubric Equations are written correctl, thoroughl eplained, and accurate. Graphs are effective and labeled correctl. The folder is well organized and provides useful information. Equations are essentiall correct with minor errors in calculations. Reasoning and eplanations are essentiall correct, but should be more thorough. Graphs ma contain minor errors in scale or ma be labeled incorrectl. The folder presents clear information, but needs to be better organized. Equations are inaccurate and the folder lacks organization. Graphs could be neater and more accurate. Eplanations lack detail. Major elements of the project are incomplete or missing. Your Evaluation of Project Evaluate our work, based on the Scoring Rubric. Teacher s Evaluation of Project Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 9