Name Class Date. Additional Vocabulary Support


 Beverly Shona Palmer
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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 nethird 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 slopeintercept 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 SlopeIntercept 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 slopeintercept 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 slopeintercept 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 slopeintercept 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 SlopeIntercept 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 slopeintercept form of each line Write an equation in slopeintercept 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 SlopeIntercept 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 SlopeIntercept 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 slopeintercept form of each line... Write an equation in slopeintercept 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 SlopeIntercept 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 SlopeIntercept 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 SlopeIntercept 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 SlopeIntercept Form The slopeintercept 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 slopeintercept 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 slopeintercept form to find b. m b Use slopeintercept form. () b Substitute for m, for, and for. b Solve for b. Substitute the slope and intercept into the slopeintercept form. m b Use slopeintercept 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) SlopeIntercept 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 slopeintercept 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 pointslope 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 pointslope 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 PointSlope 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 pointslope 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 PointSlope Form Write an equation of the line in pointslope form through the given point and with the given slope m.. (, ); m. (, ); m. (, ); m. (, ); m Graph each equation.. ( ). ( ) 7. ( ) Write an equation in pointslope form for each line (, ) (, ) (, ) (, ) (, 9) (, ) ( ) or ( ) ( ) or ( ) ( ) or 9 ( ) Write an equation in pointslope form of the line through the given points. Then write the equation in slopeintercept form.. (, ), (, ). (, ), (, ). (, ), (, ) ( ); ( ); ( ); 7. penended Write an equation of a line that has a slope of in each form. a. pointslope form b. slopeintercept 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 PointSlope Form Model the data in each table with a linear equation in slopeintercept 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 pointslope form to write the equation of a line that passes through the points (, ) and (, ) in slopeintercept form. First find the slope: 9. Then use one point in the pointslope 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 slopeintercept 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 $.. Write an equation in pointslope form that represents the cost of tickets. What is the graph of the equation?. 7.( ) Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. Cost ($) 7 Tickets
35  Practice Form K PointSlope Form Write an equation in pointslope 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. penended Write an equation in each of the following forms that has a slope of. Answers ma var. Sample: a. pointslope form b. slopeintercept 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 PointSlope Form Model the data in each table with a linear equation in slopeintercept 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 pointslope form of each line... ( ) ( ) Write an equation in pointslope form of the line that passes through the given points. Then write the equation in slopeintercept form.. (, ), (, ) 7. (, ), (, ) ( );. (, ), (, ) 9. (, ), (, 7) ( ); ( ); ( ); 9. Writing Describe how ou would use the pointslope form to write the equation of a line that passes through the points (, ) and (, ) in slopeintercept form. First, use the slope formula to find the slope: m. Then, select one of the points to write the pointslope 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 pointslope 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 PointSlope 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 pointslope 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 PointSlope Form Pointslope 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 pointslope form for the line with slope m that has intercept (a, ). m( a). Write an equation in pointslope form for the line with slope m that has intercept (, b). b m. Use our result from Eercise to write an equation in pointslope form for the line with slope that has intercept (, ). ( ). Use our result from Eercise to write an equation in pointslope form for the line with slope that has intercept (, ).. A line has intercept (a, ) and intercept (, b). a. Write an equation in pointslope form for the line using the intercept. b a ( a) b. Write an equation in pointslope 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 PointSlope Form The pointslope 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 pointslope form? m( ) Use pointslope form. () ( ) Substitute (, ) for (, ) and for m. ( ) Simplif. Problem A line passes through (, ) and (, 9). What is an equation for this line in pointslope form? What is an equation for this line in slopeintercept form? First use the two given points to find the slope. m 9 Use the slope and one point to write an equation in pointslope form. m( ) Use pointslope form ( ) Substitute (, ) for (, ) and for m. Distributive Propert Add to each side. An equation in pointslope 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 pointslope form of the line through the given points. Then write the equation in slopeintercept 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) PointSlope Form You can use the pointslope form of an equation to help graph the equation. The point given b the pointslope form provides a place to start on the graph. Plot a point there. Then use the slope from the pointslope 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 pointslope 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 gokarts. 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 etrapoint 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
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