ACCELERATED MATHEMATICS CHAPTER 7 NON-PROPORTIONAL LINEAR RELATIONSHIPS TOPICS COVERED:

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1 ACCELERATED MATHEMATICS CHAPTER 7 NON-PROPORTIONAL LINEAR RELATIONSHIPS TOPICS COVERED: Representing Linear Non-Proportional Equations Slope & -Intercept Graphing Using Slope & -Intercept Proportional vs. Non-Proportional Sstems of Linear Equations Linear Equations from Situations and Graphing Linear Equations from Tables Linear Relationships and Bivariate Data Input/Output and Functions Describing & Comparing Functions Contents modified from m.hrw.com, 8 th grade Edition online practice

2 LINEAR RELATIONSHIPS Proportional Non-Proportional Constant of Proportionalit Constant of Variation Direct Variation = mx + b = x + 4 = 4x + 7 = kx = 5x You save $5 each month. x m = slope b = -intercept You start with $5 and then save $5 each month. x

3 Verbal Examples Linear Relationships: Proportional vs. Non-Proportional Proportional: Mr. Mangham started the ear with $. Each week he earned $5. Non-Proportional: Mr. Mangham started the ear with $75. Each week he earned $5. How to tell the difference: A proportional situation alwas starts at zero (in this case $ at the first of the ear). A nonproportional situation does not start at zero (in this case $75 at the first of the ear). Table Examples Proportional: Weeks 1 4 Mone ($) Mone Weeks Non-Proportional: Weeks 1 4 Mone ($) Mone Weeks How to tell the difference: A proportional table has a constant of proportionalit in that divided b x alwas equals the same value. A non-proportional table will have different values when is divided b x. Equation Examples Proportional: = 5x Non-proportional: = 5x + 75 How to tell the difference: A proportional equation is alwas in the form = kx, where k is the unit rate or constant of proportionalit. A non-proportional equation is alwas in the form = mx + b, where m is the constant rate of change or slope. The ke difference is the added b on the end. Graph Examples Proportional: Non-Proportional: Mr. Mangham's Mone 1 4 Mr. Mangham's Mone 1 4 How to tell the difference: A proportional graph is a straight line that alwas goes through the origin. A non-proportional graph is a straight line that does not go through the origin.

4 Activit 7-1: Representing Linear Non-Proportional Relationships (4.1) You can use an equation to describe the relationship between two quantities in a real-world situation. You can use a table to show some values that make the equation true. 1. The equation = x + gives the total change,, for bowling x games at Baxter Bowling Lanes based on the prices shown. Make a table of values for this situation. x (number of games) 1 4 (cost in dollars) BAXTER BOWLING LANES $ per game $ shoe rental. Francisco makes $1 per hour doing part-time work on Saturdas. He spends $4 on transportation to and from work. The equation = 1x 4 gives his earnings, after transportation costs, for working x hours. Make a table of values for this situation. x (number of hours) (earnings in dollars). The entrance fee for Mountain World theme park is $. Visitors purchase additional $ tickets for rides, games, and food. The equation = x + gives the total cost,, to visit the park, including purchasing x tickets. x (number of tickets) (total cost in dollars) x 4. Plot the ordered pairs from the table. Describe the shape of the graph. 5. Find the rate of change between each point and the next. Is the rate constant? 6. Explain wh the relationship between number of tickets and total cost is not proportional. Cost ($) Theme Park Costs Number of Tickets 7. Would it make sense to add more points to the graph from x = to x = 1? Would it make sense to connect the points with a line? Explain.

5 Activit 7-: Representing Linear Non-Proportional Relationships (4.1) A linear equation is an equation whose solutions are ordered pairs that form a straight line when graphed on a coordinate plane. Linear equations can be written in the form = mx + b. When b, the relationship between x and is nonproportional. The diameter of a Douglas fir tree is currentl 1 inches when measured at chest height. Over the next 5 ears, the diameter is expected to increase b an average growth rate of 5 inch per ear. The equation = x + 1 gives, the diameter of the tree in inches, after x ears Complete the table. x (ears) 1 5 (diameter in inches). Plot the ordered pairs from the table and draw a line connecting the points to represent all the possible solutions.. Is this relationship linear? 4. Is it proportional? Diameter (in) Fir tree Growth Time (r) Make a table of values for each equation. 5. = x x 1 1 = x 5 8 x 8 4 8

6 Activit 7-: Representing Linear Non-Proportional Relationships (4.1) Explain wh each relationship is not proportional. 1.. x x. Complete the table for the equation and then graph the equation. = x 1 x 1 1 State whether the graph of each linear relationship is a solid line or a set of unconnected points and explain our reasoning. 4. The relationship between the number of $4 lunches ou bu with a $1 school lunch card and the mone left remaining on the card. 5. The relationship between time and the distance remaining on a -mile walk for someone walking at a stead rate of miles per hour.

7 Activit 7-4: Representing Linear Non-Proportional Relationships (4.1) Simone paid $1 for an initial ear s subscription to a magazine. The renewal rate is $8 per ear. This situation can be represented b the equation = 8x + 1, where x represents the number of ears the subscription is renewed and represents the total cost. 1. Make a table for this situation.. Draw a graph to represent the situation.. Explain wh this relationship is not proportional. 4. Does it make sense to connect the points on the graph with a solid line? Explain. Cost ($) Years 5. What is required of a proportional relationship that is not required of a general line relationship? 6. Explain how ou can identif a linear nonproportional relationship from a table? From a graph? From an equation? 7. George observes that for ever increase of 1 in the value of x, there is an increase of 6 in the corresponding value of. He claims that the relationship represented b the table is proportional. Critique George s reasoning. x Two parallel lines are graphed on a coordinate plane. How man of the parallel lines could represent proportional relationships? Explain.

8 Activit 7-5: Representing Linear Non-Proportional Relationships (4.1) Make a table of values for each equation. 1. = 4x +. x = x 4 x =.5x =.75x + 4 x 4 4 x 1 1 Make a table of values and graph the solutions of each equation. 5. = x x = x x 4 4 State whether the graph of each linear relationship is a solid line or a set of unconnected points. Explain our reasoning. 7. The relationship between the height of a tree and the time since the tree was planted. 8. The relationship between the number of $1 DVDs ou bu and the total cost.

9 Activit 7-6: Determining Slope and -intercept (4.) The graph of ever nonvertical line crosses the -axis. The -intercept is the -coordinate of the point where the graph intersects the -axis. The x-coordinate of this point is alwas. The graph represents the linear equation 1. What is the slope from (,4) to (-,6)?. What is the slope from (,4) to (6,)? = x + 4. What do ou notice about the two slopes and how the relate to the linear equation? 4. Find the value of when x = using the equation = x + 4. Describe the point on the graph that corresponds to this situation. 5. What do ou notice about this value and how it relates to the linear equation? The linear equation below is written in the slope-intercept form of an equation. Its graph is the line with slope m and -intercept b. = mx + b A linear relationship has a constant rate of change. You can find the rate of change m and the initial value b for a linear situation from a table of values. Example 1: A phone salesperson is paid a minimum weekl salar and a commission for each phone sold, as shown in the table. Determine the constant rate of change and the initial value. Number of phones sold 1 4 Weekl Income ($) $48 $6 $78 $9 The rate of change is the same as the slope. m = x = To find the initial value we need to know how much the salesperson makes when zero phones are sold. Work backwards from x = 1 to x =. The initial value is the same as the -intercept. b =

10 Activit 7-7: Determining Slope and -intercept (4.) 1. Find the slope and -intercept of the line represented b each table. x x Slope m = -intercept b = Slope m = -intercept b =. Find the slope and -intercept of the line in each graph. Slope m = -intercept b = Slope m = -intercept b = Slope m = -intercept b = Slope m = -intercept b =

11 Activit 7-8: Determining Slope and -intercept (4.) 1. Find the slope and -intercept of the line represented b each table. x x Some carpet cleaning costs are shown in the table. The relationship is linear. Find and interpret the rate of change (slope) and the initial value of for this situation. Rooms cleaned 1 4 Cost ($) The total cost to pa for parking at a state park for the da and rent a paddleboat are shown below. Number of hours 1 4 Cost ($) Find the cost to park for a da and the hourl rate to rent a paddleboat. 4. What will Lin pa if she rents a paddleboat for.5 hours and splits the total cost with a friend? Explain. Ramond s parents will pa for him to take sailboard lessons during the summer. He can take half-hour group lessons or half-hour private lessons. The relationship between cost and number of lessons is linear. Lessons 1 4 Group ($) Private ($) Find the rate of change and the initial value for the group lessons. 6. Find the rate of change and the initial value for the private lessons. 7. Compare and contrast the rates of change and the initial values.

12 Activit 7-9: Determining Slope and -intercept (4.) 1. Prove wh each relationship is not linear. x 1 4 x Your teacher asked our class to describe a real-world situation in which the -intercept is 1 and the slope is 5. Your partner gave the following description: M ounger brother originall had 1 small building blocks, but he lost 5 of them ever month since. What mistake did our partner make? How can ou change the situation to match?. John has a job parking cars. He earns a fixed salar of $ plus a fee of $5 for each car he parks. His potential earnings for a week are shown in the graph. At what point does John begin to earn more from fees that from his fixed salar? Justif our answer. Weekl earnings Cars parked Find the slope and -intercept of the line in each graph Find the slope and -intercept of the line represented b each table x x

13 Activit 7-1: Graphing Linear Nonproportional Relationships (4.) Example Graph = x 1 Step 1: What is the -intercept (b)? Plot that point. Step : What is the slope (m)? Use the slope to find a second point. Step : Draw a line through the points. Example Graph 5 = x + Step 1: What is the -intercept (b)? Plot that point. Step : What is the slope (m)? Use the slope to find a second point. Step : Draw a line through the points. Graph each equation = x + 1 = x + 4

14 Activit 7-11: Graphing Linear Nonproportional Relationships (4.) Analzing a Graph Man real-world situations can be represented b linear relationships. Example Ken has a weekl goal for the number of calories he will burn b taking brisk walks. The equation = x + 4 represents the number of calories Ken has left to burn after x hours of walking. Graph the equation = x + 4. Step 1: What is the -intercept? Step : What is the slope written as a fraction? Use the slope to locate a second point. Step : Draw a line through the two points. Calories remaining Time (h) 1. After how man hours of walking will Ken have 6 calories left to burn? After how man hours will he reach his weekl goal?. What if Ken modifies his plans b slowing his speed. The equation for the modified plan is = x + 4. Graph the new equation on the graph and compare to the previous graph.. Suppose Ken decides that instead of walking, he will jog, and jogging burns 6 calories per hour. How do ou think this would change the graph? 4. Graph each equation using the slope and -intercept. 1 = x = x + Slope = -intercept = Slope = -intercept =

15 Activit 7-1: Graphing Linear Nonproportional Relationships (4.) A friend gives ou two baseball cards for our birthda. Afterward, ou begin collecting them. You bu the same number of cards once each week. The equation = 4x + describes the number of cards,, ou have after x weeks. 1. Find and interpret the slope and the -intercept of the line that represents this situation.. Graph the equation = 4x +.. Discuss which points on the line do not make sense in this situation. Then plot three more points on the line that do make sense. Cards Weeks A spring stretches in relation to the weight hanging from it according to the equation =.75x +.5 where x is the weight in pounds and is the length of the spring in inches. 4. Graph the equation. 5. Interpret the slope and the -intercept of the line. 6. How long will the spring be if a pound weight is hung on it? 7. Will the length double if ou double the weight? Explain. Length (in) Weight (lb) 8. Identif the coordinates of four points on the line with each given slope and -intercept. A. Slope = 5, -intercept = 1 B. Slope =., -intercept =. C. Slope = 1, -intercept = 8 D. Slope = 1, -intercept = 4 9. A music school charges a registration fee in addition to a fee per lesson. Music lesson last.5 hour. The equation = 4x + represents the total cost of of x lessons. Find and interpret the slope and - intercept of the line that represents this situation.

16 Activit 7-1: Graphing Linear Nonproportional Relationships (4.) A public pool charges a membership fee and a fee for each visit. The equation = x + 5 represents the cost for x visits. 1. After locating the -intercept on the coordinate plane shown, Danielle moves up three gridlines and right one gridline. Is this a correct slope based on the information presented in the problem?. Graph the equation = x + 5. Then interpret the slope and -intercept.. How man visits to the pool can a member get for $? Cost ($) Visits 4. A students sas that the slope of a line for the equation = 15x is and the -intercept is 15. Find and correct the error. 5. Graph the lines = x, = x, and = x +. What do ou notice about the three lines? Make a conjecture based on our observation. Graph each equation using the slope and the -intercept = x 1. = x +. = x m b

17 Activit 7-14: Graphing Non-Linear Proportional Relationships (4.) 1. The equation = 15x + 1 gives our score on a math quiz, where x is number of questions ou answered correctl. a. Graph the equation. b. Interpret the slope and -intercept of the line. c. What is our score if ou answered 5 questions correctl? Proportional and Nonproportional Situations (4.4) IS THE GRAPH A STRAIGHT LINE? YES DOES IT GO THROUGH THE ORIGIN? NO Non-Linear, Non-Proportional YES Linear, Proportional NO Linear, Non-proportional. Determine if each of the following graphs represents a proportional or nonproportional relationship If it is a linear equation it can be written in the form = mx + b. If it is also proportional, then b = and it can also be written as = kx.

18 Activit 7-15: Proportional and Nonproportional Situations (4.4) 1. Determine if each of the following equations represents a proportional or nonproportional relationship. d = 65t p =.1s + n = 45 p 6 = 1d If there is not a constant rate of change in the data then it is a nonlinear, nonproportional relationship. A linear relationship is a proportional relationship when x is nonproportional. is constant. Otherwise, the linear relationship. Determine if the linear relationship represented b each table is a proportional or nonproportional relationship. x x Determine which situation is a proportional relationship and which situation is a nonproportional relationship. The cost for Test Prep Center A is $ times the number of hours that ou attend. The cost for Test Prep Center B is $5 an hour, but ou have a $1 coupon that ou can use to reduce the cost Determine if each relationship is a proportional or nonproportional situation q = p + 1 v = u t = 15d m =.75d 1

19 Activit 7-16: Proportional and Nonproportional Situations (4.4) 1.-. Determine if each relationship is a proportional or nonproportional situation. x x Describe a real-world situation where the relationship is linear and nonproportional Determine if each relationship is a proportional or nonproportional situation. Explain our reasoning. Assume all tables represent linear relationships. = x r = b + 1 x x This ear, Andrea celebrated her 1 th birthda, and her brother Carlos celebrated his 6 th birthda. Andrea noted that she was now twice as old as her brother was. Is the relationship between their ages proportional? Support our answer. Lil is considering buing books on displa three different tables. Each table has one of the following signs. 11. Each Book $1 Each Book 5% off Each Book $8 for Club Members (One-time Membership Fee: $15) What will be the total cost if Lil bus 6 books from the table whose sign indicates a nonproportional relationship? 1. Ed and Rile are ccling in the same direction on the same stright road. Ed s distance from a roadside rest area is given b d=6t. Rile s distance from the same rest area is iven b d= 4.5t+1. In each equation d is distance in miles and t is time in hours. a. Determine if each equation is proportional or non-proportional. b. When are Ed and Rile the same distance from the rest area? Show our work.

20 Activit 7-17: Proportional vs. Non-Proportional (4.4) 1. Complete the tables below and then graph each set of data. Proportional Relationship x x Rate of change (slope) = Non-Proportional Relationship x x Rate of change (slope) =. What did ou notice about the rate of change for each relationship?. What did ou notice about the graphs for the proportional and non-proportional relationship? 4. For which relationship is the rate of change the same as the ratio 5. What did ou notice about the ratio x for the each relationship? x? Proportional Linear Functions Both Non-Proportional Linear Functions Graphs will pass through the Graphs have a constant rate of origin. Graphs will not pass change (slope). through the origin. There is a constant of proportionalit. Lines are in the form of =kx. There is onl one and exactl one -value that corresponds (maps) to an given x-value. Lines are in the form of =mx+b.

21 Activit 7-18: Solving Sstems of Linear Equations b Graphing (4.5) You have learned several was to graph a linear equation in slope-intercept form. For example, ou can use the slope and -intercept or ou can find two points that satisf the equation and connect them with a line. Graph the pair of equations together on the same graph: = x and = x + What is the point of intersection of the two lines? Will that point of intersection be a solution to the first equation? Check b substituting the point into the equation. Will that point of intersection be a solution to the second equation? Check b substituting the point into the equation. An ordered pair (x,) is a solution of an equation in two variables if substituting the x- and - values into the equation results in a true statement. A sstem of equations is a set of equations that have the same variables. An ordered pair is a solution of a sstem of equations if it is a solution of ever equation in the set. Solve the sstem b graphing. = x + 4 and = x Solve the sstem b graphing. = x + and = 4x 1 Solve the sstem b graphing. = x + 5 and = x

22 Activit 7-19: Solving Sstems of Linear Equations (4.5) Show all graphs on graph paper. Show all work on notebook or graph paper. Bowl-o-Rama charges a shoe rental fee of $. and a cost per game of $.5. Bowling Pinz charges a rental fee of $4. and a cost per game of $.. 1. Let x represent the number of games plaed and let represent the total cost. Write an equation to represent the total cost at Bowl-O-Rama. Write an equation to represent the total cost at Bowling Pinz. Graph each equation and to find the solution. What does the solution represent?.. 4. Mr. Underwood runs 7 miles per week and increases his distance b 1 mile each week. Mr. Mangham runs miles per week and increases his distance b miles each week. In how man weeks will Mr. Underwood and Mr. Mangham be running the same distance? What will that distance be? Write a sstem of equations and graph to find the solution. Write an interesting, creative real-world problem that could be represented b the sstem of equations shown below. = 4x + 1 = x + 15 You have two options for our internet service: Option 1: $5 set-up fee plus $ per month Option : No set-up fee plus $4 per month In how man months will the total cost of both options be the same? Write a sstem of equations and graph to find the solution. If ou plan to cancel our internet service after 9 months, which is the cheaper option? Eight friends wanted to start a business. The will wear either a baseball cap or a shirt imprinted with their logo while working. The want to spend exactl $6 on the shirts and caps. Shirts cost $6 each and caps cost $ each. Let x represent the number of shirts and let represent the number of caps. 5. Write an equation to represent the total number of caps and shirts the will purchase. Write an equation to represent the total cost of all the shirts and caps. Rewrite both of our equations in = mx + b form. Graph each equation and to find the solution. What does the solution represent?

23 Activit 7-: Solving Sstems of Linear Equations (4.5) Solve each linear sstem b graphing. Check our answer. 1. = 1 = x 7. = x 5 = x + 4. x = 4 = x = 4x = x + 5. Two skaters are racing toward the finish line of a race. The first skater has a 5 meter lead and is traveling at a rate of 5 meters per second. The second skater is traveling at a rate of 15 meters per second. How long will it take for the second skater to pass the first skater?

24 Activit 7-1: Solving Sstems of Linear Equations (4.5) Solve each linear sstem b graphing. Check our answer. 1. = x + 4 = x 5 = x 6. = x +. x = = x 5 4. = x 6 = x + 5. Two skaters are racing toward the finish line of a race. The first skater has a 1 meter lead and is traveling at a rate of 1 meters per second. The second skater is traveling at a rate of 15 meters per second. How long will it take for the second skater to pass the first skater?