Unit 4. Linear Functions OUTLINE

Transcription

1 Unit 4 Linear Functions This unit formalizes vocabulary and processes involved in finding and analyzing attributes of linear functions. The strategies and routines that you have been developing will enhance your learning. You will examine effective communication strategies while exploring mathematics. At the end of the unit, you will look at data to see if a line can be used to represent them. Then you will create a line that fits these data the best. OUTLINE Topic 10: Understanding slope and intercepts The topic connects the rate of change of a linear function to the slope of the line that is the function s graph. You will continue to use graphs, function rules, tables, and verbal expressions as you investigate linear functions more deeply. In this topic, you will: Explore linear functions using tables, graphs, and function rules Use the connection between constant rate of change and slope to analyze and graph linear functions Use common or first differences to determine if a function is linear Explore the relationship between the x- and y- intercepts of the graph of a linear model and the situation being modeled Understand the effects of changing m or b on the graph of y = mx + b Use slope to classify lines as parallel, perpendicular, or neither Write the equation of a line in different forms (slope- intercept, standard, and point- slope forms) Topic 11: Staying motivated while solving problems In this topic you will investigate motivation in learning situations and discuss how setting goals can help you stay motivated. This theme is supported while you practice the algebraic processes of collecting like terms and applying the distributive property and solve a non- routine problem involving relationships for diagonals of polygons. In this topic, you will: Understand that maintaining motivation while engaged in learning tasks can result in more effective effort Analyze motivation while practicing the algebraic process of collecting like terms Recognize that setting goals can help maintain motivation Analyze goal setting while practicing the algebraic process of applying the distributive property Set useful goals Solve a non- routine problem by applying ideas about equations of lines

2 Topic 12: Creating linear models for data You have seen that understanding aspects of linear relationships, especially their rates of change, can be helpful in many contexts. But in many real- world situations, the data are approximately linear, rather than exactly linear. In this topic, you will explore these situations. By building on your existing knowledge and adding some new tools, you will develop a more powerful understanding of linear relationships. You will be able to solve problems in these new situations. In this topic, you will: Analyze graphs and tables using rate of change to determine whether a linear model is appropriate for the data Identify the strength and direction of correlation for approximately linear data Find a trend line to model a set of data using a manual scatterplot or a graphing calculator Write an equation for a trend line Interpret the meaning of a trend line in the context of a problem situation Transform the parent function y = x to create other linear functions

3 Topic 10: Understanding slope and intercepts 387 UNDERSTANDING SLOPE AND INTERCEPTS Lesson 10.1 Connecting rate of change and slope 10.1 OPENER Terrence is teaching his friend Teresa how to ride a skateboard. To help her practice her control of the skateboard, Terrence uses a special drill. He pulls the board as Teresa balances on it. Terrence uses a motion detector to gather data on Teresa s skateboarding. The data help them analyze this drill. Using the table and graph, make a plan for how you would move Teresa to match the data from the drill. Provide specific information about the starting point and the rate of change you would use to match the data. Start Teresa 0.7 feet from the motion detector, and pull her away from it at a rate of 0.4 feet/second CORE ACTIVITY 1. What is represented along the x- axis? What is represented along the y- axis? Time in seconds is represented along the x-axis, and distance from the motion detector in feet is represented along the y- axis. 2. Before they begin the drill, Terrence tells Teresa, I m going to pull you in a straight line at a steady rate. Use this description of the drill and the table of data created during the drill to answer the following questions. y-intercept a. What words in Terrence s description of the skateboard drill indicate that the distance- time graph will be linear? steady rate b. How can you tell by looking at the time- distance data in the table that the relationship between the variables is linear? Student responses will vary 3. Identify the y- intercept on the graph by drawing an arrow pointing to it. Label the arrow y- intercept.

4 388 Unit 4 Linear functions 4. Here are graphs for skateboard rides. a. Use arrows to label each y- intercept. Then write the y- intercept (the y- value of the point at which the graph crosses the y- axis) below each graph. Student responses may vary slightly A. B. y- intercept: 2.8 y- intercept: 2.1 C. D. y- intercept: 6.8 y- intercept: 4 b. For each graph, write the y- intercept as an ordered pair. Graph A: (0,2.8) Graph B: (0,2.1) Student responses may vary slightly Graph C: (0,6.8) Graph D: (0,4) 5. Here are examples of road signs. Using the vocabulary term slope, describe what these images are telling the driver. The slope is steep. The first sign indicates a steep upward slope and the second sign indicates a steep downward slope.

5 Topic 10: Understanding slope and intercepts Use the table provided to answer the following questions. a. Calculate the first differences for both the x- values and y- values. b. Is the change in y constant? Yes c. Is the change in x constant? Yes d. Use the slope formula to calculate the rate of change, or slope, for the line that models Teresa s drill On the graph shown, continue creating slope triangles to illustrate the rate of change using the data points from Teresa s drill. a. Which part of the slope triangle is represented by the change in y? The vertical (blue) line b. Which part of the slope triangle is represented by the change in x? The horizontal (red) line c. How are the common differences you calculated on the table shown on this graph? They are the lengths of the legs in the right triangles (the blue and red lines). 8. Fill in the blanks with the given vocabulary terms to make the statements correct. linear function slope constant rate of change a. When talking about how quickly or slowly a linear function is changing, you are discussing the function s. rate of change b. The graph of a linear function forms a straight line. The line is straight because the linear function has a constant rate of change. c. When you graph a linear function, slope refers to the steepness of the line the function makes. The slope of this line is the same as the rate of change of the linear function. The slope can be expressed as a decimal, fraction, or integer.

6 390 Unit 4 Linear functions 9. Complete this math journal for the term slope. Vocabulary term My understanding of what the term means An example that shows the meaning of the term Student responses will vary. Student responses will vary. Student responses will vary CONSOLIDATION ACTIVITY 1. Identify the numeric values in the following paragraph. Define those values in the space provided. One example is completed for you. The parents of the members of the baseball team want to raise money for new team uniforms. The parents order team baseball caps and sell them at pep rallies and games. They sell the caps for $10 each. They pay a design fee of $50, plus $2.50 for each baseball cap they order from the manufacturer. $10 Amount collected for each baseball cap sold. $50 design fee (flat fee) $2.50 fee for each hat 2. The parents want to examine how much they will spend on caps. The table and graph show what they will pay for 0 to 5 caps. Use the table and the graph to answer the following questions Number of caps < < < < < Caps Ordered Amount paid in dollars > > > > > Caps Sold a. Find the first differences in the table. b. Draw the slope triangles on the graph of the data. c. What are the vertical and horizontal lengths of the slope triangles for the line? vertical length = 2.5 horizontal length = 1 d. How do the vertical and horizontal lengths of the slope triangles connect to the problem situation? The vertical length of 2.5 represents the additional fee per hat in dollars. The horizontal length of 1 represents 1 hat. e. How do the vertical and horizontal lengths of the slope triangles connect to the first differences of the table? Their lengths are equivalent to the first differences values from the table above.

7 Topic 10: Understanding slope and intercepts The parents also want to look at how much money they will make by selling caps. A table of data is shown. Use this table to answer the following questions < < < < < Caps Sold Number of Amount caps collected in dollars > > > > > a. Graph the data on the coordinate grid shown in question 2. b. Find the first differences in the table and draw the slope triangles on the graph you constructed. c. What are the vertical and horizontal lengths of the slope triangles for the Caps sold line? vertical length = 10 units horizontal length = 1 unit d. How do the vertical and horizontal lengths of the slope triangles connect to the problem situation? 10 vertical units represent $10, the price of 1 hat. 1 horizontal unit represents one hat. e. How do the vertical and horizontal lengths of the slope triangles connect to the first differences of the table? Their lengths are equivalent in value to the first differences of the table to the left. 4. How are the slope triangles in the graph, the first differences in the tables, the rates of change in the problem situation, and the steepness of the two lines connected? They are all ways to think about, calculate, and/or demonstrate the slope of a line. 5. Using the terms slope and y- intercept, explain how the graph shows that the amount of money collected will eventually be more than the amount paid. Express your answer as a complete sentence. The y-intercept of the graph of caps ordered is higher than the y-intercept of the graph of caps sold. However, the graph of caps sold has a greater (steeper) slope than the graph of caps ordered. Eventually, the graph of caps sold will intersect the graph of caps ordered, and any cap sold after this intersection will yield a profit.

8 392 Unit 4 Linear functions HOMEWORK For the stacks of cubes shown, the sides and top of each stack are painted, but not the bottom. a. Use the situation and the data in the graph to create a table. Your table should show the relationship between the number of blocks and number of square faces painted in each tower. b. What is the y- intercept of the graph? (, ) 0 1 # blocks # painted faces c. What is the slope? m = 4 d. Explain how you found the slope. Responses will vary 2. Consider a pizza situation. a. Draw slope triangles on the graph and create a table showing the price in dollars for pizzas that have between 0 and 3 toppings. 1 2 # Toppings Price ($) 1 1 < < 1 < > 0.50 > 0.50 > 0.50 b. What is the y- intercept of the graph in question 5 and what does it represent in this problem situation? c. On the table you created in question 5, show the first differences for both the inputs and the outputs. (0,7) is the y-intercept, it represents the price of a pizza with no toppings. d. Choose two ordered pairs from the table and find the ratio of Δy Repeat the process for two different ordered pairs. Δx to calculate the slope of the line between those points. 0.50/1 = 50 cents e. Look at your answer to question 8. How can you determine if there is a constant rate of change? What is the rate of change and what does that represent in the problem situation? You can determine the rate of change is constant because the graph is linear. The rate of change of 50 cents represents the cost per topping.

9 Topic 10: Understanding slope and intercepts For each of the graphs shown, draw the slope triangles and label the length of each segment of a triangle. Then, calculate the slope of each line and explain how you calculated the slopes. a. b Slope: 1/2 Slope: -2/1 = -2 c. How did you use the slope triangles to calculate the slopes of the lines? Divide the vertical length by the horizontal length. If the line is decreasing, the slope is negative. 4. Here are two tables with statements made by two students. As a member of the classroom learning community, you want to help the students. Correct each statement and figure out where Sara and Darrell may have gone wrong so you can help them understand slope and y- intercept. x y Sara: I can tell from the table that the y- intercept of the graph will be the point ( 2, 0). Correction: The y-intercept occurs where the line crosses the y-axis. This means x must equal 0, not y. The y-intercept displayed in this table is (0,4). a b Darrell: Since the b- values go up by 8 each time, I know that the slope of the line has to be 8. Correction: To find slope, divide the change in b by the change in a. Since a is increasing by 2, the slope is 8/2 = 4.

10 394 Unit 4 Linear functions STAYING SHARP Complete the table for the function rule y = 3x Graph the function rule y = 3x 2 on these axes. Practicing algebra skills & concepts x y Complete the table representing the function rule y = x. 4. Draw a graph representing the function rule y = x. Preparing for upcoming lessons x y Convert each distance measurement to the indicated units. 6. Convert each time measurement to the indicated units. 1 mile = 1760 yards a. 7 feet in inches a. 7 days in hours Reviewing pre- algebra ideas 1 yard = 3 feet 1 foot = 12 inches 1 day = 24 hours 1 hour = 60 minutes 1 minute = 60 seconds 7 (12) = 84 inches b. 108 inches in feet 108/12 = 9 feet c. 20 feet in yards 20/3 = 6 2/3 yards 7 (24) = 168 hours b. 540 seconds in minutes 540/60 = 9 minutes c. 100 minutes in hours 100/60 = 1 2/3 hours

11 Topic 10: Understanding slope and intercepts 395 Lesson 10.2 More about rate of change and slope 10.2 OPENER 1. Choose the appropriate graph for each situation and sketch it in the box next to the description. A. Michael begins with $25, and spends $5 each week. B. Tara gains 1 friend each week. C. Mira runs to the mailbox at a constant rate, moving away from the house. D. José slowly jogs towards the finish line at a constant rate. He stops before he gets to the finish line. E. Margaret just stands and watches the sunset. 2. Looking at your graphs and situation descriptions for Tara and Michael, answer these questions: Tara Michael What is the y- intercept? (, ) 0 25 Is the rate of change constant? (Yes or No) Yes Yes Is the function increasing or decreasing? Increasing Decreasing 3. Write the name Mira, Margaret, or José next to each description. Then justify your choice. Description Name Justification This person s function is neither increasing nor decreasing. This person s rate of change is not constant. The rate of change in this person s graph is high. Margaret José Mira She is not moving. He jogs at a constant rate at first but then stops. She is running; the slope of the graph is steep.

12 396 Unit 4 Linear functions 10.2 CORE ACTIVITY Shawna earns $10 each week helping her grandmother. She decides to save her money in the bank. 1. Complete the table representing the data from Shawna s situation using the answer choices provided. Then indicate the first differences for Time in weeks and Amount saved $10 $20 $30 $40 $50 $60 a t 10t 10a Time in weeks (t) Process Amount saved (a) $10(1) $10 2 $10(2) $20 3 $10(3) $30 4 $10(4) $40 5 $10(5) $50 6 $10(6) $ t $10(7) $10 t = a 2. Create a graph representing the data in the table in question 1. Give your graph a title and label the axes. 3. On your graph from question 2, draw slope triangles to show the changes in y- values and x- values as you move from one week to the next. Use the ratio of Δy Δx Change in y/change in x = 10/1 = $10 to calculate the slope for these data. 4. How does the slope of the line connect to the rate of change in the data? Their values are equivalent 5. Use slope triangles to extend the graph in both directions. Then, find the amount saved for week 0 and for week 7. What do these two points mean in the context of the problem situation? For week 0, the week before she started working, $0 was saved. For week 7, after she had been working a total of 7 weeks, $70 was saved.

13 Topic 10: Understanding slope and intercepts On the graph, draw one slope triangle showing the change from week 1 to week 4. Draw another slope triangle showing the change from week 2 to week Using the slope formula, calculate the rate of change using the points for: a. Week 1 and Week 4 y/ x = 30/3 = b. Week 2 and Week 6 y/ x = 40/4 = 10 c. Is this statement a true statement? Justify your answer. True justifications may vary Regardless of the two points you choose on a line, the ratio of the vertical change to the horizontal change will be the same. 8. The following table shows subscript notation used in finding slope. Fill in the blanks in the table. Subscript notation t 2 a 1 a 2 a 1 t 2 t 1 a 2 a 1 / t 2 t 1 Description The t- term of the second point The a-term of the first point The a- term of the second point minus the a- term of the first point The t-term of the second point minus the t-term of the first point The ratio of the differences of the a- terms and the t- terms 9. What missing x- or y- values would create a point that falls on Shawna s extended savings plan? Verify your answers using the slope formula. a. (2,20) and (8,y 2 ) b. (7,y 2 ) and (x 1,120) Δy y 2 = 80 Δx = = 60 6 = 10 x 1 = 12 y 2 = 70 Δy Δx = = 50 5 = Compare the slope equation used for the problem situation and the slope equation as it is often written. What do you notice about the two slope equations? Responses will vary. Slope equation from the problem situation Slope equation as often written Slope = Δa Δt = a 2 a 1 Slope = Δy t! 2 t 1 Δx = y 2 y 1 x! 2 x REVIEW END-OF-UNIT ASSESSMENT Today you will review the end- of- unit assessment.

14 398 Unit 4 Linear functions HOMEWORK You were just hired to work at a local hardware store earning $200 per week. You decide to put half of your earnings into your savings account. Before you start your job, your savings account has a balance of $300. a. Create a table showing your balance from week 0 to week 4 of your new job. b. Create a graph of the data. Label the y- intercept and draw two slope triangles that have different vertical changes. Week # Balance ($) c. Find the horizontal and vertical changes for each slope triangle you drew in part b. First slope triangle Second slope triangle Vertical change Horizontal change 1 2 d. Calculate the slope from each slope triangle. Δy Δx = 100 = = 100 e. Write a rule that describes your savings, s, after t weeks on the new job. Rule: s = t

15 Topic 10: Understanding slope and intercepts Shawna has a new job working part- time in a restaurant. Shawna earns $120 each week. She decides to put all her earnings in her savings account. The savings account already has a balance of $200 that she earned while working for her grandmother. The table shows Shawna s savings, s, t weeks after starting her new job. a. Construct a graph showing Shawna s savings account balance from her start date to the end of her 6th week on the new job. Time in weeks Savings 0 $200 1 $320 2 $440 3 $560 4 $680 5 $800 6 $920 b. On the table and graph you created, show the first differences and slope triangle assuming you only have data points for week 1 and week 5. Then show the first differences and slope triangle assuming you only have the points for week 3 and week 6. c. Fill in the blanks with the correct coordinates and values. Using data for week 1 and week 5 Using data for week 3 and week 6 (x 1,y 1 ) (1,320) (3,560) (x 2,y 2 ) (5,800) (6,920) y 2 y = = 360 x 2 x = = 3 Slope 480/4 = /3 = 120 d. Write a function rule that represents Shawna s savings account balance, s, after t weeks at her new job. s = 120t e. Where do you see the slope of the graph in the function rule? Where do you see the y- intercept of the graph in the function rule? The slope is found in front of t, and the y-intercept is the 200 that is added.

16 400 Unit 4 Linear functions STAYING SHARP The equation of a line in slope- intercept form is given by y = 4x + 5. Use the table of values to find its slope. 2. Graph the function rule y = 4x + 5 on these axes. Then find the slope from the graph. Practicing algebra skills & concepts x y Preparing for upcoming lessons m = 1 ( 7) 1 ( 3) = 8 2 = 4 Use the graph to answer questions 3 and Which student has the highest grade point average? Answer with justification: Student A, because that student has the highest value on the y-axis 4. Which student was absent the fewest days? Answer with justification: Student A, because that student has the lowest value on the x-axis 5. Find the value of each perfect square. 6. Circle the square roots in the following list that are equal to a whole number. Reviewing pre- algebra ideas Square Value 1 2 = = = = = = = = = = = What is the connection between the numbers you circled in the list and your answers to Question 5? The numbers circled in the list appear as perfect squares in question 5.

17 Topic 10: Understanding slope and intercepts 401 Lesson 10.3 Rate of change and negative slope 10.3 OPENER Manuel worked all summer at an amusement park and saved $1050. At the end of the summer, he stopped putting money into his savings account and instead began making weekly withdrawals. Use the table and graph Manuel created to answer the questions. 1. Describe what is happening to the amount of money in Manuel s savings account. It is decreasing by $50 a week 2. What is the rate of change for the situation? Interpret the rate of change you found in question 2 in the context of the problem situation. It represents the amount of money Manuel takes out of the account each week CORE ACTIVITY 1. In the table, show the differences between week 2 and week Find the slope using the slope formula for the points representing week 2 and week 6. m = = = Write a function rule that represents Manuel s savings account balance, s, after w weeks. s = -50w Use your function rule to find out how much Manuel will have remaining in his savings account after: a. 10 weeks s = -50(10) = $550 b. 15 weeks s = -50(15) = $300

18 402 Unit 4 Linear functions 5. Use the graph to answer the following questions: a. What is the y- intercept? $1050 b. What is the meaning of the y- intercept in this problem situation? Write your answer in a complete sentence. The y-intercept is the amount of money that Manuel had in his savings account at the end of the summer, before he started withdrawing money. 6. Use the table of Manuel s savings to answer the following questions. a. Graph Manuel s savings account balance all the way to 25 weeks or until he has no money left, whichever comes first. b. Label the x- intercept on your graph. What ordered pair describes the x- intercept? ( 21, 0 ) c. Looking at your graph, write a complete sentence to describe the meaning of the x- intercept in this problem situation. After 21 weeks, there is no more money left in Manuel s savings account. d. According to the graph, how much money will Manuel have after 18 weeks? After 18 weeks Manuel will have $150 in his savings account. 7. Use the slope formula to calculate the slope for the following pairs of points on the graph. a. (x 1,y 1 ) = (1,1000) (x 2,y 2 ) = (10,550) m = = b. (x 1,y 1 ) = (6,750) = 50 m = (x 2,y 2 ) = (18,150) 18 6 = = 50

19 Topic 10: Understanding slope and intercepts CONSOLIDATION ACTIVITY 1. Earlier, you used the rule s = 50w to represent the situation in which Manuel spends $50 each week from his starting savings account balance of $1050. Write a rule to show what will happen if he spends $100 each week. s = -100w How would you write this function rule in function notation? s(w) = -100w Use your function rule to determine how much Manuel will have remaining in his savings account after: a. 5 weeks b. 10 weeks c. 11 weeks $550 $50 -$50 (not possible) 4. What do your answers for questions 3b and 3c tell you about Manuel s savings account balance? His account balance will be 0 at some point between 10 and 11 weeks. 5. a. This graph shows Manuel s old spending habits. Draw a graph representing Manuel s new spending habits on the same grid. Draw one slope triangle connecting two points and find the slope of the new graph. b. Identify the x- intercept on the graph for the line that represents Manuel s new spending habits. Then, in a complete sentence, indicate what this point means in the problem situation. m = -300/3 = -100 The x-intercept represents the point in time at which Manuel s savings account is empty (has 0 dollars). 6. Compare the two graphs of Manuel s savings. How are the two graphs the same? How are they different? (You may want to look at the y- intercept, the x- intercept, and the slope.) How is this information related to Manuel s different spending habits? Because both graphs start out with the same amount of money in the savings account, they both have the same y-intercept. They both have negative slopes because Manuel is withdrawing money. Because Manuel withdraws $100 dollars a week in the second graph instead of $50, the second graph has a steeper slope and smaller x-intercept.

20 404 Unit 4 Linear functions 7. The following rules represent saving and spending plans. (s represents the amount of savings in dollars; w represents the number of weeks.) For each rule, write a verbal representation describing what might be happening. Then sketch a graph of the rule. Verbal representation Sketch of graph s = w Start with $1500 and spend $100 a week s = 25w Start with $1100 and deposit $25 a week s = w Start with $500 and deposit $75 a week s = 50w Start with $1400 and spend $50 a week 8. Look back at the rules given in question 7 to answer the following questions. a. Which rule(s) represent savings plans? Which rule(s) represent spending plans? Rules 1 and 4 represent spending plans and rules 2 and 3 represent savings plans. b. Which rule shows the highest rate of savings? c. Which rule shows the highest rate of spending? How do you know? How do you know? Rule 3 shows the highest rate of savings because it has the steepest positive slope. d. Which rule shows the most extreme weekly change? e. Which rule shows the highest initial savings balance? How do you know? Which rule shows the lowest initial savings balance? Rule 1 shows the most extreme weekly change because it has the steepest slope. Rule 1 shows the highest rate of spending because it has the steepest negative slope. Rule 1 shows the highest initial savings balance because it has the largest y-intercept. Rule 3 shows the lowest initial savings balance because it has the smallest y-intercept.

21 Topic 10: Understanding slope and intercepts 405 HOMEWORK Draw a line that passes through the points (10,2) and (4,8). Determine the slope of the line. 2. Sketch a line passing through the point (1,3) with a slope of 2. m = Using specific numbers, describe a situation in which a savings account starts with a positive balance and increases quickly at a constant rate. Use a verbal description, a rule, a table, and a graph. Verbal description Rule Table Graph Student responses will vary; Answer must have a positive y-intercept and positive slope 4. Using specific numbers, describe another situation in which a savings account has a positive starting balance and decreases slowly at a constant rate. Use a verbal description, a rule, a table, and a graph. Verbal description Rule Table Graph Student responses will vary; Answer must have a positive y-intercept and a negative slope

22 406 Unit 4 Linear functions STAYING SHARP 10.3 Practicing algebra skills & concepts 1. When a school fundraiser begins, there are 11 people present. Each minute, 3 more people arrive. Use this information to complete the table. x represents the number of minutes since the fundraiser began and y represents the number of people present. x y The situation described in Question 1 is a linear relationship. Therefore, its graph would be a line. a. What is the slope of the graph of this relationship? m = 3 b. What is the y- intercept of the graph of this relationship? y-intercept = Explain how the graph of y = 0.5x (the solid line on the graph) differs from the graph of y = x (the dashed line). 4. Explain how the graph of y = 2x (the solid line on the graph) differs from the graph of y = x (the dashed line). Preparing for upcoming lessons y = 0.5x has a slope that is less steep than y = x y = -2x has a negative slope while y = x has a positive slope. y = -2x also has a steeper slope than y = x. 5. Convert each distance measurement to the indicated units. 6. Convert each time measurement to the indicated units. 1 mile = 1760 yards a. 2 feet, 3 inches in inches a. 2 hours, 20 minutes in minutes Reviewing pre- algebra ideas 1 yard = 3 feet 1 foot = 12 inches 1 day = 24 hours 1 hour = 60 minutes 2(12) + 3 = 27 inches b. 2 feet, 3 inches in feet 2 + ( 3 / 12 ) = 2.25 feet c. 2 feet, 3 inches in yards 2(60) + 20 = 140 minutes b. 2 hours, 20 minutes in hours 2 + ( 20 / 60 ) = 2 1 / 3 hours c. 2 hours, 20 minutes in days 1 minute = 60 seconds 2.25/3 =.75 yards 2 1 / 3 24 = 7 / 72 days

23 Topic 10: Understanding slope and intercepts 407 Lesson 10.4 m, b, and the graph of y = mx + b 10.4 OPENER 1. Fill in this chart to analyze the three savings account rules you recently developed. a = 10t + 0 Shawna s = 50w Manuel 1 s = 100w Manuel 2 Slope y- intercept Does the function have an increasing or decreasing rate of change? Increasing Decreasing Decreasing 2. If you graphed the functions, which graph, Manuel 1 or Manuel 2, would be steeper? How do you know by looking at the function rules? Manuel 2 would be steeper because it has a larger slope CORE ACTIVITY 1. The line drawn on the grid has a slope, m, equal to 1. Graph and label lines with slopes equal to 3, 0.5, 1, and 0. The lines should pass through the origin. m = -1 m = 3 m = 0 m = Fill in the blanks to complete the statements using the terms provided. horizontal less steep steeper falls vertical rises a. When m is positive, the graph of the line. rises b. When m is negative, the graph of the line. falls c. When m is 0, the line is. horizontal d. When m is larger than 1, the graph is steeper than the graph of y = x. e. When m is between 0 and 1, the graph is less steep than the graph of y = x. f. When m is - 5, the graph is steeper than the graph of y = 2x. g. When m is 1, the graph is less steep than the graph of y = 3x.

24 408 Unit 4 Linear functions 3. Complete this math journal for the term parallel lines. My understanding of what An example that shows Vocabulary term the term means the meaning of the term Responses will vary Responses will vary Responses will vary 4. Sketch two other lines parallel to y = x. Responses will vary, lines must have the same slope. Two examples provided. 5. Work with your partner to write a conjecture about the slopes of parallel lines in box a. You will use box b to revise your conjecture based on class discussion. a. Responses will vary b. Responses will vary 6. Based on what you have discovered about parallel lines, go back to the math journal in question 3 and refine the definition you wrote.

25 Topic 10: Understanding slope and intercepts On the following graphs, label the four lines l 1, l 2, l 3, and l 4. Fill in the chart with the y- intercepts. Do not fill in the slopes yet. Line l 1 l 2 l 3 l 4 y- intercept (0,3) (0,-4) (0,4) (0,-2) Slope / 2-1 / 2 l 3 l 2 l 1 l 4 8. Approximate the slope for each pair of parallel lines. Add the slopes to the table in question Create two graphs, one showing a pair of parallel lines with a positive slope and the other showing a pair of parallel lines with a negative slope. Label your lines l 1, l 2, l 3, and l 4. Responses will vary 10. Complete the table from the graphs you drew for question 9. Line l 1 l 2 l 3 l 4 y- intercept Responses will vary Responses will vary Responses will vary Responses will vary slope Responses will vary Responses will vary Responses will vary Responses will vary y = mx + b Responses will vary Responses will vary Responses will vary Responses will vary 11. Now that you have investigated the slopes of parallel lines, go back to question 5 and, if necessary, add to your conjecture about parallel lines. 12. Using complete sentences, describe the effects that the slope, m, and the y- intercept, b, have on parallel lines. For two lines to be parallel, they must have the same slope (measure of steepness) and different y-intercepts. The y-intercepts determine how far apart the parallel lines are.

26 410 Unit 4 Linear functions 10.4 CONSOLIDATION ACTIVITY 1. Graph each pair of lines. Then, determine whether each set of lines is parallel. (Circle Yes or No. ) Provide a justification for each of your choices, using the words slope and/or y- intercept. a. y =.75x + 1 and y =.75x 2 b. y =.5x + 3 and y = 1.25x 5 Are these lines parallel? Yes No How do you know by looking at the graph? They travel in the same direction at the same rate and they don t intersect. How do you know by looking at the equation of the line? They have the same slope and different y-intercepts. Are these lines parallel? Yes No How do you know by looking at the graph? They travel at different rates and they intersect. How do you know by looking at the equation of the line? They have different slopes. 2. Find the equation of the line that is graphed. Then, construct a line on the same graph that is parallel to the existing line. What is the equation of your new line? 3. Graph the function y = 3x + 5. On the same grid, graph the function that describes the effect of decreasing the y- intercept of the original function by 7. y = 2x - 3 y = 2x 4. What is the equation of the line that is parallel to the line y = x 4 but has a y- intercept 9 units above? y = - 1 / 3 x What is the equation of the line that passes through the point (0, - 1) and is parallel to the line represented by the equation y = 2x 4? y = 2x 1

27 Topic 10: Understanding slope and intercepts 411 HOMEWORK On the grid, draw two parallel lines with slopes of. Then complete the table. l 1 Responses will vary, but the slope of each line should be ¾. Line l 1 l 2 l 2 y- intercept 3 0 slope ¾ ¾ y = mx + b y = ¾ x + 3 y = ¾ x 2. For questions a- h, place a circle around the pairs of algebraic rules whose graphs are parallel lines. a. y = 2x 9 b. y = x + 9 c. y = - 3x + 1 d. - 4x 6 = y y = 5 2x y = 0.5x 2 y = - 3x y = 21 4x e. y = x + 6 f. y = x + 7 g. y = 11 h x = y y = x + 6 y = x 8 y = 11x y = On the grid, sketch a line that passes through the intercepts (4,0) and (0,- 2). Sketch a second line parallel to the first line with a y- intercept of (0,4). At what point does the second line cross the x- axis? (-8,0) 4. Calculate the slopes of both lines to confirm your answer and write the equations for the lines in slope- intercept form, y = mx + b. slope of line one: = slope of line two: 2 4 = ( 8) = 4 8 = 1 2 Equation of line one: y = ½ x -2 Equation of line two: y = ½ x + 4

28 412 Unit 4 Linear functions STAYING SHARP This graph represents data for a building expansion project. 2. Examine the graph in question 1. Practicing algebra skills & concepts What is the slope of the line? m = 3 What is the y- intercept of the line? (0, 15) Explain the meaning of the slope of the graph in the context of the building situation. Every month 3 stories are added to the skyscraper. Explain the meaning of the y- intercept of the graph in the context of the situation. Before the project started, the building was 15 stories. 3. Data for eight students are plotted on the graph. Which statement about the data is most accurate? Preparing for upcoming lessons a. The trend (or pattern) is roughly linear and increasing (the more absences, the higher the grade point average tends to be). b. The trend (or pattern) is roughly linear and decreasing (the more absences, the lower the grade point average tends to be). c. There is no real pattern to the data. 4. Explain why you chose your answer in question 3. The line demonstrates that there is a negative slope, indicating that as absences increases, grade point average decreases. 5. Which is larger, 9 or? Show evidence. 6. Which point on the number line best represents the position of!! 5? Reviewing pre- algebra ideas = 3 and 9 > 3; so 9 > 3 = 9. So, 9 is larger than 9. Which is larger, 7 or? Show evidence. > which means that 3 > then 7 > 3 >. So, 7 is larger than. Answer with supporting evidence: Point C best represents the square root of five on the number line since 4 < 5 < 9 2 < 5 < 3

29 Topic 10: Understanding slope and intercepts 413 Lesson 10.5 Slope and perpendicular lines 10.5 OPENER 1. Use your protractor to measure the four angles created at the intersection of the two lines in the graph. Write the angle measurement at each angle. Label one line l 1 and the other l 2. l 2 l l 90º 90º 90º 90º 2. Write a sentence to describe the relationship of these two lines. Use as many of these math vocabulary terms as you can: right angles, intersection, lines, and perpendicular. Lines l 1 and l 2 are perpendicular, meaning the angles formed where they intersect are all right angles CORE ACTIVITY 1. Complete this math journal for the term perpendicular lines. Vocabulary term My understanding of what the term means A visual example that shows the meaning of the term Responses will vary Responses will vary Responses will vary

30 414 Unit 4 Linear functions 2. Label the two perpendicular lines l 1 and l 2. Draw one slope triangle on each line. a. What do you notice about these triangles? The two triangles are similar. l l b. Use the slope triangles to determine the slope of each line. Slope l 1: 4 Slope l 2: -1/4 l 2 3. Label the two perpendicular lines l 3 and l 4. Draw one slope triangle on each line. a. What do you notice about these triangles? l 4 l 3 b. Use the slope triangles to determine the slope of each line. Slope l 3: 2 Slope l 4: -1/2 c. Do you think that, for any two perpendicular lines representing linear functions, one line will always have a positive slope and one line will have a negative slope? Why? Student responses will vary. 4. Fill in the chart using the graphs in questions 2 and 3. Line l 1 l 2 l 3 l 4 y- intercept slope 4-1/4 2-1/2

31 Topic 10: Understanding slope and intercepts Work with your partner to write a conjecture about the slopes of perpendicular lines in box a using the vocabulary term opposite reciprocal. You will use box b to revise your conjecture based on class discussion. a. Responses will vary b. Responses will vary 6. Fill in the blanks to complete the statements. a. When m of the original line is 2, m of the parallel line is. 2 b. When m of the original line is, m of the parallel line is. c. When m of the original line is 2, m of the perpendicular line is. d. When m of the original line is, m of the perpendicular line is. -2 ½ -½ 10.5 CONSOLIDATION ACTIVITY 1. Fill in the chart to find the slope of a line perpendicular to each original line. Slope of original line Slope of line perpendicular to original line 6-8/35-1/10 7/37 2. Write the equation of the line that passes through the point (0,2) and is perpendicular to the line y = x 4. y = -4/3 x Janna is asked to determine the slope of a line that is perpendicular to the line y = 5x 10. She responds that the slope is - 5. Is Janna correct? If so, how do you know? If not, help Janna understand why she is not correct and what the correct answer should be. When two lines are perpendicular, the products of their slops should be -1. For Janna, 5(-5) = -25, not -1, so she is incorrect. Janna remembered that perpendicular lines slopes have opposite signs, but she forgot to take the reciprocal. The correct slope is -1/5.

32 416 Unit 4 Linear functions 4. Three lines are described in the first column of the table. Use the information to complete the chart. Then graph all three lines on the grid provided. Slope, m y- intercept, b y = mx + b Original line, l 1 2/3-4 y = x 4 Line l 2, perpendicular to the original and passing through the point (0,6) Line l 3, parallel to the original and passing through the origin -3/2 6 y = -3/2 x + 6 2/3 0 y = 2/3 x l 3 l 1 l 2 5. Is l 3 perpendicular to l 2? Justify your answer. l 3 is perpendicular to l 2 because it has a slope that is the reciprocal of l 2 s slope with the opposite sign.

33 Topic 10: Understanding slope and intercepts 417 HOMEWORK On the grid, draw a line with a slope of and label it l 1. Then sketch a second line perpendicular to the first line and label it l 2. Finally, complete the table. Line l 1 l 2 y- intercept 0 4 Slope 3/4-4/3 y = mx + b y = ¾ x y = - 4 / 3 x For questions a- h, place a circle around the pairs of algebraic rules whose graphs are perpendicular lines. a. y = 2x 9 y = 5 x b. y = x + 9 y = 0.5x 2 c. y = - 3x + 1 y = 3x d. x 6 = y y = 21 + x e. y = x + 6 f. y = x + 7 g. y = 5 h x = y y = x + 6 y = x 8 y = x y = - 0x On the graph, sketch one line that passes through the intercepts (4,0) and (0,- 2). Sketch another line perpendicular to this line with a y- intercept of (0,4). Where does the second line cross the x- axis? (0,2) 4. Use slope calculations to prove that the lines you sketched in question 3 are perpendicular lines. Line one: m = = 2 4 = 1 2 Line two: m = = 4 2 = 2

34 418 Unit 4 Linear functions STAYING SHARP 10.5 Practicing algebra skills & concepts 1. Write the equation of the line in slope- intercept form. y = x 2. Answer these questions about the situation represented by the graph in Question 1. a. How many spectators are left 7 minutes after the end of the game? 60 spectators b. When does the last spectator leave? (How many minutes after the end of the game?) 13 minutes after the end of the game 3. Explain how the graph of y = x + 3 (the solid line) differs from the graph of y = x (the dashed line). 4. Explain how the graph of y = x 4 (the solid line) differs from the graph of y = x (the dashed line). Preparing for upcoming lessons y = x +3 has the same slope as y = x, but it has a different y-intercept (it is shifted up 3 units) 5. Find each total distance. y = x 4 has the same slope as y = x, but it has a different y-intercept (it is shifted down 4 units) 6. Find each total time. 1 mile = 1760 yards a. 21 feet + 5 yards a. 3 days + 31 hours Reviewing pre- algebra ideas 1 yard = 3 feet 1 foot = 12 inches 1 day = 24 hours 1 hour = 60 minutes (21/3) + 5 = = 12 yards OR (3) = = 36 feet b. 32 inches + 1 foot + 2 yards (12) + 2(3)(12) = = 116 inches 3(24) + 31 = 103 hours b. 3 hours + 71 minutes seconds 3(60)(60) + 71(60) = 15,180 seconds 1 minute = 60 seconds

35 Topic 10: Understanding slope and intercepts 419 Lesson 10.6 Intercepts and standard form 10.6 OPENER Use the graph to answer the following questions. y-intercept x-intercept 1. Identify the point on the line that represents the y- intercept. What are the coordinates of this point? (0,6) 2. Identify the point on the line that represents the x- intercept. What are the coordinates of this point? (4,0) 3. Use the x- and y- intercepts to calculate the slope of the line = 6 4 = 3 2 m = -3/2 4. What is the function rule that represents this line? y = - 3 / 2 x CORE ACTIVITY 1. Gas flows into Alex s tank at a steady rate. Explain the relationship between the height of the float in Alex s gas tank and the number of gallons in the tank. As the height of the float increases, the number of gallons in the tank increases. They both increase at constant rates. 2. Here is a graph modeling the relationship between the height of the float in Alex s tank and the number of gallons of gas in the tank. a. What is the y- intercept of this graph? Explain what the y- intercept means in this situation. The y-intercept is (0, 1.5). This indicates that when the height of the float is 0 inches, there are 1.5 gallons in the tank. b. What is the slope of this graph? Explain what the slope means in this situation. The slope of the graph is For every inch the float increases, there are 1.05 more gallons in the tank. c. Use the points (0, 1.5) and (10, 12) from the graph to find the slope of the line. Then use the slope to build a function rule that represents the relationship between g, the number of gallons in the tank, and h, the height of the float

36 420 Unit 4 Linear functions 3. Label the y- intercept and x- intercept on the graph. What are the coordinates of these two points and what do the values of these intercepts represent in the context of the club s profit? y- intercept: (0,-200) x-intercept x- intercept: (100,0) y-intercept 4. What is the slope of the line? What does the slope mean in this situation? The slope of the line is 2. For every bar sold $2 are made in profit. 5. In your own words, explain how you can find the y- intercept when you are given a function rule in slope intercept form. You can substitute 0 for x and solve for y. 6. Write the steps for finding the y- intercept for the linear equation 15x 5y = 30, written in the standard form. Justify each step. Step 15x 5y = 30 Original linear equation Justification 15(0) 5y = 30 x = 0 at the y- intercept. 0 5y = 30 15(0) = 0-5y -5 = 30-5 Divide each side by -5 to isolate y y = = Now, write the steps for finding the x- intercept for the linear equation 15x 5y = 30, written in the standard form. Justify each step. Step 15x 5y = 30 15x 5(0) = 30 Original linear equation y = 0 at the x intercept Justification 15x 0 = = 0 15x 15 = Divide each side by 15 to isolate x x = = 2

37 Topic 10: Understanding slope and intercepts List the two intercepts as points and graph the line. 2 x- intercept: (, ) 0 2 y- intercept: (, ) Use the intercepts and the slope formula to find the slope of the line. Check your answer by drawing a slope triangle and calculating the ratio of the vertical change to the horizontal change = 6 2 = 3 m = 3 Δy Δx = 6 2 = 3 m = CONSOLIDATION ACTIVITY 1. Determine the x- intercept and the y- intercept for each linear equation below. Write the intercepts as coordinate pairs. Then, use these points to calculate the slope of the line. x- intercept y- intercept slope a. 5x 3y = - 15 (-3,0) (0,5) b. x + 4y = 8 (8,0) (0,2) c. - 3x + 2y = 9 (-3,0) (0, 9 / 2 ) d. 5x + 4y = 2 ( 2 / 5,0) (0, 1 / 2 ) 1 e. -! 3 x 2! 3 y = - 4 (12,0) (0,6) ( 3) = = 2 8 = ( 3) = = = = = = = 6 12 = 1 2

38 422 Unit 4 Linear functions 2. Consider the linear equation 3x 2y = - 6. Determine the x- intercept and the y- intercept and use them to graph the line and find the slope of the line. x-intercept: (-2,0) y-intercept: (0,3) slope: 3/2 3. Earlier in this course, you worked on the Bike and Skateboard Problem. Uncle Eddie repairs skateboards and bicycles and needs to order more wheels for his shop. He orders a total of 48 wheels. a. Define the variables in this problem. s = # of skateboards, b = # of bicycles b. Write a linear equation in standard form that represents the number of skateboards, s, and bikes, b, in the shop if 48 wheels are needed. 2b + 4s = 48 c. Graph the equation you wrote in part b. What are the intercepts of the graph? s 12 Intercepts: (24,0) and (0,12) 8 4 Note: Students may have graphed b on the vertical axis and s on the horizontal axis. If so their answers on questions 3c and 3d should reflect that choice d. What do the intercepts mean in the context of the problem? b The point (0,12) represents the number of skateboards in the shop if there are 0 bikes, and the point (24,0) represents the number of bikes in the shop if there are 0 skateboards.

39 Topic 10: Understanding slope and intercepts 423 HOMEWORK Use your number sense to find the values for x and y that satisfy the equations. a. 2x = 8 x = 4 b. 4y = 8 y = 2 c. - 6x = - 30 x = 5 d. - 5y = - 30 y = 6 e. - 3x = 0 x = 0 f. 4y = 0 y = 0 g. 4x = 2 x = 1/2 h. - 6y = 2 y = -1/3 2. Consider the linear equation 5x + 2y = Determine the x- intercept and the y- intercept and use them to graph the line and find the slope of the line. (, ) 0-10 (, ) -4 0 m = 0 ( 10) 4 0 = 10 4 = Suppose you have a line with an x- intercept of (2,0) and a y- intercept of (0,- 6). a. Explain how you could use this information to help you write a linear equation in slope- intercept form. Responses will vary b. Write the equation of the line in slope- intercept form = 6 2 = 3 y = 3x - 6 c. Sketch a graph of the line. d. Use the graph to write the equation of the line in standard form. -6x + 2y = -12