Essential Question: How can you represent a linear function in a way that reveals its slope and a point on its graph?

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1 COMMON CORE y - y 1 x - x 1 Locker LESSON 6. Point-Slope Form Name Class Date 6. Point-Slope Form Common Core Math Standards The student is expected to: COMMON CORE A-REI.D.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane Also A-CED.A. Mathematical Practices COMMON CORE MP. Reasoning Language Objective Explain to a partner how to write a linear function in point-slope form. ENGAGE Essential Question: How can you represent a linear function in a way that reveals its slope and a point on its graph? Possible answer: You can determine the slope m of the graph of the function and the coordinates ( x 1, y 1 ) of a point on the graph, then write the equation y - y 1 = m (x - y 1 ), which is called the point-slope form of the equation. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and how the concept of slope applies to snowboarding and can be used to predict the snowboarder s height on the slope at any one specific time. Then preview the Lesson Performance Task. Essential Question: How can you represent a linear function in a way that reveals its slope and a point on its graph? Explore Deriving Point-Slope Form Resource Locker Suppose you know the slope of a line and the coordinates of one point on the line. How can you write an equation of the line? A line has a slope m of 4, and the point (, 1) is on the line. Let (x, y) be any other point on the line. Substitute the information you have in the slope formula. Use the Multiplication Property of Equality to get rid of the fraction. Reflect y - y 1 m = x - x 1 (x - ) = (y - ) 1. Discussion The equation that you derived is written in a form called point-slope form. The equation y = x + 1 is in slope-intercept form. How can you rewrite it in point-slope form? The slope is. The y-intercept is 1, so the point (0, 1) is on the line. You can use the Explain 1 Creating Linear Equations Given Slope and a Point Point-Slope Form The line with slope m that contains the point ( x 1, y 1 ) can be described by the equation. Example 1 Write an equation in point-slope form for each line. Slope is 3.5, and (-3, ) is on the line. y - = 3.5 (x - (-3) ) y - = 3.5 (x + 3) Substitute. y = x - (x - ) ( ) = y (x - ) x method of the Explore, that is, substitute the known values in the slope formula, write the equation without a fraction, and simplify the result. y - 1 = (x - 0) Slope is 0, and (-, -1) is on the line. (-1) 0 (-) y - = ( x - ) Substitute. y + 1 = 0 Module 6 49 Lesson Name Class Date 6. Point-Slope Form Essential Question: How can you represent a linear function in a way that reveals its slope and a point on its graph? A-REI.D.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane Also A-CED.A. Explore Deriving Point-Slope Form Resource Suppose you know the slope of a line and the coordinates of one point on the line. How can you write an equation of the line? A line has a slope m of 4, and the point (, 1) is on the line. Let (x, y) be any other point on the line. Substitute the information you have in the slope formula. Use the Multiplication Property of Equality to get rid of the fraction. Reflect (x - ) = ( m = y - = x - y - (x - ) ) x - (x - ) = (y - ) 1. Discussion The equation that you derived is written in a form called point-slope form. The equation y = x + 1 is in slope-intercept form. How can you rewrite it in point-slope form? Explain 1 Creating Linear Equations Given Slope and a Point Point-Slope Form The line with slope m that contains the point ( x 1, y 1 ) can be described by the equation. Example 1 Write an equation in point-slope form for each line. Slope is 3.5, and (-3, ) is on the line. y - = 3.5 (x - (-3) ) Substitute. y - = 3.5 (x + 3) The slope is. The y-intercept is 1, so the point (0, 1) is on the line. You can use the method of the Explore, that is, substitute the known values in the slope formula, write the equation without a fraction, and simplify the result. y - 1 = (x - 0) Slope is 0, and (-, -1) is on the line. (-1) 0 (-) 1 0 y - = ( x - ) Substitute. y + = Module 6 49 Lesson HARDCOVER PAGES 03 1 Turn to these pages to find this lesson in the hardcover student edition. 49 Lesson 6.

2 Reflect. Communicate Mathematical Ideas Suppose that you are given that the slope of a line is 0. What is the only additional information you need to write an equation of the line? Explain. The y-coordinate y 1 of any point on the line. A line with slope 0 is horizontal. Every point has the same y-coordinate, and the equation of the line is y = y 1. EXPLORE Deriving Point-Slope Form Your Turn Write an equation in point-slope form for each line. 3. Slope is 6, and (1, ) is on the line. 4. Slope is 1, and 3 (-3, 1) is on the line. Explain Creating Linear Models Given Slope and a Point You can write an equation in point-slope form to describe a real-world linear situation. Then you can use that equation to solve a problem. Example y - = 6 (x - 1) Solve the problem using an equation in point-slope form. Paul wants to place an ad in a newspaper. The newspaper charges $10 for the first lines of text and $3 for each additional line of text. Paul s ad is 8 lines long. How much will the ad cost? Let x represent the number of lines of text. Let y represent the cost in dollars of the ad. Because lines of text cost $10, the point (, 10) is on the line. The rate of change in the cost is $3 per line, so the slope is 3. Write an equation in point-slope form. y - 1 = 1 (x + 3) 3 INTEGRATE TECHNOLOGY Students have the option of completing the activity either in the book or online. CONNECT VOCABULARY Point out that the names slope-intercept and point-slope tell students what types of information they will use to express a linear relationship. EXPLAIN 1 Creating Linear Equations Given Slope and a Point y 10 = 3 (x ) Substitute 3 for m, for x 1, and 10 for y 1. To find the cost of 8 lines, substitute 8 for x and solve for y. y - 10 = 3 (8 - ) Substitute y 10 = 18 y = 8 The cost of 8 lines is $8. INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Show students that they can graph a line starting at any point on the line if they know the slope, by counting vertically and horizontally from that point. Show how point-slope form comes from the slope formula, and show how it also simplifies to slope-intercept form. Module 6 50 Lesson PROFESSIONAL DEVELOPMENT Integrate Mathematical Practices This lesson provides an opportunity to address Mathematical Practice MP., which calls for students to reason abstractly and quantitatively. Students learn how point-slope form is related to slope-intercept form, and how to decide which form works best in a situation. QUESTIONING STRATEGIES Could two different equations in point-slope form represent the same line? Yes; choosing two different points on a line would result in two different equations in point-slope form, although the slope would be the same. Point-Slope Form 50

3 EXPLAIN Paul would like to shop for the best price to place the ad. A different newspaper has a base cost of $15 for 3 lines and $ for every extra line. How much will an 8-line ad cost in this paper? Creating Linear Models Given Slope and a Point INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP. Remind students that once they have the point-slope equation of a line, they can find the value of y if given a value of x, or vice versa, by substituting and solving for the other variable. QUESTIONING STRATEGIES If the independent variable is the number of items and the dependent variable is the weight of the items in pounds, what would the equation in point-slope form represent? y minus the weight in pounds of a number of items would be equal to the product of the weight in pounds per item and the quantity x minus the number of items. EXPLAIN 3 Creating Linear Equations Given Two Points QUESTIONING STRATEGIES Can you write a point-slope equation for a vertical line? Explain. No; because a vertical line has rise, but no run, it has undefined slope. y - 15 = (x - 3 ) Substitute. y - = ( - ) Substitute for x. y - 15 = 10 Simplify the right side. y = 5 Solve for y. The cost of 8 lines is $ 5. Reflect 5. Analyze Relationships Suppose that you find that the cost of an ad with 8 lines in another publication is $18. How is the ordered pair (8, 18) related to the equation that represents the situation? How is it related to the graph of the equation? The ordered pair is a solution of the equation. It represents a point on the graph of the equation. Your Turn 6. Daisy purchases a gym membership. She pays a signup fee and a monthly fee of $11. After 4 months, she has paid a total of $59. Use a linear equation in point-slope form to find the signup fee. y 59 = 11 (x 4) y 59 = 11 (0 4) y = 15 The signup fee is $15. Explain 3 Creating Linear Equations Given Two Points You can use two points on a line to create an equation of the line in point-slope form. There is more than one such equation. Example Write an equation in point-slope form for each line. (, 1) and (3, 4) are on the line. Let (, 1) = ( x 1, y 1 ) and let (3, 4) = ( x, y ). Find the slope of the line by substituting the given values in the slope formula. y - y 1 m = x - x 1 = = 3 3 You can choose either point and substitute the coordinates in the point-slope form. y - 1 = 3 (x - ) Substitute 3 for m, for x 1, and 1 for y 1. Or: y - 4 = 3 (x - 3) Substitute 3 for m, 3 for x 1, and 4 for y 1. Module 6 Lesson Can you write a point-slope equation for a horizontal line? Explain. Yes; the slope would be 0, so the equation would be in the form y - y 1 = 0. COLLABORATIVE LEARNING Small Group Activity Have students work with in groups of three. Have each student write the coordinates for a point and the slope of a line. Remind students to keep their values reasonable. Students pass their data to the right. The next student writes the point-slope form for the line. When students are finished, pass the papers to the right once more. Now, students match the form to the data; when they are satisfied it is correct, they graph the line. Lesson 6.

4 (1, 3) and (, 3) are on the line. Let (1, 3) = ( x 1, y 1 ) and let (, 3) = ( x, y ). Find the slope of the line by substituting the given values in the slope formula. y - y 1 m = x - x 1 Reflect 3-3 = - 1 = Choose either point and substitute the coordinates in the point-slope form. y - 3 = 0 (x - 1 ) Substitute 0 for m, 1 for x 1, and 3 for y 1. Or: y - y = m (x - x ) y - 3 = 0 (x - ) Substitute 0 for m, for x, and 3 for y. 7. Given two points on a line, Martin and Minh each found the slope of the line. Then Martin used ( x 1, y 1 ) and Minh used ( x, y ) to write the equation in point-slope form. Each student s equation was correct. Explain how they can show both equations are correct. Martin can show that ( x, y ) is a solution of his equation, and Minh can show that Your Turn Write an equation in point-slope form for each line. 8. (, 4) and (3, 1) are on the line. 9. (0, 1) and (1, 1) are on the line. (y - 1) = -3 (x - 3) or (y - 4) = -3 (x - ) (y - 1) = 0 (x - 0) or (y - 1) = 0 (x - 1) Explain 4 Creating a Linear Model Given Two Points In a real-world linear situation, you may have information that represents two points on the line. You can write an equation in point-slope form that represents the situation and use that equation to solve a problem. Example 4 0 ( x 1, y 1 ) is a solution of hers. Solve the problem using an equation in point-slope form. An animal shelter asks all volunteers to take a training session and then to volunteer for one shift each week. Each shift is the same number of hours. The table shows the numbers of hours Joan and her friend Miguel worked over several weeks. Another friend, Lili, plans to volunteer for 4 weeks over the next year. How many hours will Lili volunteer? Volunteer Weeks worked Hours worked Joan 6 15 Miguel 10 3 Image Credits: Mila Supinskaya/Shutterstock AVOIDING COMMON ERRORS Students sometimes substitute for the variables in the opposite order, assuming that x comes first as in the ordered pair (x, y). Remind them that the value of y is substituted on the left side of the equation, while the value of x is substituted on the right side. EXPLAIN 4 Creating Linear Models Given Two Points QUESTIONING STRATEGIES How do you determine the intercepts given two points? Use the points to find the slope. Replace y with 0 and solve for x. Solve the slope-intercept form for y. Module 6 5 Lesson DIFFERENTIATE INSTRUCTION Auditory Cues Students will benefit from saying the point-slope equation aloud when writing it. They might say, for example: y minus y one equals m times the quantity x minus x one, y minus y sub one equals m times the quantity x minus x sub one, y minus y one equals the product of m and x minus x one. Be sure they state it in a way that emphasizes the distribution of m. Point-Slope Form 5

5 Analyze Information Identify the important information. Joan worked for 6 weeks for a total of 15 hours. Miguel worked for 10 weeks for a total of 3 hours. Lili will work for 4 weeks. Formulate a Plan To create the equation, identify the two ordered pairs represented by the situation. Find the slope of the line that contains the two points. Write the equation in point-slope form. Substitute the number of weeks that Lili works for x to find y, the number of hours that Lili works. Let x represent the number of weeks worked and y represent the number of hours worked. The points (6, 15) and (10, 3) are on the line. Substitute the coordinates in the slope formula to find the slope. y - y 1 m = x - x m = 10-6 m = Next choose one of the points and find an equation of the line in point-slope form. y - 15 = (x - 6 ) Substitute for m, 6 for x 1, and 15 for y 1. Or: y - y = m (x - x ) y - 3 = (x - 10 ) Substitute for m, 10 for x, and 3 for y. Finally, substitute 4 in the equation to find y. y - 15 = (x - 6 ) Substitute for m, 6 for x 1, and 15 for y y - = ( - ) Substitute for x. y - = ( ) y = Or: y - 3 = (x - 10 ) Substitute for m, 10 for x, and 3 for y y - = ( - ) Substitute for x y - = ( ) 3 y = 14 Lili will work a total of hours. Module 6 53 Lesson LANGUAGE SUPPORT Connect Vocabulary Explain to students how some English words are formed from two hyphenated words, such as real-world, family-size, and in this lesson, point-slope. The hyphen in these words links the two source words together. Together, point-slope is one compound word that describes the form of a linear equation. Contrast this with the slope-intercept form of a line, pointing out that each compound adjective describes the information in its equation. 53 Lesson 6.

6 Justify and Evaluate The ordered pair (, ) is a solution of both equations obtained using the 4 given information. y - 15 = (x - 6 ) Substitute for m, 6 for x 1, and 15 for y = ( - ) Substitute for x and for y. 36 = 36 Or: y - 3 = (x - 10 ) Substitute for m, 10 for x, and 3 for y. 4 - = ( - ) Substitute for x and for y = 8 The answer makes sense because the rate of change in the number of hours is the slope,. Because Lili will work 14 more weeks than Miguel, she will work (14) 3 + hours, or hours. Your Turn Solve the problem using an equation in point-slope form. 10. A gas station has a customer loyalty program. The graph shows the amount y dollars that two members paid for x gallons of gas. Use an equation in point-slope to find the amount a member would pay for gallons of gas m = = A member would pay $61 for gallons of gas. y - 10 = 3 (x - 5) y - 10 = 3 ( - 5) y = A roller skating rink offers a special rate for birthday parties. On the same day, a party for 10 skaters cost $107 and a party for 15 skaters cost $137. How much would a party for 1 skaters cost? m = = 6 A party for 1 skaters would cost $119. Cost ($) y = 6 (x - 10) y = 6 (1-10) y = 119 y (5, 10) 5 0 (18, 49) Amount (gal) x Module 6 54 Lesson Point-Slope Form 54

7 ELABORATE QUESTIONING STRATEGIES How is the point-slope form related to the slope formula? The point-slope form is just the slope formula with the denominator (x-values) moved to the other side of the equal sign. INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Ask students to describe the graph of a linear equation with a point-slope form of y - 3 = 3 (x - 1) without simplifying the equation. Students should recognize that the slope is 3 and the graph goes through (1, 3), so the graph is the line with slope 3 going through the origin, or y = 3x. Elaborate 1. Can you write an equation in point-slope form that passes through any two given points in a coordinate plane? No; you can t write a linear equation given two points with the same x-coordinate. A line through two such points is vertical and has no slope. 13. Compare and contrast the slope-intercept form of a linear equation and the point-slope form. Possible answers: Both forms of the equation reveal the slope. The point-slope form explicitly reveals a point ( x 1, y 1 ) on the line. The slope-intercept form reveals a point on the line, but not explicitly. It is the point (0, b) where the graph intersects the y-axis. Both can be used fairly easily to graph the function. You can plot (0, b) or ( x 1, y 1 ), and then use the slope to plot a second point. The slope-intercept form can be used to graph an equation on a graphing calculator. 14. Essential Question Check-In Given a linear graph, how can you write an equation in point-slope form of the line? Find the slope m (if it is defined) by identifying two points on the line and using the slope formula. Then substitute the slope and the coordinates of one of the points in the point-slope form. Evaluate: Homework and Practice SUMMARIZE THE LESSON How do you write linear equations in point-slope form if you know a point and the slope, or if you know two points? The point-slope form,, uses the x- and y-coordinates of a point and the slope. If you are given two points, use the slope formula to find the slope. Substitute the slope and the coordinates of a given point into the point-slope form. 1. Is the equation y + 1 = 7 (x + ) in point-slope form? Justify your answer. Yes; the equation is equivalent to y - (-1) = 7 (x - (-) ). Write an equation in point-slope form for each line.. Slope is 1 and (-, -1) is on the line. 3. Slope is -, and (1, 1) is on the line. y + 1 = 1 (x + ) y - 1 = (-) (x - 1) 4. Slope is 0, and (1, ) is on the line. 5. Slope is 1, and (1, ) is on the line. 4 y - = 0 (x - 1) y - = ( 1 4 )(x - 1) 6. (1, 6) and (, 3) are on the line. 7. (-1, 1) and (1, -1) are on the line. 3 6 m = 1 = 3 y y 1 = m (x x 1 ) y 6 = ( 3) (x 1) or y 3 = ( 3) (x ) m = (-1) (-1) = - = -1 y y 1 = m (x x 1 ) Online Homework Hints and Help Extra Practice y 1 = (-1) (x + 1) or y + 1 = (-1) (x - 1) Module 6 55 Lesson 55 Lesson 6.

8 8. (7, 7) and ( 3, 7) are on the line. 9. (0, 3) and (, 4) are on the line. 7 7 m = 0 ( 3) 7 = 10 = 0 m = = y y 1 = m (x x 1 ) y 7 = 0 (x 7) or y 7 = 0 (x + 3) Solve the problem using an equation in point-slope form. 10. An oil tank is being filled at a constant rate. The depth of the oil is a function of the number of minutes the tank has been filling, as shown in the table. Find the depth of the oil one-half hour after filling begins. 5-3 m = 10-0 = 1 5 y y 1 = m (x x 1 ) y 3 = 1 (x 0) or y 5 = 1 (x 10) 5 5 y 3 = 1 (30 0) 5 y 3 = 6 y = 9 One-half hour after filling begins, the depth of the oil is 9 feet. y y 1 = m (x x 1 ) y 3 = ( 1 ) (x - 0) or y - 4 = ( 1 )(x - ) Time (min) Depth (ft) James is participating in a 5-mile walk to raise money for a charity. He has received $00 in fixed pledges and raises $0 extra for every mile he walks. Use a point-slope equation to find the amount he will raise if he completes the walk. y - 00 = 0 (5-0) y - 00 = 100 y = 300 If he finishes the race, James will raise $300. EVALUATE ASSIGNMENT GUIDE Concepts and Skills Explore Deriving Point-Slope Form Example 1 Creating Linear Equations Given Slope and a Point Example Creating Linear Models Given Slope and a Point Example 3 Creating Linear Equations Given Two Points Example 4 Creating A Linear Model Given Two Points Practice Exercise 1 Exercises 5 Exercises 10 1, 16 0 Exercises 6 9, 1 Exercises 13 15, Keisha is reading a 35-page book at a rate of 5 pages per day. Use a point-slope equation to determine whether she will finish reading the book in 10 days. y - 35 = -5(10-0) y - 35 = -50 y = 75 No; she will still have 75 pages left to read after 10 days. INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP. Encourage students to check their answers to make sure they are reasonable. When possible, students should estimate answers before working out the problems. Module 6 56 Lesson Exercise Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices 1 3 Strategic Thinking MP.3 Logic 5 1 Recall of Information MP.5 Using Tools 6 9 Skills/Concepts MP.5 Using Tools Skills/Concepts MP.4 Modeling 16 3 Strategic Thinking MP.4 Modeling Skills/Concepts MP.4 Modeling Point-Slope Form 56

9 AUDITORY CUES To write the equation of a line in point-slope form, students need to know the slope and one point on the line. Students might state the point-slope form as follows: For a given point, y minus the y-coordinate equals the slope times the quantity x minus the x-coordinate. AVOID COMMON ERRORS Remind students to write both the subtraction symbol and negative symbol when they substitute negative values in the point-slope form. 13. Lizzy is tiling a kitchen floor for the first time. She had a tough time at first and placed only 5 tiles the first day. She started to go faster, and by the end of day 4, she had placed 35 tiles. She worked at a steady rate after the first day. Use an equation in point-slope form to determine how many days Lizzy took to place all of the 100 tiles needed to finish the floor. m = = 3 = 10 (100 5) =10 (x 1) 14. The amount of fresh water left in the tanks of a nineteenth-century clipper ship is a linear function of the time since the ship left port, as shown in the table. Write an equation in point-slope form that represents the function. Then find the amount of water that will be left in the ship s tanks 50 days after leaving port. Time (days) 95 =10x =10x =10x 10.5 = x Lizzy took 10.5 days to place all the tiles. Amount (gal) Image Credits: Fine Art Photographic Library/Corbis m = = = -45 y = -45(50-1) y = -05 y = gallons of water will be left after 50 days. Module 6 57 Lesson Exercise Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices Strategic Thinking MP.4 Modeling 1 Skills/Concepts MP.6 Precision 3 3 Strategic Thinking MP.3 Logic 4 3 Strategic Thinking MP.6 Precision 57 Lesson 6.

10 15. At higher altitudes, water boils at lower temperatures. This relationship between altitude and boiling point is linear. At an altitude of 1000 feet, water boils at 10 F. At an altitude of 3000 feet, water boils at 06 F. Use an equation in point-slope form to find the boiling point of water at an altitude of 6000 feet m = = = y - 10 = ( ) y = 00 The boiling point at 6000 feet is 00 F. 16. In art class,tico is copying a detail from a painting. He paints slowly for the first few days, but manages to increase his rate after that. The graph shows his progress after he increased his rate. How many square centimeters of his painting will he finish in 5 days after the increase in rate? m = = = 45 y = 45 (5-3) y = 65 He will finish 65 cm of his painting in 5 days after the increase in rate. Area (cm ) y (3, 175) (8, 400) Day since rate increase x INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Suggest to students that when they are given two points, they use both points to write equations in point-slope form. They can then check that the equations are equivalent by writing them in slope-intercept form. COGNITIVE STRATEGIES Encourage students to look for ways to make problems easier to solve. For example, in some problems, any two points can be used to find the slope, but some pairs may be easier to work with than others. 17. A hot air balloon in flight begins to ascend at a steady rate of 10 feet per minute. After 1.5 minutes, the balloon is at an altitude of 150 feet. After 3 minutes, it is at an altitude of 330 feet. Use an equation in point-slope form to determine whether the balloon will reach an altitude of 500 feet in 4 minutes m = = = 10 y = 10 (4-1.5) y = 450 The balloon will not reach an altitude of 500 feet in 4 minutes. 18. A candle burned at a steady rate. After 3 minutes, the candle was 11. inches tall. Eighteen minutes later, it was inches tall. Use an equation in point-slope form to determine the height of the candle after hours. m = = = y = (10-3) y = 9 After hours, the candle will be 9 inches tall. Image Credits: Royalty- Free/Corbis Module 6 58 Lesson Point-Slope Form 58

11 JOURNAL In their journal, have students compare and contrast writing linear equations from a point and the slope, and from two points. 19. Volume A rectangular swimming pool has a volume capacity of 160 cubic feet. Water is being added to the pool at a rate of about 0 cubic feet per minute. Determine about how long it will take to fill the pool completely if there were already about 100 gallons of water in the pool. Use the fact that 1 cubic foot of space holds about 7.5 gallons of water. 100 gal 7.5 gal/f t = 160 f t = x = 0x x = 100 It will take about 100 minutes to fill the pool completely. 0. Multi-Step Marisa is walking from her home to her friend Sanjay s home. When she is 1 blocks away from Sanjay s home, she looks at her watch. She looks again when she is 8 blocks away from Sanjay s home and finds that 6 minutes have passed. a. What do you need to assume in order to treat this as a linear situation? That Marisa in walking at a fixed rate. b. Identify the variables for the linear situation and identify two points on the line. Explain the meaning of the points in the context of the problem. x represents the number of minutes since Marisa first looked at her watch, and y represents the number of blocks she is from Sanjay s home. The point (0, 1) indicates that when Marisa first looked at her watch, she was 1 blocks from Sanjay s home. The point (6, 8) indicates that 6 minutes after she first looked at her watch she was 8 blocks from Sanjay s home. c. Find the slope of the line and describe what it means in the context of the problem. m = = - ; the slope indicates that for every minute Marisa 3 walks, the distance to Sanjay s home decreases by 3 block. d. Write an equation in point-slope form for the situation and use it to find the number of minutes Marisa takes to reach Sanjay s home. Show your work. 0-1 = - (x - 0) 3 18 = x Marisa takes 18 minutes to reach Sanjay s home. 1. Match each equation with the pair of points used to create the equation. a. y - 10 = 1 (x + ) c (0, 0), (-1, 1) b. y - 0 = 1 (x - 0) b (1, 1), (-1, -1) c. y - 3 = -1(x + 3) a (-, 10), (0, 1) d. y - 3 = 0 (x - ) d (1, 3), (-3.5, 3) Module 6 59 Lesson 59 Lesson 6.

12 H.O.T. Focus on Higher Order Thinking. Explain the Error Carlota wrote the equation y + 1 = ( x 3 ) for the line passing through the points ( 1, 3 ) and (, 9 ). Explain and correct her error. Carlota replaced x 1 in the point-slope form with y 1 and vice versa. A correct equation using the point (-1, 3 ) is y - 3 = (x + 1). A correct equation using the point (, 9) is y - 9 = (x - ). 3. Communicate Mathematical Ideas Explain why it is possible for a line to have no equation in pointslope form or to have infinitely many, but it is not possible that there is only one. If the slope is undefined, there is no equation of the line in point-slope form. Otherwise, there are at least two equations in point-slope form for every pair of points on the line. 4. Persevere in Problem Solving If you know that A 0 and B 0, how can you write an equation in point-slope form of the equation Ax + By = C? Possible answer: Find the y-intercept. Ax + By = C y = -Ax + C B C When x = 0, y = B. Then the point ( C A, 0 ) is on the line, and an equation in point-slope form is y - 0 = -A B ( C x - A ). Find the x-intercept. Ax + By = C x = -By + C A C When y = 0, x = A. CONNECT CONTEXT Some students may be confused by what appear to be two different uses of the term slope: the mountain has a constant slope and Alberto snowboards down the slope. Explain that a slope on a graph is a measure of how steep the line or hill is. A slope may also mean a stretch of ground that forms an incline, as on a hill. INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 It may help students to make a sketch of the situation in the Lesson Performance Task. Students should find that the bottom of the slope is not on the horizontal axis; it is along h = Lesson Performance Task Alberto is snow boarding down a mountain with a constant slope. The slope he is on has an overall length of 1560 feet. The top of the slope has a height of 4600 feet, and the slope has a vertical drop of 600 feet. It takes him 4 seconds to reach the bottom of the slope. a. If we assume that Alberto s speed down the slope is constant, what is his height above the bottom of the slope at 10 seconds into the run? b. Alberto says that he must have been going 50 miles per hour down the slope. Do you agree? Why or why not? a. m = 0 4 = = 5 (h h 1 ) = 5 (t t 1 ) Solve for t = 10: h = 5 (10) = 4350 h 4600 = 5 (t 0) h = 5t So at 10 seconds, Alberto s height is at 4350 feet, which is 350 feet above the bottom of the slope ft b. Alberto s average speed = = 65 ft/sec, which is approximately 44 miles per hour. 4 sec Module 6 60 Lesson EXTENSION ACTIVITY Have students research the length (cm) of a snowboard required for a given rider s weight in pounds for several different weights, and then determine if the graph of (weight, length) appears to be linear. Students should find that the graph of the (weight, length) points is roughly linear, although the points do not all lie on the same line. Scoring Rubric points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Point-Slope Form 60