So far you have worked with linear equations in intercept form, y a bx. y x. x 1 b

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1 LESSON. PLANNING LESSON OUTLINE One da: 0 min Eample 0 min Investigation min Sharing min Closing 0 min Eercises MATERIALS Calculator Note A Sketchpad demonstration Point- Slope Form, optional LESSON. Success breeds confidence. BERYL MARKHAM EXAMPLE Point-Slope Form of a Linear Equation So far ou have worked with linear equations in intercept form, a b. When ou know a line s slope and -intercept, ou can write its equation directl in intercept form. But what if ou don t know the -intercept? One method that ou might remember from our homework is to work backward with the slope until ou find the -intercept. But ou can also use the slope formula to find the equation of a line when ou know the slope of the line and the coordinates of onl one point on the line. Since the time Beth was born, the population of her town has increased at a rate of approimatel 80 people per ear. On Beth s 9th birthda the total population was nearl 07,60. If this rate of growth continues, what will be the population on Beth s 6th birthda? TEACHING This lesson shows that the slopeintercept form of the equation of a line can be found from two points without having to find the -intercept. INTRODUCTION The reference to homework is to Lesson., Eercise 7b. You can use the Sketchpad demonstration Point-Slope Form as an introduction to this lesson. EXAMPLE This eample derives the pointslope form of the equation of a line. If the form happened to arise during Lesson., ou ma not need to spend much time on the eample. Advise students who find it confusing that the investigation will make it clearer. Encourage critical thinking b asking some questions. [Ask] Do ou think the situation is realistic? Do populations grow at a constant rate? [Some do, but students ma realize that population growth is often eponential up to a limit.] LESSON OBJECTIVES Solution Because the rate of change is approimatel constant, a linear equation should model this population growth. Let represent time in ears since Beth s birth, and let represent the population. In the problem, ou are given one point, (9, 0760). An other point on the line will be in the form (, ). So let (, ) represent a second point on the line. You also know that the slope is 80. Now use the slope formula to find a linear equation. Learn the point-slope form of an equation of a line Write equations in point-slope form that model real-world data b Slope formula. Slope = 80 point (, ) point (9, 0760) Time (r) 07, Substitute the coordinates of the point (9, 0760) for,,and the slope 80 for b. Because we know onl one point, we use (, ) to represent an other point. NCTM STANDARDS CONTENT Number Algebra Geometr Measurement Data/Probabilit Population PROCESS Problem Solving Reasoning Communication Connections Representation CHAPTER Fitting a Line to Data

2 Step Step Step Now solve the equation for b undoing the subtraction and division. 07,60 80( 9) Multipl b ( 9) to undo the division. 07,60 80( 9) Add 07,60 to undo the subtraction. The equation 07,60 80( 9) is a linear equation that models the population growth. To find the population on Beth s 6th birthda, substitute 6 for. 07,60 80( 9) Original equation. 07,60 80(6 9) Substitute 6 for.,600 Use order of operations. The model equation predicts that the population on Beth s 6th birthda will be,600. The equation 07,60 80( 9) is a linear equation, but it is not in intercept form. This equation has its advantages too because ou can clearl identif the slope and one point on the line. Do ou see the slope of 80 and the point (9, 0760) within the equation? This form of a linear equation is appropriatel called the point-slope form. Point-Slope Form If a line passes through the point, and has slope b, the point-slope form of the equation is b Investigation The Point-Slope Form for Linear Equations Silo and Jenn conducted an eperiment in which Jenn walked at a constant rate. Unfortunatel, Silo recorded onl the data shown in this table. Find the slope of the line that represents this situation. 0.6 m/s Elapsed Distance to time (s) walker (m) Write a linear equation in point-slope form using the point (,.6) and the slope ou found in Step ( ) Write another linear equation in point-slope form using the point (6,.8) and the slope ou found in Step ( 6) [Alert] Some students ma be confused about choosing (, ) to represent an point on the line. The ma not et grasp the idea that the equation relates coordinates of eactl those points ling on the line. Help them keep in mind the goal of coming up with such an equation. [Alert] A few students ma be confused about multipling b ( 9). Remind them that the can consider ( 9) as a single number. Resist simplifing 07,60 80( 9) to 00, Although the equations are equivalent, the latter is not in point-slope form. Students will learn the distributive propert in Lesson.. You might ask students to go through the derivation again, using,,and b instead of the numbers. Emphasize that,,and b represent constants, whereas and represent variables. Also note that the coordinate is being subtracted from. Guiding the Investigation One Step Direct students attention to the Water Temperature table on page 6 and ask them to find a line of fit without finding the -intercept. LESSON. Point-Slope Form of a Linear Equation

3 Step There appears to be onl one line, which implies that the equations are equivalent. Step Step Enter the equation from Step into Y and the equation from Step into Y on our calculator, and graph both equations. What do ou notice? Look at a table of Y- and Y-values. What do ou notice? What do ou think the results mean? Now that ou have some practice at writing point-slope equations, tr using a point-slope equation to fit data. [0, 0,,, 0, ] Step The Y- and Y-values are equivalent; again, this implies that the two seemingl different equations are equivalent. Step 7 Answers will var. Using (9, ) and (6, 0), the slope is. Step 6 Step 7 Step 8 The table shows how the temperature of a pot of water changed over time as it was heated. Define variables and plot the data on our calculator. Describe an patterns ou notice. Choose a pair of points from the data. Find the slope of the line between our two points. Write an equation in point-slope form for a line that passes through our two points. Graph the line. Does our equation fit the data? Water Temperature Time (s) Temperature ( C) Step 6 Let represent time in seconds, and let represent temperature in degrees Celsius. The data set appears to have a linear pattern. Step 9 Compare our graph to those of other members of our group. Does one graph show a line that is a better fit than the others? Eplain. Answers will var. Because the data are in such a tight linear pattern, there ma not appear to be one better line of fit the will all be prett good. [0, 00, 0, 0, 00, 0] Steps 7 and 8 Each member of the group should be encouraged to select a different pair of points. Suggest that each group graph all of their lines on one calculator for eas comparison. Step 8 Using the slope from Step 7, one possibilit is 0.8( 9). EXERCISES Practice Your Skills If ou look back at the investigation, ou will notice that ou found the point-slope form of a line even though ou had onl points (but not a slope) to start with. This is possible because ou can still use the point-slope form when ou know two points on the line; there s just one additional step. What is it? You must calculate the slope using the two points. You will need our graphing calculator for Eercises,,, 9, and 0.. Name the slope and one point on the line that each point-slope equation represents. a. ( ) a ; (, ) b..9 (.) ; (.,.9) c..7( 7) a.7; (7, ) d..8(.).8; (., ). Write an equation in point-slope form for a line, given its slope and one point that it passes through. a. Slope ; point (, ) ( ) b. Slope ; point (, ) ( ) Step 9 [Ask] How could ou have wisel selected points in order to find the line of best fit to begin with? [Choose points neither close together nor too far apart that appear to lie on a line that passes near most of the data.] SHARING IDEAS Choose students to present several different equations that have the same graphs. [Ask] Is there a wa to tell that the equations have the same graphs without actuall graphing them? Encourage all ideas. You don t need to answer this question. Students will get more eperience identifing equivalent equations in Lesson.. Assessing Progress You can assess students understanding of input and output variables and their abilit to find the slope of a line through two points and to graph data points and lines on a graphing calculator. 6 CHAPTER Fitting a Line to Data

4 . A line passes through the points (, ) and (, ). a. Find the slope of this line. a b. Write an equation in point-slope form using the slope ou found in a and the point (, ). a ( ) c. Write an equation in point-slope form using the slope ou found in a and the point (, ). ( ) d. Verif that the equations in b and c represent the same line. Enter the equations into Y and Y on our calculator, and compare their graphs and tables. The graphs coincide, and the tables are identical.. APPLICATION This table shows a linear relationship between actual temperature and approimate wind Wind Chill with Wind Speed of 0 mi/h chill temperature when the wind Temperature ( F) speed is 0 mi/h. a. Find the rate of change of the Wind chill ( F) data (the slope of the line). b. Choose one point and write an equation in point-slope form to model the data. c. Choose another point and write another equation in point-slope form to model the data d. Verif that the two equations in b and c represent the same line. Enter the equations into Y and Y on our calculator, and compare their graphs and tables. e. What is the wind chill temperature when the actual temperature is 0 F? What does this represent in the graph?. F; this is the graph s -intercept.. Pla the BOWLING program at least four times. [ See Calculator Note A for instructions on how to pla the game. ] Each time ou pla, write down an equations ou tr and how man points ou score. Reason and Appl 6. The graph at right is made up of linear segments a, b, and c. Write an equation in point-slope form for the line that contains each segment. 7. A quadrilateral is a polgon with four sides. Quadrilateral ABCD is graphed at right. a. Write an equation in point-slope form for the line containing each segment in this quadrilateral. Check our equations b graphing them on our calculator. b. What is the same in the equations for the line through points A and D and the line through points B and C? What is different in these equations? a c. What kind of figure does ABCD appear to be? Do the results from 7b have anthing to do with this? a ABCD appears to be a parallelogram because each pair of opposite sides is parallel; the equal slopes in 7b mean that AD and BC are parallel. AB and DC are parallel because the both have slope. 6a. 0. ( 0) or. ( ) 6b.. 0.7( ) or 0.7( ) 6c. ( ) or ( 6) Eercise 7 [ELL] Students might not know the term parallelogram. Ask them to describe the figure if the cannot name it. Students will stud the slopes of parallel and perpendicular lines in Lesson.. If ou need to cover these topics earlier, ou ma want to cover Lesson. immediatel after Lesson.. 0 a B b c 6 A 7a. AD: 0.( ) or 0.( ) BC: 0.( ) or 0.( ) AB: ( ) or ( ) DC: ( ) or ( ) 7b. The slopes are the same; the coordinates of the points are different. C D Closing the Lesson As needed, sa that if ou know two points on a line, ou can find an equation for that line without finding the -intercept. The form is called the point-slope form. BUILDING UNDERSTANDING Students work with the pointslope form of linear equations to model real-world data. ASSIGNING HOMEWORK Essential, 8, 9 Performance assessment, 8 Portfolio 0 Journal 6, 7 Group, 8, 0 Review Helping with the Eercises Eercise [Alert] Some students ma forget that the -coordinate of the point is being subtracted from. Therefore, in b, the -coordinate of the point must be negative. Eercises d and d These eercises foreshadow Lesson.. a. 6..; the data are eactl linear, so an two points will give this slope. b. Answers will var. Using the point (, ), the equation is.( ). c. Answers will var. Using the point (0,.), the equation is..( 0). d. The two equations should give the same graph and table. LESSON. Point-Slope Form of a Linear Equation 7

5 Eercise 8 Letters or packages weighing more than oz are subject to a different rate schedule. Therefore, the possible -values for these data are restricted to whole numbers from to. Research current postal rates through Bring up the idea of step functions. [Ask] How could ou graph this relationship accuratel? Cost ($) Weight (oz) 8a. The data appear linear. 8. APPLICATION The table shows postal rates for first-class U.S. mail in the ear 00. a. Make a scatter plot of the data. Describe an patterns ou notice. b. Find the slope of the line between an two points in the data. What is the real-world meaning of this slope? a c. Write a linear equation in point-slope form that models the data. Graph the equation to check that it fits our data points. d. Use the equation ou wrote in 8c to find the cost of mailing a 0 oz letter. $. Postal Rates e. What would be the cost of mailing a. oz Weight not eceeding (oz) Cost ($) letter? A 9. oz letter? f. The equation ou found in 8c is useful for modeling this situation. Is the graph of this equation, a continuous line, a correct model for the situation? Eplain wh or wh not. a APPLICATION The table below shows fat grams and calories for some breakfast sandwiches Nutrition Facts Total fat (g) Calories Breakfast sandwich Arb s Bacon n Egg Croissant 6 0 Burger King Croissanwich with 9 0 Sausage, Egg & Cheese Carl s Jr. Sunrise Sandwich 6 (U.S. Postal Service, Hardee s Countr Steak Biscuit 60 [0, 6,,0,.,0.] 8b. $0./oz; this is the cost per additional ounce after the first. 8c. Answers will var. Using the point (, 0.7), the equation is ( ). 8e. The rates are given for weights not eceeding the given weights, so a letter weighing. oz would cost the same as a oz letter, or $.06; a letter weighing 9. oz would cost the same as a 0 oz letter, or $.. 8f. Answers will var. A continuous line includes points whose -values are not whole numbers and whose -values are not possible rates. 9a. The data are approimatel linear. Jack in the Bo Sourdough 6 Breakfast Sandwich McDonald s Sausage McMuffin 8 0 with Egg Sonic Sausage, Egg & 6 70 Cheese Toaster Subwa Ham & Egg Breakfast 0 Deli Sandwich ( [Data sets: FFFAT, FFCAL] a. Make a scatter plot of the data. Describe an patterns ou notice. b. Select two points and find the equation of the line that passes through these two points in point-slope form. Graph the equation on the scatter plot. c. According to our model, how man calories would ou epect in a Hardee s Countr Steak Biscuit with grams of fat? 0 9.( ) 70.; approimatel d. Does the actual data point representing the Hardee s Countr Steak Biscuit lie 70 calories above, on, or below the line ou graphed in 9b? Eplain what the point s location means. The actual data point lies above the graph of 0 9.( ); if a point lies above the line, the sandwich has more calories than the model predicts. 9b. Answers will var. Using the points (8, 0) and (, 0), the equation is 0 9.( ). Eercise 9g Some students ma misinterpret this to mean that all fat-free foods have 89 calories. Warn them that man factors influence calories. [0,,, 0, 60, 0] [0,,, 0, 60, 0] 8 CHAPTER Fitting a Line to Data

6 ...6 e. Check each breakfast sandwich to find if its data point falls above, on, or below our line. Answers will var. Using 0 9.( ) as a model, three points are above the line, two points are on the line, and three points are below the line. f. Based on our results for 9d and e, how well does our line fit the data? g. If a sandwich has 0 grams of fat, how man calories does our equation predict? Does this answer make sense? Wh or wh not? Answers will var. Using 0 9.( ), approimatel 89 calories; this makes sense, because not all calories in food come from fat. 0. APPLICATION This table shows the amount of trash produced in the U.S. Trash Production United States in 990 and 99. Amount of trash a. Let represent the ear, and let represent the amount of trash in Year (million tons) millions of tons for that ear. Write an equation in point-slope form for the line passing through these two points. a b. Plot the two data points and graph the equation ou found 99 in 0a. a (Environmental Protection Agenc, c. In 000, million tons of trash were produced in the United States. Plot this data point on the same graph ou made in 0b. Do ou think the linear equation ou found in 0a is a good model for these data? Eplain wh or wh not. a This table shows more data about the amount of trash produced in the United States. d. Add these data points to our graph. Adjust the window as necessar. e. Do ou think the linear equation found in 0a is a good model for this larger data set? Eplain wh or wh not. f. Find the equation of a better-fitting line. g. Use our new equation from 0f to predict the amount of trash produced in 00. Review. APPLICATION The volume of a gas is.0 L at 80 K. The volume of an gas is directl proportional to its temperature on the Kelvin scale (K). a. Find the volume of this gas when the temperature is 0 K.. L b. Find the temperature when the volume is. L. 80 K. Find the slope of the line through the first two points given. Assume the third point is also on the line and find the missing coordinate. a. (, ) and (, ); (, ) b. (, ) and (, ); (, ) c. (0, ) and (, ); ( 0, ) undefined Year. Write the equation represented b this balance. Then solve the equation for using the balancing method. a Eercise Remind students of direct variations k. You might sa that Kelvin units are the same size as Celsius degrees but that 0 K is at about 7 C. It s called absolute zero because electrons at that temperature can t move. The word degrees is not used with the Kelvin scale. Eercise Encourage a variet of approaches. Students might graph, step over unit at a time, draw slope triangles, or make calculator tables. U.S. Trash Production Amount of trash (million tons) (Environmental Protection Agenc, [Data sets: TRYR, TRAMT] Answers will var. Using ( 990), 7 million tons. Eercise Encourage students who are struggling to model the process using the balance. The ma need to draw the steps. Don t rush students into solving equations with s on both sides. The will see man of this tpe of problem in Chapter in the contet of solving sstems of equations. 9f. Answers will var. The line 0 9.( ) appears to be a good fit. Eercise 0 In 0e, a good model is one that allows accurate predictions. Part of the goal of 0g is to show that basing a model on a small set of data can lead to wild predictions. [Ask] In what ear does our equation predict that there were zero million tons of trash? [Using.8( 99), the ear would be 876.] Is this possible? 0a. 0.8( 990) or.8( 99) 0b and 0c. [9, 00,, 8, 0, 0] The point (000, ) is somewhat close to the line, but the predicted value is too low. 0d. [9, 00,, 8, 0, 0] 0e. The data are generall linear, but the line doesn t fit them ver well; a line with a steeper slope would be a better fit. 0f. Answers will var. 00.7( 990) gives a reasonable fit. [9, 00,, 8, 0, 0] See page 7 for the answer to Eercise. LESSON. Point-Slope Form of a Linear Equation 9

7 LESSON., PAGES Step Time Minivan Sports car Pickup (min) (mi) (mi) (mi) a. b [, 0,, 0, 0, 0] [, 0,, 0, 0, 0] LESSON., PAGE 7 Step a Start atthemmarkandwalktowardthem mark (awa from the motion sensor) at m/s. Step b Start at the. m mark and walk toward the 0 m mark (toward the motion sensor) at m/s. Step c Start at the m mark and walk toward the m mark (awa from the motion sensor) at m/s for s. Then walk toward the 0 m mark (toward the motion sensor) at m/s. Step a Distance (m) 6 Time (s) Step c Distance (m) 6 Time (s) LESSON., PAGE 9. 7 Original equation Subtract from both sides. Combine like terms. Subtract from both sides. Combine like terms. Divide both sides b. Reduce. LESSON., PAGE 9 Step 7 It is reasonable.although the combined data are not a simple average of male and female, the line should still lie between the two. Step 8 Answers will var. The intercept-form method first gives a parallel line that has to be raised or lowered based on estimation (weakness); however,that method involves adjusting the line to a fit (strength). The point-slope method immediatel gives ou a line (strength), but the line must go through points and could possibl benefit from adjusting (weakness). The point-slope method also increases the chance that different people will get the same equation of best fit. LESSON., PAGE 78 6d. Graphing windows will var. The one shown is [0, 000, 000, 000, 000, 6000]. Additional Answers Step b Distance (m) 6 Time (s) 7b. [0, 7000, 00, 000, 000, 000] ADDITIONAL ANSWERS 7