LESSON 12 Lines of Best Fit LEARNING OBJECTIVES Toda I am: using scatterplots and lines of best fit to make predictions. So that I can: learn to write equations of lines of best fit. I ll know I have it when I can: write the equation of a line with two points. In Lesson 11, ou graphed and analzed a variet of scatterplots. In this lesson, ou ll further explore how to write the equation of a line to describe the scatterplot. Your equations will allow ou to make predictions about the data. Opening Exercise Reading Graphs 1. What data is this graph displaing? Tpheerafarge 2. Are ou surprised b this data? Explain. Yes for 12 ear olds emails 0 Source: CrackSAT.net Who is Dann/Shutterstock.com 171
172 Module 1 Descriptive Statistics Exploration: Writing an Equation of a Line The equation of a line can be in the form = mx + b, where m represents the slope of the line and I b represents the -intercept. We ll start with a strong positive linear scatterplot. The line of best fit has been drawn in for ou. I Source: http://dev.phsicslab.org/document.aspx?doctpe 3&filename IntroductorMathematics_DataAnalsisMethods.xml 3. Choose two points on the line of best fit and then determine the slope between these two lines. Y X z Y z O 6 2 Il A. M Points: ( ) and ( ) B. Slope using m two points: m 2 1 = ----------------------- x x 2 1 12 C. What does this slope mean in terms of the data? The distance increasesb 2.5 m ever second 5 2 2 m/sec 4. What is the estimated -intercept for the line? 6 5. Write the equation of the best fit line using mx b, where m slope and b -intercept. 2 5 X t 6
Unit 2 Scatterplots and Lines of Best Fit Lesson 12 Lines of Best Fit 173 6. The slope of the line shown is 2.5. Did ou get that value? If not, how much did our slope differ from this value? Wh do ou think there might be a difference? Oh eah 7. The -intercept generated using the actual data points and a calculator or computer program 0 was 6. How much did our -intercept differ from this value? Wh do ou think there might be a difference? Oh eah 8. You can use our equation to find other points on the line. A. Use the graph to estimate the distance when time is 7.5 seconds. Y 2.5 G 5 16 X 24.75 M B. Use our equation from Exercise 5 to estimate the distance when time is 7.5 seconds. C. Which model (graph or equation) would ou use to estimate the distance at 10 seconds? What is that value? Which model is more accurate? Equation is more accurate D. How much time would ou estimate would go b for 15 meters? Which model did ou use? Wh? 15 2.5 6 O G G 9 2.5 X 3.6 s Y
2.5 2.5 X 174 Module 1 Descriptive Statistics Practice Writing & Graphing Equations of Lines For Exercise 9 14, use the given information to write an equation of the line. Then graph the line. mxtb 9. Slope 3; -intercept 4 m b 10. Slope ½; -intercept 2 11. Slope 1; -intercept ¼ 3X 4 9 2 2 x 14 T 2 m 4 3T b i of i I 12. Two points on the line 13. Two points on the line (4, 4) and (8, 5) with a ( 3, 9) and (0, 5) with a -intercept of 3. -intercept of 5. Fx m I 9291 9 5 9 4 3 s 14. Two points on the line ( 2, 0) and (4, 6) with a -intercept of 2. X 2 I 14 b E
Unit 2 Scatterplots and Lines of Best Fit Lesson 12 Lines of Best Fit 175 Practice Writing Equations of Lines from Graphs For Exercise 15 20 determine the slope and -intercept of the line. Then write the equation of the line. 15. 16. 17. g M O 0,2 3 F m 3z Slope 3 2 02 5 Slope Slope -intercept -intercept -intercept 3 x 12 18. 19. 20. 5 th Slope -intercept Slope -intercept I 1 X 11 Slope -intercept
176 Module 1 Descriptive Statistics Point-Slope Form of a Line The equation ou wrote in Exercise 3 is in slope-intercept form or mx b. Point-slope is another ver useful form of a linear equation. For point-slope we need an point on the line and the slope. When no -intercept is available, we can use a different form of the equation of a line. This new form comes from the slope formula. 21. How did the get from the slope equation to the point-slope equation? 22. Suppose two points on the line in Exercise 3 are (1, 8.5) and (7, 23.5). A. Use the point-slope equation to write the equation of the line. Find the slope: Use the formula. You ma use either point for the equation: B. Discuss with our partner how ou could rearrange this equation to isolate the and get it in slope-intercept form. slope = m = 2 1 x x Gaga 2 1 or m = 2 1 x2 x1 2 1 ( 2 1) O Point-Slope Equation of a Line: = mx x or = mx x m Yz Y mcx2 x 92 Y TX i mcx m 23j5 5 lg5 25 2s5 ( ) 1 1 x.it CX sapoint m slope 8.5 2.5 X l C. Rewrite our equation in Part A in slope-intercept form. 0 D. How does this equation compare to the one ou wrote in Exercise 5? point slope slope intercept 8.5 2.5 X 1 ee 8.5 2.5 2.5 8 5 18.5
o Unit 2 Scatterplots and Lines of Best Fit Lesson 12 Lines of Best Fit 177 23. What information do ou need to find the equation of a line? List all possibilities. If I have... Then I can use... Y slope m intercepts Y mxtb slopeandapoint or mcx x two points ix is apoint Practice Writing and Graphing Linear Equations For Exercise 24 26, write the equation of the line in point-slope form. Then graph the line. X 91 24. Slope 3; Point ( 4, 7) 25. Slope ½; Point ( 5, 0) 26. Slope 1; Point ( ½, 4) 17 31 4 t
178 Module 1 Descriptive Statistics Practice Writing and Graphing Linear Equations For Exercise 27 29 write the slope and equation for the line. Then graph the line. 27. Point ( 1, 2); Point (3, 7) Slope: 57 7 2 9,4 1 28. Point ( 3, 5); Point (5, 0) Slope: 29. Point (8, 0); Point (4, 4) Slope: I E E Writing Equations of Lines Lesson Summar If ou have two points on our line, ou can use the Point-Slope Equation given b = mx x, where m slope and (x 1, 1 ) is a point on the line. ( ) 1 1 If ou have one point and the slope of our line, ou can use the Slope-Intercept Equation given b mx b, where m slope and b -intercept.
Unit 2 Scatterplots and Lines of Best Fit Lesson 12 Lines of Best Fit 179 NAME: PERIOD: DATE: Homework Problem Set 1. Graph the following lines. A. = 2 x 4 B. 3x 5 3 2. Write the equation of the line in slope-intercept form. A. B.
180 Module 1 Descriptive Statistics 3. A. Wh is mx b called slope-intercept form and 1 = m( x x1) is called pointslope form? B. If ou wanted to change an equation from point-slope to slope-intercept form, what would ou do? 4. Change 3 2(x 1) to slope-intercept form. 5. Write the equation of the line in point-slope form. Then change the equation to slope-intercept form and graph the equation to see if our calculations were correct. A. (5, 1) and m 2 B. (2, 2) and (10, 2) Point-Slope: Slope-Intercept:
Unit 2 Scatterplots and Lines of Best Fit Lesson 12 Lines of Best Fit 181 6. Write the equation in both point-slope and slope-intercept forms. A. (2, 5) and (5, 1) B. (0, 5) and m = 1 3 Point-slope form: Slope-intercept form: 7. Write the equation of the line represented b the data in the table. Write our answer in slopeintercept and point-slope forms. x 2 1 0 1 2 3 9 7 5 3 1 1 Point-slope form: Slope-intercept form:
182 Module 1 Descriptive Statistics 8. Write the equation of the line in point-slope and slope-intercept forms. Point-slope form: Slope-intercept form: Spiral REVIEW Lines of Best Fit 9. Rachel misunderstood how a line of best fit works. Her line is shown below. What might have Rachel been considering when she drew this line? 10. Josh drew the line of best fit as shown below. Explain to Josh wh his line is not appropriate for this data.