MAT 171. August 22, S1.4 Equations of Lines and Modeling. Section 1.4 Equations of Lines and Modeling

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1 MAT 171 WebAdvisor: Dr. Claude Moore, CFCC Session 1 introduces the Course, CourseCompass, and Chapter 1: Graphs, Functions, and Models. This session is available in CourseCompass. Read the Announcements to find Session 1. If the CFCC website is not available, you may access the following with links: Campus Cruiser : I suggest that you view the examples that were not worked in your section. If you have questions or need assistance, please contact me or go to the Math Lab (S606) or the Learning Lab (L Building). Dr. Moore Slope Intercept Equations of Lines f (x) = mx + b or y = mx + b with slope m and y intercept b representing the point (0, b). Point Slope Equations of Lines The point slope equation of the line with slope m passing through (x 1, y 1) is y y 1 = m(x x 1). Parallel Lines Vertical lines are parallel. Nonvertical lines are parallel if and only if they have the same slope and different y intercepts. Perpendicular Lines Two lines with slopes m 1 and m 2 and are perpendicular if and only if the product of their slopes is 1: m 1 m 2 = 1. Lines are also perpendicular if one is vertical (x = a) and the other is horizontal (y = b). Mathematical Models When a real world problem can be described in mathematical language, we have a mathematical model. Three fairly commonly used nonlinear functions are (1) quadratic, (2) cubic, and (3) exponential. 124/2. Find the slope and the y intercept of the graph of the Curve Fitting See the TI tutorial Modeling Statistical Modeling with TI 83 Calculator linear, quadratic, and other regressions. Linear regression a procedure that can be used to model a set of data using a linear function. On some graphing calculators with the DIAGNOSTIC feature turned on, a constant r between 1 and 1, called the coefficient of linear correlation, appears with the equation of the regression line. 1

2 124/4. Find the slope and the y intercept of the graph of the 124/5. Find the slope and the y intercept of the graph of the 124/6. Find the slope and the y intercept of the graph of the 124/14. Write a slope intercept equation for a line with the given characteristics. m = 3/8, passes through (5, 6) This problem is left as an exercise for the students. Answer: 124/18. Write a slope intercept equation for a line with the given characteristics. m = 2/3, passes through ( 4, 5) 2

3 124/24. Write a slope intercept equation for a line with the given characteristics. passes through ( 5, 0) and (0, 4/5) 124/32. Find a linear function g given g( 1/4) = 6 and g(2) = 3. Then find g( 3). 124/26. Write a slope intercept equation for a line with the given characteristics. passes through ( 13, 5) and (0, 0) 124/34. Find a linear function g given h( 3) = 3 and h(0) = 2. Then find h( 6). 125/38. Determine whether the pair of lines is parallel, perpendicular, or neither: y = (3/2)x 8 and y = x This problem is left as an exercise for the students. Answer: h(x) = ( 1/3)x + 2; h( 6) = 4 3

4 125/40. Determine whether the pair of lines is parallel, perpendicular, or neither: 2x 5y = 3 and 2x + 5y = 4 125/44. Write a slope intercept equation for a line passing through the given point that is parallel to the given line. Then write a second equation for a line passing through the given point that is perpendicular to the given line. ( 1, 6) f(x) = 2x /46. Write a slope intercept equation for a line passing through the given point that is parallel to the given line. Then write a second equation for a line passing through the given point that is perpendicular to the given line. ( 4, 5) 2x + y = 4 In Exercises 51 56, determine whether the statement is true or false.true or False statement? 125/54. y = 2 and x = 3/4 intersect at ( 3/4, 2). This problem is left as an exercise for the students. Given: L 2 through ( 4, 5); L 1 : 2x + y = 4 125/56. 2x + 3y = 4 and 3x 2y = 4 are perpendicular. Answer: parallel L 2 : 2x + y = 13 and perpendicular L 2 : x 2y = 6 4

5 126/66. Sheep and Lambs. The number of sheep and lambs on farms in the United States has declined in recent years. Model the data given in the table on the following page with a linear function and estimate the number of sheep and lambs on farms in 2008 and in Answers may vary depending on the data points used. 127/68. Study Time versus Grades. A math instructor asked her students to keep track of how much time each spent studying a chapter on functions in her algebra trigonometry course. She collected the information together with test scores from that chapters test. The data are listed in the following table. a) Use a graphing calculator to model the data with a linear function. b) Predict a students score if he or she studies 24 hr, 6 hr, and 18 hr. c) What is the correlation coefficient? How confident are you about using the regression line to predict function values? See 4:LinReg(ax+b) at TI Tutorials on Important Links webpage or go directly to See the next page for answers using the TI calculator. See 4:LinReg(ax+b) at TI Tutorials on Important Links webpage or go directly to S 1. S 2. S 3. a) Use a graphing calculator to model the data with a linear function. Ans: The linear equation shown in S2 is y = 0.07x (rounded off). b) Predict a students score if he or she studies 24 hr, 6 hr, and 18 hr. Ans: 24 hr yields 83.7%; 6 hr 82.4%; 18 hr 83.3% c) What is the correlation coefficient? Ans: The correlation coefficient is r = S 4. How confident are you about using the regression line to predict function values? Ans: Not very confident. In non technical terms, we can be only 0.4% confident that the number of hours studied correlate with the performance score. This is from the coefficient of determination from S3, r 2 = = 0.4%. 5