171S1.4 Equations of Lines and Modeling
January 12, 2012
MAT 171 Dr. Claude Moore, CFCC
Section 1.4 Equations of Lines and Modeling
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 : http://my.cfcc.edu
• WebAdvisor: http://reg.cfcc.edu
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
Slope­Intercept Equations of Lines
f (x) = mx + b or y = mx + b
with slope m and y­intercept b representing the point (0, b).
Point­Slope Equations of Lines
The point slope equation of the line with slope m passing through (x1, y1) is y ­ y1 = m(x ­ x1) .
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 m1 and m2 and are perpendicular if and only if the product of their slopes is ­1: m1 m2 = ­1. Lines are also perpendicular if one is vertical (x = a) and the other is horizontal (y = b).
Use the program "Linear Function: Graph, Forms of Equation; Parallel & Perpendicular Lines & Equations" http://cfcc.edu/mathlab/geogebra/parallel_perpendicular.html
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171S1.4 Equations of Lines and Modeling
January 12, 2012
Section 1.4 Equations of Lines and Modeling
124/2. Find the slope and the y­ intercept of the graph of the linear equation. Then write the equation of the line in slope intercept form.
Mathematical Models
When a real­world 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.
Curve Fitting
Slope, equation, intercepts.
http://cfcc.edu/mathlab/geogebra/linear_2pts.html
See the TI tutorial Modeling – Statistical Modeling with TI­83 Calculator – linear, quadratic, and other regressions.
Slope between two points. http://home.wavecable.com/
~svcmath/pearson/precalci/fig102.html
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.
Section 1.4 Equations of Lines and Modeling
124/5. Find the slope and the y­ intercept of the graph of the linear equation. Then write the equation of the line in slope intercept form.
Section 1.4 Equations of Lines and Modeling
124/4. Find the slope and the y­ intercept of the graph of the linear equation. Then write the equation of the line in slope intercept form.
Slope, equation, intercepts.
http://cfcc.edu/mathlab/geogebra/linear_2pts.html
Slope, equation, intercepts.
http://cfcc.edu/mathlab/geogebra/linear_2pts.html
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171S1.4 Equations of Lines and Modeling
Section 1.4 Equations of Lines and Modeling
124/6. Find the slope and the y­ intercept of the graph of the linear equation. Then write the equation of the line in slope intercept form.
January 12, 2012
Section 1.4 Equations of Lines and Modeling
124/14. Write a slope intercept equation for a line with the given characteristics. m = ­3/8, passes through (5, 6)
Slope, equation, intercepts.
http://cfcc.edu/mathlab/geogebra/linear_2pts.html
124/18. Write a slope intercept equation for a line with the given characteristics. m = 2/3, passes through (­4, ­5)
This problem is left as an exercise for the students. Answer: Section 1.4 Equations of Lines and Modeling
124/24. Write a slope intercept equation for a line with the given characteristics. passes through (­5, 0) and (0, 4/5)
Section 1.4 Equations of Lines and Modeling
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)
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171S1.4 Equations of Lines and Modeling
January 12, 2012
Section 1.4 Equations of Lines and Modeling
124/34. Find a linear function h given h(­3) = 3 and h(0) = 2. Then find h(­6).
Section 1.4 Equations of Lines and Modeling
125/38. Determine whether the pair of lines is parallel, perpendicular, or neither: y = (3/2)x ­ 8 and y = 8 + 1.5x
This problem is left as an exercise for the students. Answer: h(x) = (­1/3)x + 2; h(­6) = 4
Section 1.4 Equations of Lines and Modeling
Section 1.4 Equations of Lines and Modeling
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 + 9
Parallel & perpendicular to line through point. http://home.wavecable.com/~svcmath/pearson/precalci/fig106.html
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171S1.4 Equations of Lines and Modeling
January 12, 2012
Section 1.4 Equations of Lines and Modeling
In Exercises 51­56, 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). Section 1.4 Equations of Lines and Modeling
125/46. 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
This problem is left as an exercise for the students. Given: L2 through (­4, ­5); L1: 2x + y = ­4
Answer: parallel L2: 2x + y = ­13 and perpendicular L2: x ­ 2y = 6
125/56. 2x + 3y = 4 and 3x ­ 2y = 4 are perpendicular.
Section 1.4 Equations of Lines and Modeling
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 2013. Answers may vary depending on the data points used.
Section 1.4 Equations of Lines and Modeling
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.
See 4:LinReg(ax+b) at TI Tutorials on Important Links webpage or go directly to http://cfcc.edu/faculty/cmoore/TI83LinReg(ax+b).htm
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 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 http://cfcc.edu/faculty/cmoore/TI83LinReg(ax+b).htm
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171S1.4 Equations of Lines and Modeling
January 12, 2012
Answer for 127/68 on previous slide.
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 + 82.00 (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 = 0.063559.
S 4.
How confident are you about using the regression line to predict function values?
Ans: Not very confident. In non­technical 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, r2 = 0.0040398098 = 0.4%.
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