171S1.4p Equations of Lines and Modeling
August 29, 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.
Mathematica Interactive Figures
are available through Tools for Success, Activities and Projects in CourseCompass. You may access these through CourseCompass or from the Important Links webpage. You must Login to MML to use this link.
Section 1.4 Equations of Lines: Point­Slope Form
Section 1.4 Equations of Lines: Parallel; Perpendicular
Aug 19­7:45 AM
Aug 28­9:30 AM
GeoGebra Programs for Mathematics Courses http://cfcc.edu/mathlab/geogebra/
This shows the Linear Function: Graph Two Points ­ Graph of the Linear Functon and Equation in Slope­
intercept form from two points. Enter the values of x1 & y1 for Point A(x1, y1) and the values of x2 & y2 for Point B(x2, y2). Check the boxes to Show Line and to Show Equation. Click the recycle icon to reset the graph and values. Hold down Shift key and use mouse to move grid. Hold down Shift key and use mouse wheel to zoom in and out.
http://cfcc.edu/mathlab/geogebra/linear_enter2pts.html
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 (x 1, 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 m 1 and m 2 and are perpendicular if and only if the product of their slopes is ­1: m 1 m2 = ­1. Lines are also perpendicular if one is vertical (x = a) and the other is horizontal (y = b).
http://cfcc.edu/mathlab/geogebra/linear_enter2pts.html
Aug 27­8:27 PM
Aug 23­9:58 PM
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171S1.4p Equations of Lines and Modeling
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
See the TI tutorial Modeling – Statistical Modeling with TI­83 Calculator – linear, quadratic, and other regressions.
August 29, 2012
Section 1.4 Equations of Lines and Modeling
117/2 . 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.
Working from a graph to write the equation is somewhat subjective. We first estimate two points (subjective). Then we use those two points to find the slope and write the equation. I chose the two points (0, 2) and (4, ­1). We get m = ­0.75 and y = ­0.75x + 2.
The graph and equation below were generated using the GeoGebra program available at http://cfcc.edu/mathlab/geogebra/linear_enter2pts.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. The constant r2 between 0 and 1, called the coefficient of determination. The larger value for the coefficient of determination the better fit is the equation.
Aug 23­10:14 PM
Section 1.4 Equations of Lines and Modeling
117/4 . 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.
Working from a graph to write the equation is somewhat subjective. We first estimate two points (subjective). Then we use those two points to find the slope and write the equation. I chose the two points (0, ­1) and (3, 1). We get m = 2/3 and y = (2/3)x ­ 1.
The graph and equation below were generated using the GeoGebra program available at http://cfcc.edu/mathlab/geogebra/linear_enter2pts.html
Aug 20­1:08 PM
Aug 20­1:08 PM
Section 1.4 Equations of Lines and Modeling
117/5 . 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.
Working from a graph to write the equation is somewhat subjective. We first estimate two points (subjective). Then we use those two points to find the slope and write the equation. This graph appears to be a horizontal line passes through the point (0, ­3). Therefore, we get m = 0 and y = ­3.
The graph and equation below were generated using the GeoGebra program available at http://cfcc.edu/mathlab/geogebra/linear_enter2pts.html
Aug 20­1:08 PM
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171S1.4p Equations of Lines and Modeling
Section 1.4 Equations of Lines and Modeling
117/6 . 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.
Working from a graph to write the equation is somewhat subjective. We first estimate two points (subjective). Then we use those two points to find the slope and write the equation. I chose the two points (0, 0) and (4, 4). We get m = 1 and y = x.
August 29, 2012
Section 1.4 Equations of Lines and Modeling
117/14 . Write a slope intercept equation for a line with the given characteristics: m = ­3/8, passes through (5, 6)
117/18 . Write a slope intercept equation for a line with the given characteristics: m = 2/3, passes through (­4, ­5)
The graph and equation below were generated using the GeoGebra program available at http://cfcc.edu/mathlab/geogebra/linear_enter2pts.html
Aug 20­1:08 PM
Aug 20­1:08 PM
Section 1.4 Equations of Lines and Modeling
117 /32 . Find a linear function g(x) given g(­1/4) = ­6 and g(2) = 3. Then find g(­3).
Section 1.4 Equations of Lines and Modeling
117/24 . Write a slope intercept equation for a line with the given characteristics: passes through (­5, 0) and (0, 4/5)
117/26 . Write a slope intercept equation for a line with the given characteristics: passes through (­13, ­5) and (0, 0)
Aug 20­1:08 PM
Aug 20­1:08 PM
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171S1.4p Equations of Lines and Modeling
August 29, 2012
Section 1.4 Equations of Lines and Modeling
117/ 34. Find a linear function h(x) given h(­3) = 3 and h(0) = 2. Then find h(­6).
Section 1.4 Equations of Lines and Modeling
138 /38 . Determine whether the pair of lines is parallel, perpendicular, or neither: y = (3/2)x ­ 8 and y = 8 + 1.5x
Aug 20­1:08 PM
Aug 20­1:08 PM
Section 1.4 Equations of Lines and Modeling
Section 1.4 Equations of Lines and Modeling
117 /40. Determine whether the pair of lines is parallel, perpendicular, or neither: 2x ­ 5y = ­3 and 2x + 5y = 4 Aug 20­1:08 PM
118 /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
Aug 20­1:08 PM
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171S1.4p Equations of Lines and Modeling
Section 1.4 Equations of Lines and Modeling
118 /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
Aug 20­1:08 PM
August 29, 2012
Section 1.4 Equations of Lines and Modeling
In Exercises 51­56, determine whether the statement is true or false.True or False statement?
118 /54 . y = 2 and x = ­3/4 intersect at (­3/4, 2). 118 /56. 2x + 3y = 4 and 3x ­ 2y = 4 are perpendicular.
Aug 20­1:08 PM
Section 1.4 Equations of Lines and Modeling
Section 1.4 Equations of Lines and Modeling
119 /68 . a) Use a graphing calculator to fit a regression line to the data in Exercise 61.
b) Estimate the number of Internet users worldwide in 2013 and compare the value with the result found in Exercise 61.
c) Find the correlation coefficient for the regression line and dtermine whether the line fits the data closely.
120 /70 . a) Use a graphing calculator to fit a regression line to the data in Exercise 66.
b) Estimate the average credit­card debt per U.S. household in 2014 and compare the result with the estimate found with the model in Exercise 66.
c) Find the correlation coefficient for the regression line and dtermine whether the line fits the data closely.
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
From #61: y = 165.2x + 454.6
2013: x = 12 yields y = 2437.
The "best fit" straight line (linear regression) is y = 171x + 431 (rounded off).
For 2013, x = 12 yields the projected (estimated) y = 2482 compared to the above y = 2437. The regression estimate is best because it uses all of the x­ and y­values.
Aug 20­1:08 PM
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
From #66: y = 522.5x + 4822
2014: x = 22 yields y = 16,317.
The "best fit" straight line (linear regression) is y = 494x + 4146 (rounded off).
For 2014, x = 22 yields the projected (estimated) y = 15,021 compared to the above y = 16,317. The regression estimate is best because it uses all of the x­ and y­
values.
Aug 20­1:08 PM
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