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This 9 page Class Notes was uploaded by Leanne Hessel PhD on Sunday October 25, 2015. The Class Notes belongs to MATH 1650 at University of North Texas taught by Jason Snyder in Fall. Since its upload, it has received 33 views. For similar materials see /class/229123/math-1650-university-of-north-texas in Mathematics (M) at University of North Texas.
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Date Created: 10/25/15
Math 1650 Lecture Notes 110 Jason Snyder PhD Lines 110 Lines The Slope of a Line For a line we will de ne the run to be the distance we move to the right and the rise to be the corresponding distance that the rises or falls The slope of the line is the ratio of rise to run rise 510 e p run The following pictures illustrate some applications of slope Pitch of a roof Grade of a road I X Slope E Slope m If a line lies in the xyplane then the rise is the change in the ycoordinate while the run is the change in the Xcoordinate quot 2 l 2 I p Rise RISCI 39 clnnuc in l 113 change Ill i N in H W Cl 1 I 391I39Llll1LIL 39 L 39 lncuulwcl IMMUNE l 0 l I gt r 39 Run Run Page 1 of 9 Math 1650 Lecture Notes 1 10 Jason Snyder PhD Lines This gives the following de nition of slope Slope ofa line The slope ofa nonvertical line that passes through the points AX1y1 and BX2y2 is given by rise yz yl run x2 26139 The slope ofa vertical line is undefined I Example 1 I Finding the Slope of a Line through Two Points Find the slope of the line that passes through the points 21 and 88 Equations of Lines Now let s nd an equation of the line passing through a given point AX1y1 and has slope m A point BXy with Xi X1 lies on this line if and only if the slope of the line through A and B is equal to m that is y Y1 X X1 By multiplying both sides by XX1 we obtain PointSlope Form ofthe Equation ofa Line An equation of the line that passes through the point X1y1 and has slope m is yy1 mac m Page 2 of 9 Math 1650 Lecture Notes 110 Jason Snyder PhD Lines I Example 2 I Finding the Equation of a Line with Given Point and Slope I a Find an equation of the line which passes through the point 1 4 with slope b Sketch the graph of the line Page 3 of 9 Math 1650 Lecture Notes 1 10 Jason Snyder PhD Lines I Example 3 Finding the Equation of a Line through Two Given Points I Find an equation of a line through the points 35 and 2 7 Now suppose a non vertical line has slope m and passes through the point Ob here we call b the y intercept of the line We can therefore write out the point slope form of this line it is y b mx 0 Solving this equation for y yields ymxb SlopeIntercept Form ofthe Equation ofa Line An equation of the line that has slope m and yintercept b is given by y mx b I Example 4 I Lines in Slope Intercept Form a Find the equation of the line with slope 5 and y intercept 10 b Find the slope and y intercept of the line 4y 8X 1 I O Page 4 of 9 Math 1650 Lecture Notes 11o Jason Snyder PhD Lines If a line is horizontal then its slope is m 0 if it is vertical its slope is unde ned However we can still write an equation for a vertical line Vertical and Horizontal Lines An equation of the vertical line passing through the point ab is X 1 An equation of the horizontal line passing through the point ab is y b M 39 General Equation of a Line The graph of every linear equation Ax By C 0 A B not both zero is a line Conversely every line is the graph ofa linear equation I Example 5 I Graphing a Linear Equation Sketch the graph of the equation 3X 2y 12 0 Page 5 of 9 Math 1650 Lecture Notes 110 Jason Snyder PhD Lines Parallel and Perpendicular Lines Parallel Lines Two nonvertical lines are parallel if and only ifthey have the same slope Any two vertical lines are parallel proof Let the lines 11 and 19 in the following gure have slopes m1 and mg If the lines are parallel then the right triangles ABC and DEF are similar so that dBC dEF m1 dA C dDF quot1239 Conversely if the slopes are equal then the triangles will be similar so LBAC LEDF therefore the lines are parallel I Example 6 I Finding the Equation of a Line Parallel to a Given Line I Find an equation of the line passing through the point 52 that is parallel to the line 2x 6y 12 o Page 6 of 9 Math 1650 Lecture Notes 1 10 Jason Snyder PhD Lines Perpendicular Lin es Two limes with slopes m1 and m2 are perpendicular if and only if m1m2 1 that is their slopes are negative reciprocals 1 m2 m1 Also a horizontal line is perpendicular to a vertical line Proof Suppose lines 11 and 12 intersect at the origin 0 furthermore suppose they are perpendicular Ifwe let their slopes be 7m and ma then there equations would be y mg and y max Notice that the point A1m1 is on 11 and the point B1m2 is on 12 By the Pythagorean Theorem and its converse CA J OB if and only if 610Al2 d0B12 6104 B This by the distance formula becomes 1m12 1m22 1 12m2 m12 2m12 m22 77122 2m2m1 m12 2 2m2m1 77117712 1 Example 7 I Perpendicular Lines Show that the points P33 Q817 and R1 15 are the vertices ofa right triangle Page 7 of 9 Math 1650 Lecture Notes 1 10 Jason Snyder PhD Lines I Example 9 Finding an Equation of a Line Perpendicular to a Given Line I Find an equation of the line that is perpendicular to the line 4X 2y 7 O and passes through the origin Applications Slope as Rate of Change I Example 10 I Slope as Rate of Change A dam is built on a river to create a reservoir The water level w in the reservoir is given by the equation W 4515 28 Where tis the number of years since the dam was constructed and w is measured in feet What do the slope and w intercept of this line represent Page 8 of 9 Math 1650 Lecture Notes 1 10 Jason Snyder PhD Lines I Example 1 1 Linear Relationship between Temperature and Elevation I a As dry air moves upward it eXpands and cool If the gound temperature is 20 C and the temperature at a height of 1 km is 10 C eXpress the temperature T in 0C in terms of height h in kilometers Assume that the relationship between T and h is linear b What is the temperature at a height of 25 km Homework Due 2 36 even 412 52 even 541 60 68 Page 9 of 9