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Any questions on the Section 3.3 homework?PowerPoint PresentationSection 3.4: The Slope of a LineSlide 4Slide 5Slide 6Slide 7Slide 8Slide 9Example from today’s homework:Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Watch out for this problem:Slide 23Slide 24Slide 25Slide 26Reminder:Slide 28Any questions on the Section 3.3 homework?Now pleaseCLOSE YOUR LAPTOPSand turn off and put away your cell phones.Sample Problems Page Link(Dr. Bruce Johnston)Section 3.4: The Slope of a LineSlope of a line:Informally, slope is the tilt of a line. It is the ratio of vertical change to horizontal change, or121212xxxxyychangexchangeyrunrisemFind the slope of the line through (4, -3) and (2, 2).If we let (x1, y1) be (4, -3) and (x2, y2) be (2, 2), then2542)3(2m252423 mExampleNote: If we let (x1, y1) be (2, 2) and (x2, y2) be (4, -3), then we get the same result.Find 2 points on the graph, then use those points in the slope formula.Given the graph of a line, how do you find the slope?8642-2-4-6-8-10-5510Slope = -4 – 2 = -6 = 3 = 3 0 – 2 -2 1Which points do you use?It’s your choice, but it’s much easier if you pick points whose x- and y-coordinates are both integers. (2, 2)(0, -4)Slope-intercept form of a line•y = mx + b •has a slope of m and has a y-intercept of (0, b).•This form is useful for graphing, since you have a point and the slope readily visible.Find the slope and y-intercept of the line –3x + y = -5.•First, we need to solve the linear equation for yBy adding 3x to both sides, y = 3x – 5.•Once we have the equation in the form of y = mx + b, we can read the slope and y-intercept. slope is 3 y-intercept is (0,-5)ExampleFind the slope and y-intercept of the line 2x – 6y = 12.•First, we need to solve the linear equation for y.-6y = -2x + 12 (subtract 2x from both sides)Exampley = x – 2 (divide both sides by –6)31• Since the equation is now in the form of y = mx + b,  slope is 1/3  y-intercept is (0,-2)Example from today’s homework:For any 2 points, the y values will be equal to the same real number.The numerator in the slope formula = 0 (the difference of the y-coordinates), but the denominator  0 (two different points would have two different x-coordinates).So the slope = 0.For any 2 points, the x values will be equal to the same real number.The denominator (x2 – x1) in the slope formula = 0so the slope is undefined (since you can’t divide by 0).Example from today’s homework:•If a line moves up as it moves from left to right, the slope is positive.•If a line moves down as it moves from left to right, the slope is negative.•Horizontal lines have a slope of 0.•Vertical lines have undefined slope (or no slope).Summary of relationship between graphs of lines and slopeTwo lines that never intersect are called parallel lines.•Parallel lines have the same slope•unless they are vertical lines, which have no defined slope.•Vertical lines are also parallel to each other, even though their slope is undefined.Two lines that intersect at right angles are called perpendicular lines.•Two nonvertical perpendicular lines have slopes that are negative reciprocals of each other.•The product of their slopes will be –1.•Horizontal and vertical lines are perpendicular to each other.Determine whether the following lines are parallel, perpendicular, or neither.-5x + y = -6 and x + 5y = 5•First, we need to solve both equations for y.• In the first equation, y = 5x – 6 (add 5x to both sides)•In the second equation,5y = -x + 5 (subtract x from both sides)Exampley = x + 1 (divide both sides by 5)51The first equation has a slope of 5 and the second equation has a slope of , so the lines are perpendicular.51Determine whether the following lines are parallel, perpendicular, or neither.-x + 2y = -2 and 2x = 4y + 3•In the first equation,2y = x – 2 (add x to both sides)Example• In the second equation, 4y = 2x – 3 (subtract 3 from both sides)y = x – 1 (divide both sides by 2)21Both lines have a slope of , so the lines are parallel.21 y = x – (divide both sides by 4)2143Example from today’s homework:Web site that shows the effect of changing slope and y-intercept: http://www.mathsisfun.com/graph/straight_line_graph.htmlA quick way to check if you goofed on a negative sign:Graph the two points and see if the slope should be positive or negative.Watch out for this problem:Also, make sure you type a 0 before decimals < 1. (e.g. type 0.08, not just .08.)Hint: To start this problem, convert both distances to the same units.Practical application of slope to construction:Web site that shows how to calculate the pitch of a roof:http://roofgenius.com/Roof-Pitch-xamples.asp Pitch of a roof = rise/runJust make sure you measure both rise and run in the same units (e.g. feet, meters, inches)Problem: Is it safe to walk on this roof?Note: You will be given sheet containing formulas to use on quizzes/tests. You can print off a copy of the formula sheet to use while you do your homework and practice quizzes/tests by clicking on the “Formula sheet” menu button. It is a good idea to print a copy of this sheet (or keep a yellow copy that we’ve handed out in class) to use while you do your homework assignments and practice quizzes/tests so you can get used to the location of the formulas and know which ones will be available to you on the sheet. (You will be given a clean copy of the sheet to use at each quiz and test, so don’t plan to rely on any notes you might write on your copy of the formula sheet...)Note: The homework for this section shouldn’t take too long, so you might want to look ahead at section 3.5 on slope-intercept equations of lines. (The homework for that section is a longer assignment so you might want to get a head start…)Reminder:This homework assignment on Section 3.4 is due at the start of next class period.You may now OPEN your LAPTOPSand begin working on the homework


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UW Stout MATH 110 - The Slope of a Line

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