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Slope of a Line

Section 1.3. Slope of a Line. Section 1.3. Slide 2. Introduction. Comparing the Steepness of Two Objects. Two ladders leaning against a building. Which is steeper?

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Slope of a Line

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  1. Section 1.3 Slope of a Line

  2. Section 1.3 Slide 2 Introduction Comparing the Steepness of Two Objects Two ladders leaning against a building. Which is steeper? We compare the vertical distance from the base of the building to the ladder’s top with the horizontal distance from the ladder’s foot to the building.

  3. Section 1.3 Slide 3 Introduction Comparing the Steepness of Two Objects Ratio of vertical distance to the horizontal distance: Latter A: Latter B: So, Latter B is steeper.

  4. Section 1.3 Slide 4 Property of Comparing the Steepness of Two Objects Comparing the Steepness of Two Objects Property To compare the steepness of two objects such as two ramps, two roofs, or two ski slopes, compute the ratio for each object. The object with the larger ratio is the steeper object.

  5. Section 1.3 Slide 5 Comparing the Steepness of Two Roads Comparing the Steepness of Two Objects Example Road A climbs steadily for 135 feet over a horizontal distance of 3900 feet. Road B climbs steadily for 120 feet over a horizontal distance of 3175 feet. Which road is steeper? Explain. These figures are of the two roads, however they are not to scale Solution

  6. Section 1.3 Slide 6 Comparing the Steepness of Two Roads Comparing the Steepness of Two Objects Solution Continued A: = = ≈ B: = = ≈ vertical distance horizontal distance 135 feet 3900 feet 0.035 1 vertical distance horizontal distance 120 feet 3175 feet 0.038 1 • Road B is a little steeper than road A

  7. Section 1.3 Slide 7 Comparing the Steepness of Two Roads Finding a Line’s Slope Definition The grade of a road is the ratio of the vertical to the horizontal distance written as a percent. What is the grade of roads A? Ratio of vertical distance to horizontal distance is for road A is 0.038 = 0.038(100%) = 3.8%. Example Solution

  8. Section 1.3 Slide 8 Slope of a Non-vertical Line Finding a Line’s Slope We will now calculate the steepness of a non-vertical line given two points on the line. Let’s use subscript 1 to label x1 and y1 as the coordinates of the first point, (x1, y1). And x2 and y2 for the second point, (x2, y2). Run: Horizontal Change = x2 – x1 Rise: Vertical Change = y2 – y1 The slope is the ratio of the rise to the run. Pronounced x sub 1 and y sub 1 Pronounced x sub 1 and y sub 1

  9. Section 1.3 Slide 9 Slope of a Non-vertical Line Finding a Line’s Slope Definition Let (x1, y1) and (x2, y2) be two distinct point of a non-vertical line. The slope of the line is vertical change horizontal change rise run y2 – y1 x2 – x1 m = = = In words: The slope of a non-vertical line is equal to the ratio of the rise to the run in going from one point on the line to another point on the line.

  10. Section 1.3 Slide 10 Slope of a Non-vertical Line Finding a Line’s Slope Definition A formula is an equation that contains two or more variables. We will refer to the equation a as the slope formula. (graphical) Sign of rise or run run is positive run is negative rise is positive rise is negative Direction (verbal) goes to the right goes to the left goes up goes down

  11. Section 1.3 Slide 11 Finding the Slope of a Line Finding a Line’s Slope Example Find the slope of the line that contains the points (1, 2) and (5, 4). (x1, y1) = (1, 2) (x2, y2) = (5, 4). Solution

  12. Section 1.3 Slide 12 Finding the Slope of a Line Finding a Line’s Slope Warning A common error is to substitute the slope formula incorrectly: Correct Incorrect Incorrect

  13. Section 1.3 Slide 13 Finding the Slope of a Line Finding a Line’s Slope Example Find the slope of the line that contains the points (2, 3) and (5, 1). Solution By plotting points, the run is 3 and the rise is –2.

  14. Section 1.3 Slide 14 Definition Increasing and Decreasing Lines Increasing: Positive Slope Decreasing: Negative Slope Positive rise Positive run negative rise positive run m = m = = Positive slope = negative slope

  15. Section 1.3 Slide 15 Finding the Slope of a Line Increasing and Decreasing Lines Example Find the slope of the line that contains the points (– 9 , –4) and (12, –8). Solution – • The slope is negative • The line is decreasing

  16. Section 1.3 Slide 16 Comparing the Slopes of Two Lines Increasing and Decreasing Lines Example Find the slope of the two lines sketched on the right. Solution For line l1 the run is 1 and the rise is 2.

  17. Section 1.3 Slide 17 Comparing the Slopes of Two Lines Increasing and Decreasing Lines Solution Continued Note that the slope of l2 is greater than the slope of l1, which is what we expected because line l2 looks steeper than line l1. For line l2 the run is 1 and the rise is 4.

  18. Section 1.3 Slide 18 Investigating Slope of a Horizontal Line Horizontal and Vertical Lines Example Find the slope of the line that contains the points (2, 3) and (6, 3). Solution Plotting the points (above) and calculating the slope we get The slope of the horizontal line is zero, no steepness.

  19. Section 1.3 Slide 19 Investigating the slope of a Vertical Line Horizontal and Vertical Lines Example Find the slope of the line that contains the points (4, 2) and (4, 5). Solution Plotting the points (above) and calculating the slope we get The slope of the vertical line is undefined.

  20. Section 1.3 Slide 20 Property Horizontal and Vertical Lines Property A horizontal line has slope of zero (left figure). A vertical line has undefined slope (right figure).

  21. Section 1.3 Slide 21 Finding Slopes of Parallel Lines Parallel and Perpendicular Lines Definition Two lines are called parallel if they do not intersect. Example Find the slopes of the lines l1 and l2 sketch to the right.

  22. Section 1.3 Slide 22 Finding Slopes of Parallel Lines Parallel and Perpendicular Lines Solution Both lines the run is 3, the rise is 1 The slope is, • It makes sense that the nonvertical parallel lines have equal slope • Since they have the same steepness

  23. Section 1.3 Slide 23 Property Parallel and Perpendicular Lines Property If lines l1 and l2 are nonvertical parallel lines on the same coordinate system, then the slopes of the lines are equal: m1 = m2 Also, if two distinct lines have equal slope, then the lines are parallel. Definition Two lines are perpendicular is they intercepts at a 90o angle.

  24. Section 1.3 Slide 24 Finding Slopes of Perpendicular Lines Parallel and Perpendicular Lines Example Find the slopes of the perpendicular lines l1 and l2. Solution • The slope of line l1 is m1 = 2/3 and l2 is m2 = –3/2

  25. Section 1.3 Slide 25 Property Parallel and Perpendicular Lines Property If lines l1 and l2 are nonvertical perpendicular lines, then the slope of one line is the opposite of the reciprocal of the slope of the other line: Also, if the slope of one line is the opposite of the reciprocal of another line’s slope, then the lines are perpendicular.

  26. Section 1.3 Slide 26 Finding Slopes of Parallel Lines Parallel and Perpendicular Lines Example Line l1 has a slope of • If l2 has is parallel to l1, find the slope of l2. • If l3 is perpendicular to l1, find the slope of l3. • 1. The slopes of l1 and l2 are equal, so l2 has a slope • 2. The slope of l3 is the opposite of the reciprocal of . or Solution

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