Slope of a Line
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. Slide 2
Ratio of vertical distance to the horizontal distance: 8 feet 2 Latter A: = 4 feet 1 8 feet 4 Latter B: = 2 feet 1 So, Latter B is steeper. Introduction Comparing the Steepness of Two Objects Slide 3
Property of Comparing the Steepness of Two Objects Property Comparing the Steepness of Two Objects To compare the steepness of two objects such as two ramps, two roofs, or two ski slopes, compute the ratio vertical distance horizontal distance for each object. The object with the larger ratio is the steeper object. Slide 4
Comparing the Steepness of Two Roads 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. Solution Comparing the Steepness of Two Objects These figures are of the two roads, however they are not to scale Slide 5
Comparing the Steepness of Two Roads Comparing the Steepness of Two Objects Solution Continued A: = vertical distance = 135 feet horizontal distance 3900 feet B: = vertical distance = 120 feet horizontal distance 3175 feet 0.035 1 0.038 1 Road B is a little steeper than road A Slide 6
Comparing the Steepness of Two Roads Definition The grade of a road is the ratio of the vertical to the horizontal distance written as a percent. Example What is the grade of roads A? Solution Finding a Line s Slope Ratio of vertical distance to horizontal distance is for road A is 0.038 = 0.038(100%) = 3.8%. Slide 7
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 x 1 and y 1 as the coordinates of the first point, (x 1, y 1 ). And x 2 and y 2 for the second point, (x 2, y 2 ). Run: Horizontal Change = x 2 x 1 Rise: Vertical Change = y 2 y 1 Pronounced x sub 1 1 and y y sub 1 The slope is the ratio of the rise to the run. Slide 8
Slope of a Non-vertical Line Definition Finding a Line s Slope Let (x 1, y 1 ) and (x 2, y 2 ) be two distinct point of a non-vertical line. The slope of the line is vertical change rise m = = = horizontal change run y 2 y 1 x 2 x 1 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. Slide 9
Slope of a Non-vertical Line Definition A formula is an equation that contains two or more variables. We will refer to the equation a y2 y1 m = as the slope formula. x 2 x 1 Sign of rise or run Direction (verbal) run is positive run is negative rise is positive rise is negative Finding a Line s Slope goes to the right goes to the left goes up goes down (graphical) Slide 10
Example Finding the Slope of a Line Find the slope of the line that contains the points (1, 2) and (5, 4). Solution (x 1, y 1 ) = (1, 2) (x 2, y 2 ) = (5, 4). 4 2 2 1 m = = = 5 1 4 2 Finding a Line s Slope Slide 11
Warning Finding the Slope of a Line Finding a Line s Slope A common error is to substitute the slope formula incorrectly: Correct Incorrect Incorrect y y y y x x m= m= m= x x x x y y 2 1 2 1 2 1 2 1 1 2 2 1 Slide 12
Example Finding the Slope of a Line Find the slope of the line that contains the points (2, 3) and (5, 1). Solution m rise 2 2 = = = run 3 3 Finding a Line s Slope By plotting points, the run is 3 and the rise is 2. Slide 13
Definition Increasing and Decreasing Lines Increasing: Positive Slope Decreasing: Negative Slope Positive rise m = Positive run = Positive slope negative rise m = positive run = negative slope Slide 14
Example Finding the Slope of a Line Find the slope of the line that contains the points ( 9, 4) and (12, 8). Solution 8 ( 4) 8 4 4 4 m + = = = = 12 9 12 + 9 21 21 ( ) The slope is negative The line is decreasing Increasing and Decreasing Lines Slide 15
Comparing the Slopes of Two Lines Example Find the slope of the two lines sketched on the right. Solution Increasing and Decreasing Lines For line l 1 the run is 1 and the rise is 2. rise 1 m = = = run 2 2 Slide 16
Comparing the Slopes of Two Lines Solution Continued Increasing and Decreasing Lines For line l 2 the run is 1 and the rise is 4. rise 4 m = = = 4 run 1 Note that the slope of l 2 is greater than the slope of l 1, which is what we expected because line l 2 looks steeper than line l 1. Slide 17
Investigating Slope of a Horizontal Line Example Find the slope of the line that contains the points (2, 3) and (6, 3). Solution Horizontal and Vertical Lines Plotting the points (above) and calculating the slope we get 3 3 0 m = = = 0 6 2 4 The slope of the horizontal line is zero, no steepness. Slide 18
Investigating the slope of a Vertical Line Example Find the slope of the line that contains the points (4, 2) and (4, 5). Solution Horizontal and Vertical Lines Plotting the points (above) and calculating the slope we get 5 2 3 m = =, division by zero is undefined. 4 4 0 The slope of the vertical line is undefined. Slide 19
Property Property Horizontal and Vertical Lines A horizontal line has slope of zero (left figure). A vertical line has undefined slope (right figure). Slide 20
Finding Slopes of Parallel Lines Definition Two lines are called parallel if they do not intersect. Example Parallel and Perpendicular Lines Find the slopes of the lines l 1 and l 2 sketch to the right. Slide 21
Finding Slopes of Parallel Lines Solution Parallel and Perpendicular Lines Both lines the run is 3, the rise is 1 rise 1 The slope is, m = = run 3 It makes sense that the nonvertical parallel lines have equal slope Since they have the same steepness Slide 22