Section 1.3 Slope of a Line

Comparing the Steepness of Two Objects 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. Section 1.3 Slide 2

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

Property Property of 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 for each object. The object with the larger ratio is the steeper object. Section 1.3 Slide 4

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 Section 1.3 Slide 5

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 Section 1.3 Slide 6

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 Section 1.3 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. Pronounced x sub 1 and y sub 1 Pronounced x sub 1 and y sub 1 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. Section 1.3 Slide 8

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. Section 1.3 Slide 9

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 Section 1.3 Slide 10

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 Section 1.3 Slide 11

A common error is to substitute the slope formula incorrectly: 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 Section 1.3 Slide 12

Find the slope of the line that contains the points (2, 3) and (5, 1). 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. Section 1.3 Slide 13

Increasing: Positive Slope Decreasing: Negative Slope 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 Section 1.3 Slide 14

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 Section 1.3 Slide 15

Find the slope of the two lines sketched on the right. 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. Section 1.3 Slide 16

Comparing the Slopes of Two Lines Increasing and Decreasing Lines Solution Continued For line l2 the run is 1 and the rise is 4. 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. Section 1.3 Slide 17

Find the slope of the line that contains the points (2, 3) and (6, 3). 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. Section 1.3 Slide 18

Find the slope of the line that contains the points (4, 2) and (4, 5). 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. Section 1.3 Slide 19

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

Two lines are called parallel if they do not intersect. 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. Section 1.3 Slide 21

Both lines the run is 3, the rise is 1 The slope is, 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 Section 1.3 Slide 22