Section 1 3 Slope of a Line Introduction

Section 1. 3 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. Section 1. 3 Lehmann, Intermediate Algebra, 4 ed Slide 2

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 Lehmann, Intermediate Algebra, 4 ed Slide 3

Property of 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. Section 1. 3 Lehmann, Intermediate Algebra, 4 ed 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. Solution • These figures are of the two roads, however they are not to scale Section 1. 3 Lehmann, Intermediate Algebra, 4 ed 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 0. 035 1 B: = vertical distance = 120 feet ≈ horizontal distance 3175 feet 0. 038 1 • Road B is a little steeper than road A Section 1. 3 Lehmann, Intermediate Algebra, 4 ed 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. Example What is the grade of roads A? Solution Ratio of vertical distance to horizontal distance is for road A is 0. 038 = 0. 038(100%) = 3. 8%. Section 1. 3 Lehmann, Intermediate Algebra, 4 ed 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 Pronounced x sub y sub 1 1 1 and y sub 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 The slope is the ratio of the rise to the run. Section 1. 3 Lehmann, Intermediate Algebra, 4 ed Slide 8

Slope of a Non-vertical Line Finding a Line’s Slope Definition 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. Section 1. 3 Lehmann, Intermediate Algebra, 4 ed 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. Sign of rise or run Direction (verbal) run is positive goes to the right run is negative goes to the left rise is positive goes up rise is negative goes down Section 1. 3 Lehmann, Intermediate Algebra, 4 ed (graphical) 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). Solution (x 1, y 1) = (1, 2) (x 2, y 2) = (5, 4). Section 1. 3 Lehmann, Intermediate Algebra, 4 ed Slide 11

Finding the Slope of a Line Finding a Line’s Slope Warning A common error is to substitute the slope formula incorrectly: Correct Section 1. 3 Incorrect Lehmann, Intermediate Algebra, 4 ed Slide 12

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 Lehmann, Intermediate Algebra, 4 ed Slide 13

Definition Increasing and Decreasing Lines Increasing: Positive Slope Decreasing: Negative Slope Positive rise m= Positive run = Positive slope Section 1. 3 negative rise m= positive run = negative slope Lehmann, Intermediate Algebra, 4 ed 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 Lehmann, Intermediate Algebra, 4 ed Slide 15

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 l 1 the run is 1 and the rise is 2. Section 1. 3 Lehmann, Intermediate Algebra, 4 ed Slide 16

Comparing the Slopes of Two Lines Increasing and Decreasing Lines Solution Continued For line l 2 the run is 1 and the rise is 4. 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. Section 1. 3 Lehmann, Intermediate Algebra, 4 ed Slide 17

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 Lehmann, Intermediate Algebra, 4 ed Slide 18

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 Lehmann, Intermediate Algebra, 4 ed Slide 19

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 Lehmann, Intermediate Algebra, 4 ed Slide 20

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 l 1 and l 2 sketch to the right. Section 1. 3 Lehmann, Intermediate Algebra, 4 ed Slide 21

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 Lehmann, Intermediate Algebra, 4 ed Slide 22