Slope of a Line Section 1.3

Lehmann, Intermediate Algebra, 3ed Section 1.3Slide 2 Introduction 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. Comparing the Steepness of Two Objects

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

Lehmann, Intermediate Algebra, 3ed Section 1.3 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. Slide 4 Property of Comparing the Steepness of Two Objects Property Comparing the Steepness of Two Objects

Lehmann, Intermediate Algebra, 3ed Section 1.3 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 Slide 5 Comparing the Steepness of Two Roads Example Solution Comparing the Steepness of Two Objects

Lehmann, Intermediate Algebra, 3ed Section 1.3 A: = = ≈ B: = = ≈ Slide 6 Comparing the Steepness of Two Roads vertical distance horizontal distance 135 feet 3900 feet vertical distance horizontal distance 120 feet 3175 feet Road B is a little steeper than road A Solution Continued Comparing the Steepness of Two Objects

Lehmann, Intermediate Algebra, 3ed Section 1.3 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(100%) = 3.8%. Slide 7 Comparing the Steepness of Two Roads Definition Solution Example Finding a Line’s Slope

Lehmann, Intermediate Algebra, 3ed Section 1.3 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. Slide 8 Slope of a Non-vertical Line 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 Finding a Line’s Slope

Lehmann, Intermediate Algebra, 3ed Section 1.3 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 Slide 9 Slope of a Non-vertical Line vertical change horizontal change rise run y 2 – y 1 x 2 – x 1 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. Definition Finding a Line’s Slope

Lehmann, Intermediate Algebra, 3ed Section 1.3 A formula is an equation that contains two or more variables. We will refer to the equation a Slide 10 Slope of a Non-vertical Line as the slope formula. 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 (graphical) Definition Finding a Line’s Slope

Lehmann, Intermediate Algebra, 3ed Section 1.3 Find the slope of the line that contains the points (1, 2) and (5, 4). (x 1, y 1 ) = (1, 2) (x 2, y 2 ) = (5, 4). Slide 11 Finding the Slope of a Line Example Solution Finding a Line’s Slope

Lehmann, Intermediate Algebra, 3ed Section 1.3 A common error is to substitute the slope formula incorrectly: Slide 12 Finding the Slope of a Line Correct Incorrect Incorrect Warning Finding a Line’s Slope

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

Lehmann, Intermediate Algebra, 3ed Section 1.3 Increasing: Positive Slope Decreasing: Negative Slope Slide 14 Definition Positive rise Positive run m = = Positive slope negative rise positive run m = = negative slope Increasing and Decreasing Lines

Lehmann, Intermediate Algebra, 3ed Section 1.3 Find the slope of the line that contains the points (– 9, –4) and (12, –8). Slide 15 Finding the Slope of a Line The slope is negative The line is decreasing – Example Solution Increasing and Decreasing Lines

Lehmann, Intermediate Algebra, 3ed Section 1.3 Find the slope of the two lines sketched on the right. Slide 16 Comparing the Slopes of Two Lines For line l 1 the run is 1 and the rise is 2. Example Solution Increasing and Decreasing Lines

Lehmann, Intermediate Algebra, 3ed Section 1.3 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 Comparing the Slopes of Two Lines For line l 2 the run is 1 and the rise is 4. Solution Continued Increasing and Decreasing Lines

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

Lehmann, Intermediate Algebra, 3ed Section 1.3 Find the slope of the line that contains the points (4, 2) and (4, 5). Slide 19 Investigating the slope of a Vertical Line Plotting the points (above) and calculating the slope we get The slope of the vertical line is undefined. Example Solution Horizontal and Vertical Lines

Lehmann, Intermediate Algebra, 3ed Section 1.3 A horizontal line has slope of zero (left figure). A vertical line has undefined slope (right figure). Slide 20 Property Horizontal and Vertical Lines

Lehmann, Intermediate Algebra, 3ed Section 1.3 Two lines are called parallel if they do not intersect. Slide 21 Finding Slopes of Parallel Lines Find the slopes of the lines l 1 and l 2 sketch to the right. Definition Example Parallel and Perpendicular Lines

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

Lehmann, Intermediate Algebra, 3ed Section 1.3 If lines l 1 and l 2 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. Slide 23 Property Two lines are perpendicular is they intercepts at a 90 o angle. Property Definition Parallel and Perpendicular Lines

Lehmann, Intermediate Algebra, 3ed Section 1.3 Find the slopes of the perpendicular lines l 1 and l 2. Slide 24 Finding Slopes of Perpendicular Lines The slope of line l 1 is m 1 = 2/3 and l 2 is m 2 = –3/2 Example Solution Parallel and Perpendicular Lines

Lehmann, Intermediate Algebra, 3ed Section 1.3 If lines l 1 and l 2 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. Slide 25 Property Parallel and Perpendicular Lines

Lehmann, Intermediate Algebra, 3ed Section 1.3Slide 26 Finding Slopes of Parallel Lines Line l 1 has a slope of 1.If l 2 has is parallel to l 1, find the slope of l 2. 2.If l 3 is perpendicular to l 1, find the slope of l 3. 1.The slopes of l 1 and l 2 are equal, so l 2 has a slope 2.The slope of l 3 is the opposite of the reciprocal of. or Example Solution Parallel and Perpendicular Lines