Chapter 1 Linear Equations and Linear Functions

1.3 Slope of a Line

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.

Comparing the Steepness of Two Objects Ratio of vertical distance to the horizontal distance: Ladder A: Ladder B: So, Ladder B is steeper.

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.

Example: Comparing the Steepness of Two Roads 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 Sketches of the two roads are shown below. Note that the distances are not drawn to scale.

Solution Calculate the approximate ratio of the vertical distance to the horizontal distance for each road: Road B is a little steeper because Road B’s ratio is greater than Road A’s.

Grade of a Road The grade of a road is the ratio of the vertical distance to the horizontal distance, written as a percentage.

Slope of a nonvertical line Definition Let (x1, y1) and (x2, y2) be two distinct points of a nonvertical line. The slope of the line is In words, the slope of a nonvertical 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).

Slope A formula is an equation that contains two or more variables. We refer to the equation as the slope formula. Here we list the directions associated with the signs of rises and runs:

Example: Finding the Slope of a Line Find the slope of the line that contains the points (1, 2) and (5, 4).

Solution Using the slope formula, where (x1, y1) = (1, 2) and (x2, y2) = (5, 4), we have

Solution By plotting the points, we find that if the run is 4, then the rise is 2. So, the slope is which is our result from using the slope formula.

Slope Warning It is a common error to substitute into the slope formula incorrectly.

Example: Finding the Slope of a Line Find the slope of the line that contains the points (2, 3) and (5, 1).

Solution By plotting the points, we find that if the run is 3, then the rise is –2 . So, the slope is which is our result from using the slope formula.

Increasing and Decreasing Lines An increasing line has positive slope A decreasing line has negative slope

Example: Finding the Slope of a Line Find the slope of the line that contains the points (–9, –4) and (12, –8).

Solution Since the slope is negative, the line is decreasing.

Example: Comparing the Slopes of Two Lines Find the slopes of the two lines sketched at the right. Which line has the greater slope? Explain why this makes sense in terms of the steepness of a line.

Solution For line l1, if the run is 1, the rise is 2. So,

Solution For line l2, if the run is 1, the rise is 4. So,

Solution Note that the slope of line l2 is greater than the slope of line l1, which is what we would expect because line l2 looks steeper than line l1.

Example: Investigating the Slopes of a Horizontal Line Find the slope of the line that contains the points (2, 3) and (6, 3).

Solution We plot the points (2, 3) and (6, 3) and sketch the line that contains the points. So, the slope of the horizontal line is zero, because such a line has “no steepness.”

Example: Investigating the Slopes of a Vertical Line Find the slope of the line that contains the points (4, 2) and (4, 5).

Solution We plot the points (4, 2) and (4, 5) and sketch the line that contains the points. Since division by zero is undefined, the slope of the vertical line is undefined.

Slopes of Horizontal and Vertical Lines A horizontal line has slope equal to zero. A vertical line has undefined slope.

Parallel Lines Two lines are called parallel if they do not intersect.

Example: Finding Slopes of Parallel Lines Find the slopes of the parallel lines l1 and l2.

Solution For both lines, if the run is 3, the rise is 1. So, the slope of both lines is It makes sense that nonvertical parallel lines have equal slope, since parallel lines have the same steepness.

Slopes of Parallel Lines 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.

Perpendicular Lines Two lines are called perpendicular if they intersect at a 90° angle.

Example: Finding Slopes of Perpendicular Lines Find the slopes of the perpendicular lines l1 and l2.

Solution We see that the slope of line l1 is and the slope of line l2 is

Slopes of Perpendicular Lines 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.

Example: Finding Slopes of Parallel and Perpendicular Lines A line l1 has slope 1. If line l2 is parallel to line l1, find the slope of line l2. 2. If line l3 is perpendicular to line l1, find the slope of line l3.

Solution 1. The slopes of lines l2 and l1 are equal, so line l2 has slope 2. The slope of line l3 is the opposite of the reciprocal of or