8.2 Lines and Their Slope Part 2: Slope

Slope The measure of the “steepness” of a line is called the slope of the line. Slope is internationally referred to as “gradient.” One way to measure the gradient of a line is to compare the vertical change in the line (the rise) to the horizontal change (the run) while moving along the line from one point to another. The letter m is used to denote slope/gradient.

Using the Gradient Formula If x1 ≠ x2, the gradient of the line through the distinct point (x1, y1) and (x2, y2) is Find the gradient of the line that passes through the points (2, -1) and (-5, 3).

 Find the gradient of the line that passes through the points (-4, 3) and (-3, 4).

Vertical and Horizontal Lines A vertical line has an equation of the form x = a, where a is a real number, and its gradient is undefined. Using the gradient formula, you will get 0 in the denominator! A horizontal line has an equation of the form y = b, where b is a real number, and its gradient is 0. Using the gradient formula, you will get 0 in the numerator!

Positive and Negative Gradients A line with a positive gradient rises from left to right. A line with a negative gradient falls from left to right. A horizontal line has a gradient of 0. A vertical line has undefined gradient.

Graphing a Line Using Gradient and a Point Graph the line that has gradient 2/3 and passes through the point (-1, 4).

Graph the line that has gradient -3/2 and passes through the point (-1, -2).

Parallel and Perpendicular Lines Gradient of Parallel Lines Two nonvertical lines with the same gradient are parallel. Two nonvertical parallel lines have the same gradient. Any two vertical lines are parallel. Gradient of Perpendicular Lines If neither is vertical, two perpendicular lines have gradients that are opposite reciprocals (their product is -1). Two lines with gradients that are opposite reciprocals are perpendicular. Every vertical line is perpendicular to every horizontal line.

Determining Whether Two Lines Are Parallel Determine whether the lines L1, through (-2, 1) and (4, 5), and L2, through (3, 0) and (0, -2), are parallel.

Determining Whether Two Lines Are Perpendicular Determine whether the lines L1, through (0, -3) and (2, 0), and L2, through (-3, 0) and (0, -2), are perpendicular.

Determine whether the lines L1, through (0, -7) and (2, 3), and L2, through (0, -3) and (1, -2), are parallel, perpendicular, or neither.

Average Rate of Change Gradient of a line is the ratio of the vertical change (y) to the horizontal change (x). In real-life situations, gradient gives the average rate of change of the dependent variable (y) per the independent variable (x).

Finding Average Rate of Change The Decline in Purchasing Power of the Dollar Find the average rate of change in the purchasing power of the dollar during the decade. 1 .75 .50 .25 $.581 Purchasing power $.766 1990 1992 1994 1996 1998 2000 Year