1. 8.2 Lines and Their Slope Part 2: Slope

2. 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.

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

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

5. 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!

6. 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.

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

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

9. 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.

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

11. Determining Whether Two Lines Are Perpendicular Determine whether the lines L 1, through (0, -3) and (2, 0), and L 2, through (-3, 0) and (0, -2), are perpendicular.

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

13. 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).