Slope and Rate of Change § 1.4 Slope and Rate of Change

Slope of a Line Slope of a Line

Slope of a Line Slope of a Line x y Positive Slope: m > 0 Lines with positive slopes go upward as x increases. Negative Slope: m < 0 x y Lines with negative slopes go downward as x increases.

Finding the Slope Given Two Points Example: Find the slope of the line through (4, – 3 ) and (2, 2). If we let (x1, y1) = (4, – 3) and (x2, y2) = (2, 2), then Note: If we let (x1, y1) be (2, 2) and (x2, y2) be (4, – 3), then we get the same result.

Slope-Intercept Form of a Line When a linear equation in two variables is written in the slope-intercept form, y = mx + b m is the slope and (0, b) is the y-intercept of the line. y = 3x – 4 The slope is 3. The y-intercept is (0, -4).

Slope-Intercept Form of a Line Example: Find the equation of the line with slope and y-intercept (0, 5). slope: y-intercept:

Slopes of Horizontal and Vertical Lines Slope of a Line x y Horizontal Line: m = 0 Horizontal lines have a slope of 0. Vertical Line: Undefined slope or no slope x y Vertical lines have an undefined slope.

Slope of a Horizontal Line x y (0, 3) y = 3 Any two points of a horizontal line will have the same y-values. This means that the y-values will always have a difference of 0 for all horizontal lines. All horizontal lines have a slope 0.

Slope of a Vertical Line x y (– 3, 0) Any two points of a vertical line will have the same x-values. This means that the x-values will always have a difference of 0 for all vertical lines. All vertical lines have undefined slope. x = – 3

Parallel Lines Two lines in the same plane are parallel if they do not intersect. x y Slope m1 m1 = m2 Slope m2

Parallel Lines Example: Determine whether the line 6x + 2y = 9 is parallel to – 3x – y = 3. Find the slope of each line. 6x + 2y = 9 – 3x – y = 3 The slopes are the same so the lines are parallel.

Perpendicular Lines Two lines are perpendicular if they lie in the same plane and meet at a 90◦ (right) angle. The product of the slopes of two perpendicular lines is – 1. x y Slope m2 m1m2 =  1 or m1 = Slope m1

Perpendicular Lines Example: Determine whether the line x + 3y = – 15 is perpendicular to – 3x + y = – 1 . Find the slope of each line. x + 3y = – 15 – 3x + y = – 1 The slopes are negative reciprocals so the lines are perpendicular.

Average Rate of Change Example: Becky decided to take a bike ride up a mountain trail. The trail has a vertical rise of 90 feet for every 250 feet of horizontal change. In percent, what is the grade of the trail? The grade of the trail is given by The grade of the trail is The slope of a line can also be interpreted as the average rate of change. It tells us how fast y is changing with respect to x.