3 Chapter Chapter 2 Graphing

Slope and Rate of Change Section 3.4 Slope and Rate of Change

Finding the Slope of a Line Given Two Points of the Line Objective 1 Finding the Slope of a Line Given Two Points of the Line

Slope Slope of a Line The slope m of the line containing the points (x1, y1) and (x2, y2) is given by

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

Example (cont) (4, –3 ) and (2, 2) Rise –5 Run 2

Helpful Hint When finding slope, it makes no difference which point is identified as (x1, y1) and which is identified as (x2, y2). Just remember that whatever y-value is first in the numerator, its corresponding x-value is first in the denominator.

Example Find the slope of the line through (–2, 1) and (3, 5). Graph the line.

Slope of Lines Positive Slope Line goes up to the right x y Lines with positive slopes go upward as x increases. m > 0 Negative Slope Line goes downward to the right x y Lines with negative slopes go downward as x increases. m < 0

Finding the Slope of a Line Given Its Equation Objective 2 Finding the Slope of a Line Given Its Equation

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 y-intercept is (0, -4). The slope is 3.

Example Find the slope and y-intercept of the line whose equation is The slope is 5/9. The y-intercept if (0, 3)

Example Find the slope of the line –3x + 2y = 11. Solve the equation for y. The slope of the line is 3/2.

Example Find the slope of the line –y = 6x – 7. Solve the equation for y not negative y. The slope of the line is ‒6.

Finding Slopes of Horizontal and Vertical Lines Objective 3 Finding Slopes of Horizontal and Vertical Lines

Slope of a Horizontal Line x y Zero Slope Horizontal Line Horizontal lines have a slope of 0. m = 0 For any two points, the y values will be equal to the same real number. The numerator in the slope formula = 0 (the difference of the y-coordinates), but the denominator ≠ 0 (two different points would have two different x-coordinates).

Example Find the slope of the line y = 3. Knowing that y = 3, all solutions must have a y-value of 3. Use the points (5, 3) and (7, 3). The slope of the line y = 3 is 0. All horizontal lines have the slope of 0.

Slope of a Vertical Line x y A vertical line has an undefined slope. Undefined Slope Vertical Line m is undefined. For any two points, the x values will be equal to the same real number. The denominator in the slope formula = 0 (the difference of the x-coordinates), but the numerator ≠ 0 (two different points would have two different y-coordinates). So the slope is undefined (since you can’t divide by 0).

Example Find the slope of the line x = –2. Knowing that x = –2 , all solutions must have a x-value of –2 . Use the points (–2, 6) and (–2, 8) . The slope of the line x = –2 is undefined. All vertical lines have undefined slopes.

Slopes of Parallel and Perpendicular Lines Objective 4 Slopes of Parallel and Perpendicular Lines

Parallel Lines Two lines that never intersect are called parallel lines. Parallel lines have the same slope. (Unless they are vertical lines, which have no slope.) Vertical lines are also parallel. x y

Perpendicular Lines Two lines that intersect at right angles are called perpendicular lines. Two nonvertical perpendicular lines have slopes that are negative reciprocals of each other. The product of their slopes will be –1. x y slope a Horizontal and vertical lines are perpendicular to each other.

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.

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.

Example Determine whether the following lines are parallel, perpendicular, or neither. –5x + y = –6 x + 5y = 5 Solve both equations for y. –5x + y = –6 5y = –x + 5 y = 5x – 6 The first equation has a slope of 5 and the second equation has a slope of , so the lines are perpendicular.

Slope as a Rate of Change Objective 5 Slope as a 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.

Example Find the grade of the road: The grade of the road is 15%.