Algebra 1 Section 6.2

Definition The slope of a line, m, is the ratio of the rise (vertical change) to the corresponding run (horizontal change). Slope describes the incline or steepness of the line.

Slope The coordinates of two points on a line can be used to find the rise and the run. Using A(3, 1) and B(5, 4): rise = 4 – 1 = 3 run = 5 – 3 = 2

Example 1 Choose two convenient points: A(-1, 3) and B(1, 2). Find the rise (change in y). Find the run (change in x). rise = -1 run = 2 slope = - 1 2

Slope There is a general formula for the slope between two arbitrary points, (x1, y1) and (x2, y2): m = rise run = y2 – y1 x2 – x1

Example 2 Find the slope of the line passing through (2, -3) and (4, 1). m = y2 – y1 x2 – x1 1 -(-3) 4 - 2 = 4 2 = = 2

Example 2 The slope is the same regardless of which point is considered as the first. m = y2 – y1 x2 – x1 -3 - 1 2 - 4 = -4 -2 = = 2

Finding the Slope of a Line When given the graph, determine the rise and run from one point to another point on the line. Then write the ratio of rise over run and reduce if possible.

Finding the Slope of a Line When given two points, label the points P1(x1, y1) and P2(x2, y2). Then use the slope formula, m = y2 – y1 x2 – x1

Example 3 Find the slope of the horizontal line y = 4, using the points (2, 4) and (7, 4). m = y2 – y1 x2 – x1 4 – 4 7 – 2 = 5 = = 0

Example 3 The slope of a horizontal line is always zero. m = y2 – y1 x2 – x1 4 – 4 7 – 2 = 5 = = 0

Example 4 Find the slope of the vertical line x = 3, using the points (3, -2) and (3, 4). m = y2 – y1 x2 – x1 4 -(-2) 3 - 3 = 6 = The slope is undefined.

Example 4 The slope of a vertical line is always undefined. m = y2 – y1 x2 – x1 4 -(-2) 3 - 3 = 6 = The slope is undefined.

Example 5 Then move one unit down and three units to the right, to (2, 1). First, plot the point (-1, 2). Draw a line through the points. You can also move one unit up and three units to the left. y x

Slope The sign of the slope indicates whether the line slants up or down moving from left to right. The absolute value of the slope indicates the “steepness” of the line.

Slope positive slope

Slope negative slope

Slope slope = 0 (horizontal)

slope is undefined (vertical)

Slope There are many different notations for slope. m = rise run = y2 – y1 x2 – x1 = Δy Δx = change in y change in x

Slope The Greek letter delta, Δ, is used to symbolize the change in a quantity. This emphasizes that slope measures a rate of change. Rates of change compare two changing quantities.

Example 6 The speed is found by dividing the change in distance by the change in time. For part (a), the change in distance is 70 mi, and the change in time is 2 hr. Therefore, speed is 35 mi/hr.

Homework: pp. 242-244