Section 2.3 The Slope of a Line

Objectives Slope Slope-Intercept Form of a Line Interpreting Slope in Applications

SLOPE The slope of the line passing through the points (x1, y1) and (x2, y2) is where x1 ≠ x2. That is, slope equals rise over run.

Example Find the slope of the line passing through the points (7, 3) and (2, 2). Plot these points and graph the line. Interpret the slope. Solution Graph the line passing through these points. The slope indicates that the line rises 1 unit for every 5 units of run.

Slope A line with positive slope rises from left to right. A line with negative slope falls from left to right. A horizontal line has a zero slope. A line with undefined slope is a vertical line.

Example Find the slope of the line passing through each pair of points, if possible. a. (3, 4), (−1, 4) b. (−4, 2), (−4, 5) Solution a. b. undefined

Example Sketch a line passing through the point (1, 2) and having slope 3/4. Solution Start by plotting (1, 2). The slope is ¾ which means a rise (increase) of 3 and a run (horizontal) of 4. The line passes through the point (1 + 4, 2 + 3) = (5, 5).

SLOPE-INTERCEPT FORM The line with slope m and y-intercept b is given by y = mx + b, the slope-intercept form of a line.

Example For the graph write the slope-intercept form of the line. Solution The graph intersects the y-axis at 0, so the y-intercept is 0. The graph falls 3 units for each 1 unit increase in x, the slope is –3. The slope intercept-form of the line is y = –3x .

Example For the graph shown, write the slope-intercept form of the line. Solution The graph passes through (0, 1), so the y-intercept is 1. Because the graph rises 5 units for each unit increased in x, the slope is 5. The slope-intercept form is y = 5x + 1.

Example The points listed on the table all lie on a line. a. Find the missing value in the table. b. Write the slope-intercept form of the line. Solution a. The line passes through (−4, 9) and (2, −3). For each unit increased in x, y decreases by 2. Therefore the missing value is 5. b. The slope-intercept form is y = −2x + 1. x y −4 9 2 −3 ? −9

Example The distance y in miles that a boat is from the dock on a fishing expedition after x hours is shown below. a. Find the y-intercept. What does the y-intercept represent? Solution a. The y-intercept is 35, so the boat is initially 35 miles from the dock.

Example (cont) The distance y in miles that a boat is from the dock on a fishing expedition after x hours is shown below. b. The graph passes through the point (4, 15). Discuss the meaning of this point. Solution b. The point (4, 15) means that after 4 hours the boat is 15 miles from the dock.

Example (cont) The distance y in miles that a boat is from the dock on a fishing expedition after x hours is shown below. c. Find the slope of the line. Interpret the slope as a rate of change. Solution c. The slope is –5. The slope means that the boat is going toward the dock at 5 miles per hour.

Example When a street vendor sells 40 tacos, his profit is $24, and when he sells 75 tacos, his profit is $66. a. Find the slope of the line passing through the points (40, 24) and (75, 66) b. Interpret the slope as a rate of change. Solution b. Profit increases on average, by $1.20 for each additional taco sold.