Slope and Rates of Change 5-3

Vocabulary Slope- of a line is a measure of its steepness and is the ratio of rise to run. Rate of change- The ratio of two quantities that change, such as slope.

The slope of a line is a measure of its steepness and is the ratio of rise to run: y If a line rises from left to right, its slope is positive. If a line falls from left to right, its slope negative. x Run Rise

Tell whether the slope is positive or negative. Then find the slope. Additional Example 1A: Identifying the Slope of the Line The line rises from left to right. The slope is positive.

Tell whether the slope is positive or negative. Then find the slope. Additional Example 1A Continued The rise is 3. The run is 3. slope = rise run = 3333 =1 3 3

Tell whether the slope is positive or negative. Then find the slope. Additional Example 1B: Identifying the Slope of the Line The line falls from right to left. The slope is negative –2 y x

Tell whether the slope is positive or negative. Then find the slope. Additional Example 1B Continued The rise is 2. The run is -3. slope = rise run = –2 y x

Check It Out: Example 1A Tell whether the slope is positive or negative. Then find the slope.

Check It Out: Example 1B (0, –4) ‏ (–2, 4) ‏ 8 –2 Tell whether the slope is positive or negative. Then find the slope.

You can graph a line if you know its slope and one of its points.

Use the slope  and the point (1, –1) to graph the line. Additional Example 2A: Using Slope and a Point to Graph a Line 2121 From point (1, 1) move 2 units down and 1 unit right, or move 2 units up and 1 unit left. Mark the point where you end up, and draw a line through the two points. y x – –2 = or rise run ● ●

You can write an integer as a fraction by putting the integer in the numerator of the fraction and a 1 in the denominator. Remember!

Use the slope and the point (–1, –1) to graph the line. Additional Example 2B: Using Slope and a Point to Graph a Line 1212 From point (–1, –1) move 1 unit up and 2 units right. Mark the point where you end up, and draw a line through the two points. y x – –2 = rise run 1 2 ●

Use the slope – and the point (2, 0) to graph the line. Check It Out: Example 2A 2323 From point (2, 0) move 2 units down and 3 units right, or move 2 units up and 3 unit left. Mark the point where you end up, and draw a line through the two points. y x – –2 = or rise run

Use the slope and the point (–2, 0) to graph the line. Check It Out: Example 2B 1414 From point (–2, 0) move 1 unit up and 4 units right. Mark the point where you end up, and draw a line through the two points. y x – –2 = rise run 1 4

The ratio of two quantities that change, such as slope, is a rate of change. A constant rate of change describes changes of the same amount during equal intervals. A variable rate of change describes changes of a different amount during equal intervals. The graph of a constant rate of change is a line, and the graph of a variable rate of change is not a line.

The ratio of two quantities that change, such as slope, is a rate of change. A constant rate of change describes changes of the same amount during equal intervals. A variable rate of change describes changes of a different amount during equal intervals. The graph of a constant rate of change is a line, and the graph of a variable rate of change is not a line.

Tell whether each graph shows a constant or variable rate of change. A.B. Additional Example 3: Identifying Rates of Change in Graphs The graph is nonlinear, so the rate of change is variable. The graph is linear, so the rate of change is constant.

Check It Out: Example 4 The graph shows the distance a jogger travels over time. Is he traveling at a constant or variable rate. How fast is he traveling? Time (min) ‏ Distance (mi) ‏ Time (min) ‏ Distance (mi) ‏

The graph is a line, so the jogger is traveling at a constant rate of speed. slope (speed) = rise (distance) ‏ run (time) ‏ The amount of distance is the rise, and the amount of time is the run. You can find the speed by finding the slope. Check It Out: Example 4 Continued