Slope and Rates of Change 5 3 Vocabulary

* 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 Run Rise x If a line rises from left to right, its slope is positive. If a line falls from left to right, its slope negative.

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

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

Additional Example 1 B: Identifying the Slope of the Line Tell whether the slope is positive or negative. Then find the slope. y 2 – 2 0 – 2 2 x The line falls from right to left. The slope is negative.

Additional Example 1 B Continued Tell whether the slope is positive or negative. Then find the slope. y 2 – 2 0 – 2 -3 2 2 x The rise is 2. The run is -3. slope = rise = 2 run -3

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

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

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

Additional Example 2 A: Using Slope and a Point to Graph a Line 1 Use the slope 2 and the point (1, – 1) to graph the line. rise = -2 or 2 run 1 -1 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 4 2 ● – 4 – 2 0 – 2 – 4 x 2 ● 4

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

Additional Example 2 B: Using Slope and a Point to Graph a Line 2 Use the slope 1 and the point (– 1, – 1) to graph the line. rise = 1 run 2 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 4 2 – 4 – 2 0 – 2 – 4 ● x 2 4

Check It Out: Example 2 A Use the slope – 2 3 and the point (2, 0) to graph the line. rise = -2 or 2 run 3 -3 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 4 2 – 4 – 2 0 – 2 – 4 x 2 4

Check It Out: Example 2 B 4 Use the slope 1 and the point (– 2, 0) to graph the line. rise = 1 run 4 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 4 2 – 4 – 2 0 – 2 – 4 x 2 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.

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

6 6 5 5 Distance (mi) 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? 4 3 2 1 7 14 21 28 35 Time (min) 7 4 1 3 2 1 7 14 21 28 35 Time (min)

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