5 4 Rates of Change and Slope Vocabulary

5 -4 Rates of Change and Slope Vocabulary rate of change slope

5 -4 Rates of Change and Slope The ratio of two quantities that change, such as distance and time, is a rate of change. In a proportional relationship, you can use two points on a graph to find the rate of change.

5 -4 Rates of Change and Slope Additional Example 1: Using a Graph to Find Rate of Change The graph shows the distance a monarch butterfly travels over time. What is the butterfly’s rate of change? The relationship is proportional because the graph is a straight line that goes through the origin. Use two points on the graph to find the rate of change.

5 -4 Rates of Change and Slope Additional Example 1 Continued Choose two, such as (1, 20) and (2, 40). rate of change = change in distance change in time – 20) = 20 = (40 (2 – 1) 1 The rate of change the butterfly travels at is 20 miles per hour.

5 -4 Rates of Change and Slope Check It Out: Example 1 The graph shows the distance a loggerhead turtle travels over time. What is the loggerhead turtle’s rate of change? Distance (mi) y 60 50 40 30 20 10 0 x 2 4 6 8 Time (hr) Rate of change = 15 ; 1 The rate of change of the loggerhead turtle is 15 miles per hour.

5 -4 Rates of Change and Slope 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 variable rate of change is not a straight line.

5 -4 Rates of Change and Slope Additional Example 2: Identifying Rates of Change in Graphs Tell whether each graph shows a constant or variable rate of change. A. B. The graph is not a line, so the rate of change is variable. The graph is a line, so the rate of change is constant.

5 -4 Rates of Change and Slope Check It Out: Example 2 Tell whether each graph shows a constant or variable rate of change. A. The graph is a line so the rate of change is constant. B. The graph is not a line so The rate of change is variab

5 -4 Rates of Change and Slope The slope of a line is the rate of change between any two points on the line. y Run Rise x The change in y is sometimes called the rise and the change in x is sometimes called the run. If a line rises from left to right, its slope is positive. If a line falls from left to right, its slope negative. In linear functions slope is a constant rate of change.

5 -4 Rates of Change and Slope Additional Example 3 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.

5 -4 Rates of Change and Slope Additional Example 3 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

5 -4 Rates of Change and Slope Additional Example 3 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.

5 -4 Rates of Change and Slope Additional Example 3 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

5 -4 Rates of Change and Slope Check It Out: Example 3 Tell whether the slope is positive or negative. Then find the slope. negative; slope = - 4.

5 -4 Rates of Change and Slope Additional Example 4 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

5 -4 Rates of Change and Slope Additional Example 4 B: Using Slope and a Point to Graph a Line Use the slope 1 2 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

5 -4 Rates of Change and Slope Check It Out: Example 4 Use the slope and the point to graph the line on the given coordinate plane. A. 1 ; (2, 1) 2 y 4 4 a 2 B. 2 ; (3, 0) 3 – 4 – 2 0 – 2 – 4 x 2 4 4 b