5 6 Slope and Rates of Change Warm

5 -6 Slope and Rates of Change Warm Up Problem of the Day Lesson Presentation Lesson Quizzes

5 -6 Slope and Rates of Change Warm Up Graph the linear function. y = 2 x + 4

5 -6 Slope and Rates of Change Problem of the Day What two 3 -digit numbers have a product of 19, 019? 133 and 143

5 -6 Slope and Rates of Change Learn to determine the slope of a line and to recognize constant and variable rates of change.

5 -6 Slope and Rates of Change Vocabulary slope rate of change

5 -6 Slope and Rates of Change 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.

5 -6 Slope and Rates of Change 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.

5 -6 Slope and Rates of Change 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

5 -6 Slope and Rates of Change 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.

5 -6 Slope and Rates of Change 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

5 -6 Slope and Rates of Change Check It Out: Example 1 A Tell whether the slope is positive or negative. Then find the slope. The line does not point upward or downward so it is not positive or negative.

5 -6 Slope and Rates of Change Check It Out: Example 1 A Continued Tell whether the slope is positive or negative. Then find the slope. 2 M(1, – 1) N(3, – 1) The rise is 0. The run is 2. slope = rise = 0 2 run

5 -6 Slope and Rates of Change Check It Out: Example 1 B Tell whether the slope is positive or negative. Then find the slope. The line falls from left to right. (– 2, 4) – 2 The slope is negative. 8 (0, – 4) The rise is 8. The run is – 2. slope = rise = 8 = – 4 – 2 run

5 -6 Slope and Rates of Change You can graph a line if you know its slope and one of its points.

5 -6 Slope and Rates of Change 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

5 -6 Slope and Rates of Change 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.

5 -6 Slope and Rates of Change 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

5 -6 Slope and Rates of Change 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 ●

5 -6 Slope and Rates of Change 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 ● 2 x 4

5 -6 Slope and Rates of Change 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.

5 -6 Slope and Rates of Change 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.

5 -6 Slope and Rates of Change Check It Out: Example 3 Tell whether each graph shows a constant or variable rate of change. A. y B. 4 2 – 4 – 2 0 – 2 2 4 x – 4 The graph is nonlinear, so the rate of change is variable. y 4 2 – 4 – 2 0 – 2 2 4 x – 4 The graph is linear, so the rate of change is constant.

5 -6 Slope and Rates of Change Additional Example 4: Using Rate of Change to Solve Problems The graph shows the distance a monarch butterfly travels overtime. Tell whether the graph shows a constant or variable rate of change. Then find how fast the butterfly is traveling.

5 -6 Slope and Rates of Change Additional Example 4 Continued The graph is a line, so the butterfly 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) = 20 miles slope (speed) = rise run (time) 1 hour The butterfly travels at a rate of 20 miles per hour.

5 -6 Slope and Rates of Change 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)

5 -6 Slope and Rates of Change 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) = 1 mi slope (speed) = rise run (time) 7 min The jogger travels at a rate of 1 mile every 7 minutes.

5 -6 Slope and Rates of Change Lesson Quizzes Standard Lesson Quiz for Student Response Systems

5 -6 Slope and Rates of Change Lesson Quiz: Part I 1. Tell whether the slope is positive or negative. Then find the slope. Negative; -1

5 -6 Slope and Rates of Change Lesson Quiz: Part II 1 2. Use the slope 2 and the point (– 2, – 3) to graph the line.

5 -6 Slope and Rates of Change Lesson Quiz: Part III 3. Tell whether the graph shows a constant or variable rate of change. variable

5 -6 Slope and Rates of Change Lesson Quiz for Student Response Systems 1. Tell whether the slope is positive or negative. Then identify the slope. A. positive; 1 B. positive; 2 C. negative; – 1 D. negative; – 2

5 -6 Slope and Rates of Change Lesson Quiz for Student Response Systems 2. Use the slope 1 and the point (– 2, – 3) to 4 identify the graph of the line. A. B.

5 -6 Slope and Rates of Change Lesson Quiz for Student Response Systems 3. Which of the following graphs represents a variable rate of change? A. B.