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12-7

Rates of Change. 12-7. Course 2. Warm Up. Problem of the Day. Lesson Presentation. Rates of Change. 12-7. Course 2. Warm Up Tell whether the function in each table could be linear or is nonlinear. A. B. nonlinear. could be linear. Rates of Change. 12-7. Course 2.

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12-7

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  1. Rates of Change 12-7 Course 2 Warm Up Problem of the Day Lesson Presentation

  2. Rates of Change 12-7 Course 2 Warm Up Tell whether the function in each table could be linear or is nonlinear. A. B. nonlinear could be linear

  3. Rates of Change 12-7 Course 2 Problem of the Day What two 3-digit numbers have a product of 19,019? 133 and 143

  4. Rates of Change 12-7 Course 2 Learn to recognize constant and variable rates of change.

  5. Rates of Change 12-7 Course 2 If you drop a bowling ball and a baseball from the same height at the same time, which do you think would hit the ground first? Without air resistance, both would hit the ground at the same time. Gravity pulls all objects to the ground at the same rate, no matter what their sizes or weights. As each ball drops, it falls faster and faster due to gravity. The rate of change, the change in the ball’s speed as it falls, is a change that can be measured.

  6. Rates of Change 12-7 Course 2 A constant rate of change means that something changes by the same amount during equal intervals. A graph that has a constant rate of change is a line, and the rate of change is the same as the slope of the line. y 4 2 x 4 –4 –2 2 0 –2 –4

  7. Rates of Change 12-7 Course 2 Additional Example 1: Identifying Constant Rates of Change Tell in which intervals of x the graph shows constant rates of change. Look for parts of the graph that are line segments. Identify the intervals of x for each line segment. The graph shows a constant rate of change in the interval x = 0 to x = 3.

  8. Rates of Change 12-7 Course 2 Insert Lesson Title Here Try This: Example 1 Tell in which intervals of x the graph shows constant rates of change. y 4 Look for parts of the graph that are line segments. 2 x 0 –2 –4 2 4 Identify the intervals of a x for each line segment. –2 –4 The graph shows a constant rate of change in the interval x = –3 to x = 1.

  9. Rates of Change 12-7 Course 2 A variable rate of change means that something changes by a different amount during equal intervals. Variable rates of change are also called nonlinear rates of change, because a graph that represents a variable rate of change is not a line.

  10. Rates of Change 12-7 Course 2 Additional Example 2: Identifying Rates of Change Tell whether each graph shows a constant or variable rate of change. The graph is nonlinear, so the rate of change is variable. The graph is linear, so the rate of change is constant.

  11. Rates of Change 12-7 Course 2 Insert Lesson Title Here Try This: Example 2 Tell whether each graph shows a constant or variable rate of change. y y 4 4 2 2 x x 4 –4 –2 2 0 4 –4 –2 2 0 –2 –2 –4 –4 The graph is linear, so the rate of change is constant. The graph is nonlinear, so the rate of change is variable.

  12. Rates of Change 12-7 Remember! slope = Course 2 rise run

  13. Rates of Change 12-7 Course 2 Additional Example 3: Using Rate of Change to Solve Problems The graph shows the distance a monarch butterfly travels at a constant rate of speed. How fast is the butterfly traveling?

  14. Rates of Change 12-7 Course 2 Additional Example 3 Continued The graph is linear. You can see that the same amount of distance (rise) is traveled during equal time intervals (run), so the butterfly is traveling at a constant rate of speed. The slope of the graph is rise ÷ run, or distance ÷ time. Since speed = distance ÷ time, you can find the speed by finding the slope. Use two points on the line such as (1, 20) and (2, 40), to find the slope. rise run 20miles 1 hour slope = = The butterfly travels at a rate of 20 miles per hour.

  15. Rates of Change 12-7 6 5 7 4 1 Distance (mi) 3 7 2 1 1 7 14 21 28 35 Time (min) Course 2 Insert Lesson Title Here Try This: Example 3 The graph shows the distance a jogger travels over time. Is he traveling at a constant or variable rate. How fast is he traveling? 6 5 4 Distance (mi) 3 2 1 7 14 21 28 35 Time (min)

  16. Rates of Change 12-7 Course 2 Insert Lesson Title Here Try This: Example 3 The graph is linear. You can see that the same amount of distance (rise) is traveled during equal time intervals (run), so the jogger is traveling at a constant rate of speed. Since speed = distance ÷ time, you can find the speed by finding the slope. Use two points on the line such as (1, 7) and (2, 14) to find the slope. rise run 1 mi 7 min = Slope = The jogger travels at a rate of 1 mile every 7 minutes.

  17. Rates of Change 12-7 Course 2 Insert Lesson Title Here Lesson Quiz 1. Tell whether the graph shows a variable or constant rate of change. variable rate of change 2. During a flood, a river rose 0.5 inch per minute. In another flood, the river rose 2 feet per hour. Which rate of change was higher? 0.5 inch/minute

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