Warm Up Problem of the Day Lesson Presentation Lesson Quizzes

Warm Up Graph the linear function. y = 2x + 4

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

Learn to determine the slope of a line and to recognize constant and variable rates of change.

Vocabulary slope rate of change

The slope of a line is a measure of its steepness and is the ratio of rise to run: y Run rise Rise slope = x run 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 1A: 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 1A 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 run = 3 1

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

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

Check It Out: Example 1A 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.

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

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

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

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

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!

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

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

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

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.

Tell whether each graph shows a constant or variable rate of change. Check It Out: Example 3 Tell whether each graph shows a constant or variable rate of change. A. B. y y 4 4 2 2 x x –4 –2 2 4 –4 –2 2 4 –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.

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.

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. rise (distance)‏ run (time)‏ 20 miles 1 hour slope (speed) = = The butterfly travels at a rate of 20 miles per hour.

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? 7 14 21 28 35 Time (min)‏ 1 2 3 4 5 6 Distance (mi)‏ 6 5 4 Distance (mi)‏ 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. rise (distance)‏ run (time)‏ 1 mi 7 min slope (speed) = = The jogger travels at a rate of 1 mile every 7 minutes.

Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems

Lesson Quiz: Part I 1. Tell whether the slope is positive or negative. Then find the slope. Negative; -1

Lesson Quiz: Part II 1 2 2. Use the slope and the point (–2, –3) to graph the line.

Lesson Quiz: Part III 3. Tell whether the graph shows a constant or variable rate of change. variable

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

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

Lesson Quiz for Student Response Systems 3. Which of the following graphs represents a variable rate of change? A. B.