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Proportional Relationships and Graphing Functions

Learn how to identify and graph proportional relationships, find rates of change, and interpret slope. Includes practice quizzes.

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Proportional Relationships and Graphing Functions

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  1. Warm Up Problem of the Day Lesson Presentation Lesson Quizzes

  2. Warm Up Tell whether y = 2x + 4 is a proportional relationship. Then graph the function. no

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

  4. Sunshine State Standards MA.7.A.1.4 Graph proportional relationships and identify the unit rate as the slope of the related linear function.

  5. Vocabulary rate of change slope

  6. 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.

  7. 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.

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

  9. 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? y 60 50 40 Distance (mi) 30 20 10 x 0 2 4 6 8 15 1 Rate of change = ; Time (hr) The rate of change of the loggerhead turtle is 15 miles per hour.

  10. 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.

  11. 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.

  12. Check It Out: Example 2 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.

  13. The slope of a line is the rate of change between any two points on the line. y Run change in y Rise slope = x change in 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.

  14. Additional Example 3A: 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.

  15. 3 3 rise run slope = = = 1 Additional Example 3A Continued Tell whether the slope is positive or negative. Then find the slope. 3 3 The rise is 3. The run is 3.

  16. y 2 x 0 –2 2 –2 Additional Example 3B: Identifying the Slope of the Line Tell whether the slope is positive or negative. Then find the slope. The line falls from right to left. The slope is negative.

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

  18. Check It Out: Example 3 Tell whether the slope is positive or negative. Then find the slope. negative; slope = - 4.

  19. rise run 2 -1 -2 1 = or Additional Example 4A: Using Slope and a Point to Graph a Line 21 Use the slope  and the point (1, –1) to graph the line. y 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 0 –4 –2 2 4 –2 ● –4

  20. rise run 1 2 = Additional Example 4B: Using Slope and a Point to Graph a Line 12 Use the slope and the point (–1, –1) to graph the line. y 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 ● 0 –4 –2 2 4 –2 –4

  21. 1 2 2 3 (3, 0) (2, 1) ; ; Check It Out: Example 4 Use the slope and the point to graph the line on the given coordinate plane. y A. 4 4a 2 x 0 –4 –2 2 4 4b –2 B. –4

  22. Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems

  23. Lesson Quiz: Part I 1. Tell whether the graph shows a constant or variable rate of change. variable

  24. Lesson Quiz: Part II 2. Tell whether the slope is positive or negative. Then find the slope. negative; -1

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

  26. 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

  27. 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.

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

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