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Graphing Proportional Relationships: Warm Up, Problem of the Day, Lesson Presentation, Lesson Quizzes

This lesson focuses on identifying and graphing proportional relationships. It includes a warm-up activity, a problem of the day, a lesson presentation, and quizzes to assess understanding.

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Graphing Proportional Relationships: Warm Up, Problem of the Day, Lesson Presentation, Lesson Quizzes

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

  2. Warm Up Identify the quadrant that contains each point. 1.(6, –4) 2. (5, 3) 3. (–5, –2) IV I III

  3. Problem of the Day Graph the ordered pairs form the table. What letter do the points form? V

  4. Sunshine State Standards MA.7.A.1.4 Graph proportional relationships…

  5. Vocabulary linear equation linear function

  6. y 6 4 Miles 2 x 0 0 2 4 Hours The table shows how far a kayak travels down a river if the kayak is moving at a rate of 2 miles per hour. Notice for all ordered pairs in the table for every 1 hour increase in time, the miles traveled increases by 2. These ordered pairs are in proportion. 1 2 3 8 2 4 3 6 = = = If the ordered pairs are in proportion, then the data represents a proportional relationship. When you graph a proportional relationship, the result is a line that passes through the origin.

  7. Additional Example 1: Graphing Proportional Relationships Graph the linear function y = 4x. Make a table. 0 4 8 12 Proportional relationships pass through (0, 0). Graph the ordered pairs (0, 0), (1, 4), (2, 8), (3, 12).

  8. Additional Example 1 Continued y 12 (3, 12) 10 Place each ordered pair on the coordinate grid and then connect the points with a line. 8 (2, 8) 6 4 (1, 4) The graph is a straight line that passes through the origin. 2 x (0, 0) 0 2 4 6 8 10 Check 1 4 2 8 3 12 The ordered pairs are proportional. = =

  9. Check It Out: Example 1 Graph y = 15x. 0 15 30 45 y 60 40 20 x 0 2 4 6 8

  10. A linear equation is an equation whose graph is a line. The solutions of a linear equation are the points that make up its graph. Linear equations and linear graphs can be different representations of linear functions. A linear function is a function whose graph is a nonvertical line.

  11. Some relationships are linear but not proportional. If the ordered pairs in a linear function are not all proportional then it is not a proportional relationship. These non-proportional relationships do not pass through the origin on a graph.

  12. Additional Example 2: Identify Proportional Relationships Tell whether the function is a proportional relationship. Then graph the function. A. y = –2x Make a table. 2 0 –2 –4 –6 1 –2 2 –4 3 –6 –1 2 = = = The ordered pairs are proportional and the graph passes through (0, 0). y = –2x is a proportional relationship.

  13. Check It Out: Example 2 Tell whether y = 10x – 1is a proportional relationship. Then graph the function. –1 9 19 29 39 The ordered pairs are not proportional, and the graph does not pass through (0, 0). y = 10x –1is not a proportional relationship.

  14. Additional Example 3: Earth Science Application The fastest-moving tectonic plates on Earth move apart at a rate of 15 centimeters per year. Write a linear function that describes the movement of the plates over time. Graph the relationship. Is this a proportional relationship? Justify your answer. Let x represent the input, which is the time in years. Let y represent the output, which is the distance in centimeters the plates move apart. distance in cm = 15 cm/yr time in years  y = 15 x  The function is y = 15x. Yes, the graph goes through the origin

  15. Additional Example 3 Continued Make a function table. Include a column for the rule. Input Rule Output y Multiply the input by 15. 15(x) x 15(0) 0 0 1 15(1) 15 30 15(2) 2 45 15(3) 3

  16. y 100 80 60 40 20 0 2 4 8 10 12 Additional Example 3 Continued Graph the ordered pairs (0, 0), (1, 15), (2, 30), and (3, 45) from your table. Connect the points with a line. Check Use the ordered pairs (1, 15), (2, 30), and (3, 45) to see if the relationship is proportional. Centimeters 2 30 3 45 1 15 = = The ordered pairs are proportional and the graph passes through (0, 0). y = 15x is a proportional relationship. x Years

  17. Check It Out: Example 3 The outside temperature is increasing at the rate of 6 °F per hour. When Reid begins measuring the temperature, it is 52 °F. Write a linear function that describes the outside temperature over time. Graph the relationship. Is this a proportional relationship? Justify your answer. y = 6x + 52, where x is the number of hours and y is the temperature. The ordered pairs are not proportional and the graph does not pass through (0, 0). y = 6x + 52is not a proportionalrelationship.

  18. 100 80 60 40 0 2 4 6 8 Check it Out: Example 3 Continued Temperature Hours

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

  20. Lesson Quiz: Part I Tell whether each function is a proportional relationship. Then graph the function. 1.y = 3x – 4 2.y = –x 3.y = 2x y = –x no y = 3x – 4 yes yes y = 2x

  21. Lesson Quiz: Part II 4. The temperature of a liquid is decreasing at a rate of 12 °F per hour. Susan begins measuring the liquid at 200 °F. Write a linear function that describes the change in temperature over time. Then make a graph to show the temperature over 5 hours. y = 200 – 12x;no, the graph does not go through the origin.

  22. Lesson Quiz for Student Response Systems 1. Tell whether the linear function y = 2x is a proportional relationship. A. yes B. no

  23. Lesson Quiz for Student Response Systems 2. Tell whether the graph of the given linear function is a proportional relationship. A. yes B. no

  24. Lesson Quiz for Student Response Systems 3. Larry has 150 cents in his piggy bank. He puts 20 cents into it everyday. Identify a linear function that describes the amount in the piggy bank over time. Is this a proportional relationship? A. y = 20x; yes B.y = –20x; yes C. y = 150 + 20x; no D.y = 150 – 20x; no

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