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graphs and proportinal relationships quick check

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Sure! Here are some quick check questions for graphs and proportional relationships:

Questions:

  1. Graph Interpretation:

    • A graph shows a straight line passing through the origin with a slope of 3. What is the relationship between the variables?

  2. Identifying Proportional Relationships:

    • Are the following pairs of quantities proportional? Why or why not?
      • Pair 1: (2, 4) and (3, 6)
      • Pair 2: (1, 2) and (2, 5)

  3. Finding the Equation:

    • If a proportional relationship is represented by the line on a graph that passes through points (2, 8) and (4, 16), what is the equation of the line in the form \(y = kx\)?

  4. Slope Calculation:

    • Calculate the slope of the line that passes through the points (1, 2) and (4, 5).

  5. Graphing Proportional Relationships:

    • If you were to graph the proportional relationship \(y = 2x\), what are the coordinates of the points when \(x = 0, 1, 2, 3\)?

  6. Understanding Non-Proportional Relationships:

    • Explain why the relationship represented by the equation \(y = x + 3\) is not proportional.

Answers:

  1. Answer: The relationship between the variables is proportional because the line passes through the origin, and the slope of 3 indicates that for every 1 unit increase in x, y increases by 3 units.

  2. Answer:

    • Pair 1: Yes, they are proportional because the ratio \(\frac{4}{2} = \frac{6}{3} = 2\).
    • Pair 2: No, they are not proportional because \(\frac{2}{1} \neq \frac{5}{2}\).

  3. Answer: The equation of the line is \(y = 4x\) (since the ratio of \(y\) to \(x\) is constant: \(8/2 = 4\) and \(16/4 = 4\)).

  4. Answer: The slope \(m\) is calculated as \(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 2}{4 - 1} = \frac{3}{3} = 1\).

  5. Answer: The coordinates are:

    • \(x = 0\), \(y = 2(0) = 0\) → (0, 0)
    • \(x = 1\), \(y = 2(1) = 2\) → (1, 2)
    • \(x = 2\), \(y = 2(2) = 4\) → (2, 4)
    • \(x = 3\), \(y = 2(3) = 6\) → (3, 6)

  6. Answer: The relationship \(y = x + 3\) is not proportional because it does not pass through the origin (when \(x = 0\), \(y = 3\)). Proportional relationships must always pass through the origin and maintain a constant ratio between \(y\) and \(x\).

Feel free to use these questions to test understanding on the topic of graphs and proportional relationships!