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graphs and proportinal relationships quick check
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GPT-4o mini
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!
### 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!
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