HW: Graphing Proportional Relationships - Regina Sapienza ... - Free Printable

Problem Analysis

The task involves determining whether the relationships shown in the tables are proportional, identifying the constant of proportionality, writing the equation for each relationship, graphing the points, and connecting them with a straight line. Finally, we need to observe and describe the graphs of proportional relationships.

#### Key Concepts:
1. Proportional Relationship: A relationship between two variables \( x \) and \( y \) is proportional if the ratio \( \frac{y}{x} \) is constant for all pairs of values. This constant is called the constant of proportionality.
2. Equation: For a proportional relationship, the equation is of the form \( y = kx \), where \( k \) is the constant of proportionality.
3. Graph: The graph of a proportional relationship is a straight line that passes through the origin (0, 0).

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Step-by-Step Solution

#### Problem 1
| \( x \) | \( y \) |
|---------|---------|
| 2 | 4 |
| 3 | 6 |
| 4 | 8 |
| 5 | 10 |
| 6 | 12 |

1. Check for Proportionality:
- Calculate the ratio \( \frac{y}{x} \) for each pair:
\[
\frac{4}{2} = 2, \quad \frac{6}{3} = 2, \quad \frac{8}{4} = 2, \quad \frac{10}{5} = 2, \quad \frac{12}{6} = 2
\]
- The ratio is constant (\( k = 2 \)), so the relationship is proportional.

2. Constant of Proportionality:
- \( k = 2 \)

3. Equation:
- \( y = 2x \)

4. Graph:
- Plot the points: (2, 4), (3, 6), (4, 8), (5, 10), (6, 12).
- Draw a straight line through these points, extending it to pass through the origin (0, 0).

#### Problem 2
| \( x \) | \( y \) |
|---------|---------|
| 1 | 6 |
| 2 | 8 |
| 3 | 10 |
| 4 | 12 |
| 5 | 14 |

1. Check for Proportionality:
- Calculate the ratio \( \frac{y}{x} \) for each pair:
\[
\frac{6}{1} = 6, \quad \frac{8}{2} = 4, \quad \frac{10}{3} \approx 3.33, \quad \frac{12}{4} = 3, \quad \frac{14}{5} = 2.8
\]
- The ratio is not constant, so the relationship is not proportional.

2. Constant of Proportionality:
- Not applicable (since it is not proportional).

3. Equation:
- Not applicable (since it is not proportional).

4. Graph:
- Plot the points: (1, 6), (2, 8), (3, 10), (4, 12), (5, 14).
- Draw a straight line through these points, but note that it does not pass through the origin.

#### Problem 3
| \( x \) | \( y \) |
|---------|---------|
| 3 | 2 |
| 6 | 4 |
| 9 | 6 |
| 12 | 8 |

1. Check for Proportionality:
- Calculate the ratio \( \frac{y}{x} \) for each pair:
\[
\frac{2}{3} \approx 0.67, \quad \frac{4}{6} \approx 0.67, \quad \frac{6}{9} \approx 0.67, \quad \frac{8}{12} \approx 0.67
\]
- The ratio is constant (\( k \approx 0.67 \)), so the relationship is proportional.

2. Constant of Proportionality:
- \( k = \frac{2}{3} \)

3. Equation:
- \( y = \frac{2}{3}x \)

4. Graph:
- Plot the points: (3, 2), (6, 4), (9, 6), (12, 8).
- Draw a straight line through these points, extending it to pass through the origin (0, 0).

#### Problem 4
| \( x \) | \( y \) |
|---------|---------|
| 3 | 1 |
| 4 | 2 |
| 5 | 3 |
| 6 | 4 |

1. Check for Proportionality:
- Calculate the ratio \( \frac{y}{x} \) for each pair:
\[
\frac{1}{3} \approx 0.33, \quad \frac{2}{4} = 0.5, \quad \frac{3}{5} = 0.6, \quad \frac{4}{6} \approx 0.67
\]
- The ratio is not constant, so the relationship is not proportional.

2. Constant of Proportionality:
- Not applicable (since it is not proportional).

3. Equation:
- Not applicable (since it is not proportional).

4. Graph:
- Plot the points: (3, 1), (4, 2), (5, 3), (6, 4).
- Draw a straight line through these points, but note that it does not pass through the origin.

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Observations About Proportional Relationships

- Graphs of Proportional Relationships:
- The graph of a proportional relationship is a straight line that passes through the origin (0, 0).

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Final Answer

\[
\boxed{\text{straight line, origin}}
\]