3) How does the pattern of change between the two variables (such as distance (d) and time ( t )) in a linear relationship show up in a graph? (9pts) For example

In a linear relationship between two variables, such as distance (\(d\)) and time (\(t\)), the pattern of change is consistently proportional. This consistent proportionality manifests in a graph as a straight line. Here's how this appears and what it signifies: ### **Graphical Representation** 1. **Axes Setup**: - **Horizontal Axis (x-axis)**: Typically represents the independent variable, which in this case could be time (\(t\)). - **Vertical Axis (y-axis)**: Represents the dependent variable, such as distance (\(d\)). 2. **Straight Line**: - The data points plotted on the graph will lie along a straight line. - This line can be described by the equation \(d = mt + b\), where: - \(m\) is the slope of the line, indicating the rate of change of distance with respect to time (e.g., speed). - \(b\) is the y-intercept, representing the distance when time (\(t\)) is zero. 3. **Slope and Interpretation**: - **Positive Slope**: Indicates that as time increases, distance increases proportionally. - **Zero Slope**: Implies no change in distance regardless of time (the line is horizontal). - **Negative Slope**: Suggests that distance decreases as time increases (though this is less common in distance-time scenarios). 4. **Consistency**: - Each unit increase in time corresponds to a consistent unit increase in distance, maintaining the straight-line form without curves or bends. ### **Example** **Scenario**: Suppose a car travels at a constant speed of 60 km/h. - **Equation**: \(d = 60t\) - Here, \(m = 60 \, \text{km/h}\) (slope) and \(b = 0\) (y-intercept). - **Graph Interpretation**: - **At \(t = 0\) hours**, \(d = 0\) km. The graph starts at the origin (0,0). - **At \(t = 1\) hour**, \(d = 60\) km. - **At \(t = 2\) hours**, \(d = 120\) km. - Plotting these points and connecting them forms a straight line with a slope of 60, demonstrating that for each additional hour, the distance increases by 60 km consistently. ### **Key Takeaways** - **Linear Relationship**: A straight-line graph indicates a linear relationship where one variable changes at a constant rate relative to the other. - **Predictability**: Such a graph allows for easy prediction of one variable based on the other using the linear equation. - **Proportional Change**: The consistent slope reflects proportional change—doubling the time doubles the distance, and so on. By observing a straight line on a distance-time graph, one can quickly determine the nature of the relationship and quantify variables like speed or rate of change.