The image you provided includes a problem where you're asked to graph the distance traveled by Long over time, find the slope of the line, and explain what the slope represents. Here’s how we can break it down:

Step 1: Data from the Table

The table lists time (in minutes) and corresponding distance (in miles):

• At 0 minutes, the distance is 0 miles.
• At 12 minutes, the distance is 3 miles.
• At 20 minutes, the distance is 5 miles.

Step 2: Plotting the Graph

You've already plotted the points on the graph:

• (0, 0)
• (12, 3)
• (20, 5)

These points form a straight line.

Step 3: Finding the Slope

The slope of a line is calculated as the change in the y-values (distance) divided by the change in the x-values (time). Using two points on the line: $\text{Slope} = \frac{y_2 - y_1}{x_2 - x_1}$ Let’s use the points (0, 0) and (12, 3): $\text{Slope} = \frac{3 - 0}{12 - 0} = \frac{3}{12} = \frac{1}{4} \text{ miles per minute}$ Alternatively, using (12, 3) and (20, 5): $\text{Slope} = \frac{5 - 3}{20 - 12} = \frac{2}{8} = \frac{1}{4} \text{ miles per minute}$

Step 4: Interpretation of the Slope

The slope of $\frac{1}{4}$ means that Long is traveling at a constant rate of 0.25 miles per minute. This represents Long's speed.

Summary

• Slope: $\frac{1}{4}$ miles per minute.
• Interpretation: The slope indicates Long's speed.

Would you like more details or have any questions? Here are five related questions to further your understanding:

1. What would the graph look like if Long had increased their speed halfway through the trip?
2. How would the slope change if the distance covered in 20 minutes was 8 miles instead of 5 miles?
3. Can you calculate the total distance Long would cover in 1 hour at this speed?
4. How would you describe the relationship between time and distance in this scenario?
5. What happens to the slope if the time intervals increase but the distance remains the same?

Tip: Remember, the slope of a line on a distance-time graph always represents speed. The steeper the line, the faster the speed.