Mario leaves his house at 12:00 noon and drives straight west for 15 minutes toward his friend Dominic’s house at 40 miles per hour. At 12:15pm, Mario realizes he forgot his phone and turns around and drives straight back home at 40 miles per hour. At 12:30pm, Mario looks for his phone in his house for 5 minutes. At 12:35pm, Mario leaves his house again and heads straight west to Dominic’s house at 45 miles per hour. He arrives at Dominic’s house after 20 minutes of driving. He stays at Dominic’s house for the rest of the afternoon. Assume Mario drives at a constant speed on each leg of his journey. a) (8 points) Sketch a possible graph of a relation satisfying all of the following properties:  Let the horizontal axis represent the number of minutes that have passed since Mario left his house.  Let the vertical axis represent the distance that Mario is from his house in miles.  Let the relation be all points (x,y) such that x is a time in minutes after Mario left his house and y is the distance that Mario from his house is at time x. 

To sketch a graph representing Mario's journey, we need to follow each segment of his trip step by step and plot the corresponding points on the graph where the horizontal axis represents time (in minutes since he left his house) and the vertical axis represents the distance from his house (in miles).

### Step-by-Step Breakdown:
1. **From 12:00 pm to 12:15 pm:**
   - Mario drives straight west at 40 miles per hour for 15 minutes.
   - Distance covered: \( \text{Distance} = \text{Speed} \times \text{Time} = 40 \, \text{mph} \times \frac{15}{60} \, \text{hours} = 10 \, \text{miles} \).
   - At 12:15 pm, Mario is 10 miles west of his house.
   - Points: \( (0, 0) \) to \( (15, 10) \).

2. **From 12:15 pm to 12:30 pm:**
   - Mario realizes he forgot his phone and turns around to head back home at 40 miles per hour.
   - He drives back 10 miles in 15 minutes.
   - At 12:30 pm, Mario is back at his house (0 miles from his house).
   - Points: \( (15, 10) \) to \( (30, 0) \).

3. **From 12:30 pm to 12:35 pm:**
   - Mario spends 5 minutes searching for his phone at home.
   - During this time, he does not move from his house.
   - Points: \( (30, 0) \) to \( (35, 0) \).

4. **From 12:35 pm to 12:55 pm:**
   - Mario leaves his house again, driving straight west at 45 miles per hour for 20 minutes.
   - Distance covered: \( \text{Distance} = 45 \, \text{mph} \times \frac{20}{60} \, \text{hours} = 15 \, \text{miles} \).
   - At 12:55 pm, Mario arrives at Dominic's house, which is 15 miles west of his house.
   - Points: \( (35, 0) \) to \( (55, 15) \).

### Graph Sketch:
- **Horizontal Axis (x-axis)**: Represents time in minutes since 12:00 noon.
- **Vertical Axis (y-axis)**: Represents the distance in miles that Mario is from his house.

Here’s a detailed description of the graph:

- Start at the origin (0,0).
- Draw a straight line from (0,0) to (15,10), indicating Mario's drive west to Dominic's house.
- Draw another straight line from (15,10) to (30,0), indicating Mario's return trip home.
- From (30,0) to (35,0), draw a horizontal line indicating the 5 minutes Mario spends at home.
- Draw a final straight line from (35,0) to (55,15), indicating Mario's second drive to Dominic's house.

This graph visually represents Mario’s journey with respect to time and his distance from home.

### Do you want details or have any questions?

### Related Questions:
1. How would the graph change if Mario drove at different speeds during each leg of his journey?
2. How could you calculate the total distance Mario traveled during the entire trip?
3. What would the graph look like if Mario stopped for a break on the way to Dominic’s house?
4. How would you represent the speed of Mario's journey on a separate graph?
5. What are the implications of changing the speed on the time it takes Mario to reach Dominic’s house?

### Tip:
When sketching graphs of real-world situations, it’s essential to carefully track time intervals and corresponding distances to ensure accuracy.

Learn how to sketch a graph representing Mario's journey based on his movements throughout the afternoon. Understand the relationship between time and distance as Mario drives to and from his friend Dominic's house, with detailed points plotted for clarity.

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