Using the version of the story in part (c), graph the relation where x is the time since Mario left his house 

measured in minutes and y is the speed of Mario’s car at time x labeled in miles per hour. Label at least 3 important 

points which show your reasoning on this problem

To graph the relation where the horizontal axis (\(x\)) represents the time since Mario left his house (measured in minutes) and the vertical axis (\(y\)) represents the speed of Mario's car (measured in miles per hour), we'll need to plot how his speed changes over time based on the journey described.

### Steps for Graphing:

1. **From 12:00 pm to 12:15 pm (0 to 15 minutes):**
   - Mario drives west at a constant speed of 40 miles per hour.
   - The speed remains constant during this time.
   - **Important Point:** At \(x = 0\) and \(x = 15\), \(y = 40\) mph.

2. **From 12:15 pm to 12:30 pm (15 to 30 minutes):**
   - Mario turns around and drives back home at the same speed of 40 miles per hour.
   - The speed remains constant at 40 mph but the direction changes (not shown in speed, only distance).
   - **Important Point:** At \(x = 30\), \(y = 40\) mph.

3. **From 12:30 pm to 12:35 pm (30 to 35 minutes):**
   - Mario is at home searching for his phone, so his speed drops to 0 mph.
   - **Important Point:** At \(x = 35\), \(y = 0\) mph.

4. **From 12:35 pm to 12:55 pm (35 to 55 minutes):**
   - Mario leaves his house again, driving west at a constant speed of 45 miles per hour.
   - The speed is constant at 45 mph during this time.
   - **Important Point:** At \(x = 55\), \(y = 45\) mph.

### Graph Description:

- **Horizontal Axis (x-axis):** Represents the time in minutes since Mario left his house.
- **Vertical Axis (y-axis):** Represents the speed of Mario’s car in miles per hour.

#### Graph Plot:
- **Segment 1:** From \(x = 0\) to \(x = 15\), draw a horizontal line at \(y = 40\) mph.
- **Segment 2:** From \(x = 15\) to \(x = 30\), continue the horizontal line at \(y = 40\) mph (same speed, opposite direction, but speed graph is unchanged).
- **Segment 3:** At \(x = 30\), the speed drops to 0 mph. Draw a vertical drop from \(y = 40\) mph to \(y = 0\) mph.
- **Segment 4:** From \(x = 30\) to \(x = 35\), draw a horizontal line at \(y = 0\) mph (Mario is searching for his phone).
- **Segment 5:** At \(x = 35\), the speed increases to 45 mph. Draw a vertical jump from \(y = 0\) mph to \(y = 45\) mph.
- **Segment 6:** From \(x = 35\) to \(x = 55\), draw a horizontal line at \(y = 45\) mph.

### Labeled Points:
1. **Point A:** \((0, 40)\) mph - Mario starts his journey at 40 mph.
2. **Point B:** \((30, 0)\) mph - Mario stops at his house, reducing speed to 0 mph.
3. **Point C:** \((35, 45)\) mph - Mario resumes his journey at a new speed of 45 mph.

These points highlight key moments where Mario's speed changes and can be used to discuss the behavior of the graph.

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

### Related Questions:
1. How would the graph change if Mario’s speeds were different on each leg of his journey?
2. What would the graph look like if Mario accelerated or decelerated instead of driving at constant speeds?
3. How could you represent Mario’s acceleration on a different graph?
4. What effect would a longer or shorter time spent at home have on the graph?
5. What other factors could be considered to make this graph more detailed or realistic?

### Tip:
When interpreting a speed-time graph, remember that horizontal segments represent constant speed, while vertical changes indicate an instantaneous change in speed.

Explore how to graph the speed of Mario's car over time, measured in minutes, from the moment he leaves his house. Learn about constant speed segments and sudden stops, illustrated with key points like Mario's initial speed of 40 mph, stopping at home, and resuming at 45 mph.

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