Martina turns on a treadmill to start walking. She takes 1 minute to gradually increase her speed and then walks at the same speed for 4 minutes. She increases her speed twice more during the 15 minutes she is on the treadmill, as shown on the graph below. Which best describes the function of Martina's walking speed over the interval [0,15] ？a piecewise function with separate pieces of linear functions onlya single linear function with three points of discontinuitya piecewise function with separate pieces of linear and step functionsa step function for the entire interval

1. From 0 to 1 minute, the speed increases linearly.
2. From 1 to 5 minutes, the speed remains constant (step function).
3. From 5 to 6 minutes, the speed increases linearly.
4. From 6 to 10 minutes, the speed remains constant (step function).
5. From 10 to 11 minutes, the speed increases linearly.
6. From 11 to 15 minutes, the speed remains constant (step function).

Supplemental Knowledge

Piecewise functions are composed of multiple sub-functions defined over an interval. These sub-functions may be linear, quadratic or any other kind of function; step functions are one type where values remain constant within an interval while abrupt changes take place at its endpoints.
In analyzing the graph provided:

1. Initial Increase (0 to 1 minute): The speed increases linearly from 0 to about 2 miles per hour.
2. Constant Speed (1 to 5 minutes): The speed remains constant at about 2 miles per hour.
3. First Increase (5 to 6 minutes): The speed increases linearly again.
4. Constant Speed (6 to 10 minutes): The speed remains constant at about 3 miles per hour.
5. Second Increase (10 to 11 minutes): Another linear increase in speed.
6. Constant Speed (11 to 15 minutes): The speed remains constant at about 4 miles per hour.

Real-Life Connections

Learning how functions represent real-life scenarios can be extremely useful. For instance, understanding how your speed changes over time when exercising on a treadmill will allow you to optimize it and achieve greater results from each workout session.
Imagine Martina is training for a marathon and uses this treadmill routine regularly. By studying her speed graph, she can spot areas where her stamina needs improvement or where her pace could be adjusted to improve endurance.

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