Lesson

In this investigation we are going to use rational functions to determine how long it would take you to run a marathon.

- To explore rational functions in real life.
- To practice graphing rational functions.
- To practice exploring characteristics of rational functions.

- Tape
- Stopwatch
- Paper
- Different colored pens
- Measuring Tape
- Internet

You can work with a partner for this investigation.

- The average marathon is about $26$26 miles long. The following function indicates how long it would take someone to run a marathon with respect to the rate at which they run: $t=\frac{26}{r}$
`t`=26`r` where $t$`t`is the time it takes you to run the marathon (in hours), and $r$`r`is the rate at which you run (in miles per hour). - Fill in the table of values for the given function.
- Time how long it takes you to run $50$50 feet. You may want to use your tape to mark a starting and ending location.
- Divide the distance you ran ($50$50 feet) by the total amount of time it took you to find your rate.
- Convert the rate you just found into miles per hour instead of feet per second.
- Draw an arrow to indicate where you fall on the graph you have created.

- In what quadrant(s) does portion of the function that you graphed lie?
- If you were plotting the entire function of $t=\frac{26}{r}$
`t`=26`r` miles on the coordinate plane what quadrant(s) would your graph lie in? - Does it make sense to graph the entire function for this scenario? Why or why not?
- What is the domain of the function that you graphed?
- What is the range of the function that you graphed?
- What are the vertical and horizontal asymptotes of the graph that you created? Draw them in on your graph using a dotted line drawn by a different coloured pen from the one you previously used.
- Interpret the horizontal and vertical asymptotes in terms of the situation. Why does it make sense that there are asymptotes?
- How long would it take someone to run the marathon approximately if they were running at a rate of $38$38 miles per hour? Is this plausible? Why or why not? You can use the internet to look up any information that may be relevant to answering this question.
- Compare with a friend! If you both were able to maintain your rate for the entire marathon who would run the marathon faster?

Display the graphs of linear and non-linear functions and connect the structure of the functions with their graphs

Apply graphical methods in solving problems