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New Zealand
Level 7 - NCEA Level 2

Running a Marathon (Investigation)


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.

  1. 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=26r 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).
  2. Fill in the table of values for the given function.
  3. 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.
  4. Divide the distance you ran ($50$50 feet) by the total amount of time it took you to find your rate.
  5. Convert the rate you just found into miles per hour instead of feet per second.
  6. Draw an arrow to indicate where you fall on the graph you have created.


  1. In what quadrant(s) does portion of the function that you graphed lie?
  2. If you were plotting the entire function of $t=\frac{26}{r}$t=26r miles on the coordinate plane what quadrant(s) would your graph lie in?
  3. Does it make sense to graph the entire function for this scenario? Why or why not?
  4. What is the domain of the function that you graphed?
  5. What is the range of the function that you graphed?
  6. 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.
  7. Interpret the horizontal and vertical asymptotes in terms of the situation. Why does it make sense that there are asymptotes?
  8. 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.
  9. 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

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