# Rational Functions

A rational expression has the form p/q, where p and q are polynomials and q is not 0.

A rational function has the form f(x) = p/q, where p and q are polynomials and q is not 0.

These are two examples of rational functions:

Here is a problem involving a rational function.

Problem:Suppose you need to make a rectangular dog pen with an area of 9 square meters. What dimensions would you make the pen so that its perimeter (or fence) is as small as possible?

- As a first step to solving this problem, the rational function may be used to represent the perimeter, where P is the perimeter and w is the width, in meters.
- Try 4 different sets of dimensions, then compare the perimeters of your 4 sets of dimensions with a partner.
- Suppose you plotted the width versus Perimeter of many guesses. What do you think the graph will look like?
- Use the interctive below to explore this problem.

- What happens to the perimeter of the pen as you increase the width from 1 m to 9 m?

- What happens to the area of the pen as you increase the width from 1 m to 9 m?
- Now watch the diagram of the pen as you do this. How does the shape of the pen change as you increase the width from 1 m to 9 m?

- Why is 9 m the largest width? Could the width be larger than 9 m? Explain why or why not.

- Choose two points on the graph that have the same perimeter. What do you notice about the length and width values for these points? How are the rectangles for these points similar? How are they different?

- Click on the trail button and move the slider from 1 m to 9 m. Describe the shape of the graph.

- What is the minimum perimeter of the pen? How does this value relate to the graph?

- What width results in the minimum perimeter? How does this value relate to the graph?

- What shape of pen results in the minimum perimeter? Is this shape a rectangle?

- Suppose you had a space of 16 square metres for your pen. Describe how the graph will change.

- What was the shape of the graph of width versus perimeter? Were you surprised by this? Why or why not?

- What is the shape of the pen that minimizes the amount of fencing needed?

- Suppose you want to created a rectangular pen for your dog that has an area of 30 square metres. What dimensions would minimize the amount of fencing you will need for the pen?

See the matching Ontario Mathematics Curriculum expectations.