# Quadratic Functions

Use the interactive below to explore the following problem:

**Problem:** Imagine that you have 18 metres of fencing to create a rectangular pen for your dog. What dimensions would result in the largest area for the pen?

- Move the slider to change the width of the pen.
- Watch the diagram of the pen as you do this.
- Notice how the area (plotted on the grid) changes.
- When you use the trail feature to plot the width versus area, what was the shape of your graph? Were you surprised by this? Why or why not?

- If you plotted the width versus the area of your different guesses, what do you predict the shape of the graph will be?
- Check the Trail checkbox to see the curve as the slider moves.

- How does the shape of the pen change as you increase the width from 0 m to 9 m?
- What happens to the area of the pen as you increase the width from 0 m to 9 m?
- What happens to the perimeter of the pen as you increase the width from 0 m to 9 m?
- Why is 9 m the largest width? Could the width be larger than 9 m? Explain why or why not.
- What point on your graph represents the width that produces the maximum area? the minimum area?
- What is the maximum area of the pen? How does this value relate to the graph?
- Choose two points on the graph that have the same area. 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?

**Extensions**

- Suppose you were given an extra 7 metres of fencing so you now have 25 metres in total. What dimensions would result in the largest area for the pen?

- If you had 30 metres of fencing, what dimensions would result in the largest area for the pen?

See the matching Ontario Mathematics Curriculum expectations.