# Graphs of Changing Areas

##### Stage: 5 Challenge Level:

The graph below shows the curve $y=\frac{10}{x}$.

Imagine $x$ and $y$ are the length and width of a rectangle.
Each point on the curve represents a rectangle - what property do these rectangles share?

What symmetry does the graph have? How do you know?
What happens to the graph as $x$ gets very large? How do you know?

You could plot graphs of other curves such as $y=\frac{5}{x}$ or $y=\frac{20}{x}$.
How would these graphs relate to the one above? Would the graphs intersect? How do you know?

Rectangles of equal perimeter can be represented graphically by the line $y=\frac{1}{2}P-x$ where P is the perimeter.
Would you expect the line $y=\frac{1}{2}P-x$ to intersect with the curve $y=\frac{10}{x}$ for all values of P?

How can you use the graph to find the smallest possible perimeter of a rectangle with an area of $10$?

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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