Inverse relationships are all about one thing doing the opposite of the other. When one variable gets bigger, the other gets smaller, and vice versa. Here’s the graph of a typical inverse relationship between x and y:

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First of all, notice that you can’t actually plot the
graph right at x = 0, because y gets too big to fit on the graph. That’s why
the line doesn’t go all the way to the y-axis. Looking more to the right side
of the graph, the line almost flattens out. This is because as you get to
large values of ‘x’, _{} becomes very small and doesn’t
change much – for instance the difference between _{} and _{} is only 0.002!

You can get inverse square _{} or inverse cubic _{} relationships
too. They are more *exaggerated* – they get to very large values of ‘y’
much more quickly as the graph approaches the y-axis, and flatten out much more
quickly as you head towards large values of ‘x’.

If you include the negative side of the x-axis, the graphs look like this:

The _{} graph becomes negative on the left
side of the y-axis because when you divide 1 by a negative number, you get a
negative answer. However, the _{} graph stays positive on the left
side of the y-axis, because of the square bit – the negative x values are being
squared to become positive numbers. 1 divided by a positive number gives you a
positive number.

The graph of _{} is mirror imaged across *both*
the x-axis *and* the y-axis. The graph of _{} is mirror imaged *only* across
the y-axis.

When y varies inversely as x, we can write down this proportionality statement:

_{}

When y varies inversely* *as x^{2}, we can
write down this proportionality statement:

_{}

There is also a ‘k’ constant for inverse relationships. If I want to turn the proportionality sign into an equals sign, I replace the ‘1’ with a ‘k’, like this:

_{}