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Identify Characteristics of Hyperbolas


Imagine a sequence of rectangles drawn so that their heights are decreasing while their widths are increasing. They have been constructed so that their areas are all the same.

We could write $hw=A$hw=A where $h$h and $w$w are variables and $A$A is a constant.

Equivalently, we could draw a graph in the Cartesian plane that shows the points whose product $xy$xy is a particular constant $k$k. We obtain the graph of a relation $xy=k$xy=k.

Try exploring this applet, where the top slider changes the area of the rectangle, and the bottom slider changes one dimension of the rectangle. The width changes automatically to ensure that the area of the rectangle stays constant.

While this applet only shows the hyperbola in the first quadrant, you can observe how changing the value of $k$k changes the shape of $y=\frac{k}{x}$y=kx in a dynamic way. If we look at all the quadrants, we obtain the full hyperbola - below is the graph of the relation $xy=2$xy=2, and you may be more familiar seeing it written as $y=\frac{2}{x}$y=2x


This graph gives more than just the dimensions of rectangles with area $2$2. It also gives the points $xy=2$xy=2 when $x$x and $y$y can be negative, even though we do not normally allow negative lengths and widths of geometrical objects.

Notice that numbers that multiply together to give $2$2 have to be both positive or both negative. For this reason, the graph has points in the first and third quadrants of the Cartesian plane. on the other hand, an equation like $xy=-2$xy=2 would have a graph with points in the second and fourth quadrants where exactly one of $x$x or $y$y is negative and the other positive. the diagram below shows the graph of $xy=-2$xy=2 in red.


So in blue we have the graph $y=\frac{2}{x}$y=2x and in red we have $y=\frac{-2}{x}$y=2x.  

With $y=k/x$y=k/x, if we consider it written as $xy=k$xy=k then it's easier to see that as $x$x increases $y$y has to decrease, and if $y$y increases then $x$x has to decrease.

Neither of the two variables can ever be zero. They can be as close to zero as we wish but not zero itself. We say the horizontal and vertical axes are asymptotes to the graph, which correspond to $x=0$x=0 and $y=0$y=0.  


Example 1

Compare the graphs of $y=1/x$y=1/x, $y=4/x$y=4/x and $y=100/x$y=100/x.

We see that the point where $x=y$x=y in each case moves away from the centre $(0.0)$(0.0) as the value of the constant increases.

For $y=1/x$y=1/x it is the point $(1,1)$(1,1) or the point $(-1,-1)$(1,1).

For$y=4/x$y=4/x it is $(2,2)$(2,2) or $(-2,-2)$(2,2).

And for $y=100/x$y=100/x the points are $(10,10)$(10,10) and $(-10,-10)$(10,10).


In fact, for $y=k/x$y=k/x, the points where $x=y$x=y are $\pm\left(\sqrt{k},\sqrt{k}\right)$±(k,k).

We begin to suspect that increasing $k$k is equivalent to enlarging the diagram.

Suppose we take any point $(a,b)$(a,b) that satisfies the equation $y=1/x$y=1/x. We shift our attention to another point in the plane with coordinates $\left(a\sqrt{k},b\sqrt{k}\right)$(ak,bk). This point does not satisfy $y=1/x$y=1/x but it does satisfy $y=k/x$y=k/x

This shows that the graphs of $y=1/x$y=1/x and $y=k/x$y=k/x are different only by the scale factor $\sqrt{k}$k.

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