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What effect does varying a and b in the standard equation of a hyperbola centred at the origin have on the graph of the hyperbola?

We now know that the standard equations of hyperbolas centred at the origin are: x2/ a2 - y2/ b2 = 1 if the hyperbola opens left and right - x2/ a2 + y2/ b2 = 1 if the hyperbola opens up and down.

Using the action figure and question box below, let's explore what happens to the graph of a hyperbola as we vary a and b in these equations.

Some of the questions below will refer to the angles a (alpha) and b (beta). These angles are labelled for no purpose other than to give a convenient way to ask questions regarding the slope of the asmyptotes. Notice that as a increases, the slope of Line 1 decreases, and as b increases, the slope of Line 1 increases.

Note that the action figure has two buttons at the bottom left corner which allow you to switch between hyperbolas that open right and left (x2 - y2 = 1) and hyperbolas that open up and down (-x2 + y2 = 1).

	x2	-	y2	= 1	and	-	x2	+	y2	= 1
	a2		b2				a2		b2

Now, explore how the asymptotes of the function can be described by using the standard equation of the hyperbola centred at the origin.