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The standard equation of a hyperbola centred at the origin is x2/ a2 - y2/ b2 = 1 when the hyperbola opens left and right, and -x2/ a2 + y2/ b2 = 1 when the hyperbola opens up and down.

Notice that these equations are similar to the standard equation of an ellipse centred at the origin: x2 + y2 = 1. a2 b2

When we translated an ellipse from being centred at the origin to being centred at any other point (h, k), the standard equation became: (x - h)2 + (y - k)2 = 1. a2 b2

When the centre of a hyperbola is translated to a point (h, k), the standard equation changes in the same way as the standard equation for an ellipse.

The standard equation for a hyperbola centred at (h, k) and opening horizontally is: (x - h)2 - (y - k)2 = 1. a2 b2 The standard equation for a hyperbola centred at (h, k) and opening vertically is: - (x - h)2 + (y - k)2 = 1. a2 b2

The following activities will explore the standard equation of a hyperbola centred at a point (h, k). Note that when you translate the red hyperbola, a faint black hyperbola will remain in the original position. It is only a reference that is there to aid you in understanding how it was translated.

	(x - h)2	-	(y - k)2	= 1	or	-	(x - h)2	+	(y - k)2	= 1
	a2		b2				a2		b2

Now, let's look at the how varying a and b effect the graph of a hyperbola centered at (h, k).