Page 2

PARABOLA Menu

MAIN MENU

We know that when B = 0, A 0 and D 0, the equation Ax2 + By2 + Cx + Dy + F = 0 represents a parabola that opens up or down.

Notice though, that the equation Ax2 + By2 + Cx + Dy + F = 0 does not give us enough information to draw a picture of a parabola.

How can we rewrite the equation Ax2 + By2 + Cx + Dy + F = 0 in a way that provides more useful information about the graph of a parabola?

Just as with circles and ellipses, there is a standard equation for parabolas which provides useful information about their graphs. This equation looks like this: y - k = a(x - h)2 where (h, k) is the vertex of the parabola and a describes important information about the shape of the parabola.

As before, to change from general form to this new standard form, you will use a method called Completing the Square.

In the following examples we will use this method to rewrite equations from the form Ax2 + Cx + Dy + F = 0 to the form y - k = a(x - h)2.

Summary of Steps

STEP 1:	Collect like terms together.
STEP 2:	Move constant terms to the right hand side of the equal sign.
STEP 3:	Factor out the coefficient of x2 from each x term. Factor out the coefficient of y2 form each y term.
STEP 4:	Complete the squares if necessary. Add numbers to both sides of the equation.
STEP 5:	Simplify.
STEP 6:	Divide and rearrange to get an equation of the form (y - k) = a(x - h)2.
NOTE:	If we are dealing with a parabola that has its vertex at the origin, we only need to use Steps 1 and 5.
In the following questions, you will practise converting the equations of parabolas from general form to standard form.