9. Graph 4x2 + 9y2 - 16x + 18y = 11

We first need to complete the squares on the x and y terms. Factor out a 4 from the x terms and a 9 from the y terms

4(x2 - 4x) + 9(y2 + 2y) = 11

Looking ahead we can see that the completed squares on the left will be

4(x - 2)2 + 9(y + 1)2

because if we square the binominals we get

4(x2 - 4x + 4) + 9(x2 + 2y + 1)

and if we then remove the parentheses we get

4x2 - 16x + 16 + 9y2 + 18y + 9

= 4x2 + 9y2 - 16x + 18y + 25

So we need to add 25 to both sides of the original equation to get

4(x - 2)2 + 9(y + 1)2 = 36

Now, to get a 1 on the right, we divide both sides by 36.

We recognize this as the equation of the ellipse whose major semi-axis is horizontal and has length 3 and whose minor semi-axis is vertical and has length 2 centered at (2, -1).