1. ## help with quadric surfaces

I'm trying to do some quadric surface sketching for my multivariable calc class, but I'm getting confused when there are equations that don't match the forms that i've "memorized" at all. so for example these two:

1. (y^2)-2y+5z^2=(x^2)-1
2. y=(x^2)-2x+3

in class, we've done a few similar problems where we completed the square for x^2, y^2, and z^2. but in #1, it looks like i could only complete the square for y and all that does is make the equation look even weirder to me. #2 seems like it might be similar. i can't find anything like these two in my book or notes, can somebody please set me straight here?

2. Hello,
Originally Posted by rajeesh733
1. (y^2)-2y+5z^2=(x^2)-1
Completing the square for $y$ works well :

\begin{aligned}y^2-2y+5z^2&=x^2-1 \\
y^2-2y+1-1+5z^2&=x^2-1\\
(y-1)^2+5z^2&=x^2\\
Y^2+Z^2&=X^2
\end{aligned}

for $X=x$, $Y=y-1$ and $Z=\sqrt{5}z$. What curve is described by $Y^2+Z^2=X^2$ ?

2. y=(x^2)-2x+3
$y=x^2-2x+3\implies (x-1)^2=y-2\implies Y=X^2$ for $X=x-1$, $Y=y-2$ and $Z=z$. The equation $Y=X^2$ doesn't depend on $Z$ so the curve is a cylinder. To find which kind of cylinder it is, note that in the X,Y plane $Y=X^2$ is the equation of a parabola so the curve we're looking for is a parabolic cylinder.