Let's analyze and identify the surfaces for #11 and #19:

Problem #11

Equation:
$x = y^2 + 4z^2$

Identification:
This equation resembles the general form of an elliptic paraboloid in the form $x = ay^2 + bz^2$, where the cross-sections parallel to the $yz$-plane are parabolas, and those parallel to the $xy$- and $xz$-planes are ellipses. Since both $y^2$ and $4z^2$ are positive, the surface opens along the $x$-axis.

Description of Surface:

• It is an elliptic paraboloid that opens along the $x$-axis.
• For fixed values of $x$, the cross-sections in the $yz$-plane are ellipses.
• For fixed values of $y$ or $z$, the surface will appear as parabolic curves.

Problem #19

Equation:
$y = z^2 - x^2$

Identification:
This equation matches the form of a hyperbolic paraboloid. Hyperbolic paraboloids typically have equations of the form $y = z^2 - x^2$ or $z = x^2 - y^2$, where one variable is expressed as the difference of squares of the other two.

Description of Surface:

• This is a hyperbolic paraboloid, which has a saddle shape.
• The cross-sections parallel to the $yz$-plane (for fixed $x$) are parabolas opening upwards or downwards.
• The cross-sections parallel to the $xy$-plane (for fixed $z$) are hyperbolas.

Let me know if you'd like more details on either of these or have any questions.

Here are five related questions to deepen understanding:

1. How would the orientation change if $y$ were the leading variable in Problem #11?
2. What is the effect on the shape of the surface if the coefficients of $y^2$ and $z^2$ change in elliptic paraboloids?
3. Why is a hyperbolic paraboloid often referred to as a "saddle surface"?
4. Can you determine the traces for a specific plane (e.g., $y = k$) for each surface?
5. What happens to the shape if the terms in Problem #19 are rearranged to $y = x^2 - z^2$?

Tip: When sketching quadratic surfaces, always start by identifying traces in the coordinate planes to understand the cross-sectional shapes.