Let's analyze the given equation and answer the questions step by step.

Given equation:

$z = 3 - y^2$

Step 1: Identify the surface type

• This equation represents a parabolic cylinder because:
  • The equation does not contain $x$, meaning the surface extends infinitely along the $x$-axis.
  • The equation is quadratic in $y$ and linear in $z$, which confirms a parabolic shape in the $yz$-plane.

Thus, the correct answer is:
✅ Parabolic Cylinder

Step 2: Cross-section equations

1. Cross-section at $z = 0$ using $x$ and $y$
   Set $z = 0$ in the given equation: $0 = 3 - y^2$ Solving for $y$: $y^2 = 3$ $y = \pm \sqrt{3}$ Since the equation does not depend on $x$, the cross-section is two horizontal lines:
   $y = \pm \sqrt{3}$

2. Cross-section at $y = 0$ using $x$ and $z$
   Set $y = 0$ in the given equation: $z = 3 - 0^2$ $z = 3$ Since $x$ is not involved, this represents a horizontal line: $z = 3$

3. Cross-section at $x = 0$ using $y$ and $z$
   Since the original equation does not depend on $x$, setting $x = 0$ does not change it: $z = 3 - y^2$ This represents a parabola opening downward in the $yz$-plane.

Final Answers:

1. Surface Type: Parabolic Cylinder
2. Equation for cross-section at $z = 0$: $y = \pm \sqrt{3}$
3. Equation for cross-section at $y = 0$: $z = 3$
4. Equation for cross-section at $x = 0$: $z = 3 - y^2$

Would you like further clarification on any of these points?

Related Questions:

1. How does the equation change if the parabola opens along the $x$-axis instead?
2. What would happen if we had $x^2 + y^2 = z$ instead?
3. How does shifting the vertex of the parabola affect the equation?
4. Can you describe how the surface looks in three dimensions?
5. What happens if we introduce a coefficient in front of $y^2$?

Tip: Always check which variable is missing in the equation—this helps determine which axis the surface extends along.