Question

Describe and sketch the surface of equation \(z = 1 - {y^2}\).

Short answer

The surface of equation \(z = 1 - {y^2}\) is described and sketched.

Step-by-step solution

Represent the Surface of Equations

The equation\(z = 1 - {y^2}\)represents\(\left\{ {(x,y,z)\mid z = 1 - {y^2}} \right\}\)and does not involve\(x\)-coordinates. Hence, set the\(x\)-plane equation as\(x = k\), where\(k\)is a constant.

Therefore, the resultant curve of equation \(z = 1 - {y^2}\) forms a parabolic cylinder with infinitely many shifted duplicates of the same parabolic cylinder. The rulings in the representation of \(z = 1 - {y^2}\) are parallel to the \(x\) axis.

The sketch of equation \(z = 1 - {y^2}\) is shown below

Thus, the surface of equation \(z = 1 - {y^2}\) is described and sketched.