Question

What does the equation \(z=x^{2}\) represent in the \(x z\) -plane? What does it represent in three-space?

Short answer

The equation \(z=x^{2}\) represents an upwards-opening parabola in the \(xz\)-plane. In three-space, this equation represents a parabolic cylinder.

Step-by-step solution

Plot in the \(xz\)-plane

To understand what \(z=x^{2}\) represents in the \(xz\)-plane, we must first plot this equation. The \(x^{2}\) equation suggests a parabola shape. Indeed, in the \(xz\)-plane, this equation gives rise to a parabola that opens upwards.

Interpretation in the \(xz\)-plane

This upwards-opening parabola represents all the possible values of \(x\) and \(z\) satisfying the equation \(z=x^{2}\) in two dimensions.

Visualizing in Three-space

When it comes to visualizing this equation in three-space, there's an additional dimension compared to the previous scenario. Here, all the points \((x, y, z)\), where \(y\) can be any real number, and \(z = x^{2}\), will make up the graph of the equation in three dimensions.

Interpretation in three-space

The graph in three-space visually represents a surface that is parabolic in the \(xz\)-plane and constant along the \(y\)-axis. This surface is usually referred to as a parabolic cylinder because it's like a parabola extended along a third dimension