7.2 Vertical Slices

We can take a vertical slice of a function \( z \depends (x, y) \) by fixing either of its input variables.

Important.

The vertical slices of a function \( z \depends (x, y) \) are graphs.

By fixing a \( y \)-value, we find a \( (z \depends x) \)-graph.
By fixing an \( x \)-value, we find a \( (z \depends y) \)-graph.

Let's take a look at how we vertically slice a graph.

We'll look at a slice \( \{ {y = 0} \} \) that fixes \( y \), and then we'll look at a slice \( \{ {x = 0} \} \) that fixes \( x \).

\( \{ {y = 0} \} \)-slice.

By fixing \( y \), we see how \( z \) depends on \( x \). The slice lives in an \( (x, z) \)-coordinate plane.

We see the slice by standing in front of the surface and looking along the \( y \)-axis.

\( \{ {x = 0} \} \)-slice.

By fixing \( x \), we see how \( z \) depends on \( y \). This slice lives in a \( (y, z) \)-coordinate plane.

We see the slice by standing to the right of the surface and looking along the \( x \)-axis.

We'll often take \( \{ {x = 0} \} \) and \( \{ {y = 0} \} \) slices of a surface because they help us understand what's happening for inputs near the origin \( (0, 0) \). We can take other vertical slices by fixing other \( x \) or \( y \)-values. We'll take lots of slices later in this chapter when we study the bowl, the parabolic saddle, the warp saddle, and the thick parabola. As you may have guessed, it is also possible to horizontally slice a surface. Horizontal slices are important enough that they get a section of their own!