Problem 8 - Precalculus Mathematics for Calculus

Question

Describe and sketch the surface represented by the given equation. $$y=-2$$

Step-by-Step Solution

Answer

The surface is a horizontal plane, parallel to the \( xz \)-plane, at \( y = -2 \).

Step 1: Identify the Type of Surface

The equation given is in the form of a constant plane equation. Specifically, it is in the form \( y = c \), where \( c \) is a constant. This indicates that the surface described is a plane parallel to the \( xz \)-plane.

Step 2: Interpret the Equation

The equation \( y = -2 \) tells us that for every point on this surface, the \( y \)-coordinate is always \(-2\). There are no restrictions on the \( x \) or \( z \) values. Thus, the surface extends infinitely in both the \( x \) and \( z \) directions.

Step 3: Visualize the Surface

To visualize this, imagine a flat plane situated at \( y = -2 \). This plane cuts through the \( x \) and \( z \) axes, and it is parallel to the \( xz \)-plane at this \( y \)-level.

Step 4: Sketch the Surface

Draw the three-dimensional Cartesian coordinate system. On this system, locate the point where \( y = -2 \) on the \( y \)-axis. From this point, draw a horizontal plane that extends parallel to the \( xz \)-plane, which is often represented as a rectangle or a parallelogram in sketches.

Key Concepts

Planes in 3D Space

Understanding planes in 3D space can initially seem complex, but the concept revolves around simple geometry. A plane is essentially a flat, two-dimensional surface extending infinitely in 3D space. When dealing with equations such as \( y = -2 \), you're visualizing a "slice" of space.
In 3D coordinate geometry:

• The equation \( y = -2 \) represents a plane that is parallel to the \( xz \)-plane.
• Every point on this plane has a \( y \)-coordinate of \(-2\), but the \( x \) and \( z \) coordinates can be any real number.

This means that the surface does not have any boundaries in the \( x \) or \( z \) directions. Instead, it extends forever across those axes, "floating" at the level where \( y = -2 \). When visualizing or sketching such a plane, think of a sheet of paper floating parallel to the \( xz \) plane, anchored at the specific height of \(-2\) along the \( y \)-axis.

Cartesian Coordinate System

The Cartesian coordinate system is a way to visually and mathematically represent the position of points in space. This system uses three perpendicular axes: the \( x \)-, \( y \)-, and \( z \)-axes.
In this system:

• The \( x \)-axis runs horizontally.
• The \( y \)-axis runs vertically.
• The \( z \)-axis is perpendicular to both, often visualized coming "out of" the plane formed by the other two.

In the context of the equation \( y = -2 \), only the \( y \)-value is fixed at \(-2\), while the \( x \) and \( z \) axes remain free. This means any point meeting these criteria lies on our plane. When sketching this in the Cartesian system, the focus is on accurately depicting how the plane interacts with the axes, giving us an intuitive grasp of its shape and orientation.

Visualizing Surfaces

Visualizing surfaces in a 3D space helps to understand the spatial relationships between different elements of geometry. To effectively sketch and understand these surfaces, a basic visualization strategy involves interpreting the coordinate plane locations and behaviors.
For the surface defined by \( y = -2 \):

• First, imagine the \( y \)-axis. Find the point \( -2 \), which is below the origin if you consider positive \( y \) values.
• From this point, envision a sheet—or plane—that stretches outwards indefinitely.
• This sheet is flat and parallel to the \( xz \)-plane, maintaining consistent height \( y = -2 \).

Drawing this on paper involves sketching a three-dimensional graph and mapping where this infinite plane would sit relative to the other axes. This visualization demonstrates how all \( y \)-values remain consistent while \( x \) and \( z \) vary freely.