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Magazine Chapter 11: Problem 4
Name and sketch the graph of each of the following equations in three-space. $$ z^{2}=3 y $$
Short Answer
Expert verified
The graph is a parabolic cylinder, \( z^2 = 3y \), extending along the x-axis.
Step by step solution
01
Identify the Type of Surface
The given equation is \( z^2 = 3y \). Recognize this as a parabolic cylinder in three-space. The equation represents a parabola that opens along the y-axis due to the square term \( z^2 \), while the lack of an x-term indicates that it is constant for every x-value, hence forming a cylinder.
02
Analyze the Shape in 2D
To understand the surface, first consider the equation \( z^2 = 3y \) in the yz-plane (where x=0). This is a parabola opening in the positive y-direction with vertex at the origin (0,0).
03
Extend to Three Dimensions
In three dimensions, the two-dimensional parabola from the yz-plane extends infinitely along the x-axis, forming a parabolic cylinder. Every cross-section parallel to the yz-plane remains a parabola.
04
Sketch the Graph
Start by sketching the parabola \( z^2 = 3y \) on the yz-plane. Then, extend the parabola along the x-axis to create the cylindrical shape. Each plane parallel to yz and at a different value of x will also show a parabola of the same shape.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Parabolic Cylinder
A parabolic cylinder is a fascinating geometrical structure that emerges from quadratic equations. It's called a cylinder not because it looks like a round pipe but because it retains its shape along one of the axes, specifically the x-axis in this case. The defining characteristic of a parabolic cylinder comes from its equation, such as \( z^2 = 3y \). Here鈥檚 what makes it unique:
- The squared term \( z^2 \) signifies that its cross-section is a parabola.
- Since the equation lacks an x-term, the parabola is stretched infinitely along the x-axis.
- Each slice parallel to the yz-plane is identical, showing the same parabolic shape.
Three-space
Three-space, or 3D space, allows for greater exploration of complex shapes than two-dimensional space. It's better understood with coordinates (x, y, z), representing width, depth, and height.
In the context of three-space, the equation \( z^2 = 3y \) helps us visualize a shape that extends along the x-axis. Think about how:
In the context of three-space, the equation \( z^2 = 3y \) helps us visualize a shape that extends along the x-axis. Think about how:
- Each point in space is represented by its three coordinates, making shapes more dynamic.
- We can understand the complex unity of shapes, like cylinders, as they stretch through different planes.
- Three-space enables precise sketching and better realization of mathematical surfaces.
Graph Sketching
Graph sketching is a visual method to bring equations to life. For mathematicians and students alike, it is an essential skill to interpret abstract formulas into visual representations.
Consider our parabolic cylinder example, \( z^2 = 3y \):
Consider our parabolic cylinder example, \( z^2 = 3y \):
- Begin by sketching its 2D form in the yz-plane \( z^2 = 3y \), visualizing it as a basic parabola.
- Recognize the absence of an x-term which implies stretching the shape along the x-axis to form the complete three-dimensional structure.
- Imagine slicing through the surface at any x-value, still seeing a parabolic 2D slice.
Mathematical Surfaces
Mathematical surfaces encompass a wide range of shapes residing in three-space. They include everything from simple planes to more complicated forms like our parabolic cylinder.
- Surfaces are represented by equations, which define their shape and extent.
- Understanding these surfaces requires recognizing their inherent properties, like symmetry, curvature, and how they interact with axes.
- Our parabolic cylinder case reveals a continuous surface stretched infinitely across one axis, showcasing the interaction of algebra with geometry.
