Problem 38 - College Algebra and Calculus: An Applied Approach

Question

Describe the traces of the surface in the given planes. $$ \begin{aligned} &y=x^{2}+z^{2}\\\ &x y \text { -plane, } y=1, y z \text { -plane } \end{aligned} $$

Step-by-Step Solution

Answer

The surface \(y = x^2 + z^2\) intersects the xy-plane in the parabola \(y = x^2\), the plane at \(y = 1\) in the circle \(1 = x^2 + z^2\), and the yz-plane as \(y = z^2\).

Step 1: Intersection with the xy plane

In the xy-plane, \(z = 0\) because there is no z-coordinate involved. Therefore, substitute \(z = 0\) into the equation \(y = x^2 + z^2\), which simplifies to \(y = x^2\). This is the equation of a parabola with its vertex at the origin in the xy-plane.

Step 2: Intersection where y=1

For the plane at \(y = 1\), substitute \(y = 1\) into the the equation. This gives the equation for the xz-plane intersecting the surface as \(1 = x^2 + z^2\). This is the equation of a circle with radius 1, centered at the origin in the xz-plane.

Step 3: Intersection with the yz plane

In the yz-plane, \(x = 0\) because there is no x-coordinate involved. Substitute \(x = 0\) into the equation \(y = x^2 + z^2\), resulting in \(y = z^2\). This is the equation of a parabola with its vertex at the origin in the yz-plane.

Key Concepts

Conic Sections

When studying the shapes of graphs in math, especially in a 3D coordinate system, one comes across special curves known as conic sections. These are the curves obtained when a plane intersects the surface of a cone. Depending on the angle and position of the intersection, the shape could be a circle, an ellipse, a parabola, or a hyperbola.

Let's look at the exercise where the surface equation given is y = x^2 + z^2. Intersecting this surface with different planes gives us different 2D traces. For instance, when intersecting with the xy-plane, and thus setting z = 0, the resulting shape is a parabola. Parabolas are one of the four classic conic sections and they have a single axis of symmetry with a shape like a U or an inverted U. The given parabola is upright and has a vertex at the origin of the xy-plane.

3D Coordinate System

Understanding the 3D coordinate system is essential when dealing with equations representing surfaces. It is made up of three axes: the x-axis, y-axis, and z-axis, which are all perpendicular to each other, forming a three-dimensional space. Every point in this system can be represented by a set of three coordinates: \(x, y, z\).

In our exercise, the 3D graph of the surface reveals how the equation behaves in spatial terms. By setting one of the coordinates, say x or z, to zero, we can investigate the trace of the surface on the plane defined by the remaining two axes. This method is how the given exercise provides the parabolic and circular traces on the xy-plane and xz-plane, respectively.

Plane Intersections

The concept of plane intersections comes into play when a flat surface, or 'plane', cuts through a solid shape. Where the two meet, they reveal a 'trace' or cross-section, which can be represented by its own equation. In 3D space, these intersections can reveal much about the nature of the solid shape.

In the exercise, the plane y = 1 intersects with the surface defined by the equation y = x^2 + z^2, resulting in a circle. Intersecting the surface with the yz-plane reduces the original equation to a parabola. This neatly demonstrates how different planes slicing through the same 3D surface can produce various 2D shapes, which can provide additional insights into the nature and symmetry of the surface.