Problem 37 - College Algebra and Calculus: An Applied Approach

Question

Describe the traces of the surface in the given planes. $$ x^{2}-y-z^{2}=0 \quad x y \text { -plane, } y=1, y z \text { -plane } $$

Step-by-Step Solution

Answer

The traces of the surface \(x^{2}-y-z^{2}=0\) are given by the equations \(x^{2}=y\) in the xy-plane, \[x^{2}=1+z^{2}\] for the plane \(y=1\), and \( y=-z^{2}\) in the yz-plane.

Step 1: Trace in the xy-plane

In the xy-plane, z equals zero. So substitute zero for \(z\), we get the equation in the xy-plane.\[x^{2}-y-0^{2}=0 \rightarrow x^{2}=y\] This is the equation of a parabola that opens upwards in the xy-plane.

Step 2: Trace in the plane y=1

For the plane \(y=1\), substitute \(1\) for every \(y\) in the equation, we have \[x^{2}-1-z^{2}=0\]\[x^{2}=1+z^{2}\]This is the equation of a hyperbola in the xz-plane where \(y=1\).

Step 3: Trace in the yz-plane

In the yz-plane, \(x\) equals zero. Substitute zero for \(x\) in the equation, we get \[0^{2}-y-z^{2}=0\]\[ -y=z^{2}\]\[ y=-z^{2}\] This is the equation of a downward parabola in the yz-plane.

Key Concepts

Parabola

A parabola is a U-shaped curve found in many mathematical contexts. It is described by the equation \(y = ax^2 + bx + c\). Parabolas can open upwards or downwards. The orientation is determined by the sign of the leading coefficient:

• If \(a\) is positive, the parabola opens upwards.
• If \(a\) is negative, it opens downwards.

They are symmetrical with respect to their vertical axis, known as the axis of symmetry. Parabolas also have a vertex, which is the highest or lowest point on the curve.

In our exercise, when examining the xy-plane with the equation \(x^2 = y\), the graph represents a parabola that opens upwards. This is straightforward, as when \(z=0\), the quadratic form \(x^2 = y\) leaves us with a classic parabolic equation.

Hyperbola

A hyperbola is another familiar conic section related to the parabola. It is a set of all points in the plane where the difference of the distances to two fixed points (foci) is constant. Hyperbolas are characterized by their unique, open branched structure.

The standard form of a hyperbola's equation is:

• Horizontal: \(\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\)
• Vertical: \(\frac{(y - k)^2}{b^2} - \frac{(x - h)^2}{a^2} = 1\)

Hyperbolas have two disconnected curves, which are mirror images of each other. Each of these curves is called an "arm". In the given exercise, substituting \(y=1\) provides the equation \(x^2 = 1 + z^2\), representing a hyperbola in the xz-plane. This reflects the property of hyperbolas being open curves in either horizontal or vertical orientations.

Coordinate Planes

Coordinate planes help in visualizing different traces of objects in 3D space.

• The **xy-plane** is where \(z = 0\), which traditionally represents the first layer or surface at the base of the space.


• The **yz-plane** is defined where \(x = 0\), allowing us to see how objects extend along the vertical and depth axes without lateral spread.


• The **xz-plane** involves setting \(y = 0\), covering how objects spread horizontally and depth-wise, excluding height.

In the exercise step solutions:- We explored the trace in the **xy-plane** leading to a parabola where the object appears flat.- The **plane of y=1** in combination with the other variables illustrated a hyperbola, showcasing the varied structure as we introduce three-dimensional constraints.- Finally, the **yz-plane** trace led to another parabola, but downward facing; this shows how the shape can change depending on axis orientation.