Question

In Exercises \(11-18,\) identify and sketch the quadric surface. Use a computer algebra system to confirm your sketch. $$ x^{2}-y+z^{2}=0 $$

Short answer

The given equation \(x^{2}-y+z^{2}=0\) represents a hyperbolic paraboloid, the sketch opens upward along x-axis and downward along z-axis with the vertex at the origin. Use a computer algebra system to verify the sketch.

Step-by-step solution

Identifying the type of quadric surface

Given the equation \(x^{2}-y+z^{2}=0\), is a hyperbolic paraboloid. The standard equation for a hyperbolic paraboloid is \(x^{2}/a^{2} -y/b + z^{2}/c^{2} = 1\). For the given equation, we observe that a=1, b=1 and c=1.

Sketching the surface

To sketch a hyperbolic paraboloid, locate the vertex, which is at the origin (0, 0, 0) in this case. Draw parabolas in the xz-plane and yz-plane. The graph opens upward along x-axis, and downward along z-axis, as y value is negative.

Using a computer algebra system to confirm the sketch

There are several free online graphing tools, such as grapher in mac, and online 1, 2, 3D graphing calculators. Enter the equation \(x^{2}-y+z^{2}=0\) into the software and compare the computer generated graph with your sketch to confirm.

Key concepts

Hyperbolic Paraboloid

The hyperbolic paraboloid is a fascinating and complex surface encountered in higher mathematics, specifically within the family of quadric surfaces. Its distinct saddle shape makes it stand out among other quadrics. This shape results from the cross-sections of the surface being parabolas that open in opposite directions along different axes.

In the context of the given equation, \(x^{2}-y+z^{2}=0\), we deal with a hyperbolic paraboloid because the equation can be reorganized to fit the form \(\frac{x^{2}}{a^{2}} - \frac{y}{b} + \frac{z^{2}}{c^{2}} = 1\), where \(a\), \(b\), and \(c\) are constants that determine the 'width' and 'depth' of the parabolas. Since all constants equate to 1, the hyperbolic paraboloid exhibits a standard saddle geometry centered at the origin (0, 0, 0).

By understanding the hyperbolic paraboloid's nature and structure, students can better visualize how the surface extends infinitely in both the positive and negative directions along the x- and z-axes, while curving away from the y-axis.

Computer Algebra System

A Computer Algebra System (CAS) is a powerful tool that greatly enhances a student's ability to understand complex mathematical concepts. By allowing for the manipulation of mathematical expressions in a symbolic form, CAS can provide clear insights into the underlying structures of equations and problems.

For instance, in our exercise, a CAS can be employed to confirm the sketch of our hyperbolic paraboloid. Students can turn to programs such as Mathematica, Maple, or even free tools available online, to enter the equation \(x^{2}-y+z^{2}=0\) and receive a precise visualization of the surface. This confirmation process not only assists in ensuring the accuracy of one’s manual sketches but also deepens the student’s comprehension of the shape and properties of the hyperbolic paraboloid.

3D Graphing

3D graphing is an essential skill when it comes to visualizing and understanding multidimensional mathematical concepts. It brings equations to life by allowing the representation of functions with a three-dimensional perspective. For quadric surfaces like the hyperbolic paraboloid, 3D graphing is particularly helpful as it showcases the unique features of these surfaces that cannot be fully appreciated in two dimensions.

In educational contexts, utilizing 3D graphing aids students in bridging the gap between abstract equations and their tangible geometric counterparts. The graph for the equation \(x^{2}-y+z^{2}=0\) can be plotted using various graphing calculators or software, revealing the intricate shape of the hyperbolic paraboloid, with its curves and contours vividly outlined against the three axes.

Identifying Quadric Types

Identifying quadric types is a fundamental skill in multivariable calculus and geometry. Quadric surfaces come in various forms, such as ellipsoids, hyperboloids, paraboloids, and more, each defined by a specific quadratic equation. Recognizing the type of quadric involved in an equation is crucial for understanding its geometry and behavior.

The equation from our exercise, \(x^{2}-y+z^{2}=0\), characterizes a hyperbolic paraboloid. To properly identify such surfaces, one should assess the signs and coefficients of the squared terms and the presence of any linear or constant terms. A general insight for students is that the signs '+ - +' or '- + -' in the squared terms suggest a hyperbolic paraboloid, while other combinations of signs lead to different quadrics. Mastery in identifying quadric surfaces enables students to tackle more complex problems and build an intuitive grasp of three-dimensional shapes.