Skip to Content

Graph and Characteristics of z Equals 1 Minus y Squared

Home | Calculus 3 | Cylinders and quadric surfaces | Graph and Characteristics of z Equals 1 Minus y Squared

Using the equation z=1y2z = 1 - y^2, sketch the graph and determine its characteristics.

In this problem, we are tasked with sketching the graph of the equation z equals one minus y squared and discussing its characteristics. This equation represents a three-dimensional quadratic surface, commonly known as a parabolic cylinder. The concept behind this problem is understanding how quadratic equations in multiple variables translate into three-dimensional surfaces.

When visualizing z as a function of y, while x is held constant, it is useful to recognize this set of planes parallel to the x-axis in the xz-plane, each containing a parabola. This leads to intriguing insights into the shape and orientation of our surface in space, specifically forming a parabolic sheet opening in the negative z-direction across the y-axis.

Key characteristics to look out for include symmetry, as the parabola is symmetric about the yz-plane, and the vertex of the parabola can be identified as lying on the line z equal to one, y equal to zero for all x. Understanding these properties is crucial in visualizing surface behavior in multivariable calculus, as it aids in anticipating the surface's context when combined with more complex systems.

Posted by Gregory 4 months ago

Related Problems

For the cone represented by the equation x24+y29=z216\displaystyle \frac{x^2}{4} + \frac{y^2}{9} = \frac{z^2}{16}, determine the intersection traces with the coordinate planes.

For a circular paraboloid given by z=x2+y2z = x^2 + y^2, determine its axis of symmetry and describe the shape of its traces in the coordinate planes.

Find the equation of a quadric surface using the general form Ax2+By2+Cz2+Dxy+Exz+Fyz+Gx+Hy+Iz+J=0Ax^2+By^2+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Iz+J=0.

Find the traces on the xy, xz, and yz planes for the quadric surface given by the function 2x2+9y2+18z2=182x^2 + 9y^2 + 18z^2 = 18.