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Numerade Educator

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Problem 5 Easy Difficulty

Describe and sketch the surface.

$ z = 1 - y^2 $

Answer

On the $y z$ plane, this is parabola that opens towards the negative $z$ direction.
$\ln 3 \mathrm{D}$ , it is an parabolic cylinder that extends in directions parallel to the
$x$ -axis.

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Video Transcript

let's do it. Okay, so we want to look at the graph of Z equals one minus y squared. Uh The first thing I want to notice is uh there is no variable X. So Z equals one minus y squared regardless of what X is. Uh So Z equals one minus y squared. Uh You can almost think of this as you know, a function of one variable as opposed to Z being a function of two variables X. And Y. Let me give you an idea what this looks like for a fixed x value. Here. You see uh the three dimensional figure uh graphical representation of this functions equals one minus y squared. If you notice it kind of has this parabolic shape and that parabolic shape extends uh down the length of uh you know this this figure. Now this green axis is your y axis, the red axis is your X axis and the blue vertical axis is your Z axis. So I want to show you what a cross section mm for a particular X value would look like. So right now you're looking at the green uh Y axis and a vertical blue Z axis. So you're really right now uh Right now it's kind of as if X was a fixed value and you're just looking at Z being a function of uh Y Z equals one minus y squared. Here's your y axis. Here is use the axis for a particular X value. You can see for a particular X value. Uh we have a parabolic shape and that's because Z equals one minus y squared Uh is a parabola. If you graft this without X. If you just used a uh two dimensional plane and you graft uh Y is a function of x. Y equals one minus x squared. You would have a problem. So Z equals one minus y squared for a fixed value of X really is justice parabola. And so since Z equals one minus y squared, Okay, for any X, that's why you get this parabola for a fixed uh X or that same parabolic shape extends all the way up and down the entire X axis because the equals one minus y squared regardless of what X is because it's not mentioned. So a cross section uh perpendicular uh two D X axis coming from straight up and down parallel to the Z axis of your picture, taking a vertical plane and slicing right into this figure. You would get that parabola that we just saw when we looked at it uh straight on. So is the equals one minus square is a three dimensional surface. A cross section is basically a parabola because Z equals one minus y squared for a fixed number of x is the equation of a parabola. And so you this surface can be described as a cross section is a is a parabola. And uh that parabola shape keeps its shape and extends all the way up and down the X axis. Give you a couple different views of what this looks like. Mhm. Okay. Okay. And, yeah. Yeah. Yeah.