Graphing a level surface f(x,y,x)=c

[bfusco]

Homework Statement

sketch the graph f(x,y,z)=x² + (1/4)y² - z, c=1

The Attempt at a Solution

while i don't expect anyone to be able to graph it for me for i think obvious reasons, i have no idea how to interpret any of the information given to even attempt a graph.

usually i would try to figure out how it looks in an 2d x,y plane and add the z dimension, but i can't do that with this. please help

 

[Mark44]

Homework Statement

sketch the graph f(x,y,z)=x² + (1/4)y² - z, c=1

The Attempt at a Solution

while i don't expect anyone to be able to graph it for me for i think obvious reasons, i have no idea how to interpret any of the information given to even attempt a graph.

usually i would try to figure out how it looks in an 2d x,y plane and add the z dimension, but i can't do that with this. please help

Graph the surface x² + (1/4)y² - z = 1 in three dimensions.

 

[bfusco]

Graph the surface x² + (1/4)y² - z = 1 in three dimensions.

yea i don't know how, is what I am saying. i even tried to look online for a 3d grapher that would allow for inputs of f(x,y,z) but i couldn't find any, so is there any way to interpret the info from the equation. like in the x direction is the graph x^2, in the y direction does it look like the graph x/4..etc?

 

[Mark44]

Have you spent any time sketching graphs of quadric surfaces?

The thing to do in this problem is to recognize what sorts of shapes you get in various planes, not axes. For example, plane z = 0 intersects the surface in an ellipse. In other words, the cross section of the surface in the x-y plane is the ellipse x^2 + (1/4)y^2 = 1.
In the plane z = -1, the cross section is the graph of the equation x^2 + (1/4)y^2 = 0, which is a degenerate ellipse that consists of only a single point.

In other planes that are parallel to the x-y plane, you get different ellipses.

In the x-z plane (where y = 0) what shape do you get?

In the y-z plane (where x = 0) what shape do you get?

If you get a few cross sections at various places, you can start to get an idea of what the overall surface looks like.

 

[LCKurtz]

yea i don't know how, is what I am saying. i even tried to look online for a 3d grapher that would allow for inputs of f(x,y,z) but i couldn't find any, so is there any way to interpret the info from the equation. like in the x direction is the graph x^2, in the y direction does it look like the graph x/4..etc?

You could solve your equation for z to get it in the form z = g(x,y) if that's what your 3d grapher needs.