How do you draw the level curves of f(x,y) = x^2+y?

Answer 1

Hello,

First, let's recall what level curves are. They're going to be cross sections of the graph of z = f(x,y) taken with constant values of z. So, let's set z = f(x,y) = c where c is some arbitrary constant. Then we have

f(x,y) = c

x^2 +y = c

y = -x^2 + c

So our level curves are just going to be the concave down parabolas y=-x^2 offset vertically by our constant c.

For example, a few level curves would be

y = -x^2

y= -x^2 + 1

y = -x^2 - 1

You can see them plotted on wolframalpha here:

https://www.wolframalpha.com/input/?i=contour+f(x,y)+%3D+x%5E2+%2By

Hope this helps,

Philip