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Exam Review Drawing Contours

mbanawan

I’m having trouble drawing contours from equations like question 7 on the exam review asks. We went over that specific question in class but I’m still a bit lost on how to go about drawing contours from an equation. Anyone have any tips or steps to follow?

mark

Keep in mind that there are three basic types of functions that I might ask you to draw contours for:

  • Lines, which have the form f(x,y) = ax+by,
  • Ellipses, which have the form f(x,y) = ax^2 + by^2 where a and b have the same sign, and
  • Hyperbolas, which have the form f(x,y) = ax^2 + by^2, where a and b have different signs.

Here’s a closer look at ellipses:

Problem #7 on the review sheet is a little variation of

F(x,y) = 3x^2 + y^2.

Again, you should recognize this as one of the basic forms so you should know that the level curves are ellipses. Once you’re there, think about one of the contours by setting the function equal to some easy number - like 1. That gets you to

3x^2 + y^2 = 1.

At this point, identifying the ellipse should be a simple matter of identifying the intercepts. Well,

  • If x=0, then y=\pm1 and
  • if y=0, then x=\pm1/\sqrt{3}

That means that your parabola goes through these four points:

desmos-graph

I guess you should be able to draw the ellipse through those points.

Of course, problem #7 actually asks you to deal with

f(x,y) = 3(x-2)^2 + (y+1)^2.

This is just a shift of the last function, though. That is, since

F(x,y) = 3x^2 + y^2,

our new function f(x,y) is just

f(x,y) = F(x-2,y+1)

So, our origin is shifted to the point where

\begin{aligned} x-2 &= 0 \\ y+1 &= 0 \end{aligned}

i.e., x=2, y=-1. So, just shift your ellipses over to there!