Purchase Solution

Level Curves

Not what you're looking for?

Ask Custom Question

Sketch the graph:
z= f(x,y) where f(x,y)= x^2 + xy ; c=0,1,2,3,-1,-2,-3

Please show/describe all steps AND explain why it is a hyperbola.

Also, sketch the graph:
z=f(x,y) where f(x,y)= x/y ; c= 0,1,2,3,-1,-2,-3
Please show/describe all steps.

Purchase this Solution

Solution Summary

This solution is comprised of a detailed explanation to sketch the graph.

Solution Preview

Hello,

I have attached the explanation below as a txt file which should make the 2x2 matrices easier to read. A sketch of the hyperbola is also attached as a jpeg file - if opened in Internet Explorer please maximise (by clicking in lower right corner of image).

I have tried to use your equation to give a general method for recognizing conic sections, so you can use the same method to sketch similar looking equations in the future, so apologies if it is a little long winded for the given question.

We have the function z = f(x,y) = x^2 + xy, so the level curve at height z = c
is given by:

x^2 + xy - c = 0 (1)

This is an equation of the form:

A x^2 + B xy + c y^2 + F x + G y + H = 0 (2)

(the general form for a conic section)

with A=1, B=1, C=F=G=0, and H=-c

We know the standard form for an hyperbola is:

x^2/(a^2) - y^2/(b^2) = 1 (3)

and that this standard form would tell us useful properties of the hyperbola in terms of

a and b:

i) There is a directrix at y = (a/b)x and also at y = -(a/b)x

(These are the asymptotes to which the two branches of the hyperbola tend towards.)

ii) The hyperbola cuts the x axis at x = a

Both of which we can see directly from the equation in standard form.

If we take a hyperbola in its standard form (3) and imagine rotating about the origin
through angle theta, and then apply a translation by (x,y), you can see we still have an

hyperbola of exactly the same shape - just at a different place and orientation in the xy

plane. After the transformations, its equation ...

Purchase this Solution


Free BrainMass Quizzes
Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts