# Graphing Level Curve F(x,y)=1 for x^2-y^2: Circle or Hyperbola?

• dgoudie
In summary, a level curve is a curve on a graph that represents points where a function has a constant value. To graph a level curve, you set the function equal to a constant value and plot points where the function is equal to that value. If the level curve is a circle, it means the function has a constant value at all points equidistant from the origin. A hyperbola can be identified if the level curve can be rewritten in a specific equation and has two branches extending infinitely in opposite directions. It is possible for a function to have multiple level curves, each representing a different constant value.
dgoudie
[SOLVED] Level Curve

## Homework Statement

For the given equation sketch the level curve F(x,y)=1

F(x,y)= x^2-y^2

## The Attempt at a Solution

Would this: x^2-y^2=1 still be a circle? What will the minus change?

No it's not a circle. It's a hyperbola. Remember conic sections? Even if you don't, you can still start sketching it by putting numbers in. The would let you figure out it's not a circle pretty fast.

## 1. What is a level curve?

A level curve is a curve on a graph that represents points where a function has a constant value. It can be thought of as a "slice" of the function at a specific value.

## 2. How do you graph a level curve?

To graph a level curve, you first need to set the function equal to a constant value. In this case, we are setting f(x,y)=1. Then, you plot points on the graph where the function is equal to 1. In this example, it would result in a circle or a hyperbola, depending on the values of x and y.

## 3. What does it mean if the level curve is a circle?

If the level curve is a circle, it means that the function has a constant value of 1 at all points equidistant from the origin. This can also be represented as a circle equation, such as (x-a)^2 + (y-b)^2 = r^2.

## 4. How can you determine if the level curve is a hyperbola?

A hyperbola is represented by the equation (x-h)^2/a^2 - (y-k)^2/b^2 = 1. If the level curve can be rewritten in this form, then it is a hyperbola. Additionally, a hyperbola will have two "branches" that extend infinitely in opposite directions.

## 5. Can a function have multiple level curves?

Yes, a function can have multiple level curves, each representing a different constant value. In the case of f(x,y)=1, there would be infinitely many level curves, each representing a different circle or hyperbola.

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