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.
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.
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.
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.
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.