Final answer:
The level curve for the function f(x,y) = xy when c = 3 is a hyperbola represented by the equation y = 3/x. It includes points where the product of x and y is equal to 3, approaching the axes as asymptotes. This graph differs from the straight line of y = x and the parabola of y = (x - 2)^2, as well as from the exponential curves y = e^x and y = e^-x.
Step-by-step explanation:
To sketch the level curve of the function f(x,y) = xy for c = 3, we are looking for all the points (x, y) where the product of x and y is equal to 3. This generates the equation xy = 3. Solving for y, we get y = 3/x. This is a hyperbola, and it will include all points where the product of the x-coordinate and the y-coordinate equals 3. The curve will approach but never touch the x-axis or y-axis, indicating an asymptotic behavior towards these axes.
To draw this curve on a coordinate plane, plot several points that satisfy the equation (such as (1,3), (3,1), (-1,-3), (-3,-1)) and make sure to draw the curve so that it shows the right asymptotic approach towards the x-axis and y-axis.
Examples of Other Graphs
For the equations y = x and y = (x - 2)2, these represent a straight line with a 45-degree angle with the axes, and a parabola opening upwards with vertex at (2,0), respectively. The exponential functions y = ex, y = e-x, each represent exponential growth and decay respectively. The graph of y = ex will be a curve increasing rapidly as x gets larger, and the graph of y = e-x will be a curve rapidly decreasing as x gets larger.