match each function with its contour plot. click on a graph to make it larger. darker areas represent lower…

match each function with its contour plot. click on a graph to make it larger. darker areas represent lower elevations and lighter areas represent higher elevations.\n1. $f(x,y)=x - y^{2}$\n2. $f(x,y)=y - x^{2}$\n3. $f(x,y)=y + x^{2}$\n4. $f(x,y)=x + y^{2}$

Answer

Explanation:

Step1: Analyze (f(x,y)=x - y^{2})

The function (z=x - y^{2}) is a parabolic - cylinder opening in the positive (x) - direction. When (y = 0), (z=x) (a straight line with slope 1), and as (y) varies, the surface curves downwards in the (y) - direction. This matches the contour plot where the lighter (higher elevation) areas are on the right - hand side as (x) increases. So (f(x,y)=x - y^{2}) matches plot B.

Step2: Analyze (f(x,y)=y - x^{2})

The function (z=y - x^{2}) is a parabolic - cylinder opening in the positive (y) - direction. When (x = 0), (z=y) (a straight line with slope 1), and as (x) varies, the surface curves downwards in the (x) - direction. This matches the contour plot where the lighter (higher elevation) areas are on the upper - hand side as (y) increases. So (f(x,y)=y - x^{2}) matches plot A.

Step3: Analyze (f(x,y)=y + x^{2})

The function (z=y + x^{2}) is a parabolic - cylinder opening in the positive (y) - direction. Since the (x^{2}) term is positive, the surface curves upwards in the (x) - direction. The lighter (higher elevation) areas are on the upper - hand side as (y) increases and the surface is higher for non - zero (x) values compared to when (x = 0). So (f(x,y)=y + x^{2}) matches plot C.

Step4: Analyze (f(x,y)=x + y^{2})

The function (z=x + y^{2}) is a parabolic - cylinder opening in the positive (x) - direction. Since the (y^{2}) term is positive, the surface curves upwards in the (y) - direction. The lighter (higher elevation) areas are on the right - hand side as (x) increases and the surface is higher for non - zero (y) values compared to when (y = 0). So (f(x,y)=x + y^{2}) matches plot D.