Question

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.

1. $f(x,y)=x - y^{2}$
2. $f(x,y)=y - x^{2}$
3. $f(x,y)=y + x^{2}$
4. $f(x,y)=x + y^{2}$

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.

Answer:

1. B
2. A
3. C
4. D