Answer
(a) For $f\left( {x,y} \right) = 3x + 4y$, the contour map matches Figure (B)
(b) For $g\left( {x,y} \right) = {x^3} - y$, the contour map matches Figure (A)
(c) For $h\left( {x,y} \right) = 4x - 3y$, the contour map matches Figure (C)
(d) For $k\left( {x,y} \right) = {x^2} - y$, the contour map matches Figure (D)
Work Step by Step
(a) We have $f\left( {x,y} \right) = 3x + 4y$. The horizontal trace is the line $c=3x+4y$. The contour map matches Figure (B).
(b) We have $g\left( {x,y} \right) = {x^3} - y$. The horizontal trace is the curve $c=x^3-y$. The contour map matches Figure (A).
(c) We have $h\left( {x,y} \right) = 4x - 3y$. The horizontal trace is the line $c=4x-3y$. The contour map matches Figure (C).
(d) We have $k\left( {x,y} \right) = {x^2} - y$. The horizontal trace is the parabola $c = {x^2} - y$. The contour map matches Figure (D).
