Match the function to its contour diagram
Advanced level — Class exercise, Section 12.3, Contour Diagrams
Match each function (a)–(i) with its contour diagram (I)–(IX). The numbers on the curves are the values of the function. Useful questions: Where is the center of symmetry? Which way do the parabolas or hyperbolas open? Are the values bounded? Do they repeat?
- (a) \(f(x,y) = x^2 + 4y^2\)
- (b) \(f(x,y) = (x-1)^2 + y^2\)
- (c) \(f(x,y) = x - y^2\)
- (d) \(f(x,y) = x^2 - y\)
- (e) \(f(x,y) = \sin x\)
- (f) \(f(x,y) = e^{-(x+y)^2}\)
- (g) \(f(x,y) = -xy\)
- (h) \(f(x,y) = y^2 - x^2\)
- (i) \(f(x,y) = \dfrac{1}{1 + x^2 + y^2}\)
Answer key
- (a) \(f(x,y) = x^2 + 4y^2\) → (IV). Ellipses centered at the origin, twice as wide (in x) as they are tall, values increasing outward.
- (b) \(f(x,y) = (x-1)^2 + y^2\) → (VII). Circles, but centered at (1, 0) rather than the origin: the bowl of x2 + y2 shifted one unit in the x-direction.
- (c) \(f(x,y) = x - y^2\) → (II). Solving x − y2 = c gives x = y2 + c: parabolas opening to the right, values increasing to the right.
- (d) \(f(x,y) = x^2 - y\) → (VI). Here y = x2 − c: parabolas opening upward, stacked vertically, values increasing downward.
- (e) \(f(x,y) = \sin x\) → (IX). No y, so the contours are vertical lines; the values rise and fall periodically as you move along the x-axis, and the lines bunch up where sin x crosses 0 (steepest) and spread out near ±1 (flat).
- (f) \(f(x,y) = e^{-(x+y)^2}\) → (V). Depends only on x + y, so the contours are lines of slope −1. The largest value 1 is on the line y = −x, and the values fall off symmetrically on both sides: a ridge, not a plane.
- (g) \(f(x,y) = -xy\) → (VIII). Hyperbolas with the axes as asymptotes, like xy, but the signs are swapped: negative in the first and third quadrants, positive in the second and fourth.
- (h) \(f(x,y) = y^2 - x^2\) → (I). Hyperbolas with the diagonals y = ±x as asymptotes; positive values open up and down (around the y-axis), negative values open left and right.
- (i) \(f(x,y) = \dfrac{1}{1 + x^2 + y^2}\) → (III). Circles centered at the origin with values between 0 and 1, decreasing outward, and the circles spread out far from the origin, where the bump has flattened.