Match the function to its graph
Zen level — Class exercise, Section 12.2, Graphs and Surfaces
Without a computer or calculator, match each equation (a)–(i) with its graph (I)–(IX). Useful questions: Is z ever negative? What happens on the x-axis and y-axis? Is the graph symmetric about the z-axis? What happens far from the origin?
- (a) \(z = xy\,e^{-(x^2+y^2)}\)
- (b) \(z = \cos\!\left(\sqrt{x^2+y^2}\right)\)
- (c) \(z = \sin y\)
- (d) \(z = -\dfrac{1}{x^2+y^2}\)
- (e) \(z = \cos^2 x\,\cos^2 y\)
- (f) \(z = \dfrac{\sin(x^2+y^2)}{x^2+y^2}\)
- (g) \(z = \cos(xy)\)
- (h) \(z = |x|\,|y|\)
- (i) \(z = (2x^2+y^2)\,e^{\,1-x^2-y^2}\)
Drag a graph to rotate it. In every graph the x-axis starts out pointing toward you.
Answer key
- (a) \(z = xy\,e^{-(x^2+y^2)}\) → (II). Zero along both axes (factor xy), positive in the first and third quadrants, negative in the second and fourth, and it dies out far from the origin because of the exponential. Four lobes: two up, two down.
- (b) \(z = \cos\!\left(\sqrt{x^2+y^2}\right)\) → (V). Depends only on the distance r from the z-axis, so it is rotationally symmetric; z = cos r ripples between −1 and 1 without dying out. Value 1 at the origin.
- (c) \(z = \sin y\) → (IX). No x in the formula, so every cross-section with x fixed is the same sine wave: a corrugated sheet whose ridges run parallel to the x-axis.
- (d) \(z = -\dfrac{1}{x^2+y^2}\) → (VIII). Rotationally symmetric, always negative, and it goes to −∞ at the origin: a bottomless funnel pointing down, flattening toward z = 0 far away.
- (e) \(z = \cos^2 x\,\cos^2 y\) → (VII). A product of two non-negative factors, so 0 ≤ z ≤ 1: a regular grid of bumps, with z = 0 along the lines x = ±π/2, ±3π/2, … and y = ±π/2, …. Value 1 at the origin.
- (f) \(z = \dfrac{\sin(x^2+y^2)}{x^2+y^2}\) → (III). Rotationally symmetric with a central peak (z → 1 at the origin), and the ripples shrink as you move out because of the denominator. Compare with (b), whose ripples keep their size.
- (g) \(z = \cos(xy)\) → (I). Equals 1 along both axes (where xy = 0) and oscillates faster and faster as |xy| grows, so the ridges are hyperbolas xy = const that bunch up toward the corners.
- (h) \(z = |x|\,|y|\) → (IV). Zero along both axes and non-negative everywhere, rising toward the four corners; the absolute values put sharp creases along the axes (unlike the smooth saddle z = xy).
- (i) \(z = (2x^2+y^2)\,e^{\,1-x^2-y^2}\) → (VI). Zero at the origin, positive elsewhere, and decaying far away: a crater. On the x-axis it reaches 2 at x = ±1, on the y-axis only 1 at y = ±1, so the rim is higher on the x-axis.