Match the function to its graph

Advanced 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? Where is z = 0? Is a variable missing? Are the cross-sections straight lines, parabolas, or something else? Is the surface smooth, or does it have creases?

(a) \(z = y^2 - x^2\)
(b) \(z = -e^{-(x^2+y^2)}\)
(c) \(z = \dfrac{1}{1 + x^2 + y^2}\)
(d) \(z = \sin x\)
(e) \(z = x^2 + y\)
(f) \(z = e^{-(x+y)^2}\)
(g) \(z = -xy\)
(h) \(z = \sqrt{x^2 + y^2}\)
(i) \(z = |x| + |y|\)

Drag a graph to rotate it. In every graph the x-axis starts out pointing toward you.

(I)

(II)

(III)

(IV)

(V)

(VI)

(VII)

(VIII)

(IX)

Answer key

(a) \(z = y^2 - x^2\) → (III). A saddle that falls along the x-axis (cross-sections with y fixed are downward parabolas) and rises along the y-axis. It is zero on the diagonal lines y = ±x, whereas the saddle in (g) is zero on the axes.
(b) \(z = -e^{-(x^2+y^2)}\) → (VIII). Always negative, equal to −1 at the origin, and it rises toward the xy-plane far away: the bump of Example 2 reflected across the xy-plane, a dimple.
(c) \(z = \dfrac{1}{1 + x^2 + y^2}\) → (V). Always positive, largest (= 1) at the origin, circular symmetry, and it decays to 0 far away: a bump above the xy-plane. It cannot blow up, because the denominator is never less than 1.
(d) \(z = \sin x\) → (II). No y in the formula, so every cross-section with y fixed is the same sine wave: a corrugated sheet whose ridges run parallel to the y-axis.
(e) \(z = x^2 + y\) → (IX). Cross-sections with y fixed are the parabola z = x² + b, all the same shape, shifted up as y increases; cross-sections with x fixed are lines of slope 1. A trough that is tilted to climb in the y-direction.
(f) \(z = e^{-(x+y)^2}\) → (IV). Depends only on x + y, so it is constant along every line x + y = c: a single ridge running along the diagonal y = −x, where z = 1, falling off to 0 on either side. Always positive, and no circular symmetry.
(g) \(z = -xy\) → (VII). Zero on both axes, negative in the first and third quadrants, positive in the second and fourth: a smooth saddle whose high ridge runs over the diagonal y = −x. Compare (a), which is zero on the diagonals instead.
(h) \(z = \sqrt{x^2 + y^2}\) → (I). Circular symmetry (depends only on the distance from the z-axis), zero at the origin, and it grows linearly: the vertical cross-sections through the origin are V shapes, not parabolas. A cone with its point at the origin.
(i) \(z = |x| + |y|\) → (VI). Also zero at the origin and growing linearly, but the horizontal slices are squares (|x| + |y| = c is a diamond), not circles, and there are creases over the two axes: an inverted pyramid.