Match the function to its graph
Intermediate level — Class exercise, Section 12.2, Graphs and Surfaces
Without a computer or calculator, match each equation (a)–(i) with its graph (I)–(IX). These are the basic surfaces of Section 12.2. Useful questions: Is a variable missing from the equation? Does the surface open up or down? Is it flat? Is it symmetric about the z-axis? Is it the graph of a function at all?
- (a) \(z = x^2+y^2\)
- (b) \(z = 5 - x^2 - y^2\)
- (c) \(z = e^{-(x^2+y^2)}\)
- (d) \(z = x^2 - y^2\)
- (e) \(z = 1 - x - y\)
- (f) \(z = y^2\)
- (g) \(x^2 + y^2 = 4\)
- (h) \(x^2 + y^2 + z^2 = 4\)
- (i) \(z = 3\)
Drag a graph to rotate it. In every graph the x-axis starts out pointing toward you.
Answer key
- (a) \(z = x^2+y^2\) → (III). A bowl (paraboloid) with its lowest point at the origin: every cross-section is an upward parabola, and the surface has circular symmetry (Figure 12.13).
- (b) \(z = 5 - x^2 - y^2\) → (VIII). The same bowl turned upside down, with its highest point at (0, 0, 5). Unlike (c), it keeps going down without limit as you move away from the origin (Figure 12.15).
- (c) \(z = e^{-(x^2+y^2)}\) → (VI). Always positive, equal to 1 at the origin, and it flattens out toward the xy-plane in every direction: a single bump with circular symmetry (Figure 12.17).
- (d) \(z = x^2 - y^2\) → (IX). A saddle: cross-sections with y fixed are upward parabolas, cross-sections with x fixed are downward parabolas, so the surface rises along the x-axis and falls along the y-axis (Figure 12.23).
- (e) \(z = 1 - x - y\) → (I). A linear function, so its graph is a plane (compare Figure 12.24). It slopes down in both the x-direction and the y-direction, and every cross-section is a straight line.
- (f) \(z = y^2\) → (IV). x is missing, so every cross-section with x fixed is the same parabola z = y2: a trough (parabolic cylinder) running along the x-axis. Compare Figure 12.25, where the trough runs along the y-axis.
- (g) \(x^2 + y^2 = 4\) → (VII). z is missing, so every horizontal slice is the same circle of radius 2: a circular cylinder around the z-axis (Figure 12.26). Not the graph of a function.
- (h) \(x^2 + y^2 + z^2 = 4\) → (II). All points at distance 2 from the origin: a sphere of radius 2. Not the graph of a function, since most vertical lines meet it twice.
- (i) \(z = 3\) → (V). Constant height: a horizontal plane 3 units above the xy-plane.