Sketching Surfaces & Level Curves

$z$ appears only to the first power, so the axis is the $z$-axis.

$z = k$: $\ x^2 + y^2 = 4 - k$. A circle of radius $\sqrt{4 - k}$ for $k < 4$, a point at $k = 4$, nothing for $k > 4$.

$y = 0$: $\ z = 4 - x^2$, a parabola opening down with vertex $(0, 0, 4)$.

Circles that shrink to a point as $z$ rises to $4$: a circular paraboloid opening downward.

$y^2$ has the opposite sign to $x^2$ and $z^2$, so the axis is the $y$-axis.

$y = k$: $\ x^2 + z^2 = k^2 - 4$. Nothing for $|k| < 2$, a point at $k = \pm 2$, circles of radius $\sqrt{k^2 - 4}$ beyond.

$z = 0$: $\ y^2 - x^2 = 4$, a hyperbola through $(0, \pm 2, 0)$ opening along the $y$-axis.

Two stacks of growing circles with a gap for $-2 < y < 2$: a hyperboloid of two sheets.

As $x^2 + 4y^2 - z^2 = 0$: all squared, $z^2$ has the opposite sign, and the right side is $0$. Axis: the $z$-axis.

$z = k$: $\ x^2 + 4y^2 = k^2$, an ellipse with semi-axes $|k|$ (along $x$) and $\tfrac{|k|}{2}$ (along $y$); a single point at $k = 0$.

$y = 0$: $\ z^2 = x^2$, the two lines $z = \pm x$.

Ellipses growing linearly from the origin, above and below: an elliptic cone.

$y$ is missing, so this is a cylinder parallel to the $y$-axis.

$y = k$: $\ x^2 + z^2 = 9$ for every $k$, the same circle of radius $3$.

$x = 0$: $\ z = \pm 3$, two lines parallel to the $y$-axis.

One circle repeated all along the $y$-axis: a circular cylinder.

$(x^2 - 6x) + (y^2 + 4y) + (z^2 - 2z) = 2$.

$(x - 3)^2 - 9 + (y + 2)^2 - 4 + (z - 1)^2 - 1 = 2$, so $(x - 3)^2 + (y + 2)^2 + (z - 1)^2 = 16$.

$16 > 0$: a real sphere.

Center $(3, -2, 1)$, radius $\sqrt{16} = 4$.

$x^2 + y^2 - (z^2 - 4z) = 5$.

$z^2 - 4z = (z - 2)^2 - 4$, so $x^2 + y^2 - (z - 2)^2 + 4 = 5$, that is $x^2 + y^2 = (z - 2)^2 + 1$.

The right side is at least $1$ for every $z$: every slice $z = k$ is a circle, never empty.

Circles of radius $\sqrt{(k - 2)^2 + 1}$, smallest (radius $1$) at $z = 2$: a hyperboloid of one sheet.

$z = (x^2 - 2x) + (y^2 + 6y)$.

$z = (x - 1)^2 - 1 + (y + 3)^2 - 9$, that is $z = (x - 1)^2 + (y + 3)^2 - 10$.

$z + 10 = (x - 1)^2 + (y + 3)^2 \ge 0$: slices exist only for $z \ge -10$.

Circles centered on the vertical line through $(1, -3)$, growing as $z$ increases: a paraboloid opening up.

$(x^2 + 2x) + y^2 + (z^2 - 4z) = -5$.

$(x + 1)^2 - 1 + y^2 + (z - 2)^2 - 4 = -5$, so $(x + 1)^2 + y^2 + (z - 2)^2 = 0$.

The right side is $0$: a sum of squares is $0$ only when each square is $0$.

$x = -1$, $y = 0$, $z = 2$: not a sphere, just one point.

$y - x^2 = c$.

$y = x^2 + c$: the parabola $y = x^2$ shifted up by $c$.

Every $c$ gives the same shape: no special cases.

Draw $c = -2, -1, 0, 1, 2$: identical parabolas stacked one unit apart.

$x^2 - y^2 = c$.

A hyperbola when $c \ne 0$; when $c = 0$, $(x - y)(x + y) = 0$.

$c > 0$: hyperbolas opening left and right through $(\pm\sqrt{c}, 0)$. $c = 0$: the lines $y = \pm x$. $c < 0$: hyperbolas opening up and down through $(0, \pm\sqrt{-c})$.

Draw $c = -1, 0, 1$ (and $\pm 4$); every hyperbola has the asymptotes $y = \pm x$.

$\sqrt{9 - x^2 - y^2} = c$.

Square (allowed since $c \ge 0$): $x^2 + y^2 = 9 - c^2$, a circle centered at the origin.

A square root is never negative and is at most $3$: curves only for $0 \le c \le 3$. Radius $\sqrt{9 - c^2}$; at $c = 3$ just the origin.

$c = 0, 1, 2, 3$: radii $3$, $2\sqrt{2}$, $\sqrt{5}$, $0$, bunching toward the edge where the dome is steep.

$z - x^2 - y^2 = c$.

$z = x^2 + y^2 + c$: the paraboloid $z = x^2 + y^2$ moved up by $c$.

Every $c$ gives the same shape: no special cases.

$x^2 + y^2 - z^2 = c$, that is $x^2 + y^2 = z^2 + c$.

Slices $z = k$ are circles $x^2 + y^2 = k^2 + c$ about the $z$-axis, when $k^2 + c \ge 0$.

$c > 0$: a circle at every height, a hyperboloid of one sheet. $c = 0$: a cone. $c < 0$: no slices for $|z| < \sqrt{-c}$, a hyperboloid of two sheets.

$x^2 + 4y^2 + 9z^2 = c$.

For $c > 0$ this is an ellipsoid: it meets the axes at $x = \pm\sqrt{c}$, $y = \pm\tfrac{\sqrt{c}}{2}$, $z = \pm\tfrac{\sqrt{c}}{3}$.

$c > 0$: an ellipsoid. $c = 0$: only the origin. $c < 0$: empty, since $f \ge 0$.

$x - y^2 = c$.

$x = y^2 + c$, with $z$ missing: the parabola $x = y^2 + c$ repeated at every height.

Every $c$ gives the same shape, moved $c$ units along the $x$-axis.

$x^2 + y^2 = g(z)^2$.

On the profile $z = 2y$, so at height $z \ge 0$ the radius is $g = \tfrac{z}{2}$.

$x^2 + y^2 = \tfrac{z^2}{4}$, $z \ge 0$. Check: $(0, 1, 2)$ gives $1 = \tfrac{4}{4}$. ✓

$x^2 + y^2 = g(z)^2$.

On the profile $y^2 = z - 1$, so at height $z \ge 1$ the radius is $g = \sqrt{z - 1}$.

$x^2 + y^2 = z - 1$. Check: $(0, 2, 5)$ gives $4 = 5 - 1$. ✓

$(x - a)^2 + (y - b)^2 = g^2$.

Radius $g = 2$ at every height; center $a = z$, $b = 0$.

$(x - z)^2 + y^2 = 4$. Check: at $z = 0$ this is the circle $x^2 + y^2 = 4$, center $(0, 0, 0)$. ✓