How Translations and Scaling Change a Quadric

Position and Size Are Separate Features

A standard quadric equation carries information about both where a surface sits and how large it is. Expressions such as \(x-h\) shift the surface’s position; squared denominators set length scales; and constants determine the level set being described. Reading these features separately makes it easier to compare equations and anticipate the effect on a sketch.

For a sphere, the pattern is familiar: changing the center moves the sphere, while changing the radius changes its size. The same distinction applies to ellipsoids, hyperboloids, cones, and paraboloids, although a paraboloid has a vertex rather than a center. In each case, a translated coordinate expression identifies a reference location, and the remaining terms determine how the surface extends away from it.

Translations: Moving the Reference Point

In three dimensions, translating a point by \((h,k,\ell)\) means adding those coordinate changes to its original coordinates. In a quadric equation, the corresponding translated coordinate expressions are \(x-h\), \(y-k\), and \(z-\ell\). They measure displacement from the reference point \((h,k,\ell)\).

For example, the sphere equation

describes a sphere centered at the origin with radius \(3\). Replacing the coordinate expressions by \(x+2\), \(y-1\), and \(z-3\) gives

This is still a sphere of radius \(3\), but its center is \((-2,1,3)\). The sign inside each parentheses is opposite to the corresponding center coordinate: \(x+2=x-(-2)\).

For a paraboloid, the same idea locates the vertex. For instance, \((x-1)^2+(y+2)^2=z-3\) has its vertex at \((1,-2,3)\). It does not have a center; the point identifies where the surface begins and from which its cross-sections extend.

Squared Denominators: Setting Length Scales

Consider the ellipsoid

The denominators are the squares of the three semi-axis lengths. The lengths are therefore \(2\), \(3\), and \(4\), not \(4\), \(9\), and \(16\). The ellipsoid extends twice as far from its center in the \(x\)-direction as in the \(y\)-direction, and four times as far in the \(z\)-direction as in the \(x\)-direction.

For a hyperboloid or cone, denominators also set scale parameters, but the signs determine which standard form is present. Changing a denominator does not change a plus sign into a minus sign. For a paraboloid, the coefficients and denominators determine how rapidly the surface widens as it moves away from its vertex.

Why the Translation and Scale Rules Work

Both rules follow by expressing a point on the surface relative to a reference point. Let \((u,v,w)\) be coordinates measured from the origin of a standard-form equation, and let the reference location in the sketch be \((h,k,\ell)\). A translation places the point with relative coordinates \((u,v,w)\) at \((h+u,k+v,\ell+w)\). Consequently, the relative coordinates of a point \((x,y,z)\) are \(u=x-h\), \(v=y-k\), and \(w=z-\ell\).

Now suppose each relative coordinate is also scaled by a positive factor: \(s_x\) in the \(x\)-direction, \(s_y\) in the \(y\)-direction, and \(s_z\) in the \(z\)-direction. Substituting \(u=(x-h)/s_x\), \(v=(y-k)/s_y\), and \(w=(z-\ell)/s_z\) into a standard ellipsoid gives the translated and scaled equation below.

The resulting semi-axis lengths are \(s_xa\), \(s_yb\), and \(s_zc\). The reference point changes through \((h,k,\ell)\); each length scale changes through its own factor. This substitution argument also explains why the same location-and-scale reading applies to standard forms with different sign patterns: translation changes the coordinate expressions, while scaling changes the squared length parameters.

Worked Examples: Reading Paired Equations

A Common Mistake: Confusing a Denominator with a Length

A frequent error is to read the denominator \(25\) in \((x-1)^2/25\) as a length of \(25\). The denominator is a squared length parameter, so the corresponding length is \(\sqrt{25}=5\). The same mistake can make a surface appear far larger or smaller than its equation describes.

A separate issue arises when an entire equation is multiplied by a nonzero constant. For example, \(x^2+y^2+z^2=9\) and \(2x^2+2y^2+2z^2=18\) describe the same sphere: dividing the second equation by \(2\) recovers the first. Multiplying every term on both sides by the same nonzero number does not change the solution set. By contrast, changing \(x^2+y^2+z^2=9\) to \(x^2+y^2+z^2=18\) changes the sphere’s radius from \(3\) to \(\sqrt{18}=3\sqrt{2}\).

Using the Rules in a Sketch

When comparing two standard-form equations, first locate their centers or vertices, then compare the squared denominators or other scale parameters. The position and scale changes can be read independently, while the sign pattern remains visible in the terms. Paired equations are particularly useful: they show whether a surface has merely moved, changed size, or done both.

These readings are predictions about the sketch, not substitutes for checking the equation. When a point is proposed as lying on a surface, substitution verifies that claim directly. For a broader sketch, keep each equation’s standard form intact and use its reference location and scales to place and size the surface consistently.