Greenwood press

TANGENT
143
sidewalk curves with nontangent arcs of circles
sidewalk curves with tangent arcs of circles
Smooth curves in sidewalks are created with tangent arcs of circles.
Moving belts remain tightly on spin-
ning wheels, because they are tangent
to both circles at both locations.

---

TRANSLATIONS
A translation is a shift of points over the same distance and in the same direc-
tion. When you slide a checker piece across a game board from one square to
another, you are performing a translation. A home run in baseball represents the
hitter’s four translations: home plate to first base, first to second, second to third,
and third to home. The carpet design that is produced in hundreds of yards of a
carpet roll represents many translations of a single design. Musicians who trans-
pose a piece of music down to the range of a singer are performing a translation.
Translations in the coordinate plane can be expressed by the addition of coor-
dinates. The following figure shows the translation of the plane by the translation
6 right and 2 down. The translation can be expressed as an ordered pair (6,–2),
and the transformation by addition of ordered pairs. This is shown on the draw-
ing as the movement of a triangle. Every point 
(x, y) in the preimage of the tri-
angle will be translated 6 right and 2 down to a corresponding point 
(x
′
, y
′
) in
the image triangle. This gives the equation 
(x, y) + (6,-2) = (x
′
, y
′
). Applying
this to the vertex 
(-3, 4) gives (-3, 4) + (6,-2) = (3, 2) as shown. Applied to the
vertex
(-5, 1) gives (-5, 1) + (6,-2) = (1,-1). The picture shows that the corre-
sponding vertex in the image triangle is 
(-1, 1).
Translations change equations of functions in a systematic fashion. If a graph
of a function 
y = f (x) is translated by (h, k), the resulting graph will be the
function
y − k = f(x − h). If you translate a parabola y = x
2
by
(3,-2), the
resulting graph has the equation 
y − (-2) = (x − 3)2, or y = x
2
− 6x + 7.
Translations apply to three-dimensional figures and functions in the same way as
they do in two dimensions.
The language of translations depends on the application. Translations that
represent moves of chess pieces can be indicated by the column or rank (labeled
with letters in the picture below) and rows or files (labeled with numbers).