,
,
.
Notice that when t = 0 we have
and when t = 1 we have
,
so the curve starts at 
and ends at 
.
a) Graph the Bezier curve with control points
(4,1),
(28,48),
(50,42),
and 
(40,5).
Then on the same screen, graph the line segments
.
(Exercise 27 in Section 9.1 shows how to do this.)
Notice that the middle control points
and
don't lie on the curve; the curve starts at
,
heads toward 
and 
without reaching them, and ends at
.
b) From the graph in part (a) it appears that the tangent at
passes through
and the tangent at
passes through
.
Prove it.
c) Try to produce a Bezier curve with a loop by changing the second control point in part (a).
Solution:
a)Graph the curve, and then graph the line segments using the equations
from #27 in Section 9.1:
Similar equations are used for the other two line segments.
b) The tangent of the curve at 
is:
The slope of the tangent of the curve is the same as the slope
of the line segment
, thus the tangent
at 
passes through
.
The same can be done to prove that the tangent at
passes through 
using the tangent of the curve at
.
c) A loop can be produced by placing 
to the right of the point
.
The point used below was 
(150,50).
by: nl