TITLE: What is the equation to produce this Bezier curve?
QUESTION [5 upvotes]: What is the equation for the first curve in the image?

I have this formula:

But when I substitute values, I get an image that looks like x^2.  It isn't the same shape as the image.

REPLY [9 votes]: The three points $P_0,P_1,P_2$ are the control points
of the quadratic Bezier segment. 
On the images these points are connected 
with straight lines. $P_0$ and $P_2$
are the endpoints of the curve,
$P_1$ (marked with $\times$) 
usually is not on the curve.
The formula
\begin{align} 
B(t)&=(1-t)^2P_0+2(1-t)tP_1+t^2P_2
,\quad t\in[0,1]
\end{align}
is parametric, that is there are two
expressions in terms of parameter $t$,
that define $x(t)$ and $y(t)$:
Example for $a=3$:
\begin{align} 
B_x(t)&=(1-t)^2P_{0x}+2(1-t)tP_{1x}+t^2P_{2x}
,\\
B_y(t)&=(1-t)^2P_{0y}+2(1-t)tP_{1y}+t^2P_{2y}
.
\end{align}
So for the first image we can assume that
the control points looks like
\begin{align} 
P_0&=(0,a),\quad
P_1=(0,0),\quad
P_2=(a,0),
\end{align}
and for the second one 
\begin{align} 
P_0&=(a,a),\quad
P_1=(0,0),\quad
P_2=(a,0)
\end{align}
for some constant $a$.
Note, that coordinates of the points 
are completely independent of the 
parametric range $[0,1]$.
The two images demonstrate
how the curve changes, 
when just one endpoint is moved.
Example for $a=3$:


The tangent lines on the
images illustrate 
the process of construction of the points on a quadratic Bezier curve
by means of the points on linear Bezier segments:

Connecting the control points $P_0-P_1-P_2$,
we create two linear Bezier segments: 
$P_0P_1$ with
control points $P_0$ and $P_1$,
and 
$P_1P_2$ with
control points $P_1$ and $P_0$,
For any $t\in[0,1]$, find points $u_t,v_t$
on the line segments $P_0P_1$ and $P_1P_2$ 

\begin{align} 
u_t&=P_0(1-t)+P_1(t)
,\\
v_t&=P_1(1-t)+P_2(t)
.
\end{align}
Next, find the point $w_t$
on the linear Bezier segment $u_tv_t$
with control points $u_t$ and $v_t$,
\begin{align} 
w_t&=u_t(1-t)+v_t t
,
\end{align}
and the point $w_t$ is the point 
on the quadratic Bezier curve.