With PSTricks. I show you how to put a point at any location: on the curve or not on the curve.

\documentclass[pstricks,border=12pt]{standalone}
\usepackage{pst-plot}
\usepackage{pst-eucl}
\def\f(#1){x^2+0.5}
\begin{document}
\begin{pspicture}(-2.5,-0.5)(3,6)
\psaxes[linecolor=gray,tickcolor=gray]{->}(0,0)(-2.5,-0.5)(2.5,5.5)[$x$,0][$y$,90]
\psplot[algebraic]{-2.2}{2.2}{\f(x)}
\pstGeonode[PosAngle={45,0}](0,1){F}(*1.1 {\f(x)}){A}
\end{pspicture}
\end{document}
or

\documentclass[pstricks,border=12pt]{standalone}
\usepackage{pst-plot}
\usepackage{pst-eucl}
\usepackage{mathpazo}
\def\f(#1){x^2+0.5}
\begin{document}
\begin{pspicture}(-2.5,-0.5)(3,6)
\psaxes[linecolor=gray,tickcolor=gray]{->}(0,0)(-2.5,-0.5)(2.5,5.5)[$x$,0][$y$,90]
\psplot[algebraic]{-2.2}{2.2}{\f(x)}
\pstGeonode[PosAngle=-30,PointNameSep=15pt,PointName={{\scriptstyle(0,0.5)},{\scriptstyle(1,f(1))}}]
(0,0.5){F}(*1 {\f(x)}){A}
\end{pspicture}
\end{document}
Animation

\documentclass[pstricks,border=12pt]{standalone}
\usepackage{pst-plot}
\usepackage{pst-eucl}
\usepackage{mathpazo}
\def\f(#1){x^2+0.5}
\def\Picture#1{%
\begin{pspicture}(-2.5,-0.5)(3,6)
\psaxes[linecolor=gray,tickcolor=gray]{->}(0,0)(-2.5,-0.5)(2.5,5.5)[$x$,0][$y$,90]
\psplot[algebraic]{-2.2}{2.2}{\f(x)}
\pstGeonode[PosAngle={-15,135},PointNameSep=20pt,PointName={{\scriptscriptstyle(0.5,f(0.5))},{\scriptscriptstyle(#1,f(#1))}}]
(*0.5 {\f(x)}){F}(*#1 {\f(x)}){A}
\pstLineAB[nodesep=-1,linecolor=blue]{F}{A}
\end{pspicture}}
\begin{document}
\multido{\n=2.0+-0.1}{15}{\Picture{\n}}
\multido{\n=0.7+0.1}{13}{\Picture{\n}}
\end{document}
Warning!
The point (*1 {\f(x)}){A} uses the following weird "RPN-algebraic" syntax
(*<a constant numerical value in RPN expression> {<symbolic algebraic expression in x>})
No "algebraic-algebraic" syntax such as
({<a constant numerical algebraic expression>}| {<symbolic algebraic expression in x>})
is available.
label={180:{$(0,1)$}}-- putting the coordinate in math mode. – Xenon Mar 24 '13 at 12:50