\documentclass[tikz]{standalone}
\begin{document}
\begin{tikzpicture}
\draw [thin,->] (-4,0)--(4.3,0) node[below]{$x$};
\draw[thin,->] (0,-4)--(0,4.3) node[right]{$t$};
\clip(-4,-4) rectangle (4,4);
\foreach \c in {-8,...,4}
\draw[blue, domain=-4:4,samples=1000] plot  (  {\c+.5*\x*\x},{\x});
\end{tikzpicture}
\end{document}