\documentclass[11pt,border=5pt,tikz,class=scrartcl]{standalone}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage{mathtools}
 
\usepackage{pgfplots}
\pgfplotsset{compat=1.11}
\usetikzlibrary{arrows,shapes,calc,positioning,matrix,fit}
 
\pgfplotsset{mystyle/.style={%
	y=2.75cm,x=1.5cm,
	axis lines=middle,
        xlabel=$t$,
        ytick=\empty,
        ymin=-.5,ymax=1.1,
        anchor=center,
        smooth
}}

\begin{document}
\begin{tikzpicture}[very thick,auto]
	\begin{axis} [mystyle,domain=0:6.5,xtick={0,1,...,6},name=p1,at={(0cm,0cm)}]
		\addplot [blue,ultra thick] {exp(-x)};
		\node at (axis cs:1,.7) {$f(t)$};
	\end{axis}
	\begin{axis} [mystyle,xmin=0,xmax=6.5,xtick={0,1,...,6},
	name=p2,at={(10cm,0)}]
		\addplot [red,ultra thick,domain=1:4] {2*exp(-x)} \closedcycle;
		\node at (axis cs:2,.7) {$g(t)$};
	\end{axis}
	\begin{axis} [mystyle,domain=0:6.5,xtick={0,1,...,6},xlabel=$\tau$,
	name=p3,at={(0cm,-4.5cm)}]
		\addplot [blue,ultra thick] {exp(-x)};
		\node at (axis cs:1,.7) {$f(\tau)$};
	\end{axis}
	\begin{axis} [mystyle,xmin=-6,xmax=.5,xlabel=$\tau$,xtick={-5,...,0},
	xticklabels={$-5$,$-4$,$-3$,$-2$,$-1$,$0$},
	name=p4,at={(10cm,-4.5cm)}]
		\addplot [red,ultra thick,domain=-4:-1] {2*exp(x)} \closedcycle;
		\node at (axis cs:-2,.7) {$g(-\tau)$};
	\end{axis}
	\begin{axis} [mystyle,xmin=-6.5,xmax=6.5,xlabel=$\tau$,xtick={-4,-3,...,6},
	xticklabels={$t-4$,$t-3$,$t-2$,$t-1$,$0$,$1$,$2$,$3$,$4$,$5$,$6$},
	name=p5,at={(5cm,-9cm)}]
		\addplot [blue,ultra thick,domain=0:6] {exp(-x)};
		\node at (axis cs:1,.7) {$f(\tau)$};
		\addplot [red,ultra thick,domain=-4:-1] {2*exp(x)} \closedcycle;
		\node at (axis cs:-2,.7) {$g(t-\tau)$};
		\draw [dashed,green!50!black,thick] (-4.25,.8) rectangle (-.75,-.25);
		\draw [green!50!black,thick,-stealth] (-.65,.5) -- (-.1,.5);
	\end{axis}
	\begin{axis} [mystyle,xmin=-6.5,xmax=6.5,xlabel=$\tau$,xtick={-1.5,-.5,0,.5,1,1.5,2,3,4,5,6},
	xticklabels={$t-4$,$t-3$,$0$,$t-2$,$1$,$t-1$,$2$,$3$,$4$,$5$,$6$},
	name=p5,at={(5cm,-13.5cm)}]
		\addplot [blue,ultra thick,domain=0:6] {exp(-x)};
		\addplot [red,ultra thick,domain=-1.5:1.5] {2*exp(x-2.5)} \closedcycle;
		\draw [dashed,green!50!black,thick] (-1.75,.8) rectangle (1.75,-.25);
		\draw [green!50!black,thick,-stealth] (1.85,.5) -- (2.4,.5);
	\end{axis}
	\begin{axis} [mystyle,xmin=-6.5,xmax=6.5,xlabel=$\tau$,xtick={0,1,1.5,2,2.5,3,3.5,4,4.5,5,6},
	xticklabels={$0$,$1$,$t-4$,$2$,$t-3$,$2$,$t-2$,$3$,$t-1$,$4$,$5$,$6$},
	name=p5,at={(5cm,-18cm)}]
		\addplot [blue,ultra thick,domain=0:6] {exp(-x)};
		\addplot [red,ultra thick,domain=1.5:4.5] {2*exp(x-5.5)} \closedcycle;
		\draw [dashed,green!50!black,thick] (1.25,.8) rectangle (4.75,-.25);
		\draw [green!50!black,thick,-stealth] (4.85,.5) -- (5.4,.5);
	\end{axis}
\end{tikzpicture}
\end{document}